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ℋγ*γ−1(x)=xsubscriptℋ𝛾superscript𝛾1𝑥𝑥\mathcal{H}_{\gamma*\gamma^{-1}}(x)=xcaligraphic_H start_POSTSUBSCRIPT italic_γ * italic_γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) = italic_x for all x∈π−1(γ(0))𝑥superscript𝜋1𝛾0x\in\pi^{-1}(\gamma(0))italic_x ∈ italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_γ ( 0 ) ) at which ℋγsubscriptℋ𝛾\mathcal{H}_{\gamma}caligraphic_H start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT is defined. In particular, ℋγ*γ−1(x)subscriptℋ𝛾superscript𝛾1𝑥\mathcal{H}_{\gamma*\gamma^{-1}}(x)caligraphic_H start_POSTSUBSCRIPT italic_γ * italic_γ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_x ) is defined whenever ℋγ(x)subscriptℋ𝛾𝑥\mathcal{H}_{\gamma}(x)caligraphic_H start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_x ) is defined.
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π−1(γ(0))superscript𝜋1𝛾0\pi^{-1}(\gamma(0))italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_γ ( 0 ) )
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A partial connection is similar to a connection except that the homeomorphisms ℋγsubscriptℋ𝛾\mathcal{H}_{\gamma}caligraphic_H start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT may not be defined on all of π−1(γ(0))superscript𝜋1𝛾0\pi^{-1}(\gamma(0))italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_γ ( 0 ) ). A partial connection ℋℋ\mathcal{H}caligraphic_H is a choice of a homeomorphism ℋγsubscriptℋ𝛾\mathcal{H}_{\gamma}caligraphic_H start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT for each path γ:[0,1]→B:𝛾→01𝐵\gamma\colon[0,1]\to Bitalic_γ : [ 0 , 1 ] → italic_B from some (topological) subspace of π−1(γ(0))superscript𝜋1𝛾0\pi^{-1}(\gamma(0))italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_γ ( 0 ) ) to some subspace of π−1(γ(1))superscript𝜋1𝛾1\pi^{-1}(\gamma(1))italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_γ ( 1 ) ). The subspaces may depend on the path γ𝛾\gammaitalic_γ. The homeomorphisms are again required to be independent of the parameterization of γ𝛾\gammaitalic_γ and functorial with respect to composition of paths:
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Parallel transport along a trivial path is defined on the entire fiber π−1(γ(0))superscript𝜋1𝛾0\pi^{-1}(\gamma(0))italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_γ ( 0 ) ) and is equal to the identity map.
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The monodromy of 𝒥𝑝𝑎𝑟𝑡𝑖𝑎𝑙superscript𝒥𝑝𝑎𝑟𝑡𝑖𝑎𝑙\mathcal{J}^{\text{partial}}caligraphic_J start_POSTSUPERSCRIPT partial end_POSTSUPERSCRIPT along a curve parallel to a filling slope \IfSubStrfix6:7fix8 on a toroidal endon a toroidal end of Mφ∘superscriptsubscript𝑀𝜑M_{\varphi}^{\circ}italic_M start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT is trivial. To be precise, let p𝑝pitalic_p be a point on an unstable prong incident to an end of Mφ∘superscriptsubscript𝑀𝜑M_{\varphi}^{\circ}italic_M start_POSTSUBSCRIPT italic_φ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT. For any compact subspace W⊂π−1(p)𝑊superscript𝜋1𝑝W\subset\pi^{-1}(p)italic_W ⊂ italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_p ), the monodromy of 𝒥𝑝𝑎𝑟𝑡𝑖𝑎𝑙superscript𝒥𝑝𝑎𝑟𝑡𝑖𝑎𝑙\mathcal{J}^{\text{partial}}caligraphic_J start_POSTSUPERSCRIPT partial end_POSTSUPERSCRIPT around a small enough loop γ𝛾\gammaitalic_γ based at p𝑝pitalic_p and homotopic to the Dehn filling meridian is defined on W𝑊Witalic_W and equal to the identity.
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C
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From the proof of Theorem B in [drs22] it is clear that Theorem 2.4 also holds in the case that L𝐿Litalic_L is disconnected with finitely many connected components as long as we assume that every component is non-Legendrian since the heart of the argument is purely local around the image of any connected component of L𝐿Litalic_L. Furthermore, even if L𝐿Litalic_L has countably infinitely many connected components, only finitely many components of f(L)𝑓𝐿f(L)italic_f ( italic_L ) can intersect the (compact!) support of ΦtsubscriptΦ𝑡\Phi_{t}roman_Φ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT by properness of f𝑓fitalic_f. Thus, the conclusion of Theorem 2.4 also holds in this case (still under the assumption that f is non-Legendrian on every connected component of L𝐿Litalic_L).
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These results follow from the following more general theorem about loose Legendrians which states that we can guarantee the existence of an isotopy of small energy, and even C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-approximate any given isotopy.
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We will need the following C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-close version of Theorem 2.4.
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Proof. We will explain how to adjust the arguments in the proof of Theorem 1.2 in [mur12] to prove this theorem. The C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-close part in the case of a fixed loose chart and the fact that we may choose compactly supported homotopies are consequences of the constructions in Murphy’s proof. We will first explain how to obtain these two results and then reduce the proof in the general case to these special cases.
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We will also need the following statement that allows us to approximate a formal Legendrian by a loose Legendrian.
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B
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θ*(X∧𝕃Y)superscript𝜃superscript𝕃𝑋𝑌\theta^{*}(X\wedge^{\mathbb{L}}Y)italic_θ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_X ∧ start_POSTSUPERSCRIPT blackboard_L end_POSTSUPERSCRIPT italic_Y ) and we get Fn(θ*(X∧𝕃Y))∈𝒜Jnsubscript𝐹𝑛superscript𝜃superscript𝕃𝑋𝑌superscript𝒜subscript𝐽𝑛F_{n}(\theta^{*}(X\wedge^{\mathbb{L}}Y))\in\mathcal{A}^{J_{n}}italic_F start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_θ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_X ∧ start_POSTSUPERSCRIPT blackboard_L end_POSTSUPERSCRIPT italic_Y ) ) ∈ caligraphic_A start_POSTSUPERSCRIPT italic_J start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT as below.
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H0(Jn;Fn(θ*(X∧𝕃Y)))=⨁i+j=nZ(i)⊗Z~(j)subscript𝐻0subscript𝐽𝑛subscript𝐹𝑛superscript𝜃superscript𝕃𝑋𝑌subscriptdirect-sum𝑖𝑗𝑛tensor-productsuperscript𝑍𝑖superscript~𝑍𝑗H_{0}(J_{n};F_{n}(\theta^{*}(X\wedge^{\mathbb{L}}Y)))=\bigoplus_{i+j=n}Z^{(i)}%
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H1(Jn;Fn−1(θ*(X∧𝕃Y)))=⨁i+j=n+1B(i)⊗B~(j)subscript𝐻1subscript𝐽𝑛subscript𝐹𝑛1superscript𝜃superscript𝕃𝑋𝑌subscriptdirect-sum𝑖𝑗𝑛1tensor-productsuperscript𝐵𝑖superscript~𝐵𝑗H_{1}(J_{n};F_{n-1}(\theta^{*}(X\wedge^{\mathbb{L}}Y)))=\bigoplus_{i+j=n+1}B^{%
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0→⨁i+j=nZ(i)⊗Z~(j)→Fn(Eζn)→⨁i+j=n−1B(i)⊗B~(j)→0.→0subscriptdirect-sum𝑖𝑗𝑛tensor-productsuperscript𝑍𝑖superscript~𝑍𝑗→subscript𝐹𝑛subscript𝐸subscript𝜁𝑛→subscriptdirect-sum𝑖𝑗𝑛1tensor-productsuperscript𝐵𝑖superscript~𝐵𝑗→00\rightarrow\bigoplus_{i+j=n}Z^{(i)}\otimes\widetilde{Z}^{(j)}\rightarrow F_{n%
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Hp(Jn;Fq(θ*(X∧𝕃Y)))⇒Fp+q(hocolimJnθ*(X∧𝕃Y))⇒subscript𝐻𝑝subscript𝐽𝑛subscript𝐹𝑞superscript𝜃superscript𝕃𝑋𝑌subscript𝐹𝑝𝑞subscripthocolimsubscript𝐽𝑛superscript𝜃superscript𝕃𝑋𝑌H_{p}(J_{n};F_{q}(\theta^{*}(X\wedge^{\mathbb{L}}Y)))\Rightarrow F_{p+q}(%
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A
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\in C(X)\text{ and }\mu\in\mathcal{M}(X).⟨ italic_ϕ , italic_μ ⟩ ≔ ∫ italic_ϕ roman_d italic_μ for italic_ϕ ∈ italic_C ( italic_X ) and italic_μ ∈ caligraphic_M ( italic_X ) .
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The following theorem is a consequence of the results in [Co16, BZ15, HLMXZ19] and our investigations on little Lipschitz functions.
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The main goal of this section is to establish in Lemma 8.1 a local closing lemma that produces, for a nonempty compact forward-invariant set 𝒦𝒦\mathcal{K}caligraphic_K disjoint from critical points, a periodic orbit 𝒪𝒪\mathcal{O}caligraphic_O close to 𝒦𝒦\mathcal{K}caligraphic_K in terms of its (r,θ)𝑟𝜃(r,\theta)( italic_r , italic_θ )-gap defined below. This result relies on two other forms of closing lemmas, namely, a local Anosov closing lemma (Subsection 8.2) and a (global) Bressaud–Quas closing lemma (Subsection 8.3). Even though a global version of the Anosov closing lemma for expanding Thurston maps is available in [Li18, Lemma 8.6], it only holds for sufficiently high iterations of the map and sufficiently long periodic pseudo-orbits. It is crucial in our proof of Lemma 8.1 to be able to close periodic pseudo-orbits of arbitrary length for the map itself. Therefore, we formulate and prove a local version of the Anosov closing lemma (Lemma 8.6) from scratch. It closes periodic pseudo-orbits away from critical points to avoid the more complicated combinatorics near critical points. The proof relies on considerations over combinatorial structures like tiles, flowers, and bouquets. In Subsection 8.3, we define the Bressaud–Quas shadowing property for general dynamical systems and prove that it can be passed on to related systems via the factor relation and iterations. We establish this property for expanding Thurston maps in Theorem 8.11. Lemma 8.1 is then proved in Subsection 8.4.
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We relate flowers and bouquets of similar levels in the following lemma from [Li18, Lemma 8.5], which will be crucial in the proof of Lemma 8.4 below.
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Next, we record the following strengthened version of [BZ15, Lemma 2], which follows from Lemma 2 and its proof in [BZ15].
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D
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For a Heisenberg-limited QPE algorithm with maximal runtime Tmaxsubscript𝑇T_{\max}italic_T start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT, if the size of the time samples needed is 𝒪(polylogTmax)𝒪polysubscript𝑇\mathcal{O}(\mathrm{poly}\log T_{\max})caligraphic_O ( roman_poly roman_log italic_T start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ), the sampling is considered sparse in our paper. In this work, we design a non-adaptive algorithm that only requires discrete and sparse sampling of times without sacrificing performance. Informally, our main result can be stated as follows.
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As discussed in Sec. 2.3, because we cannot always assume f≈n/N,∀f∈ℱformulae-sequence𝑓𝑛𝑁for-all𝑓ℱf\approx n/N,\ \forall f\in\mathcal{F}italic_f ≈ italic_n / italic_N , ∀ italic_f ∈ caligraphic_F, the regular compressed sensing algorithm is not guaranteed to work. Our algorithm significantly relaxes the assumption by introducing a grid-shift parameter ν𝜈\nuitalic_ν. As a simple instance, suppose the frequency support ℱℱ\mathcal{F}caligraphic_F satisfies
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Suppose the target signal satisfies an approximately-on-grid assumption and its dominant part has frequency gap N−1superscript𝑁1N^{-1}italic_N start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT. Then our compressed sensing based QPE algorithm satisfies the following conditions:
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An overview of our algorithm is described as follows. For signal vectors with size N𝑁Nitalic_N, when the frequencies are all nearly on-grid (f≈n/N,n∈ℤformulae-sequence𝑓𝑛𝑁𝑛ℤf\approx n/N,n\in\mathbb{Z}italic_f ≈ italic_n / italic_N , italic_n ∈ blackboard_Z) and the noise for each sample is bounded by a constant, the convex relaxation algorithm can recover the frequencies with only 𝒪(polylogN)𝒪poly𝑁\mathcal{O}(\operatorname{poly}\log N)caligraphic_O ( roman_poly roman_log italic_N ) samples, which satisfies the Heisenberg limit. With no prior knowledge about f𝑓fitalic_f (i.e., f𝑓fitalic_f could be off-grid), we introduce a grid shift parameter ν𝜈\nuitalic_ν such that after shifting the signal by e−i2πft→e−i2π(f−ν/N)t→superscript𝑒i2𝜋𝑓𝑡superscript𝑒i2𝜋𝑓𝜈𝑁𝑡e^{-\mathrm{i}2\pi ft}\to e^{-\mathrm{i}2\pi(f-\nu/N)t}italic_e start_POSTSUPERSCRIPT - i2 italic_π italic_f italic_t end_POSTSUPERSCRIPT → italic_e start_POSTSUPERSCRIPT - i2 italic_π ( italic_f - italic_ν / italic_N ) italic_t end_POSTSUPERSCRIPT, the dominant frequencies of the new signal become nearly on-grid. This step requires an assumption on the signal, but we will show that a wide range of signals satisfy such an assumption. For each trial of ν𝜈\nuitalic_ν, we run the compressed sensing subroutine on the data set {yt}t∈𝒯subscriptsubscript𝑦𝑡𝑡𝒯\{y_{t}\}_{t\in\mathcal{T}}{ italic_y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_t ∈ caligraphic_T end_POSTSUBSCRIPT to obtain a trial solution sνsubscript𝑠𝜈s_{\nu}italic_s start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT. The optimal ν𝜈\nuitalic_ν is the one with the smallest ‖sν‖1subscriptnormsubscript𝑠𝜈1\|s_{\nu}\|_{1}∥ italic_s start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. By searching for the optimal grid-shift parameter in a finite set 𝒱𝒱\mathcal{V}caligraphic_V, the accuracy of the dominant frequencies is 𝒪(σoff/N)𝒪subscript𝜎off𝑁\mathcal{O}(\sigma_{\mathrm{off}}/N)caligraphic_O ( italic_σ start_POSTSUBSCRIPT roman_off end_POSTSUBSCRIPT / italic_N ), where σoffsubscript𝜎off\sigma_{\mathrm{off}}italic_σ start_POSTSUBSCRIPT roman_off end_POSTSUBSCRIPT is the maximal entry of the minimal off-grid component. This quantity is related to the noise, the frequency gap, and the residual part of the signal. In terms of the maximum runtime Tmaxsubscript𝑇T_{\max}italic_T start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT, since the samples of the compressed sensing algorithm are integers in [1,N]1𝑁[1,N][ 1 , italic_N ], Tmaxsubscript𝑇T_{\max}italic_T start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT scales linearly in N𝑁Nitalic_N, and Ttotalsubscript𝑇totalT_{\text{total}}italic_T start_POSTSUBSCRIPT total end_POSTSUBSCRIPT is 𝒪(NpolylogN)𝒪𝑁poly𝑁\mathcal{O}(N\operatorname{poly}\log N)caligraphic_O ( italic_N roman_poly roman_log italic_N ).
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In this subsection, we present an overview of our algorithm for QPE using compressed sensing. The quantum part of the algorithm is represented as follows.
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B
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The same result is true also if N=1𝑁1N=1italic_N = 1 for α>β≥2,α≥3formulae-sequence𝛼𝛽2𝛼3\alpha>\beta\geq 2,\,\alpha\geq 3italic_α > italic_β ≥ 2 , italic_α ≥ 3 and (α,β)≠(3,2)𝛼𝛽32(\alpha,\beta)\neq(3,2)( italic_α , italic_β ) ≠ ( 3 , 2 ). The proof is exactly the same, the only difference is that the term in (3.8) was strictly positive since α−2>2𝛼22\alpha-2>2italic_α - 2 > 2 and KN=1/2subscript𝐾𝑁12K_{N}=1/2italic_K start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = 1 / 2, while now it is strictly positive since α−2>1𝛼21\alpha-2>1italic_α - 2 > 1 and KN=1subscript𝐾𝑁1K_{N}=1italic_K start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = 1.
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In Section 3 we prove our main results. Specifically, Theorem 3.2 provides existence of an optimal set for small mass under some technical assumptions on g𝑔gitalic_g. Then, in Theorem 3.6, we observe that this abstract result can be applied for instance in the cases when Theorem 1.1 ensures that the optimal measure is given by atoms in the vertices of ΔNsubscriptΔ𝑁\Delta_{N}roman_Δ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT. The existence of optimal sets is true also when Theorem 1.1 says that the optimal measure is uniformly distributed over a sphere, and this is the content of Theorem 3.8, which is valid not only for small mass but for all m𝑚mitalic_m values. Finally, Theorem 3.10 gives some technical conditions under which the minimal measures are concentrated on the vertices of ΔNsubscriptΔ𝑁\Delta_{N}roman_Δ start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT. This generalizes the results by Davies, Lim and McCann. In this case the abstract existence result of Theorem 3.2 can be applied again to show that optimal sets exist for small mass. We highlight that the hypotheses of this last theorem are stable with respect to small perturbations, showing that the existence of minimizing sets is not a feature possessed only by some specific kernels.
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Dividing by α𝛼\alphaitalic_α and β𝛽\betaitalic_β clearly makes no difference, but it is convenient so that the minimal interaction is reached at distance 1111. It is possible to apply arguments by Frank and Lieb to g2subscript𝑔2g_{2}italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and see that again, when m𝑚mitalic_m is large enough, the minimizers of (1.5) are characteristic functions of a ball. This kind of kernel has been studied by Davies, Lim and McCann in a series of papers ([12, 13, 21]), and they are able to precisely characterize the solutions of (1.6) in some cases.
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In contrast, the next theorem shows that for certain choices of the parameters α𝛼\alphaitalic_α and β𝛽\betaitalic_β the minimizer of (1.5) is the characteristic function of a set for all values of m𝑚mitalic_m.
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The last result that we present is an a-priori bound on the diameter of the support of a minimizing density, and this deserves a quick comment. When dealing with minimizing measures, the boundedness of the support is a quite standard result, and it has been proved in several different contexts (see for instance [3, 6]). As we have already noticed, for many properties (for instance the existence given by the above lemma) working with measures or with densities does not make much difference. However, the compactness of the support of minimizers is more delicate for the case of densities due to the fact that the Euler–Lagrange condition (EL𝐸𝐿ELitalic_E italic_L) for densities has an additional constraint (see [17] for the special case when g𝑔gitalic_g is given by a power-law of the form g1subscript𝑔1g_{1}italic_g start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT). As a consequence, the proof of the result below does not follow by a simple generalization of the proofs available for the case of measures. Therefore we provide a complete proof.
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C
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Let g:T→Tnormal-:𝑔𝑇normal-→𝑇g\mathrel{\mathop{\ordinarycolon}}T\to Titalic_g : italic_T → italic_T be a C𝐶Citalic_C-quasi-isometry. There exists a constant D>0𝐷0D>0italic_D > 0 and a D𝐷Ditalic_D-deep mixed subtree quasi-isometry f𝑓fitalic_f such that g𝑔gitalic_g and f𝑓fitalic_f are at bounded distance. Moreover, if g(v0)=v0𝑔subscript𝑣0subscript𝑣0g(v_{0})=v_{0}italic_g ( italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = italic_v start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, then D𝐷Ditalic_D only depends on T𝑇Titalic_T and C𝐶Citalic_C.
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The following lemma states that every quasi-isometry between a rooted tree and itself is at bounded distance from an order-preserving quasi-isometry. This extends the result of [Nai22] where this is shown for (1,C)1𝐶(1,C)( 1 , italic_C )-quasi-isometries between spherically homogeneous trees.
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By Lemma 2.8, which states that all quasi-isometries are at bounded distance from order-preserving quasi-isometries, it suffices to show the moreover part for an order-preserving quasi-isometry. So we assume in the following that g𝑔gitalic_g is order-preserving.
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In [Nai22], Nairne shows that every (1,C)1𝐶(1,C)( 1 , italic_C )-quasi-isometry between spherically homogeneous trees is at bounded distance from an order-preserving quasi-isometry. In Lemma 2.8 we extend this result and show that any C𝐶Citalic_C-quasi-isometry of a rooted tree to itself is at bounded distance from an order-preserving quasi-isometry.
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Outline. In Section 2 we introduce the relevant notation and prove some of the technical results about quasi-isometries of trees. In particular, we extend a result of [Nai22] and show that any quasi-isometry is at bounded distance from an order-preserving quasi-isometry. In Section 3 we describe mixed-subtree quasi-isometries and prove Theorem 1.1, which states that a map from a rooted tree of degree at least 3 to itself is a quasi-isometry if and only if it is at bounded distance from a mixed-subtree quasi-isometry.
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B
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Theorem 1.3 extends Shen and Wang’s work [MR4308060], in which they compute ΛΛ\operatorname{\Lambda}roman_Λ when M𝑀Mitalic_M is simply connected, and separate this case with multiply-connected domains. Our result further characterizes doubly-connected M𝑀Mitalic_M. Theorem 1.3 shows ΛΛ\operatorname{\Lambda}roman_Λ can be attained for doubly-connected domains, specifically as B1−B¯βsubscript𝐵1subscript¯𝐵𝛽B_{1}-\overline{B}_{\operatorname{\beta}}italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT up to a composition of translation and homotheties. In particular, when β=0𝛽0\operatorname{\beta}=0italic_β = 0, Theorem 1.3 also provides the sharp case of b) in Theorem 1.1.
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and equality holds if and only if ΩΩ\Omegaroman_Ω is the image of B1−B¯βsubscript𝐵1subscript¯𝐵𝛽B_{1}-\overline{B}_{\operatorname{\beta}}italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT under a composition of translation and homotheties.
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An interesting byproduct of Theorem 1.3 is a new geometric interpretation of the exponent of modulus of continuity β𝛽\operatorname{\beta}italic_β.
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where 0<β<10𝛽10<\operatorname{\beta}<10 < italic_β < 1 is the exponent of the modulus of continuity of Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, such that Ω0subscriptΩ0\Omega_{0}roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is biholomorphic to B1−B¯βsubscript𝐵1subscript¯𝐵𝛽B_{1}-\overline{B}_{\operatorname{\beta}}italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT.
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In the second equality in (3.10), we used the fact that f𝑓fitalic_f extends to a C3,αsuperscript𝐶3𝛼C^{3,\alpha}italic_C start_POSTSUPERSCRIPT 3 , italic_α end_POSTSUPERSCRIPT diffeomorphism from B¯1−Bβsubscript¯𝐵1subscript𝐵𝛽\overline{B}_{1}-B_{\beta}over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_B start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT onto its image. The regularity of the extension is an easy application of Theorem 2.1, which indicates the existence of a C3,αsuperscript𝐶3𝛼C^{3,\alpha}italic_C start_POSTSUPERSCRIPT 3 , italic_α end_POSTSUPERSCRIPT diffeomorphism f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT from the closure of the region bounded by the C3,αsuperscript𝐶3𝛼C^{3,\alpha}italic_C start_POSTSUPERSCRIPT 3 , italic_α end_POSTSUPERSCRIPT curve 𝒞𝒞\mathcal{C}caligraphic_C to B¯1subscript¯𝐵1\overline{B}_{1}over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The composition f1∘fsubscript𝑓1𝑓f_{1}\circ fitalic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∘ italic_f is then a map from B1−B¯βsubscript𝐵1subscript¯𝐵𝛽B_{1}-\overline{B}_{\beta}italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT to B1−Ksubscript𝐵1𝐾B_{1}-Kitalic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_K for some compact K⊂B1𝐾subscript𝐵1K\subset B_{1}italic_K ⊂ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT that sends ∂B1subscript𝐵1\partial B_{1}∂ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT to ∂B1subscript𝐵1\partial B_{1}∂ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. An application of the Schwarz reflection principle [MR924157]*Theorem 11.14 extends f1∘fsubscript𝑓1𝑓f_{1}\circ fitalic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∘ italic_f to a C3,αsuperscript𝐶3𝛼C^{3,\alpha}italic_C start_POSTSUPERSCRIPT 3 , italic_α end_POSTSUPERSCRIPT function on B¯1−B¯βsubscript¯𝐵1subscript¯𝐵𝛽\overline{B}_{1}-\overline{B}_{\beta}over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - over¯ start_ARG italic_B end_ARG start_POSTSUBSCRIPT italic_β end_POSTSUBSCRIPT. Composing (f1)−1superscriptsubscript𝑓11(f_{1})^{-1}( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT with this extension of f1∘fsubscript𝑓1𝑓f_{1}\circ fitalic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∘ italic_f gives us the desired C3,αsuperscript𝐶3𝛼C^{3,\alpha}italic_C start_POSTSUPERSCRIPT 3 , italic_α end_POSTSUPERSCRIPT extension of f𝑓fitalic_f.
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B
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Here we point out that (fε2,nx⋅v)∂≥0subscriptsuperscriptsubscript𝑓𝜀2⋅subscript𝑛𝑥𝑣0\left(f_{\varepsilon}^{2},\,n_{x}\cdot v\right)_{\partial}\geq 0( italic_f start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_n start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⋅ italic_v ) start_POSTSUBSCRIPT ∂ end_POSTSUBSCRIPT ≥ 0 due to (2.4) of Lemma 2.1. Integrating along time, we derive (2.13) as claimed.
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The model (1.1), with i=1𝑖1i=1italic_i = 1 or 2222, is a kinetic description of the probability distribution of a certain system of interacting particles, submitted to an external force derived from the potential ϕitalic-ϕ\phiitalic_ϕ, at time t𝑡titalic_t located at the position x𝑥xitalic_x in the physical space Ω⊂ℝdΩsuperscriptℝ𝑑\Omega\subset\mathbb{R}^{d}roman_Ω ⊂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT with the velocity v∈ℝd𝑣superscriptℝ𝑑v\in\mathbb{R}^{d}italic_v ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT. It is complemented with the proper boundary conditions corresponding to the behaviour of particles at the boundary. The boundary condition in (1.1) is prescribed as zero influx when α=β=0𝛼𝛽0\alpha=\beta=0italic_α = italic_β = 0, and it is generally governed by the balance relations between the distribution of particles at the incoming and outgoing boundaries. The operator ℒ1subscriptℒ1\mathcal{L}_{1}caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT serves as a prototype for describing collision processes of linear relaxation type, including the case of neutron transport. The operator ℒ2subscriptℒ2\mathcal{L}_{2}caligraphic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT derived from the Ornstein-Uhlenbeck velocity process captures the impact of collisions with particles of a surrounding bath. The collision operators act only on the velocity variable and conserve local mass. They are both dissipative in the velocity variable and contribute to the relaxation process towards local equilibrium. The hypocoercivity theory seeks to understand how the interaction between transport and collisions leads to time-decay convergence for inhomogeneous collisional kinetic equations such as (1.1). The (small) parameter ε∈(0,1]𝜀01\varepsilon\in(0,1]italic_ε ∈ ( 0 , 1 ] representing the ratio of the collisional mean free path and the observation length is introduced to measure the balance between the transport part v⋅∇x⋅𝑣subscript∇𝑥v\cdot\nabla_{x}italic_v ⋅ ∇ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT and the collision part ℒisubscriptℒ𝑖\mathcal{L}_{i}caligraphic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, while diffusion phenomena can be observed on long time scales of order ε−1superscript𝜀1\varepsilon^{-1}italic_ε start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT. This regime with ε→0→𝜀0\varepsilon\rightarrow 0italic_ε → 0 is often referred to as the diffusion limit.
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The dissipation property exhibited by the operator ℒisubscriptℒ𝑖\mathcal{L}_{i}caligraphic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, which acts only on the velocity variable, is formulated in the following lemma.
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The following lemma gives a weighted dissipation estimate for solutions to the homogeneous counterpart of (1.1), which will be used in Section 4 to address the initial layer correction. It is worth noting that the variable x𝑥xitalic_x involved in the lemma below serves as a parameter and therefore does not affect the relaxation process.
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We have to address the boundary term Σ2subscriptΣ2\Sigma_{2}roman_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and the initial layer term Rψsubscript𝑅𝜓R_{\psi}italic_R start_POSTSUBSCRIPT italic_ψ end_POSTSUBSCRIPT appeared in (4.8). Let us assume the compatibility condition on the initial data that finsubscript𝑓inf_{\rm in}italic_f start_POSTSUBSCRIPT roman_in end_POSTSUBSCRIPT depends only on the variable x𝑥xitalic_x in ∂Ω×ℝdΩsuperscriptℝ𝑑\partial\Omega\times\mathbb{R}^{d}∂ roman_Ω × blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT. By (2.14), we know that ψε|Σevaluated-atsubscript𝜓𝜀Σ\psi_{\varepsilon}|_{\Sigma}italic_ψ start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT | start_POSTSUBSCRIPT roman_Σ end_POSTSUBSCRIPT depends only on the variable x𝑥xitalic_x. Consequently, according to the boundary condition of fεsubscript𝑓𝜀f_{\varepsilon}italic_f start_POSTSUBSCRIPT italic_ε end_POSTSUBSCRIPT, we have
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C
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|πkπℓ|∉qℤsuperscript𝜋𝑘superscript𝜋ℓsuperscript𝑞ℤ\left\lvert\pi^{k}\pi^{\ell}\right\rvert\notin q^{\mathds{Z}}| italic_π start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_π start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT | ∉ italic_q start_POSTSUPERSCRIPT blackboard_Z end_POSTSUPERSCRIPT and |πk|≠|πℓ|superscript𝜋𝑘superscript𝜋ℓ\left\lvert\pi^{k}\right\rvert\neq\left\lvert\pi^{\ell}\right\rvert| italic_π start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT | ≠ | italic_π start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT |.
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Assume that ϕ∈L2(U(Eq(K)))0italic-ϕsuperscript𝐿2subscript𝑈subscript𝐸𝑞𝐾0\phi\in L^{2}(U(E_{q}(K)))_{0}italic_ϕ ∈ italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_U ( italic_E start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( italic_K ) ) ) start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT. Then
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Case k+ℓ≡0modv(q)𝑘ℓmodulo0𝑣𝑞k+\ell\equiv 0\mod v(q)italic_k + roman_ℓ ≡ 0 roman_mod italic_v ( italic_q ).
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In this case, first assume that 0≤k<ℓ<v(q)0𝑘ℓ𝑣𝑞0\leq k<\ell<v(q)0 ≤ italic_k < roman_ℓ < italic_v ( italic_q ). Then
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Case k+ℓ≢0modv(q)not-equivalent-to𝑘ℓmodulo0𝑣𝑞k+\ell\not\equiv 0\mod v(q)italic_k + roman_ℓ ≢ 0 roman_mod italic_v ( italic_q ) and k≡ℓmodv(q)𝑘moduloℓ𝑣𝑞k\equiv\ell\mod v(q)italic_k ≡ roman_ℓ roman_mod italic_v ( italic_q ). This is the case iff |πk|=|πℓ|superscript𝜋𝑘superscript𝜋ℓ\left\lvert\pi^{k}\right\rvert=\left\lvert\pi^{\ell}\right\rvert| italic_π start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT | = | italic_π start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT |. Then
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C
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6. Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-operator norms on ℂnsuperscriptℂ𝑛\mathbb{C}^{n}blackboard_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT
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Proposition 4.13 (7) shows that, at least when p=1,𝑝1p=1,italic_p = 1 , the hypotheses of Theorem 4.1 are not met
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F0p(G)subscriptsuperscript𝐹𝑝0𝐺F^{p}_{0}(G)italic_F start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_G ) still does not meet the hypotheses in Theorem 4.1.
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Theorem 3.4 can also be proved by either appropriately modifying the proof of Proposition 2.6.12 in [2] or by the methods described in Section 6 of [19]. We are grateful to both David Blecher and Hannes Thiel for pointing these references to us. An alternative way is to use
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We have shown that the conclusion of Theorem 4.1 can go either way when dropping the two hypotheses.
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D
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_{\sigma\otimes\eta}(\mathbf{f}).italic_I start_POSTSUBSCRIPT roman_BC ( italic_σ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT ( bold_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_J start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ( bold_f ) + italic_J start_POSTSUBSCRIPT italic_σ ⊗ italic_η end_POSTSUBSCRIPT ( bold_f ) .
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In Section 3 and Section 4, we recall the geometric side and the spectral side of the relative trace formula of [XZ23] respectively. Then the proof of Theorem 1.2 is given in Section 5, while the proof of Theorem 1.3 is given in Section 6.
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Assume that π𝜋\piitalic_π be an irreducible (H,χ−1)𝐻superscript𝜒1(H,\chi^{-1})( italic_H , italic_χ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT )-distinguished supercuspidal representation of G𝐺Gitalic_G. We keep the notations from the proof of Theorem 4.12. In particular, we have matching test functions 𝐟=⊗fv∈𝒞c∞(G¯(𝔸F¯))\mathbf{f}=\otimes f_{v}\in\mathcal{C}_{c}^{\infty}({\underline{G}}({\mathbb{A%
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The proof essentially follows from [XZ23, §4.1]. For completeness, we reproduce the proof here, after we introduce some necessary notations and tools from [BPLZZ21].
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where ρ𝜌\rhoitalic_ρ is an irreducible supercuspidal representation of GLs(D)subscriptGL𝑠𝐷\operatorname{GL}_{s}(D)roman_GL start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_D ), and n=sℓ𝑛𝑠ℓn=s\ellitalic_n = italic_s roman_ℓ. Then the discrete series case follows from this consideration and the supercuspidal case. Finally, using a classification of (H,χ−1)𝐻superscript𝜒1(H,\chi^{-1})( italic_H , italic_χ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT )-distinguished representations, the general case of Theorem 1.2 follows from the case of discrete series.
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C
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It is now clear that symmetry breaking is intimately related to the event that the control vector fields available for the motion and their Lie brackets (possibly of higher order, i.e., nested Lie brackets) provide a sufficiently rich ensemble of linearly independent vectors in the ambient space.
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The lesson of the Scallop Theorem is that the symmetry of the reciprocal motion of opening and closing the valves must be broken to achieve non-zero net displacement. A first solution was provided by Purcell’s swimmer [34], consisting of three concatenated links, with the ability to change independently the angles between each couple of adjacent links. If two independent controlled parameters are enough to avoid reciprocal motion and properly change position, the question becomes whether they are also sufficient to reach any desired position.
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When building a model for swimming, it is therefore helpful to distinguish between shape variables s𝑠sitalic_s and position variables g𝑔gitalic_g: the state of the swimmer is then described by the pair (s,g)∈𝒵𝑠𝑔𝒵(s,g)\in\mathcal{Z}( italic_s , italic_g ) ∈ caligraphic_Z, where 𝒵𝒵\mathcal{Z}caligraphic_Z is a suitable manifold. For example, for planar locomotion of Purcell’s swimmer, the center of the middle link and its orientation are the position variables and the angles that the lateral links form with the middle one are the two shape parameters.
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The controllability results obtained in Section 4 can be extended to the case of the N𝑁Nitalic_N-link swimmer; the reasoning follows that of [28, Theorem 4.2].
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To obtain this, there are various strategies, such as adding more shape parameters that can be actuated independently, so each of them is related to a control vector field: this is the case of Purcell’s three link swimmer [34], of the N𝑁Nitalic_N-link swimmer [2, 16, 28]), and when the hydrodynamic interaction between two (possibly non-controllable) swimmers is exploited [4, 27, 43].
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D
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For a colored binary tree T𝑇Titalic_T, we define a branch of T𝑇Titalic_T to be a path from the root of T𝑇Titalic_T to a leaf of T𝑇Titalic_T.
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For a colored binary tree T𝑇Titalic_T, we define a branch of T𝑇Titalic_T to be a path from the root of T𝑇Titalic_T to a leaf of T𝑇Titalic_T.
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Note that the fineness of g𝑔gitalic_g is the same as the depth of the target tree of a pair of colored binary trees obtained from the grid diagram of g𝑔gitalic_g.
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if we take one of the shortest words with length n𝑛nitalic_n, we can say that the depth is at most 4n4𝑛4n4 italic_n since each process of multiplying a generator increases the length of the branches by at most four.
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The depth of T𝑇Titalic_T is then defined as the maximum length of the branches of T𝑇Titalic_T (with each edge having a length of one).
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D
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Several factors can significantly impact the efficiency and accuracy of the approximate tensor network contraction process.
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In Section 3.2.2, we will demonstrate how the utilization of the partial contraction tree abstraction enables the straightforward extension of various contraction algorithms designed for 2D grids with different environments, including those that have not been automated in the prior work [35, 61, 14, 26].
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To begin with, the choice of contraction path plays a crucial role. Ref. [26] demonstrates that selecting different contraction paths using various heuristics can lead to substantial variations in both runtime and accuracy for different problems.
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Achieving a balance between accuracy and efficiency requires favoring different structures and sizes of the environment 𝐄^^𝐄\hat{\mathbf{E}}over^ start_ARG bold_E end_ARG for different problems. Hence, it becomes crucial to provide an automated tensor network contraction algorithm with the necessary flexibility to accommodate different environments. This flexibility enables the algorithm to adapt and optimize the contraction process according to the specific requirements of each problem.
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spanning tree of tensors around the pair of tensors to be truncated. Ref. [26] demonstrates that including a larger environment leads to more accurate contraction results for multiple problems.
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B
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Noetherian frames}\}roman_Fr bold_Grz bold_.3 = { transitive reflexive non-branching Noetherian frames }
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Interestingly, all these logics share some desirable properties, such as finite axiomatization, finite model property, decidability, etc. This stands in contrast to the known results on 𝐊𝐚𝐬superscript𝐊𝐚𝐬\mathbf{K}^{\mathbf{as}}bold_K start_POSTSUPERSCRIPT bold_as end_POSTSUPERSCRIPT [Gor20] and GL𝐚𝐬superscriptGL𝐚𝐬\textbf{GL}^{\mathbf{as}}GL start_POSTSUPERSCRIPT bold_as end_POSTSUPERSCRIPT [Ver21] that imply that these logics lack the finite axiomatizability.
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These logics have the finite model property, so each of them is the logic of all finite point-generated frames that satisfy the corresponding frame condition [BdRV01]. For instance, 𝐊𝐃𝟓𝐊𝐃𝟓\mathbf{KD5}bold_KD5 is the logic of all finite point-generated serial Euclidean frames.
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Notice that 𝐊𝟓𝐁𝐊𝟓𝐁\mathbf{K5B}bold_K5B is the logic of its finite point-generated frames, which are exactly the finite clusters and the irreflexive singletons. If φ∉𝐊𝟓𝐁𝜑𝐊𝟓𝐁\varphi\not\in\mathbf{K5B}italic_φ ∉ bold_K5B for some formula φ,𝜑\varphi,italic_φ , then either ([r],[r]×[r])⊧̸φnot-modelsdelimited-[]𝑟delimited-[]𝑟delimited-[]𝑟𝜑([r],\,[r]\times[r])\not\models\varphi( [ italic_r ] , [ italic_r ] × [ italic_r ] ) ⊧̸ italic_φ for some r∈ω𝑟𝜔r\in\omegaitalic_r ∈ italic_ω, or ({a},∅)⊧̸φ.not-models𝑎𝜑(\{a\},\varnothing)\not\models\varphi.( { italic_a } , ∅ ) ⊧̸ italic_φ .
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Fr𝐊𝐃𝟓={serial Euclidean frames};Fr𝐊𝐃𝟓serial Euclidean frames\operatorname{Fr}\mathbf{KD5}=\{\text{serial Euclidean frames}\};roman_Fr bold_KD5 = { serial Euclidean frames } ;
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B
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We investigate two scenarios: The first considers only the norm of the displacement vector ∥u(x1,x2)∥delimited-∥∥𝑢subscript𝑥1subscript𝑥2\lVert u(x_{1},x_{2})\rVert∥ italic_u ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∥ for the ensuing best fit. The second considers the full displacement field u(x1,x2)𝑢subscript𝑥1subscript𝑥2u(x_{1},x_{2})italic_u ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) instead. Both methods are compared in Section 5. They may greatly differ from either the Euclidean or logarithmic approach depending on the strength of the anisotropy, as indeed all discussed approaches deliver the correct isotropic result for an isotropic input. We believe that for our academic plane strain problem the full-field resolution and optimization represent a valuable alternative to more direct calculations. The computational burden is manageable because the FEM calculation has to be done only twice, once for the center of dilatation and once for the concentrated couple, and is applicable to any symmetry class.
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We investigate two scenarios: The first considers only the norm of the displacement vector ∥u(x1,x2)∥delimited-∥∥𝑢subscript𝑥1subscript𝑥2\lVert u(x_{1},x_{2})\rVert∥ italic_u ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∥ for the ensuing best fit. The second considers the full displacement field u(x1,x2)𝑢subscript𝑥1subscript𝑥2u(x_{1},x_{2})italic_u ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) instead. Both methods are compared in Section 5. They may greatly differ from either the Euclidean or logarithmic approach depending on the strength of the anisotropy, as indeed all discussed approaches deliver the correct isotropic result for an isotropic input. We believe that for our academic plane strain problem the full-field resolution and optimization represent a valuable alternative to more direct calculations. The computational burden is manageable because the FEM calculation has to be done only twice, once for the center of dilatation and once for the concentrated couple, and is applicable to any symmetry class.
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The paper is structured as follows. We derive the linear elastic solution for a concentrated couple and center of dilatation for planar cubic materials using Green’s functions and recall the well-known isotropic linear elastic case. We show that they can straightforwardly be used to correctly retrieve isotropic parameters via a full-field simulation in Comsol Multiphysics® and quadratic error minimization (here done in Mathematica) for both the norm-scenario and the full displacement approach. Further, our results reveal the necessary domain size for sufficient accuracy. Since both solutions decay to zero far away from the center, it becomes inconsequential whether one tries to find a matching amplitude for an additional Dirichlet boundary displacement, or simply fixes the displacement field to zero on the entire boundary.
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While for smaller domains it seems to be necessary to adjust the value of the boundary displacement correctly (which can always be done recursively), we circumvent the problem by increasing the considered domain. Therein, we change the overall size but not the radii of the applied force nor the maximum distance of points used for the fitting procedure. This allows us to reduce the effect of inaccuracies at the boundary as much as required which is attributed to Saint-Venant’s principle. Consequently, for a sufficiently large domain, we can set zero Dirichlet displacements at the outside boundary.
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We present two different procedures for finding the best approximated isotropic elasticity tensor with quadratic error minimization using Mathematica. In the first one, we only consider the radially averaged norm of the displacement ∥u(r)∥delimited-∥∥𝑢𝑟\lVert u(r)\rVert∥ italic_u ( italic_r ) ∥. In the second procedure, we take the full displacement solution u(x1,x2)𝑢subscript𝑥1subscript𝑥2u(x_{1},x_{2})italic_u ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) into account. For the former, we average the norm of the displacement ∥u(r)∥delimited-∥∥𝑢𝑟\lVert u(r)\rVert∥ italic_u ( italic_r ) ∥ over all angles111For the cubic symmetry class considered here, an average between 0∘superscript00^{\circ}0 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT and 45∘superscript4545^{\circ}45 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT would be sufficient., cf. Figure 8. Note again, that the exact magnitude of the Dirichlet boundary conditions is not known a priori as it depends on the final best approximation isotropic elasticity tensor ℂisosubscriptℂiso\mathbb{C}_{\rm iso}blackboard_C start_POSTSUBSCRIPT roman_iso end_POSTSUBSCRIPT. Therefore, we set zero Dirichlet boundary conditions on the outside and only use the data within a circle of half the size of the computed domain for the actual fitting procedure.
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B
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Since the cases of I2(3)=𝕊3subscript𝐼23subscript𝕊3I_{2}(3)={\mathbb{S}}_{3}italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 3 ) = blackboard_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and I2(4m)subscript𝐼24𝑚I_{2}(4m)italic_I start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 4 italic_m ) are to be found in [9, 20, 13], Corollary 4.7, Theorem 4.8 and Proposition 4.9 conclude the classification of finite-dimensional Nichols algebras over dihedral groups.
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The paper is structured as follows: in §2 we introduce the basic notions on Coxeter groups, racks, cocycles and Nichols algebras; in §2.4 we define the cocycles q+superscript𝑞q^{+}italic_q start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and q−superscript𝑞q^{-}italic_q start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT on the rack of reflections in an arbitrary Coxeter group and translate twist equivalence of q+superscript𝑞q^{+}italic_q start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and q−superscript𝑞q^{-}italic_q start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT into existence of a section, generalizing the strategy in [39]. The core argument is in §3 where we apply results in [38] in order to show existence of the sought section (Theorem 2.8). In §4 we collect and complement to what is known on Nichols algebras associated with the rack of reflections in finite Coxeter groups, we apply Theorem 2.8 and the main results in [8, 27] in order to list the Coxeter groups for which a conjugacy class of reflections could possibly support a finite-dimensional Nichols algebra. Last section is devoted to the completion of the classification of finite-dimensional Nichols algebra over dihedral groups.
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We wish to thank N. Andruskiewitsch for useful discussions and I. Heckenberger for pointing out the content of Remark 4.5.
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It was kindly pointed out to us by I. Heckenberger that ℬ(TD4,q±)ℬsubscript𝑇subscript𝐷4superscript𝑞plus-or-minus{\mathcal{B}}(T_{D_{4}},q^{\pm})caligraphic_B ( italic_T start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_q start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ) is infinite-dimensional as a consequence of [8]*Theorem 6.14 because the Coxeter group of type D4subscript𝐷4D_{4}italic_D start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT is solvable, non-cyclic, and generated by T𝑇Titalic_T, which has size >7absent7>7> 7. Hence, dimℬ(TDn,q±)=∞dimensionℬsubscript𝑇subscript𝐷𝑛superscript𝑞plus-or-minus\dim{\mathcal{B}}(T_{D_{n}},q^{\pm})=\inftyroman_dim caligraphic_B ( italic_T start_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_q start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ) = ∞ for any n⩾4𝑛4n\geqslant 4italic_n ⩾ 4.
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In the present paper we first focus on finitely generated Coxeter groups and a special family of racks and 2222-cocycles, that is of interest due to its relation with the cohomology of the flag variety. Here the rack is the class (or union of classes) of reflections in a Coxeter group W𝑊Witalic_W, and the 2222-cocycle is a cocycle with values in {±1}plus-or-minus1\{\pm 1\}{ ± 1 }. If W𝑊Witalic_W is a Weyl group, the quadratic approximation of the corresponding Nichols algebra is a non-commutative algebra containing the cohomology algebra of the flag variety of the corresponding algebraic group. For W=𝕊n𝑊subscript𝕊𝑛W={\mathbb{S}}_{n}italic_W = blackboard_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT this is the so-called Fomin-Kirillov algebra introduced in [21] for creating an algebraic and combinatorial framework to perform Schubert calculus. The construction was later generalized to the case of arbitrary Weyl groups in [12]. For W=𝕊3,𝕊4𝑊subscript𝕊3subscript𝕊4W={\mathbb{S}}_{3},{\mathbb{S}}_{4}italic_W = blackboard_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , blackboard_S start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT and 𝕊5subscript𝕊5{\mathbb{S}}_{5}blackboard_S start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT the Nichols algebra in question is quadratic and finite-dimensional [36, 22]. For n>5𝑛5n>5italic_n > 5 a proof of infinite-dimensionality of the Fomin-Kirillov algebras appeared in [11], but it is not known whether the Nichols algebras are quadratic. This property, as well as dimension, is preserved by twists.
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B
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Taking 𝒖−𝒗=λ𝒘𝒖𝒗𝜆𝒘\boldsymbol{u}-\boldsymbol{v}=\lambda\boldsymbol{w}bold_italic_u - bold_italic_v = italic_λ bold_italic_w, for λ>0𝜆0\lambda>0italic_λ > 0, dividing by λ𝜆\lambdaitalic_λ, passing to the limit with λ→0→𝜆0\lambda\to 0italic_λ → 0, and then exploiting the hemicontinuity property of the operator G(⋅)G⋅\mathrm{G}(\cdot)roman_G ( ⋅ ), we finally obtain G0(t)=G(𝒖(t))subscriptG0𝑡G𝒖𝑡\mathrm{G}_{0}(t)=\mathrm{G}(\boldsymbol{u}(t))roman_G start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) = roman_G ( bold_italic_u ( italic_t ) ). Uniqueness follows by taking the estimate (2.36) instead of (3.50) and following similarly as in the Part (2) of Theorem 3.5.
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The rest of the paper is organized as follows. In the next section, we discuss the functional setting of the problem described in (1.1). After defining necessary function spaces, we define the linear and nonlinear operators and show that these operators satisfy a monotonicity property for r≥3𝑟3r\geq 3italic_r ≥ 3 (see Theorems 2.5 and 2.6). Furthermore, we show that the sum of linear and nonlinear operators satisfy demicontinuous property also (Lemma 2.8). After providing an abstract formulation of the system (1.1), the existence and uniqueness of a global weak solution in the Leray-Hopf sense is examined in Section 3. The monotonicity and hemicontinuity properties of the linear and nonlinear operators as well as the Minty-Browder techniques are exploited in the proofs (Theorem 3.5). In the final section, we discuss the global strong solutions to the system (1.1). Due to technical difficulties described above, we are able to prove the regularity of the weak solutions in the class (1.3) in periodic domains only (Theorem 4.2). Moreover, using the abstract theory developed in [3, 4], we prove the existence and uniqueness of global strong solutions for the system (1.1) in the class (1.3) (Theorem 4.4).
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We point out here that the results obtained in Theorems 3.5 and 3.6 hold true in unbounded domains also as we are not using any compactness arguments to get the required results. As discussed in [24, Section 2.5], in the unbounded domain case, one needs to replace the eigenfunctions of the Stokes operator in Step (iii) in the proof of Theorem 3.5 by the eigenfunctions of the operator defined in [24, Section 2.5].
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One can follow in a similar way as in the proof of Theorem 3.5. In the critical case r=3𝑟3r=3italic_r = 3, note that the operator G(⋅)G⋅\mathrm{G}(\cdot)roman_G ( ⋅ ) is monotone (see (2.34)), and hence we provide a short proof in this case. Using the convergences given in (3.15), we take limit supremum in (3.12) to find
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As discussed in [23], one can obtain the results obtained in the previous Theorem in the following way also. Let us define
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B
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‖𝒮1−𝒜~𝒮0‖Fsubscriptnormsubscript𝒮1~𝒜subscript𝒮0𝐹\|\mathcal{S}_{1}-\tilde{\mathcal{A}}\mathcal{S}_{0}\|_{F}∥ caligraphic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - over~ start_ARG caligraphic_A end_ARG caligraphic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT
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In this paper, we apply the data-driven reduced-order modeling technique DMD [25, 27] for option pricing and compare it with the POD with respect to accuracy and speed-up over the full order models. The DMD is able to extract dynamically relevant flow features from time-resolved experimental or numerical data by generalizing the global stability modes and approximating the eigenvalues and eigenfunctions of the Koopman operator [14]. Both methods use the snapshots of the fully discretized PDEs in time. The POD solves a low dimensional model by Galerkin projection, whereas DMD is equation-free, the reduced solution is given in form of Fourier series in time and space. We would like to remark that due to its computational efficiency, the DMD is used for financial applications like in high-frequency trading [19] and stock market data analysis [8].
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Different DMD algorithms are developed for the estimation of Koopman modes, eigenvalues and amplitudes from the given set of snapshots. In this paper we consider the exact DMD algorihm in [27] and a variant of DMD algorithm in [6].
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modes. DMD is equation-free, where the solutions are given in form of Fourier series in space and time. This feature of DMD allows making future predictions.
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In all numerical tests, we have used linear dG elements in space and backward Euler in time. For the computation of the DMD modes, we used the MATLAB Toolbox Koopman mode decomposition [3]. The numerical simulations given in this paper are performed on
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B
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β(x)=xex−1=∑n=0∞βnxn.𝛽𝑥𝑥superscripte𝑥1superscriptsubscript𝑛0subscript𝛽𝑛superscript𝑥𝑛\beta(x)=\frac{x}{\mathrm{e}^{x}-1}=\sum_{n=0}^{\infty}\beta_{n}x^{n}.italic_β ( italic_x ) = divide start_ARG italic_x end_ARG start_ARG roman_e start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT - 1 end_ARG = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_x start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT .
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log(exp(t1X1)⋅…⋅exp(tkXk))=BCH(…BCH(BCH(t1X1,t2X2),t3X3)…,tnXn).⋅subscript𝑡1subscript𝑋1…subscript𝑡𝑘subscript𝑋𝑘BCH…BCHBCHsubscript𝑡1subscript𝑋1subscript𝑡2subscript𝑋2subscript𝑡3subscript𝑋3…subscript𝑡𝑛subscript𝑋𝑛\log(\exp(t_{1}X_{1})\cdot\ldots\cdot\exp(t_{k}X_{k}))=\operatorname{BCH}(%
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degX(M)!degY(M)!uasc(M)vdes(M)Gk1(u,v)⋅…⋅Gkp(u,v).\deg_{X}(M)!\deg_{Y}(M)!\,u^{\operatorname{asc}(M)}v^{\operatorname{des}(M)}G_%
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BCH(X,Y)degX,Y=(1,n)=(−1)nβn[[…[X,Y]…,Y],Y]⏟n times.\operatorname{BCH}(X,Y)_{\deg_{X,Y}=(1,n)}=(-1)^{n}\beta_{n}[[\ldots[X%
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BCH(X,Y)degX,Y=(n,1)=(−1)nβn[X,[X,…[X,⏟n timesY]…]];\operatorname{BCH}(X,Y)_{\deg_{X,Y}=(n,1)}=(-1)^{n}\beta_{n}\underbrace{[X,[X,%
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D
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Together with the 18 equations we found to ensure that the matrix mxsubscript𝑚𝑥m_{x}italic_m start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT commutes with the complex structure, we now have 24 equations on the 24-dimensional space 𝒮x,x∈Lsubscript𝒮𝑥𝑥𝐿\mathcal{S}_{x},x\in Lcaligraphic_S start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , italic_x ∈ italic_L we require to hold on the matrices, which will now in effect guarantee that they can in fact arise as the derivative (Jacobian) of some embedding as previously discussed.
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It can be easily verified that these two groups of equations are independent and do not imply (or deny) one another. Therefore, by dimensionality arguments in this linear setting of matrices. there must exist solutions to this system of equations, if fact there will be a unique solution for every x∈L𝑥𝐿x\in Litalic_x ∈ italic_L.
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We can in fact show this directly by demonstrating that these two sets (linear spaces) are equal, particularly let v∈Tx(M)∩J(Tx(M))=ηx𝑣subscript𝑇𝑥𝑀𝐽subscript𝑇𝑥𝑀subscript𝜂𝑥v\in T_{x}(M)\cap J(T_{x}(M))=\eta_{x}italic_v ∈ italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_M ) ∩ italic_J ( italic_T start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_M ) ) = italic_η start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT. Hence:
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We also note that it could be of interest to find a new formula for the Bishop invariant in terms of the angles θmsubscript𝜃𝑚\theta_{m}italic_θ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT among the set of complex tangents m∈N𝑚𝑁m\in Nitalic_m ∈ italic_N. We will leave this work for another paper as it would need an analytic sophistication that we do not make use of in our current work, which has been in the spirit of complex differential topology in nature.
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This imposes twelve linear conditions on the (real) 36-dimensional space GL(ℝ6)𝐺𝐿superscriptℝ6GL(\mathbb{R}^{6})italic_G italic_L ( blackboard_R start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT ) for each x∈L𝑥𝐿x\in Litalic_x ∈ italic_L. We thus obtain 24-dimensional real space of matrices for each x∈L𝑥𝐿x\in Litalic_x ∈ italic_L satisfying these conditions. Denote these spaces by:
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A
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Eventually (with probability 1111) there will be one iteration, with index M𝑀Mitalic_M, for which the random input XMsubscript𝑋𝑀X_{M}italic_X start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT indicates that U∉(λM,μM]𝑈subscript𝜆𝑀subscript𝜇𝑀U\notin(\lambda_{M},\mu_{M}]italic_U ∉ ( italic_λ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ]. This is the last iteration. At this point sMsubscript𝑠𝑀s_{M}italic_s start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT is known, and it is also known that τ∈(λM,μM]𝜏subscript𝜆𝑀subscript𝜇𝑀\tau\in(\lambda_{M},\mu_{M}]italic_τ ∈ ( italic_λ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ]. If sM=0subscript𝑠𝑀0s_{M}=0italic_s start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT = 0 the interval (λM,μM]subscript𝜆𝑀subscript𝜇𝑀(\lambda_{M},\mu_{M}]( italic_λ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ] is the lower half of (λM−1,μM−1]subscript𝜆𝑀1subscript𝜇𝑀1(\lambda_{M-1},\mu_{M-1}]( italic_λ start_POSTSUBSCRIPT italic_M - 1 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_M - 1 end_POSTSUBSCRIPT ], thus U>μM≥τ𝑈subscript𝜇𝑀𝜏U>\mu_{M}\geq\tauitalic_U > italic_μ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ≥ italic_τ, and the output Y𝑌Yitalic_Y is 00. Similarly, if sM=2subscript𝑠𝑀2s_{M}=2italic_s start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT = 2 the output is Y=1𝑌1Y=1italic_Y = 1, because U≤λM<τ𝑈subscript𝜆𝑀𝜏U\leq\lambda_{M}<\tauitalic_U ≤ italic_λ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT < italic_τ. If sM=1subscript𝑠𝑀1s_{M}=1italic_s start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT = 1 the interval (λM,μM]subscript𝜆𝑀subscript𝜇𝑀(\lambda_{M},\mu_{M}]( italic_λ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_M end_POSTSUBSCRIPT ] is in the middle of (λM−1,μM−1]subscript𝜆𝑀1subscript𝜇𝑀1(\lambda_{M-1},\mu_{M-1}]( italic_λ start_POSTSUBSCRIPT italic_M - 1 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_M - 1 end_POSTSUBSCRIPT ] and a final input XM+1subscript𝑋𝑀1X_{M+1}italic_X start_POSTSUBSCRIPT italic_M + 1 end_POSTSUBSCRIPT is needed to decide if U𝑈Uitalic_U is in the lower or upper quarter of (λM−1,μM−1]subscript𝜆𝑀1subscript𝜇𝑀1(\lambda_{M-1},\mu_{M-1}]( italic_λ start_POSTSUBSCRIPT italic_M - 1 end_POSTSUBSCRIPT , italic_μ start_POSTSUBSCRIPT italic_M - 1 end_POSTSUBSCRIPT ], both events being equally likely, to determine if the output Y𝑌Yitalic_Y is 1111 or 00 respectively.
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The number of iterations M𝑀Mitalic_M is, by construction, a shifted geometric random variable with parameter 1/2121/21 / 2, and thus for any m∈ℕ𝑚ℕm\in\mathbb{N}italic_m ∈ blackboard_N
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1. Since the number of iterations M𝑀Mitalic_M is a shifted geometric random variable, it is finite with probability 1111.
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The described procedure can be compared with that given in [7, section 1] to simulate a rational constant τ∈(0,1)𝜏01\tau\in(0,1)italic_τ ∈ ( 0 , 1 ): using the binary representation of τ𝜏\tauitalic_τ, which in the rational case is completely known (it is either finite or repeating), output the i𝑖iitalic_i-th binary digit where i𝑖iitalic_i is given by a shifted geometric random variable with parameter 1/2121/21 / 2. To extend this approach for irrational τ𝜏\tauitalic_τ its binary representation, which is not fully known, would have to be computed up to the i𝑖iitalic_i-th digit. That is similar to what the algorithm presented here does: it computes increasingly accurate approximations of τ𝜏\tauitalic_τ until a sufficient level of accuracy, given by the shifted geometric random variable M𝑀Mitalic_M, is achieved.
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Since M𝑀Mitalic_M is a shifted geometric variable with parameter 1/2121/21 / 2, E[M]=2E𝑀2\operatorname{E}[M]=2roman_E [ italic_M ] = 2. Inequality (13) then follows from (31).
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A
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Any finite subgroup of Sp(4,ℂ)Sp4ℂ\operatorname{Sp}(4,\mathbb{C})roman_Sp ( 4 , blackboard_C ) lies inside its maximal compact subgroup, which is the compact symplectic group Sp(2)Sp2\operatorname{Sp}(2)roman_Sp ( 2 ).
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the quotient of a finite subgroup of Sp(4,ℂ)×ℂ∗Sp4ℂsuperscriptℂ\operatorname{Sp}(4,\mathbb{C})\times\mathbb{C}^{*}roman_Sp ( 4 , blackboard_C ) × blackboard_C start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT.
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Any finite subgroup of Sp(4,ℂ)Sp4ℂ\operatorname{Sp}(4,\mathbb{C})roman_Sp ( 4 , blackboard_C ) lies inside its maximal compact subgroup, which is the compact symplectic group Sp(2)Sp2\operatorname{Sp}(2)roman_Sp ( 2 ).
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for some r≥1𝑟1r\geq 1italic_r ≥ 1 and H𝐻Hitalic_H a finite subgroup inside Spin(5)Spin5\operatorname{Spin}(5)roman_Spin ( 5 ).
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In turn Sp(2)Sp2\operatorname{Sp}(2)roman_Sp ( 2 ) is isomorphic to Spin(5)Spin5\operatorname{Spin}(5)roman_Spin ( 5 ).
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D
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ξg(x)=Πg(x)∇g(x)superscript𝜉𝑔𝑥superscriptΠ𝑔𝑥∇𝑔𝑥\xi^{g}(x)=\Pi^{\,g}(x)\nabla g(x)italic_ξ start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ( italic_x ) = roman_Π start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ( italic_x ) ∇ italic_g ( italic_x ): projection of the gradient of g𝑔gitalic_g onto the linear space spanned by
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V⊥g(x)superscriptsubscript𝑉bottom𝑔𝑥V_{\bot}^{g}(x)italic_V start_POSTSUBSCRIPT ⊥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_g end_POSTSUPERSCRIPT ( italic_x ): (d×(d−k))𝑑𝑑𝑘(d\times(d-k))( italic_d × ( italic_d - italic_k ) )-matrix formed by the (d−k)𝑑𝑘(d-k)( italic_d - italic_k ) trailing eigenvectors of Hessian of g𝑔gitalic_g
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\widehat{f}}(x)\big{]}italic_V start_POSTSUBSCRIPT ⊥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over^ start_ARG italic_f end_ARG end_POSTSUPERSCRIPT ( italic_x ) = [ italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over^ start_ARG italic_f end_ARG end_POSTSUPERSCRIPT ( italic_x ) , ⋯ , italic_V start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT over^ start_ARG italic_f end_ARG end_POSTSUPERSCRIPT ( italic_x ) ] be the matrix of the (d−k)𝑑𝑘(d-k)( italic_d - italic_k ) trailing orthonormal eigenvectors of ∇2f^(x)superscript∇2^𝑓𝑥\nabla^{2}\widehat{f}(x)∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT over^ start_ARG italic_f end_ARG ( italic_x ). We recall that the KDE has the form
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-k)}italic_V start_POSTSUBSCRIPT ⊥ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_f end_POSTSUPERSCRIPT ( italic_x ) = [ italic_V start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_f end_POSTSUPERSCRIPT ( italic_x ) , ⋯ , italic_V start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_f end_POSTSUPERSCRIPT ( italic_x ) ] ∈ blackboard_R start_POSTSUPERSCRIPT italic_d × ( italic_d - italic_k ) end_POSTSUPERSCRIPT the matrix of the trailing (d−k)𝑑𝑘(d-k)( italic_d - italic_k ) eigenvectors, we define
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(d−k)𝑑𝑘(d-k)( italic_d - italic_k ) trailing eigenvectors of Hessian of g𝑔gitalic_g
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D
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[ICadq]+[IC0,0q]delimited-[]subscriptsuperscriptIC𝑞addelimited-[]subscriptsuperscriptIC𝑞00[{\operatorname{IC}}^{q}_{\operatorname{ad}}]+[{\operatorname{IC}}^{q}_{0,0}][ roman_IC start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ad end_POSTSUBSCRIPT ] + [ roman_IC start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 , 0 end_POSTSUBSCRIPT ] by Proposition 4.2.4, Corollary 4.2.3
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The class of ICtautq⋆(ICtautq)∗⋆subscriptsuperscriptIC𝑞tautsuperscriptsubscriptsuperscriptIC𝑞taut{\operatorname{IC}}^{q}_{\operatorname{taut}}\star({\operatorname{IC}}^{q}_{%
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It remains to check that ICtautq⋆(ICtautq)∗⋆subscriptsuperscriptIC𝑞tautsuperscriptsubscriptsuperscriptIC𝑞taut{\operatorname{IC}}^{q}_{\operatorname{taut}}\star({\operatorname{IC}}^{q}_{%
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generated by the sheaves ICtautq,(ICtautq)∗subscriptsuperscriptIC𝑞tautsuperscriptsubscriptsuperscriptIC𝑞taut{\operatorname{IC}}^{q}_{\operatorname{taut}},({\operatorname{IC}}^{q}_{%
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ICtautq,(ICtautq)∗,IC0,0qsubscriptsuperscriptIC𝑞tautsuperscriptsubscriptsuperscriptIC𝑞tautsubscriptsuperscriptIC𝑞00{\operatorname{IC}}^{q}_{\operatorname{taut}},({\operatorname{IC}}^{q}_{%
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B
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ψ:ℝ→ℝ:𝜓→ℝℝ\psi:\mathbb{R}\to\mathbb{R}italic_ψ : blackboard_R → blackboard_R is the derivative of ρ𝜌\rhoitalic_ρ,
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and its derivative ψ=ρ′𝜓superscript𝜌′\psi=\rho^{\prime}italic_ψ = italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is 1-Lipschitz.
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we extend ψ𝜓\psiitalic_ψ and ψ′superscript𝜓′\psi^{\prime}italic_ψ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT to functions ℝn→ℝn→superscriptℝ𝑛superscriptℝ𝑛\mathbb{R}^{n}\to\mathbb{R}^{n}blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT → blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT by
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ψ′superscript𝜓′\psi^{\prime}italic_ψ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT the derivative of ψ𝜓\psiitalic_ψ and
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, ψ:=ρ′assign𝜓superscript𝜌′\psi:=\rho^{\prime}italic_ψ := italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and ψ′superscript𝜓′\psi^{\prime}italic_ψ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT denote the derivatives.
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C
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We use base(μ)base𝜇\operatorname{base}(\mu)roman_base ( italic_μ ) to denote the set {c1,…,ck}subscript𝑐1…subscript𝑐𝑘\{c_{1},\dots,c_{k}\}{ italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } and markarc(μ)markarc𝜇\operatorname{markarc}(\mu)roman_markarc ( italic_μ ) to denote {t1,…,tk}subscript𝑡1…subscript𝑡𝑘\{t_{1},\dots,t_{k}\}{ italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_t start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT }. A marking μ𝜇\muitalic_μ is clean if for each i∈{1,…,k}𝑖1…𝑘i\in\{1,\dots,k\}italic_i ∈ { 1 , … , italic_k }, the arc tisubscript𝑡𝑖t_{i}italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the lift of a curve ci′subscriptsuperscript𝑐′𝑖c^{\prime}_{i}italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT such that ci′subscriptsuperscript𝑐′𝑖c^{\prime}_{i}italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and cisubscript𝑐𝑖c_{i}italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT fill a complexity 1 subsurface U𝑈Uitalic_U and dU(ci′,ci)=1subscript𝑑𝑈subscriptsuperscript𝑐′𝑖subscript𝑐𝑖1d_{U}(c^{\prime}_{i},c_{i})=1italic_d start_POSTSUBSCRIPT italic_U end_POSTSUBSCRIPT ( italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = 1.
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A pants subsurface is any subsurface homeomorphic to S0,pnsuperscriptsubscript𝑆0𝑝𝑛S_{0,p}^{n}italic_S start_POSTSUBSCRIPT 0 , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT with p+n=3𝑝𝑛3p+n=3italic_p + italic_n = 3. Given a subsurface U𝑈Uitalic_U of S𝑆Sitalic_S we call a multicurve that is contained in U𝑈Uitalic_U and has maximal cardinality among multicurves on U𝑈Uitalic_U a pants decomposition of U𝑈Uitalic_U. Note, when U𝑈Uitalic_U has any annular components, then every pants decomposition of U𝑈Uitalic_U contains the core curve for each of the annular components of U𝑈Uitalic_U.
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Given a marking μ𝜇\muitalic_μ and a non-pants subsurface U𝑈Uitalic_U, Masur and Minsky defined the subsurface projection of μ𝜇\muitalic_μ as follows:
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Given a simplex μ𝜇\muitalic_μ of Xαsubscript𝑋𝛼X_{\alpha}italic_X start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT, define Uμsubscript𝑈𝜇U_{\mu}italic_U start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT to be the (possibly disconnected) subsurface filled by the curves of Xαsubscript𝑋𝛼X_{\alpha}italic_X start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT that are disjoint from μ𝜇\muitalic_μ. That is, Uμsubscript𝑈𝜇U_{\mu}italic_U start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT is the subsurface filled by the curves in lk(μ)lk𝜇\operatorname{lk}(\mu)roman_lk ( italic_μ ).
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A curve c𝑐citalic_c and subsurface U𝑈Uitalic_U are disjoint if the annulus with core curve c𝑐citalic_c is disjoint from U𝑈Uitalic_U. This extends to define the disjointness of a multicurve and a subsurface. If a multicurve μ𝜇\muitalic_μ is not disjoint from a subsurface U𝑈Uitalic_U, then we say μ𝜇\muitalic_μ and U𝑈Uitalic_U intersect. If the multicurve μ𝜇\muitalic_μ intersects a subsurface U𝑈Uitalic_U, but is disjoint from ∂U𝑈\partial U∂ italic_U, then some component of μ𝜇\muitalic_μ is contained in U𝑈Uitalic_U and we let μ∩U𝜇𝑈\mu\cap Uitalic_μ ∩ italic_U denote this subset of components of μ𝜇\muitalic_μ.
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B
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These lemmas imply that (E1,⋯,En)subscript𝐸1⋯subscript𝐸𝑛(E_{1},\cdots,E_{n})( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) is a complete exceptional sequence. Indeed, if i<j𝑖𝑗i<jitalic_i < italic_j we have Hom(Ej,Ei)=0Homsubscript𝐸𝑗subscript𝐸𝑖0\operatorname{Hom}(E_{j},E_{i})=0roman_Hom ( italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = 0 since Ejsubscript𝐸𝑗E_{j}italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is not a submodule of Eisubscript𝐸𝑖E_{i}italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT by Lemma 1.14 and Eisubscript𝐸𝑖E_{i}italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is not a quotient module of Ejsubscript𝐸𝑗E_{j}italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT by Lemma 1.15. Also, Ext(Ej,Ei)=0Extsubscript𝐸𝑗subscript𝐸𝑖0\operatorname{Ext}(E_{j},E_{i})=0roman_Ext ( italic_E start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = 0 by Lemma 1.16.
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It is easy to convert a rooted labeled forest into a complete exceptional sequence. Start with the following.
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Figure 4 illustrates the sequence of five rooted labeled forests corresponding to the following five complete exceptional sequences for A10subscript𝐴10A_{10}italic_A start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT:
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Figure 1. By Theorem A1, this figure indicates the rooted labeled forest corresponding to the complete exceptional sequence for the quiver A7subscript𝐴7A_{7}italic_A start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT:
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Let F𝐹Fitalic_F be a rooted labeled forest with associated exceptional sequence E∗=(E1,⋯,En)subscript𝐸∗subscript𝐸1⋯subscript𝐸𝑛E_{\ast}=(E_{1},\cdots,E_{n})italic_E start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = ( italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋯ , italic_E start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ). Then for any i<n𝑖𝑛i<nitalic_i < italic_n we have the following.
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A
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We are now ready to define singular composition using corks. As in the nonsingular case, we can compose two triples (S1×I,K1,C1)#s(S2×I,K2,C2)subscript𝑆1𝐼subscript𝐾1subscript𝐶1subscript#𝑠subscript𝑆2𝐼subscript𝐾2subscript𝐶2(S_{1}\times I,K_{1},C_{1})\#_{s}(S_{2}\times I,K_{2},C_{2})( italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT × italic_I , italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) # start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT × italic_I , italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) both with singular corks C1subscript𝐶1C_{1}italic_C start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and C2subscript𝐶2C_{2}italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT by removing the corks and pasting along boundaries of the now-removed corks. Since both corks were singular, each Mi\\CiM_{i}\backslash\backslash C_{i}italic_M start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT \ \ italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT contains a disk Eisubscript𝐸𝑖E_{i}italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT with two arcs on Cisubscript𝐶𝑖C_{i}italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. When composing the two manifolds, we can glue these two disks together along their arcs on the boundaries of Cisubscript𝐶𝑖C_{i}italic_C start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Then E1∪E2subscript𝐸1subscript𝐸2E_{1}\cup E_{2}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_E start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT will be an annulus A𝐴Aitalic_A in the resulting manifold M𝑀Mitalic_M. (An annulus of this type is called a reducing annulus in the literature.) We can cut along A𝐴Aitalic_A, and cap off each resulting copy of A𝐴Aitalic_A with a thickened disk to yield a manifold M′superscript𝑀′M^{\prime}italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT. Importantly, M′superscript𝑀′M^{\prime}italic_M start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is homeomorphic to the manifold obtained by performing singular composition by cutting and pasting along once-punctured annuli defined by singular curves.
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There are three important notes to make about the two types of composition detailed above. First, these do in fact correspond to all possible compositions of diagrams of virtual knots [16]. For examples of how composition with manifolds relates to the diagrammatic composition of virtual knots, the reader should refer to Section 6.
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We include a few explicit examples here. We also include Table 1 of volumes of virtual knots with corks removed to provide volumes for the examples and to allow the reader to play with different compositions without having to calculate volumes of manifolds. In the first column, if the cork is nonsingular, the nonsingular volume volns(S×I,K,C)𝑣𝑜subscript𝑙𝑛𝑠𝑆𝐼𝐾𝐶vol_{ns}(S\times I,K,C)italic_v italic_o italic_l start_POSTSUBSCRIPT italic_n italic_s end_POSTSUBSCRIPT ( italic_S × italic_I , italic_K , italic_C ) is listed and the second volume has N/A for not applicable. All of these examples are hyperbolically composable.
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In [20], Thurston proved that all classical knots fall into three disjoint categories: torus knots, satellite knots, and hyperbolic knots. All compositions of classical knots fall into the category of satellite knots and thus none are hyperbolic. So hyperbolic invariants are useless for studying composition of classical knots. However, virtual knots behave very differently. As we will see, any two non-classical tg-hyperbolic virtual knots have a composition that is a tg-hyperbolic virtual knot.
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Just as we define composition of classical knots, we can define composition of virtual knots in terms of projections. We choose a region of each projection to be the exterior region, remove an arc from an edge of that region, and then glue the two projections together at their endpoints, obtaining a projection of the composition. However, in the category of virtual knots, two virtual knots can have infinitely many distict compositions coming from complicating the original projections before composing.
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A
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Despite their effectiveness in computer applications, error-correcting codes are quite useless if we want to model consensus in biological systems.
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The communication noise studied in this type of problem can be divided into two types: uniform (or unbiased) and non-uniform (or biased). The uniform noise wants to capture errors in communications between agents in real-world scenarios, in which communication noise affects all opinions in the same way. The non-uniform communication noise instead describes the case in which opinions are affected differently from one another.
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In this context, error-correcting codes are very effective methods to reduce communication errors in computer systems [34, 40], and this is why many theoretical studies of the (majority) consensus problem assume that communication between entities occurs without error, and instead consider some adversarial behavior (e.g. byzantine fault [8]).
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The consensus problem is a fundamental problem in distributed computing [6] in which we have a system of agents supporting opinions that interact between each other by exchanging messages, with the goal of reaching an agreement on some valid opinion (i.e. an opinion initially present in the system). In particular, we focus on the majority consensus problem where the goal is to converge towards the initial majority opinion. The numerous theoretical studies in this area find justifications in many different application scenarios, ranging from social networks [41, 2], swarm robotics [5], cloud computing, communication networks [44], and distributed databases [20], to biological systems [25, 26]. As for the latter, the goal of the majority consensus problem is to model some real-world scenarios where biological entities need to communicate and agree in order to pursue some collective task. Many biological entities in different real situations perform this type of process, e.g. molecules [13], bacteria [4], flock of birds [9], school of fish [45], or social insects [27], such as honeybees [43].
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Indeed, they involve sending complicated codes through communication links, and it is reasonable to assume that biological type entities communicate between each other in a simpler way.
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D
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1(ebl+ebr)!eg!er!1subscript𝑒blsubscript𝑒brsubscript𝑒gsubscript𝑒r\frac{1}{(e_{\mathrm{bl}}+e_{\mathrm{br}})!e_{\mathrm{g}}!e_{\mathrm{r}}!}divide start_ARG 1 end_ARG start_ARG ( italic_e start_POSTSUBSCRIPT roman_bl end_POSTSUBSCRIPT + italic_e start_POSTSUBSCRIPT roman_br end_POSTSUBSCRIPT ) ! italic_e start_POSTSUBSCRIPT roman_g end_POSTSUBSCRIPT ! italic_e start_POSTSUBSCRIPT roman_r end_POSTSUBSCRIPT ! end_ARG
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Finally, we regroup the terms in the sum over (Γ,I)Γ𝐼(\Gamma,I)( roman_Γ , italic_I ) to obtain the right-hand side of (18). For this, given (Γ,I)Γ𝐼(\Gamma,I)( roman_Γ , italic_I ) we compute a triple of twisted graphs as follows:
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At this point we have described how to use Theorem 2.4 and Lemma 3.2 to expand the left-hand side of (18) into a graph sum (with insertions being strata of k𝑘kitalic_k-differentials and 1111-differentials) and how to regroup this sum using Theorem 2.4 to obtain the right-hand side of (18). We established these expansions and regroupings on the levels of the involved combinatorial objects (i.e. twisted graphs), but it remains to be verified that in the final step, all summands appear with the correct rational coefficients. Looking at the relevant formulas (18), (9) and (19) most of the factors are easily matched:444Instead of performing Step 3 in the direction described above (regrouping terms (Γ,I)Γ𝐼(\Gamma,I)( roman_Γ , italic_I )), the reader might find it easier to go in the opposite direction and expand the right-hand side of (18) into a graph sum. With this interpretation, the equations (23),(24), (25), (26) below show how the coefficient of the graph sum term (Γ,I)Γ𝐼(\Gamma,I)( roman_Γ , italic_I ) from Steps 1,2 agrees with the coefficient obtained from this inverse of Step 3.
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Due to the factor fs,t(Γℓ,Iℓ)subscript𝑓𝑠𝑡superscriptΓℓsuperscript𝐼ℓf_{s,t}(\Gamma^{\ell},I^{\ell})italic_f start_POSTSUBSCRIPT italic_s , italic_t end_POSTSUBSCRIPT ( roman_Γ start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT , italic_I start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) appearing in Lemma 3.2, any (Γ,I)Γ𝐼(\Gamma,I)( roman_Γ , italic_I ) appearing with nonzero coefficient will have to satisfy that s,t𝑠𝑡s,titalic_s , italic_t appear at two different vertices (the two central vertices depicted in blue). Using this observation, we see that we can uniquely333Strictly speaking the unique reconstruction requires the additional data of an identification of the edges of ΓssuperscriptΓ𝑠\Gamma^{s}roman_Γ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT with the edges in ΓΓ\Gammaroman_Γ incident to the non-central vertices. We will suppress this detail for now and return to it during the last part of the proof, when we match the precise coefficients and multiplicities of all involved terms. reconstruct (Γs,Is)superscriptΓ𝑠superscript𝐼𝑠(\Gamma^{s},I^{s})( roman_Γ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT , italic_I start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ) and (Γℓ,Iℓ)superscriptΓℓsuperscript𝐼ℓ(\Gamma^{\ell},I^{\ell})( roman_Γ start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT , italic_I start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ) given (Γ,I)Γ𝐼(\Gamma,I)( roman_Γ , italic_I ): we obtain ΓssuperscriptΓ𝑠\Gamma^{s}roman_Γ start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT by contracting all edges between the two central components, and we obtain ΓℓsuperscriptΓℓ\Gamma^{\ell}roman_Γ start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT by cutting all other edges and removing all other vertices.
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To conclude we must not just compare the coefficients mentioned above but also with how many different numberings each (Γ,I)Γ𝐼(\Gamma,I)( roman_Γ , italic_I ) can appear. To do this, we use the following result:
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D
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We also visualize benchmarks from previous work, either with (black, dashed) or without (black, solid) covariates. In the background, we indicate ranges of class sizes in the Project STAR protocol (gray).
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As in previous work, we consider the third grade test score to be the short term reward S𝑆Sitalic_S, and a subsequent test score to be the long term reward Y𝑌Yitalic_Y. By choosing different grades as different long term rewards, we evaluate how our methods perform over different horizons. Our variable definitions are identical to [Athey et al., 2020a], except that we use a continuous action.
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We study nonparametric causal functions of a continuous action D𝐷Ditalic_D, which may refer to intervention intensities, medical dosages, or program lengths. For example, Project STAR randomly assigned kindergarten students to various class sizes D𝐷Ditalic_D, ranging from 12 to 28 students. The short term test scores S𝑆Sitalic_S of these students were collected, however their long term earnings Y𝑌Yitalic_Y were not. Separately, researchers may have access to test scores S𝑆Sitalic_S and earnings Y𝑌Yitalic_Y for students in another school district.
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The oracle, visualized in red, is estimated from long term experimental data, i.e. joint observations of the randomized action D𝐷Ditalic_D and long term reward Y𝑌Yitalic_Y in Project STAR. Our goal is to recover similar estimates without access to long term experimental data. Figure 4 shows that the oracle curve is typically decreasing: larger class sizes appear to cause lower test scores, across horizons. In particular, the oracle estimates are nonlinearly decreasing, from positive counterfactual test scores (above average) to negative counterfactual test scores (below average). As the long term horizon increases, i.e. as the definition of Y𝑌Yitalic_Y corresponds to later grades, the oracle curves flatten: the effect of kindergarten class size on test scores appears to attenuate over time.
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We illustrate the practicality of our approach by estimating the long term dose response of Project STAR, modelling class size as a continuous action. By allowing for continuous actions and heterogeneous links, our long term dose response estimate suggests that the effects of class size are nonlinear. Using short term experimental data and long term observational data, our method measures similar long term effects as an oracle method that has access to long term experimental data.
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C
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The latter condition is implied by the former condition. Indeed, X𝑋Xitalic_X is a compactification of the torus 𝕋nsuperscript𝕋𝑛\mathbb{T}^{n}blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, hence by [AKMWo02] there is a zigzag of simple blowups relating X𝑋Xitalic_X to a toric variety Xtsubscript𝑋𝑡X_{t}italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT compactifying 𝕋nsuperscript𝕋𝑛\mathbb{T}^{n}blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. The variety Xtsubscript𝑋𝑡X_{t}italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT has the property that the volume form ω𝜔\omegaitalic_ω has at worst simple poles along Xt∖𝕋nsubscript𝑋𝑡superscript𝕋𝑛X_{t}\setminus\mathbb{T}^{n}italic_X start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∖ blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT, and this condition is maintained for all simple blowups or blowdowns supported on the compliment of 𝕋nsuperscript𝕋𝑛\mathbb{T}^{n}blackboard_T start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT. The assumption that ω𝜔\omegaitalic_ω extends to a non-vanishing volume form on all of U𝑈Uitalic_U then implies the existence of a decomposition of KX+Dsubscript𝐾𝑋𝐷K_{X}+Ditalic_K start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT + italic_D in the form required for Condition 2222.
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Condition 1111 is assumed for the non-archimedean construction of Keel-Yu, and Condition 2222 is assumed for the logarithmic construction of Gross-Siebert. We make the following assumption throughout:
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In this section, we briefly recall the mirror constructions of Gross-Siebert and Keel-Yu in [GS19] and [KY23]. As was mentioned in Remark 3.1, the logarthimic construction given in [GS19] applies to a larger class of log Calabi-Yau targets than the non-archimedean construction given in [KY23]. However, a different set of assumptions are imposed in [GS21] to construct the canonical wall structure for (X,D)𝑋𝐷(X,D)( italic_X , italic_D ). We recall the assumptions needed for these constructions, and their implications in what follows.
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For U𝑈Uitalic_U a smooth affine log Calabi-Yau containing a Zariski dense torus, the Keel-Yu mirror equals the Gross-Siebert mirror for a compactifying pair (X,D)𝑋𝐷(X,D)( italic_X , italic_D ) satisfying the condition of Theorem 1.1. In particular, the Gross-Siebert mirror extends to a family over Spec 𝕜[NE(X)]𝑆𝑝𝑒𝑐 𝕜delimited-[]𝑁𝐸𝑋Spec\text{ }\mathds{k}[NE(X)]italic_S italic_p italic_e italic_c blackboard_k [ italic_N italic_E ( italic_X ) ].
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We will begin by recalling the relationship between tropical and logarithmic moduli problems, followed by briefly recalling the mirrors constructed using non-archimedean and logarithmic geometry. By making use of the Frobenius structure theorem proven by Keel-Yu for their mirror construction, we will reduce the question of comparing the mirrors to proving Theorem 1.1. To prove this theorem, we make use of the restrictions of the targets imposed either by the Zariski dense torus assumption, or those imposed for the construction of the canonical wall structure in [GS21] to remove the toric model assumptions used elsewhere. More precisely, the assumptions allows us to define a useful class of tropical types of punctured log curve called broken line types. Moreover, by appropriately degenerating the point constraint of the log Gromov-Witten invariant of interest, we see that the tropical types of curves contributing to the invariant arise as limits of families of expansions of k𝑘kitalic_k broken lines meeting at a point. We then apply a tropical lemma, together with the non-negative boundary condition, to sufficiently restrict curves potentially contributing to the log Gromov Witten invariant with generic point constraint so that they all lie in the smooth locus of the moduli space of interest, and results from [KY23] Section 3333 will allow us to conclude Theorem 1.1.
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A
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Let d∈𝒥(∂X)𝑑𝒥𝑋d\in\mathcal{J}(\partial X)italic_d ∈ caligraphic_J ( ∂ italic_X ), u:∂X→ℝ:𝑢→𝑋ℝu:\partial X\to\mathbb{R}italic_u : ∂ italic_X → blackboard_R be a d𝑑ditalic_d-Lipschitz function and p>Hausdim(∂X,d)𝑝Hausdim𝑋𝑑p>\mathrm{Hausdim}(\partial X,d)italic_p > roman_Hausdim ( ∂ italic_X , italic_d ). We have ‖dΦ(u)‖p<∞subscriptnorm𝑑Φ𝑢𝑝||d\Phi(u)||_{p}<\infty| | italic_d roman_Φ ( italic_u ) | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT < ∞ and hence [Φ(u)]∈ℓpHcont1(X)delimited-[]Φ𝑢superscriptℓ𝑝superscriptsubscript𝐻cont1𝑋[\Phi(u)]\in\ell^{p}H_{\mathrm{cont}}^{1}(X)[ roman_Φ ( italic_u ) ] ∈ roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT roman_cont end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_X ). Moreover, if u𝑢uitalic_u is non-constant, [Φ(u)]≠0delimited-[]Φ𝑢0[\Phi(u)]\neq 0[ roman_Φ ( italic_u ) ] ≠ 0 in ℓpHcont1(X)superscriptℓ𝑝superscriptsubscript𝐻cont1𝑋\ell^{p}H_{\mathrm{cont}}^{1}(X)roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_H start_POSTSUBSCRIPT roman_cont end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_X ). In particular, ℓpH1(X)≠{0}superscriptℓ𝑝superscript𝐻1𝑋0\ell^{p}H^{1}(X)\neq\{0\}roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_X ) ≠ { 0 } for all p>Confdim(∂X,d)𝑝Confdim𝑋𝑑p>\mathrm{Confdim}(\partial X,d)italic_p > roman_Confdim ( ∂ italic_X , italic_d ).
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The main result of this section is that for large p𝑝pitalic_p, the ℓpsuperscriptℓ𝑝\ell^{p}roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-homology of a building with Weyl group W𝑊Witalic_W does not vanish in degree equal to the virtual cohomological dimension vcdℝ(W)subscriptvcdℝ𝑊\mathrm{vcd}_{\mathbb{R}}(W)roman_vcd start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT ( italic_W ) of W𝑊Witalic_W over ℝℝ\mathbb{R}blackboard_R. This is a generalization of the non-vanishing assertion shown for buildings of type PM𝑃𝑀PMitalic_P italic_M in the previous section. We begin by giving a quick review on the notion of virtual cohomological dimension for Coxeter groups. Then we introduce the Bestvina chamber and the Bestvina realization of a building following [Bes93] to obtain this non-vanishing result.
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We want to define a pushforward for functions on the boundary. For this, let 𝒜osubscript𝒜𝑜\mathcal{A}_{o}caligraphic_A start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT be the set of apartments of X𝑋Xitalic_X containing an origin o𝑜oitalic_o lying in the interior of the central Davis chamber of the retraction ρ𝜌\rhoitalic_ρ. We endow 𝒜osubscript𝒜𝑜\mathcal{A}_{o}caligraphic_A start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT with a probability measure ν𝜈\nuitalic_ν as in [Bou00, Section 2.2.3], the following proposition sums up its main properties.
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We can think of this construction as a Poisson transform defined by integrating functions on the boundary over shadows with respect to a probability measure. Here we chose a Dirac mass on an element of the shadow. The choice of this particular probability measure is not important because the function we are integrating is Lipschitz.
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Similar ideas can be applied to the first ℓpsuperscriptℓ𝑝\ell^{p}roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-cohomology group. Indeed, cochains can be seen as functions f:X(0)→ℝ:𝑓→superscript𝑋0ℝf:X^{(0)}\to\mathbb{R}italic_f : italic_X start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT → blackboard_R on vertices of X𝑋Xitalic_X whose simplicial differential is ℓpsuperscriptℓ𝑝\ell^{p}roman_ℓ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT. Here the pullback ρ∗superscript𝜌\rho^{*}italic_ρ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is naturally defined as precomposition by ρ𝜌\rhoitalic_ρ and pushforward ρ∗subscript𝜌\rho_{*}italic_ρ start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT of functions is defined by taking averages on preimages of ρ𝜌\rhoitalic_ρ. The main question here is: given f:Σ→ℝ:𝑓→Σℝf:\Sigma\to\mathbb{R}italic_f : roman_Σ → blackboard_R with a control on the infimal p>1𝑝1p>1italic_p > 1 such that ‖df‖p<∞subscriptnorm𝑑𝑓𝑝||df||_{p}<\infty| | italic_d italic_f | | start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT < ∞, what can be said about an r>1𝑟1r>1italic_r > 1 such that
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C
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Then sf¯=sf′¯⋅f¯¯𝑠𝑓⋅¯𝑠superscript𝑓′¯𝑓\overline{sf}=\overline{sf^{\prime}}\cdot\overline{f}over¯ start_ARG italic_s italic_f end_ARG = over¯ start_ARG italic_s italic_f start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ⋅ over¯ start_ARG italic_f end_ARG follows from
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If (M,S,η)𝑀𝑆𝜂\left(M,S,\eta\right)( italic_M , italic_S , italic_η ) is a factorable monoid, then the restriction
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Assume that (M,S,η)𝑀𝑆𝜂\left(M,S,\eta\right)( italic_M , italic_S , italic_η ) is a factorable monoid. The
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Let (M,S,η)𝑀𝑆𝜂\left(M,S,\eta\right)( italic_M , italic_S , italic_η ) be a factorable monoid. If (M,S+)𝑀subscript𝑆\left(M,S_{+}\right)( italic_M , italic_S start_POSTSUBSCRIPT + end_POSTSUBSCRIPT )
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(M,S,η)𝑀𝑆𝜂\left(M,S,\eta\right)( italic_M , italic_S , italic_η ) be a factorable monoid. If Condition (4.3)
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D
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Moreover, we show that the assumption that the policy does not depend on the number of samples can be made without loss of optimality, because for any problem, there exists such a policy achieving the optimal asymptotic worst-case regret (see Proposition B-4).
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Our second contribution is the development of a general framework to bound the asymptotic regret of SAA under the assumption that the distance is an integral probability metric (IPM), as defined in Section 4.1. Notably, this broad class of metrics—which includes the Kolmogorov and Wasserstein distances—can be linked to specific generating classes of functions. We leverage this connection to introduce the “approximation parameter,” a quantity that captures the complexity required to uniformly approximate the objective function using functions from the generating class of the IPM. This parameter not only provides a bound on the regret of SAA but also has a natural geometric interpretation. By utilizing standard topological results, we show that this approach enables meaningful regret bounds. Specifically, for the Kolmogorov and Wasserstein distances, our framework relates the worst-case regret of SAA to practical quantities, such as the Lipschitz constant or the total variation of the objective function. Our analysis extends the results derived in the context of “the method of probability metrics” (Rachev and Römisch, 2002). We discuss the relation of our work to these results in Remark 2.
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As mentioned in Section 1.3, the results developed in this section are conceptually close to the method of probability metrics (see e.g. Rachev and Römisch (2002)). We discuss in detail the relation to this work in Remark 2.
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We show in Section 4.3 that the upper bounds based on the approximation parameter directly imply bounds on the worst-case regret of SAA for Newsvendor under both heterogeneity types and for pricing under the Kolmogorov heterogeneity. Furthermore, we complement these results with lower bounds on the best achievable performance and show that for these three settings, SAA achieves the best possible dependence in ϵitalic-ϵ\epsilonitalic_ϵ and its asymptotic worst-case regret scales linearly with ϵitalic-ϵ\epsilonitalic_ϵ. For pricing under Wasserstein distance, the picture is starkly different: the approximation parameter becomes infinite, and hence does not allow us to derive any meaningful upper bound on the worst-case regret of SAA. As a matter of fact, we show that SAA performs extremely poorly and the asymptotic regret does not even vanish when ϵitalic-ϵ\epsilonitalic_ϵ goes to 00. Hence, the performance of SAA deteriorates considerably by slightly deviating from the i.i.d. regime for this class of problems.
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Proposition 4 translates the bounds derived on UnifℐsubscriptUnifℐ\mathrm{Unif}_{\mathcal{I}}roman_Unif start_POSTSUBSCRIPT caligraphic_I end_POSTSUBSCRIPT through the approximation parameter in Section 4.2 into bounds on the worst-case asymptotic regret of SAA. It implies that for a broad class of problems, the asymptotic worst-case regret of SAA vanishes as the radius of the heterogeneity ball goes to 00. More specifically, the asymptotic worst-case regret scales linear for the Kolmogorov distance whenever the objective function g𝑔gitalic_g has uniformly bounded variation, and for the Wasserstein distance
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A
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TpΣ⟂subscript𝑇𝑝superscriptΣperpendicular-toT_{p}\Sigma^{\perp}italic_T start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT roman_Σ start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT and TbBsubscript𝑇𝑏𝐵T_{b}Bitalic_T start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_B so the normal bundle, thus Λ+2(Σ)subscriptsuperscriptΛ2Σ\Lambda^{2}_{+}(\Sigma)roman_Λ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( roman_Σ ), is trivial. Choose a basis Z1,Z2,Z3subscript𝑍1subscript𝑍2subscript𝑍3Z_{1},Z_{2},Z_{3}italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_Z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT for TbBsubscript𝑇𝑏𝐵T_{b}Bitalic_T start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_B.
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We will use the same notation to denote the corresponding normal vector fields along ΣΣ\Sigmaroman_Σ.
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We shall use the same notation as before, but to simplify we shall assume Z𝑍Zitalic_Z is always perpendicular to ΣtsubscriptΣ𝑡\Sigma_{t}roman_Σ start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT: this will not change the volumes.
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where we use the fact that ΣΣ\Sigmaroman_Σ is totally geodesic and Ricci-flat. We conclude that, at b𝑏bitalic_b,
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The restriction to normal vector fields corresponds to the fact that ℳℳ\mathcal{M}caligraphic_M is defined modulo reparametrizations,
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A
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Due to a long manual setup time, the company lacks the flexibility to introduce new PCBs in the build process as well as in responding to unprecedented situations, like COVID-19, for which the company may have to reschedule the build process according to the changed demands. Moreover, the manual setup also leads to a suboptimal solution, requiring more trolleys to load PCB components which are difficult to manage. To address these issues, we automate the TOP in this case study, as discussed below.
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Thus, the TOP aims to find a minimum number of trolleys and stackers of common capacities to load a given set of PCB components of varying sizes (measured by the number of slots required) to build a set of PCBs in an assembly line, subject to the constraint that the total number of trolleys and stackers used for each PCB must not exceed the assembly line capacity (referred as maximum trolley limit). The assignment of components to trolleys and to stackers are exactly similar and are solely dependent on the assembly line capacity. For instance, suppose a PCB is built on an assembly line with a capacity of 16 trolleys. If a PCB requires two stackers, the remaining components must be loaded onto 14 trolleys, as two spaces are occupied by stackers (a stacker and a trolley occupy equal space on the assembly line). This information can be determined in advance based on the PCBs to be manufactured as we can precompute if we need stackers. Leveraging this structural characteristic, the problem is decomposed into two smaller and independent sub-problems: trolley assignment and stacker assignment. Consequently, a single, smaller MILP model is developed to address both assignments. For simplicity, we primarily focus on the trolley assignment (refer to Subsection 3.2), as the stacker assignment is trivial due to the manageable scale of the problem.
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A trolley and a stacker are containers used to hold components, such as registers, capacitors and diodes, required to build PCBs. Instead of plugging individual components directly into CAP machines, i.e., assigning individual components to machine slots (Gao et al. (2021)), components are loaded onto trolleys and stackers which then are plugged into the CAP machines. This helps to manage components, which are large in number and also helps to change components while building a variety of PCBs. So, the TOP is a problem of finding a minimum number of trolleys and stackers with common capacities required to load a given set of PCB components of different sizes to build a given set of PCBs in an assembly line with multiple machines. Key terms used in defining the TOP are defined in Table 1.
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We present the case study of a company having an assembly shop with multiple assembly lines, which builds a variety of PCBs in low-volume and high-mix on each assembly line. The company needs to load components required to build all PCBs on the assembly line, onto trolleys and stackers. Here, two industrial datasets from the company, corresponding to two assembly lines with two and three CAP machines, respectively, are presented, whose statistics are given in Table 3. The two datasets, named as ‘A’ and ‘B’, have 80 and 62 PCBs, each of which is built from a subset of 579 and 930 components, respectively. Most of the components, requiring up to five slots are placed onto the trolleys and bigger components are placed on the stackers. The distribution of the size of components for both datasets is given in Fig. 2. From the figure, it is clear that most of the components need one slot and a small number of components are loaded onto the stackers. Since PCBs in dataset A are built based on an assembly line equipped with two CAP machines, the assembly line’s capacity (referred to as the maximum trolley limit in the model development) is either 14 trolleys and 2 stackers, or 16 trolleys if stackers are not needed. Consequently, for any PCB in dataset A, the trolley/stacker requirements must not exceed the assembly line’s capacity. Similarly, since the PCBs in dataset B are built in an assembly line with three CAP machines so for any PCB in dataset B, the trolley/stacker requirements must not exceed 22 trolleys and 2 stackers or 24 trolleys. We set the maximum number of allowed trolleys equal to the solution obtained using the manual approach, i.e., T=28𝑇28T=28italic_T = 28 and 50 for datasets A and B, respectively. This value is very important because a very small number can make the problem infeasible and a very large number will increase the number of variables, thereby the problem will become computationally expensive.
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The company builds P𝑃Pitalic_P different types of PCBs using C𝐶Citalic_C different types of components. The components can be categorised into two categories, (i) the components which need one to five slots on a container for loading different PCB components onto the CAP machine, and are put on the trolleys, and (ii) the components needing more than five slots on the container and are loaded onto the stackers. The stackers are used for larger components due to placement efficiency. So, the components are loaded onto trolleys and stackers which are then loaded onto CAP machines. Thus, instead of loading individual components directly onto the CAP machine, which could be a very complex process, trolleys and stackers are loaded onto the CAP machine. This makes the task of loading different components for different PCBs easier and more manageable during the operation of the assembly shop (see Fig. 1 for an example of trolleys). Moreover, each component in an assembly line is loaded onto only one trolley/stacker for easy traceability. Each assembly line has a predefined capacity for the number of trolleys and stackers that it can accommodate. Additionally, it’s important to note that both trolleys and stackers occupy an equal amount of space within the assembly line configuration.
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D
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For each square-free integer d>1𝑑1d>1italic_d > 1 such that x2−dy2=−1superscript𝑥2𝑑superscript𝑦21x^{2}-dy^{2}=-1italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_d italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = - 1 has an integral solution, we denote the n𝑛nitalic_n-th smallest positive integral solution of |x2−dy2|=1superscript𝑥2𝑑superscript𝑦21|x^{2}-dy^{2}|=1| italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_d italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | = 1 by (x,y)=(fn(d),gn(d)).𝑥𝑦superscriptsubscript𝑓𝑛𝑑superscriptsubscript𝑔𝑛𝑑(x,y)=(f_{n}^{(d)},g_{n}^{(d)}).( italic_x , italic_y ) = ( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT , italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT ) .
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Every nontrivial integral solution (a,b,c)𝑎𝑏𝑐(a,b,c)( italic_a , italic_b , italic_c ) of (⋆italic-⋆\star⋆ ‣ 1) is given by
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We say that a solution (a,b,c)𝑎𝑏𝑐(a,b,c)( italic_a , italic_b , italic_c ) of (⋆⋆\star⋆ ‣ 1) is trivial if |a|=|b|.𝑎𝑏|a|=|b|.| italic_a | = | italic_b | .
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Conversely, every triple (a,b,c)𝑎𝑏𝑐(a,b,c)( italic_a , italic_b , italic_c ) of Form (1.1) or (1.2) is an integral solution of (⋆italic-⋆\star⋆ ‣ 1).
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Every nontrivial integral solution (a,b,c)𝑎𝑏𝑐(a,b,c)( italic_a , italic_b , italic_c ) of (⋆⋆\star⋆ ‣ 1) can necessarily be written in the form
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A
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\nabla^{2}_{x_{2}x_{1}}f_{2}(x)&0\end{array}\right].italic_H ( italic_x ) := [ start_ARRAY start_ROW start_CELL ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) end_CELL start_CELL ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) end_CELL end_ROW start_ROW start_CELL ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) end_CELL start_CELL ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) end_CELL end_ROW end_ARRAY ] and italic_E ( italic_x ) := [ start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x ) end_CELL end_ROW start_ROW start_CELL ∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_x ) end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ] .
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Additionally, we are going to require some control on the Taylor approximation of the gradients around the solution, as established next:
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While the assumptions of Proposition 4.4 are mostly associated with smoothness of the functions defining the problem, condition (63) is related with the diagonal dominance of H(x∗)𝐻superscript𝑥H(x^{*})italic_H ( italic_x start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ), noticing that the constant 132132\frac{1}{32}divide start_ARG 1 end_ARG start_ARG 32 end_ARG used in (63) can be improved with some small changes in the proof which we opted to omit. Inspired by the results related with the convergence of Newton’s method for optimization problems, it would be natural, however, to expect the quadratic local convergence of Algorithm 2. The key for proving this in optimization problems is first showing that the step-size t=1𝑡1t=1italic_t = 1 is always accepted for convex quadratics when α≤12𝛼12\alpha\leq\frac{1}{2}italic_α ≤ divide start_ARG 1 end_ARG start_ARG 2 end_ARG. Since smooth functions behave locally very similarly to its Taylor quadratic approximations, one can obtain that if α<12𝛼12\alpha<\frac{1}{2}italic_α < divide start_ARG 1 end_ARG start_ARG 2 end_ARG then the unitary step is accepted on a neighborhood of a solution where the second order sufficient conditions hold. However, the situation is very different for the case of NEPs where we have that the approximation of the gradient (8) can make it so that the step-size needs to be reduced in order to attain a good approximation accuracy, regardless of how close we are from a well-behaved solution. The next example illustrates this situation, indicating that it is not straightforward to obtain better than linear local convergence rates. In the example we show that arbitrarily close to the unique solution of a convex NEP, the step-size t=1𝑡1t=1italic_t = 1 may be rejected.
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Looking from the perspective of traditional optimization problems, one would expect the limitation of t𝑡titalic_t for more general cases, since this occurs on a neighborhood of a point where the gradient is Lipschitz continuous, see (28). However, NEPs can behave surprisingly differently than optimization problems. For instance, in [10] it is shown that, in constrained problems, it is not always possible to force the KKT residual to be as close to zero as desired on a neighborhood of a solution, unlike the case of optimization problems. In the case of the algorithm proposed in this work, we highlight that the approximation of the gradient done in (8) can demand step-sizes arbitrarily close to zero in order to achieve good accuracy and thus guarantee descent directions around a point, as is shown in the next example.
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The next result ensures the angle inequalities (14) and (17) hold for t𝑡titalic_t sufficiently small, provided that we require some control on the linear approximation of the gradients, as is stated in the assumption below.
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A
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_{*}(A^{\prime})=A}N_{p,q,r}^{A^{\prime}}italic_N start_POSTSUBSCRIPT italic_p , italic_q , italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT bold_italic_τ ⊢ ( italic_τ , italic_A ) end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT bold_italic_τ end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT bold_italic_τ ⊢ ( italic_τ , italic_A ) end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT bold_italic_γ → bold_italic_τ end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT bold_italic_γ end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_A end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT bold_italic_γ ⊢ ( italic_γ , italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT bold_italic_γ end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) = italic_A end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT italic_p , italic_q , italic_r end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT
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Under the prediction that the mirror algebras constructed in [GS19] and [GS21] are the algebro-geometric realization of degree zero symplectic cohomology of U𝑈Uitalic_U, this result says at least that after changing Novikov parameters, the mirror algebra is invariant under certain modifications of the boundary, which is expected under this prediction.
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Secondly, punctured invariants play a critical role in the mirror construction of Gross and Siebert. This construction takes as input a log Calabi-Yau pair (X,D)𝑋𝐷(X,D)( italic_X , italic_D ), and outputs an algebra over a monoid ring of effective curve classes in X𝑋Xitalic_X which is a candidate for the ring of functions on the mirror to (X,D)𝑋𝐷(X,D)( italic_X , italic_D ). A log étale modification X~→X→~𝑋𝑋\tilde{X}\rightarrow Xover~ start_ARG italic_X end_ARG → italic_X is another log Calabi-Yau compactificaton of U:=X∖Dassign𝑈𝑋𝐷U:=X\setminus Ditalic_U := italic_X ∖ italic_D, and in particular give a pair of distinct mirror algebras R(X,D)subscript𝑅𝑋𝐷R_{(X,D)}italic_R start_POSTSUBSCRIPT ( italic_X , italic_D ) end_POSTSUBSCRIPT and R(X~,D~)subscript𝑅~𝑋~𝐷R_{(\tilde{X},\tilde{D})}italic_R start_POSTSUBSCRIPT ( over~ start_ARG italic_X end_ARG , over~ start_ARG italic_D end_ARG ) end_POSTSUBSCRIPT. After applying the pushforward morphism H2(X~)→H2(X)→subscript𝐻2~𝑋subscript𝐻2𝑋H_{2}(\tilde{X})\rightarrow H_{2}(X)italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( over~ start_ARG italic_X end_ARG ) → italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_X ) to all monomials in the mirror algebra, we wish to identify the mirror algebras produced by the two compactifications of U𝑈Uitalic_U. The following corollary fulfills this wish:
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In the above display, the second equality follows from Proposition 10.1. This gives our desired relation of structure constants of the mirror algebra, and hence completes the proof of Corollary 1.4.
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In order to produce a tropcalization functor, we recall the category of generalized cone complexes, introduced in [ACP15], and the construction from Appendix C𝐶Citalic_C of [ACGS20b] of the tropicalization functor ΣΣ\Sigmaroman_Σ from algebraic fine log stacks to generalized cone complexes. If an Artin fan 𝒜𝒜\mathcal{A}caligraphic_A has a Zariski cover by Artin cones, we will say 𝒜𝒜\mathcal{A}caligraphic_A is a Zariski Artin fan. The following proposition follows from the proof of Proposition 2.102.102.102.10 of [ACGS20a].
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C
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K_{n}□ start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_K start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. We then use this new approach to improve the upper bound in Theorem 1.3, leading to Theorem 1.4.
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We finish the introduction by pointing out that our results also give improvements for multidimensional lexicographic-monotone array.
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Finally, we apply Lemma 3.1 to S⊆[t]d−1𝑆superscriptdelimited-[]𝑡𝑑1S\subseteq[t]^{d-1}italic_S ⊆ [ italic_t ] start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT. By the choices of t,u,ϵ^𝑡𝑢^italic-ϵt,u,\hat{\epsilon}italic_t , italic_u , over^ start_ARG italic_ϵ end_ARG we have that t≥ϵ^−g(d−1)nd−2⋅ng(d−1)nd−2𝑡⋅superscript^italic-ϵ𝑔𝑑1superscript𝑛𝑑2superscript𝑛𝑔𝑑1superscript𝑛𝑑2t\geq\hat{\epsilon}^{-g(d-1)n^{d-2}}\cdot n^{g(d-1)n^{d-2}}italic_t ≥ over^ start_ARG italic_ϵ end_ARG start_POSTSUPERSCRIPT - italic_g ( italic_d - 1 ) italic_n start_POSTSUPERSCRIPT italic_d - 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ⋅ italic_n start_POSTSUPERSCRIPT italic_g ( italic_d - 1 ) italic_n start_POSTSUPERSCRIPT italic_d - 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT, and so we obtain a monotone (d−1)𝑑1(d-1)( italic_d - 1 )-dimensional array T𝑇Titalic_T of size [u]d−1superscriptdelimited-[]𝑢𝑑1[u]^{d-1}[ italic_u ] start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT. Since B𝐵Bitalic_B is d𝑑ditalic_d-consistent, T×{a}𝑇𝑎T\times\{a\}italic_T × { italic_a } is monotone for every a∈A𝑎𝐴a\in Aitalic_a ∈ italic_A (with the same choice of direction of monotonicity). By construction, this gives a monotone d𝑑ditalic_d-dimensional array on T×A𝑇𝐴T\times Aitalic_T × italic_A.
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We say a 1111-dimensional array is consistent if it is monotone. For d≥2𝑑2d\geq 2italic_d ≥ 2, we say an array f:[N]d→ℝ:𝑓→superscriptdelimited-[]𝑁𝑑ℝf:[N]^{d}\rightarrow\mathbb{R}italic_f : [ italic_N ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → blackboard_R is d𝑑ditalic_d-consistent if
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A multidimensional array is said to be monotone if for each direction all the 1111-dimensional subarrays in that direction are increasing or decreasing.
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A
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For four ququads, the algorithm converges to the antisymmetric state |Ψ4⟩ketsubscriptΨ4|\Psi_{4}\rangle| roman_Ψ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ⟩, yielding a measure of 2324≈0.958323240.9583\frac{23}{24}\approx 0.9583divide start_ARG 23 end_ARG start_ARG 24 end_ARG ≈ 0.9583. Interestingly, while being 1111-uniform, this state is not AME. The so far only known AME(4,4)AME44\text{AME}(4,4)AME ( 4 , 4 ), a graph state Helwig (2013); Burchardt and Raissi (2020), yields a measure of 1516=0.937515160.9375\frac{15}{16}=0.9375divide start_ARG 15 end_ARG start_ARG 16 end_ARG = 0.9375.
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Still, the analysis of AME states is important for understanding quantum error correction and regarded as one of the central problems in the field Horodecki et al. (2022); Rather et al. (2022).
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Similarly, there exists a general procedure to construct AME(5,d)AME5𝑑\text{AME}(5,d)AME ( 5 , italic_d )
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such a state is then denoted by AME(n,d)AME𝑛𝑑\text{AME}(n,d)AME ( italic_n , italic_d ). Interestingly, not for all
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Finally, the recently found AME(4,6)AME46\text{AME}(4,6)AME ( 4 , 6 ) Rather et al. (2022) is not maximally entangled
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D
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Suppose that {Dk}k=1∞superscriptsubscriptsubscript𝐷𝑘𝑘1\{D_{k}\}_{k=1}^{\infty}{ italic_D start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT is a sequence of open discs in the complex plane such that ∑k=1∞r(Dk)<1superscriptsubscript𝑘1𝑟subscript𝐷𝑘1\sum_{k=1}^{\infty}r(D_{k})<1∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_r ( italic_D start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) < 1, and set K=D¯∖⋃k=1∞Dk𝐾¯𝐷superscriptsubscript𝑘1subscript𝐷𝑘K=\overline{D}\setminus\bigcup_{k=1}^{\infty}D_{k}italic_K = over¯ start_ARG italic_D end_ARG ∖ ⋃ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_D start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. Then R(K)≠C(K)𝑅𝐾𝐶𝐾R(K)\not=C(K)italic_R ( italic_K ) ≠ italic_C ( italic_K ).
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A set of the form K=D¯∖⋃k=1∞Dk𝐾¯𝐷superscriptsubscript𝑘1subscript𝐷𝑘K=\overline{D}\setminus\bigcup_{k=1}^{\infty}D_{k}italic_K = over¯ start_ARG italic_D end_ARG ∖ ⋃ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_D start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT with {Dk}k=1∞superscriptsubscriptsubscript𝐷𝑘𝑘1\{D_{k}\}_{k=1}^{\infty}{ italic_D start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT a sequence of open discs such that ∑k=1∞r(Dk)<∞superscriptsubscript𝑘1𝑟subscript𝐷𝑘\sum_{k=1}^{\infty}r(D_{k})<\infty∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_r ( italic_D start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) < ∞ is called a Swiss cheese. (Sometimes in the literature a more restrictive definition of Swiss cheese is used.) Thus the set K𝐾Kitalic_K in Theorem 1.2 is a Swiss cheese. For such a set K𝐾Kitalic_K, the following standard result gives a very useful criterion insuring that R(K)𝑅𝐾R(K)italic_R ( italic_K ) is nontrivial [23, Lemma 24.1].
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Feinstein and the author introduced a general method for constructing essential uniform algebras. Using this method they constructed an essential, natural, regular uniform algebra on the closed unit disc D¯¯𝐷\overline{D}over¯ start_ARG italic_D end_ARG [12, Theorem 1.2]. Repeating the proof of [12, Theorem 1.2] with the strongly regular uniform algebra of Theorem 1.2 above in place of McKissick’s normal uniform algebra shows that the result can be strengthened by replacing regularity by strong regularity. (By a result of Wilken [26, Lemma], every strongly regular uniform algebra is natural, so we omit mention of naturality from the statement of the theorem.)
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The first example of a nontrivial normal uniform algebra was given by Robert McKissick [21]. His example is R(K)𝑅𝐾R(K)italic_R ( italic_K ) for a certain Swiss cheese K𝐾Kitalic_K. Theorem 1.2 is thus a sharpening of McKissick’s result.
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In addition, the example in Theorem 1.1 is the first strongly regular uniform algebra known to be finitely generated; for K𝐾Kitalic_K a compact planar set, R(K)𝑅𝐾R(K)italic_R ( italic_K ) is always generated by two functions [23, Corollary 24.4].
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C
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The paper is organised as follows: RCDT, POD and interpolation methodology are introduced in section 2, and their numerical implementation with simple test cases is reported in section 3. To observe and test the errors described above and assess the capabilities of RCDT in the context of MOR, we consider a number of images and flow case studies in section 3.1 and section 4. Test cases to quantify specific errors can be found in section 3.1. These include discontinuous images of a unit circle and a circular ring and continuous Gaussian functions to observe the intrinsic error in the RCDT workflow, and the flow field given by a twin jet at different separation widths to test the interpolation in RCDT space compared to the physical.
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This work has focused on implementing and verifying the Radon-Cumulative Distribution Transform (RCDT) for image and flow capture and assessing its applicability in model order reduction (MOR) – under proper orthogonal decomposition (POD) – of high-fidelity CFD input data. RCDT and subsequent RCDT-POD MOR workflows were tested for accuracy compared against either the original input images or standard POD in physical space.
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The paper is organised as follows: RCDT, POD and interpolation methodology are introduced in section 2, and their numerical implementation with simple test cases is reported in section 3. To observe and test the errors described above and assess the capabilities of RCDT in the context of MOR, we consider a number of images and flow case studies in section 3.1 and section 4. Test cases to quantify specific errors can be found in section 3.1. These include discontinuous images of a unit circle and a circular ring and continuous Gaussian functions to observe the intrinsic error in the RCDT workflow, and the flow field given by a twin jet at different separation widths to test the interpolation in RCDT space compared to the physical.
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In section 4, instead, we focus on the complete MOR procedure starting with a simple moving Gaussian distribution, transformed into RCDT space and order-reduced using POD, compared alongside ’standard’ POD in physical space. We then test our workflow for a multi-phase fluid wave and the flow around an airfoil using high-resolution CFD data. Final discussions and future work directions are then reported in section 5.
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In this work, we utilise the peculiar properties of the RCDT to capture geometric and spatial variations within a parameterised input and use this to produce an approximate solution for system parameters in a model order reduction methodology. Initially, we investigate the properties of the RCDT with simplified test cases to gauge the strengths and weaknesses of the potential use of the transform in the ROM and CFD communities. The RCDT is then applied to a POD-based ROM workflow and later tested on a number of computational fluid dynamics (CFD) data sets. This allows for the preservation of the flow features when transformed between spaces, alongside the accuracy of ROM’s flow reconstruction at reduced order. We finally introduce interpolation and study the error in predicting flows, giving an initial qualitative gauge of RCDTs’ applicability to fluid dynamics and advection-dominated problems.
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C
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It is an ongoing endeavour to understand which graph properties lead to convergence and divergence respectively, however, since clique convergence is known to be undecidable in general [2], this investigation often restricts to certain graph classes, such as graphs of low degree [17], circular arc graphs [13], or locally H𝐻Hitalic_H graphs (e. g. locally cyclic graphs [6] or shoal graphs [4]).
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The focus of the present article is on locally cyclic graphs, that is, graphs for which the neighbourhood of each vertex induces a cycle.
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can be formalised as graphs for which each open neighbourhood is either a cycle (of length at least four) or a path graph – we shall call them locally cyclic with boundary.
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Moving on from the topologically motivated investigations, yet another route is to generalise from locally cyclic graphs of a particular minimum degree to graphs of a lower-bounded local girth (that is, the girth of each open neighbourhood is bounded from below).
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Can the results for locally cyclic graphs of minimum degree δ≥6𝛿6\delta\geq 6italic_δ ≥ 6 be generalized to graphs of local girth ≥6absent6\geq 6≥ 6?
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A
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\underline{\bf X})^{T}=\underline{\bf X}^{{\dagger}}*\underline{\bf X}.( under¯ start_ARG bold_X end_ARG ∗ under¯ start_ARG bold_X end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = under¯ start_ARG bold_X end_ARG ∗ under¯ start_ARG bold_X end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT , ( under¯ start_ARG bold_X end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ∗ under¯ start_ARG bold_X end_ARG ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT = under¯ start_ARG bold_X end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ∗ under¯ start_ARG bold_X end_ARG .
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The procedure of the computation of the GTCUR for tensor triples is summarized in Algorithm 8. Lines 6-8 can be efficiently computed in the Fourier domain and similar algorithms like Algorithm 6 can be developed for this computation. The t-RSVD of the tensor triplets (𝐗¯,𝐘¯,𝐙¯)¯𝐗¯𝐘¯𝐙(\underline{\bf X},\underline{\bf Y},\underline{\bf Z})( under¯ start_ARG bold_X end_ARG , under¯ start_ARG bold_Y end_ARG , under¯ start_ARG bold_Z end_ARG ) provides the common tensor factors 𝐋¯,𝐖¯¯𝐋¯𝐖\underline{\bf L},\,\underline{\bf W}under¯ start_ARG bold_L end_ARG , under¯ start_ARG bold_W end_ARG and extra tensor factors 𝐔¯¯𝐔\underline{\bf U}under¯ start_ARG bold_U end_ARG and 𝐕¯¯𝐕\underline{\bf V}under¯ start_ARG bold_V end_ARG. These bases can be used to sample lateral and horizontal slices of the tensor triplets (𝐗¯,𝐘¯,𝐙¯)¯𝐗¯𝐘¯𝐙(\underline{\bf X},\underline{\bf Y},\underline{\bf Z})( under¯ start_ARG bold_X end_ARG , under¯ start_ARG bold_Y end_ARG , under¯ start_ARG bold_Z end_ARG ). Since we have the common tensor factors 𝐋¯¯𝐋\underline{\bf L}under¯ start_ARG bold_L end_ARG and 𝐖¯¯𝐖\underline{\bf W}under¯ start_ARG bold_W end_ARG, the indices, which are used to sample the horizontal slices of the data tensors 𝐗¯¯𝐗\underline{\bf X}under¯ start_ARG bold_X end_ARG and 𝐙¯¯𝐙\underline{\bf Z}under¯ start_ARG bold_Z end_ARG are the same, while the indices for the selection of lateral slices of the data tensors 𝐗¯¯𝐗\underline{\bf X}under¯ start_ARG bold_X end_ARG and 𝐘¯¯𝐘\underline{\bf Y}under¯ start_ARG bold_Y end_ARG are identical. Nevertheless, the tensor bases 𝐔¯,𝐕¯¯𝐔¯𝐕\underline{\bf U},\,\underline{\bf V}under¯ start_ARG bold_U end_ARG , under¯ start_ARG bold_V end_ARG are used to sample lateral slice indices of the tensor 𝐘¯¯𝐘\underline{\bf Y}under¯ start_ARG bold_Y end_ARG and horizontal slice indices of the tensor 𝐙¯¯𝐙\underline{\bf Z}under¯ start_ARG bold_Z end_ARG. The visualization of this approach is described in Figure 2. So, the idea is to compute the t-RSVD of the tensor triplets (𝐗¯,𝐘¯,𝐙¯)¯𝐗¯𝐘¯𝐙(\underline{\bf X},\underline{\bf Y},\underline{\bf Z})( under¯ start_ARG bold_X end_ARG , under¯ start_ARG bold_Y end_ARG , under¯ start_ARG bold_Z end_ARG ) and to find the corresponding tensor factors 𝐔¯,𝐕¯,𝐙¯,𝐖¯¯𝐔¯𝐕¯𝐙¯𝐖\underline{\bf U},\,\underline{\bf V},\,\underline{\bf Z},\,\underline{\bf W}under¯ start_ARG bold_U end_ARG , under¯ start_ARG bold_V end_ARG , under¯ start_ARG bold_Z end_ARG , under¯ start_ARG bold_W end_ARG and use them to find the indices for selecting horizontal and lateral slices using the TDEIM algorithm. We should remark that the computation of the t-RSVD is demanding for the case of large-scale data tensors and the fast randomized GSVD algorithm proposed in [40] can be used for the computation of the double GSVD, which are required to compute the t-RSVD of the tensor triplets. The connection between the GTCUR for tensor triplets with the GTCUR for tensor pairs and the TCUR is described in the next theorem.
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We see that the GTSVD provides the same right tensor 𝐙¯¯𝐙\underline{\bf Z}under¯ start_ARG bold_Z end_ARG in (14)-(15) and we can use it to sample lateral slices of the data tensors 𝐗¯¯𝐗\underline{\bf X}under¯ start_ARG bold_X end_ARG and 𝐘¯¯𝐘\underline{\bf Y}under¯ start_ARG bold_Y end_ARG based on the TDEIM algorithm. As a result, the same indices can be used to sample the lateral slices for the data tensors 𝐗¯¯𝐗\underline{\bf X}under¯ start_ARG bold_X end_ARG and 𝐘¯¯𝐘\underline{\bf Y}under¯ start_ARG bold_Y end_ARG. The horizontal slices of the data tensors 𝐗¯¯𝐗\underline{\bf X}under¯ start_ARG bold_X end_ARG and 𝐘¯¯𝐘\underline{\bf Y}under¯ start_ARG bold_Y end_ARG can also be sampled using the left tensor parts 𝐔¯¯𝐔\underline{\bf U}under¯ start_ARG bold_U end_ARG and 𝐕¯¯𝐕\underline{\bf V}under¯ start_ARG bold_V end_ARG, although they do not necessarily provide identical horizontal slice indices. Following this idea, we can compute the GTSVD of the tensors 𝐗¯,𝐘¯¯𝐗¯𝐘\underline{\bf X},\,\underline{\bf Y}under¯ start_ARG bold_X end_ARG , under¯ start_ARG bold_Y end_ARG and by applying the TDEIM to the shared tensor factors 𝐔¯,𝐕¯,𝐙¯¯𝐔¯𝐕¯𝐙\underline{\bf U},\,\underline{\bf V},\,\underline{\bf Z}under¯ start_ARG bold_U end_ARG , under¯ start_ARG bold_V end_ARG , under¯ start_ARG bold_Z end_ARG, we can select indices of horizontal and lateral slices of the given data tensors. This approach is summarized in Algorithm 6. In Line 1 of Algorithm 6, we need to compute the GTSVD of two tensors and this is clearly prohibitive for large-scale tensors. However, to tackle this problem, the randomized algorithms proposed in [38] can be used. Note that Lines 10-11 in Algorithm 6 can be efficiently computed, and this is outlined in Algorithm 7.
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The MP pseudoinverse of a tensor can also be computed in the Fourier domain and this is shown in Algorithm 2.
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The basis tensors 𝐔¯¯𝐔\underline{\bf U}under¯ start_ARG bold_U end_ARG and 𝐕¯¯𝐕\underline{\bf V}under¯ start_ARG bold_V end_ARG required in Algorithm 4 can be computed very fast through the randomized truncated t-SVD [31, 32, 33]. This version can be regarded as a randomized version of the TDEIM algorithm.
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C
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For which rational numbers a𝑎aitalic_a and b𝑏bitalic_b are the slopes of the angle bisectors between two straight lines with slopes a𝑎aitalic_a and b𝑏bitalic_b rational?
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For which rational numbers a𝑎aitalic_a and b𝑏bitalic_b are the slopes of the angle bisectors between two straight lines with slopes a𝑎aitalic_a and b𝑏bitalic_b rational?
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The bisector of one of the angles and that of the supplementary angle are perpendicular to each other.
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Given two straight lines, we consider the two angles formed by them, regardless of whether they are acute or not.
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In the case when they are not parallel to the coordinate axes, if one of the slopes is rational, then so is the other, since the product of the slopes is −1.1-1.- 1 .
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C
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Consider a regular correspondence (ϕ,XAA)italic-ϕsubscriptsubscript𝑋𝐴𝐴(\phi,{}_{A}X_{A})( italic_ϕ , start_FLOATSUBSCRIPT italic_A end_FLOATSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) and let i:EndA0(X)→EndA(X):𝑖→superscriptsubscriptEnd𝐴0𝑋subscriptEnd𝐴𝑋i\colon\operatorname{End}_{A}^{0}(X)\to\operatorname{End}_{A}(X)italic_i : roman_End start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_X ) → roman_End start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_X ) denote the inclusion.
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so ψ−1(EndA0(X))⊂(ψ⊗IdB)−1(EndB0(X⊗αB))superscript𝜓1superscriptsubscriptEnd𝐴0𝑋superscripttensor-product𝜓subscriptId𝐵1superscriptsubscriptEnd𝐵0subscripttensor-product𝛼𝑋𝐵\psi^{-1}(\operatorname{End}_{A}^{0}(X))\subset(\psi\otimes\operatorname{Id}_{%
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(α1⊗IdEndA0(X),(X⊗αB)⊗ψEndA0(X))≅(i⊗IdEndA0(X),X⊗ϕEndA0(X))tensor-productsubscript𝛼1subscriptIdsuperscriptsubscriptEnd𝐴0𝑋subscripttensor-product𝜓subscripttensor-product𝛼𝑋𝐵superscriptsubscriptEnd𝐴0𝑋tensor-product𝑖subscriptIdsuperscriptsubscriptEnd𝐴0𝑋subscripttensor-productitalic-ϕ𝑋superscriptsubscriptEnd𝐴0𝑋\big{(}\alpha_{1}\otimes\operatorname{Id}_{\operatorname{End}_{A}^{0}(X)},(X%
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It follows from [Pim97, Corollary 3.7] that if ϕX(a)⊗IdY∈EndC0(X⊗Y)tensor-productsubscriptitalic-ϕ𝑋𝑎subscriptId𝑌superscriptsubscriptEnd𝐶0tensor-product𝑋𝑌\phi_{X}(a)\otimes\operatorname{Id}_{Y}\in\operatorname{End}_{C}^{0}(X\otimes Y)italic_ϕ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_a ) ⊗ roman_Id start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ∈ roman_End start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( italic_X ⊗ italic_Y ), then ϕX(a)∈EndA0(X⋅ϕY−1(EndB0(Y)))subscriptitalic-ϕ𝑋𝑎superscriptsubscriptEnd𝐴0⋅𝑋superscriptsubscriptitalic-ϕ𝑌1superscriptsubscriptEnd𝐵0𝑌\phi_{X}(a)\in\operatorname{End}_{A}^{0}(X\cdot\phi_{Y}^{-1}(\operatorname{End%
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Then (i⊗IdEndA0(X),X⊗EndA0(X))tensor-product𝑖subscriptIdsuperscriptsubscriptEnd𝐴0𝑋tensor-product𝑋superscriptsubscriptEnd𝐴0𝑋(i\otimes\operatorname{Id}_{\operatorname{End}_{A}^{0}(X)},X\otimes%
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D
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In this section, we establish some promising results concerning the values of ∀∃for-all\forall\exists∀ ∃-sentences in free group factors. Recall that a ∀∃for-all\forall\exists∀ ∃-sentence is a sentence σ𝜎\sigmaitalic_σ of the form supx→infy→φ(x→,y→)subscriptsupremum→𝑥subscriptinfimum→𝑦𝜑→𝑥→𝑦\sup_{\vec{x}}\inf_{\vec{y}}\varphi(\vec{x},\vec{y})roman_sup start_POSTSUBSCRIPT over→ start_ARG italic_x end_ARG end_POSTSUBSCRIPT roman_inf start_POSTSUBSCRIPT over→ start_ARG italic_y end_ARG end_POSTSUBSCRIPT italic_φ ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ), where φ𝜑\varphiitalic_φ is a quantifier-free formula. For a II1 factor ℳℳ\mathcal{M}caligraphic_M, we consider the ∀∃for-all\forall\exists∀ ∃-theory Th∀∃(ℳ)subscriptThfor-allℳ\operatorname{Th}_{\forall\exists}(\mathcal{M})roman_Th start_POSTSUBSCRIPT ∀ ∃ end_POSTSUBSCRIPT ( caligraphic_M ) of ℳℳ\mathcal{M}caligraphic_M, which is the function σ↦σℳmaps-to𝜎superscript𝜎ℳ\sigma\mapsto\sigma^{\mathcal{M}}italic_σ ↦ italic_σ start_POSTSUPERSCRIPT caligraphic_M end_POSTSUPERSCRIPT defined on the set of all ∀∃for-all\forall\exists∀ ∃ sentences.
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Our results concerning ∀∃for-all\forall\exists∀ ∃-sentences will follow from the existence of certain nice embeddings between some pairs of interpolated free group factors.
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In this section, we establish some promising results concerning the values of ∀∃for-all\forall\exists∀ ∃-sentences in free group factors. Recall that a ∀∃for-all\forall\exists∀ ∃-sentence is a sentence σ𝜎\sigmaitalic_σ of the form supx→infy→φ(x→,y→)subscriptsupremum→𝑥subscriptinfimum→𝑦𝜑→𝑥→𝑦\sup_{\vec{x}}\inf_{\vec{y}}\varphi(\vec{x},\vec{y})roman_sup start_POSTSUBSCRIPT over→ start_ARG italic_x end_ARG end_POSTSUBSCRIPT roman_inf start_POSTSUBSCRIPT over→ start_ARG italic_y end_ARG end_POSTSUBSCRIPT italic_φ ( over→ start_ARG italic_x end_ARG , over→ start_ARG italic_y end_ARG ), where φ𝜑\varphiitalic_φ is a quantifier-free formula. For a II1 factor ℳℳ\mathcal{M}caligraphic_M, we consider the ∀∃for-all\forall\exists∀ ∃-theory Th∀∃(ℳ)subscriptThfor-allℳ\operatorname{Th}_{\forall\exists}(\mathcal{M})roman_Th start_POSTSUBSCRIPT ∀ ∃ end_POSTSUBSCRIPT ( caligraphic_M ) of ℳℳ\mathcal{M}caligraphic_M, which is the function σ↦σℳmaps-to𝜎superscript𝜎ℳ\sigma\mapsto\sigma^{\mathcal{M}}italic_σ ↦ italic_σ start_POSTSUPERSCRIPT caligraphic_M end_POSTSUPERSCRIPT defined on the set of all ∀∃for-all\forall\exists∀ ∃ sentences.
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We then turn to investigating existential embeddings between the free group factors in Section 4, and obtain that the ∀∃for-all\forall\exists∀ ∃-theory of L(𝔽r)𝐿subscript𝔽𝑟L(\mathbb{F}_{r})italic_L ( blackboard_F start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT ) is increasing in r𝑟ritalic_r.
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If the first-order fundamental group of the free group factors is not trivial, then the ∀∃for-all\forall\exists∀ ∃-theory of all interpolated free group factors is the same.
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A
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Let 𝒫𝒫\mathcal{P}caligraphic_P be an arbitrary Prob-solvable loop and suppose that a (non-basic) state variable x∈𝒫𝑥𝒫x\in\mathcal{P}italic_x ∈ caligraphic_P has a non-polynomial L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-type update g(𝐙)𝑔𝐙g({\mathbf{Z}})italic_g ( bold_Z ), where 𝐙=(Z1,…,Zk)T𝐙superscriptsubscript𝑍1…subscript𝑍𝑘𝑇{\mathbf{Z}}=(Z_{1},\ldots,Z_{k})^{T}bold_Z = ( italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_Z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT is a vector of (basic) continuous, independent, and identically distributed random variables across iterations. That is, if fZj(n)subscript𝑓subscript𝑍𝑗𝑛f_{Z_{j}(n)}italic_f start_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_n ) end_POSTSUBSCRIPT is the pdf of the random variable Zjsubscript𝑍𝑗Z_{j}italic_Z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT in iteration n𝑛nitalic_n, then fZj(n)≡fZj(n′)subscript𝑓subscript𝑍𝑗𝑛subscript𝑓subscript𝑍𝑗superscript𝑛′f_{Z_{j}(n)}\equiv f_{Z_{j}(n^{\prime})}italic_f start_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_n ) end_POSTSUBSCRIPT ≡ italic_f start_POSTSUBSCRIPT italic_Z start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT, for all iterations n,n′𝑛superscript𝑛′n,n^{\prime}italic_n , italic_n start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT and j=1,…,k𝑗1…𝑘j=1,\ldots,kitalic_j = 1 , … , italic_k. The basic random variables Z1,…,Zksubscript𝑍1…subscript𝑍𝑘Z_{1},\ldots,Z_{k}italic_Z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_Z start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and the update function g𝑔gitalic_g satisfy conditions (A)–(E) in Section 3.1.
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For this class of Prob-solvable loops, conditions (A)–(C) in Section 3.1 hold, but (D) and/or (E) may not be fulfilled.
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Our methods can accommodate non-linear, non-polynomial updates in classes of probabilistic loops amenable to automated moment computation, such as the class of Prob-solvable loops.
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Due to the polynomial arithmetic supported in Prob-solvable loops, non-polynomial functions of random variables can be mixed via multiplication in the resulting program.
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For the class of Prob-solvable loops where all variables with non-polynomial updates satisfy these conditions, the computation of the Fourier coefficients in the PCE approximation (8) can be carried out as explained in Section 3.2.
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D
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In this section we will identify the constructible Witt theory of étale sheaves of ΛΛ\Lambdaroman_Λ-modules on SpecℝSpecℝ\mathrm{Spec\ }\mathbb{R}roman_Spec blackboard_R, for a ring ΛΛ\Lambdaroman_Λ of finite characteristic not equal to 2, as a ℤ/2ℤℤ2ℤ\mathbb{Z}/2\mathbb{Z}blackboard_Z / 2 blackboard_Z-equivariant Witt theory of ΛΛ\Lambdaroman_Λ. Also for a real projective variety f:X→Specℝ:𝑓→𝑋Specℝf:X\rightarrow\mathrm{Spec\ }\mathbb{R}italic_f : italic_X → roman_Spec blackboard_R we construct natural homomorphisms
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Wi(f∗):Wci(X,Λ)→Wci(ℝ,Λ)=Wlfi(Λ[ℤ/2ℤ]):superscript𝑊𝑖subscript𝑓→subscriptsuperscript𝑊𝑖𝑐𝑋Λsubscriptsuperscript𝑊𝑖𝑐ℝΛsubscriptsuperscript𝑊𝑖𝑙𝑓Λdelimited-[]ℤ2ℤW^{i}(f_{*}):W^{i}_{c}(X,\Lambda)\rightarrow W^{i}_{c}(\mathbb{R},\Lambda)=W^{%
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Wi(f∗):Wci(Xe´t,Λ)→Wci((Specℂ)e´t,Λ)=Wlfi(Λ):superscript𝑊𝑖subscript𝑓→subscriptsuperscript𝑊𝑖𝑐subscript𝑋´𝑒𝑡Λsubscriptsuperscript𝑊𝑖𝑐subscriptSpecℂ´𝑒𝑡Λsubscriptsuperscript𝑊𝑖𝑙𝑓ΛW^{i}(f_{*}):W^{i}_{c}(X_{\acute{e}t},\Lambda)\rightarrow W^{i}_{c}((\mathrm{%
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Wi(f∗):Wci(Xe´t,Λ)→Wci((Specℝ)e´t,Λ)=Wlfi(Λ[ℤ/2ℤ]).:superscript𝑊𝑖subscript𝑓→subscriptsuperscript𝑊𝑖𝑐subscript𝑋´𝑒𝑡Λsubscriptsuperscript𝑊𝑖𝑐subscriptSpecℝ´𝑒𝑡Λsubscriptsuperscript𝑊𝑖𝑙𝑓Λdelimited-[]ℤ2ℤW^{i}(f_{*}):W^{i}_{c}(X_{\acute{e}t},\Lambda)\rightarrow W^{i}_{c}((\mathrm{%
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Wi(f∗):Wci(Xe´t,Λ)→Wci((Specℝ)e´t,Λ)=Wlfi(Λ[ℤ/2ℤ]).:superscript𝑊𝑖subscript𝑓→subscriptsuperscript𝑊𝑖𝑐subscript𝑋´𝑒𝑡Λsubscriptsuperscript𝑊𝑖𝑐subscriptSpecℝ´𝑒𝑡Λsubscriptsuperscript𝑊𝑖𝑙𝑓Λdelimited-[]ℤ2ℤW^{i}(f_{*}):W^{i}_{c}(X_{\acute{e}t},\Lambda)\rightarrow W^{i}_{c}((\mathrm{%
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A
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We also use the notation (S±)psuperscriptsuperscript𝑆plus-or-minus𝑝(S^{\pm})^{p}( italic_S start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT to denote p𝑝pitalic_p-many positive (negative) stabilizations. A stabilization of K𝐾Kitalic_K is independent of its location in the front projection up to Legendrian isotopy. Adding a stabilization changes tb𝑡𝑏tbitalic_t italic_b by −11-1- 1 and rotation number by ±1plus-or-minus1\pm 1± 1.
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For a topological knot type 𝒦𝒦\mathcal{K}caligraphic_K, let ℒ(𝒦)/∼\mathcal{L(K)}/\simcaligraphic_L ( caligraphic_K ) / ∼ denote the set of equivalence classes of Legendrian representatives of 𝒦𝒦\mathcal{K}caligraphic_K up to Legendrian isotopy. The Cost function gives a metric on the set ℒ(𝒦)/∼\mathcal{L(K)}/\simcaligraphic_L ( caligraphic_K ) / ∼ in the following way. The map
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A (Legendrian) topological knot type is an equivalence class of (Legendrian) topological knots up to (Legendrian) topological isotopy. For a topological knot type 𝒦𝒦\mathcal{K}caligraphic_K, we use the notation ℒ(𝒦)ℒ𝒦\mathcal{L(K)}caligraphic_L ( caligraphic_K ) to denote the set of all Legendrian representatives of 𝒦𝒦\mathcal{K}caligraphic_K.
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Using the Cost function we define a graph invariant of the topological knot. We associate a graph G𝒦subscript𝐺𝒦G_{\mathcal{K}}italic_G start_POSTSUBSCRIPT caligraphic_K end_POSTSUBSCRIPT to a topological knot type 𝒦𝒦\mathcal{K}caligraphic_K in the following way. Fix the set of vertices for G𝒦subscript𝐺𝒦G_{\mathcal{K}}italic_G start_POSTSUBSCRIPT caligraphic_K end_POSTSUBSCRIPT to be ℒ(𝒦)/∼\mathcal{L(K)}/\simcaligraphic_L ( caligraphic_K ) / ∼. Two vertices [K],[K~]∈ℒ(𝒦)/∼[K],[\tilde{K}]\in\mathcal{L(K)}/\sim[ italic_K ] , [ over~ start_ARG italic_K end_ARG ] ∈ caligraphic_L ( caligraphic_K ) / ∼ are joined by an edge if and only if Cost(K,K~)=1Cost𝐾~𝐾1\text{Cost}(K,\tilde{K})=1Cost ( italic_K , over~ start_ARG italic_K end_ARG ) = 1. We refer to this graph as the graph of Legendrian representatives of 𝒦𝒦{\mathcal{K}}caligraphic_K. The Cost function graph for unknot is shown in Figure 22.
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Let 𝒦𝒦\mathcal{K}caligraphic_K be a topological knot type which is Legendrian simple. Let K𝐾Kitalic_K and K~~𝐾\tilde{K}over~ start_ARG italic_K end_ARG be Legendrian representatives of 𝒦𝒦\mathcal{K}caligraphic_K. Then
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B
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Since h=Φ2(q~,h)ℎsubscriptΦ2~𝑞ℎh=\Phi_{2}(\widetilde{q},h)italic_h = roman_Φ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( over~ start_ARG italic_q end_ARG , italic_h ), by (3.7) h(s)>0ℎ𝑠0h(s)>0italic_h ( italic_s ) > 0 for any s<−b1𝑠subscript𝑏1s<-b_{1}italic_s < - italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. This together with (3.31) implies that (3.9) holds in (−∞,−b1)subscript𝑏1(-\infty,-b_{1})( - ∞ , - italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). Multiplying (3.7) by ea0(q~,s)+(n−2)ssuperscript𝑒subscript𝑎0~𝑞𝑠𝑛2𝑠e^{a_{0}(\widetilde{q},s)+(n-2)s}italic_e start_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( over~ start_ARG italic_q end_ARG , italic_s ) + ( italic_n - 2 ) italic_s end_POSTSUPERSCRIPT and differentiating with respect to s𝑠sitalic_s we get that hℎhitalic_h satisfies (3.4) in (−∞,−b1)subscript𝑏1(-\infty,-b_{1})( - ∞ , - italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ).
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By an argument similar to the proof of Theorem 2.2 but with Lemma 3.1 replacing Lemma 2.1 in the proof we get the following theorem.
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By Corollary 3.4 and an argument similar to the proof of Corollary 2.4 we have the following result.
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By an argument similar to the proof of Proposition 2.7 but with Theorem 1.2 replacing Theorem 1.1 in the proof there we have the following result.
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By an argument similar to the proof of Theorem 1.1 but with Theorem 3.2 replacing Theorem 2.2 in the proof, we get Theorem 1.2 and the following corollary.
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A
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E.g., if one assumes that the system is governed by normal diffusive behaviour and it is started in equilibrium, then J0(t)subscript𝐽0𝑡J_{0}(t)italic_J start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_t ) grows typically as t1/4superscript𝑡14t^{1/4}italic_t start_POSTSUPERSCRIPT 1 / 4 end_POSTSUPERSCRIPT, see e.g. [14].
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Our second theorem shows that a long time is needed before energy starts getting dissipated across the system for small values of λ𝜆\lambdaitalic_λ:
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Our next theorem shows that we should not expect κ𝜅\kappaitalic_κ to scale polynomially with λ𝜆\lambdaitalic_λ,
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and presumably does not affect the fundamental characteristics of the long-time behavior of the dynamics, so that the restriction to the atomic limit should not fundamentally alter the long-time behaviour of the system.
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In Section 4, we develop a perturbative expansion in λ𝜆\lambdaitalic_λ for the observables appearing in our theorems.
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A
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Here, Pv+Pv⟂=Isubscript𝑃𝑣subscript𝑃superscript𝑣perpendicular-to𝐼P_{v}+P_{v^{\perp}}=Iitalic_P start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT + italic_P start_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_I denote the orthogonal projections onto span{v}span𝑣\mathrm{span}\{v\}roman_span { italic_v } and {v}⟂superscript𝑣perpendicular-to\{v\}^{\perp}{ italic_v } start_POSTSUPERSCRIPT ⟂ end_POSTSUPERSCRIPT respectively.
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In this section, we use the a priori estimates to give a construction scheme for the systems (1) and (5) using an iteration scheme. Before we can do this, however, we need the following Lemma which guarantees that solutions to the linear system exist (under very strong hypotheses):
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The spaces 𝔈𝔈\mathfrak{E}fraktur_E differ from the spaces in [6] denoted by the same symbols. Specifically, the 𝔈𝔈\mathfrak{E}fraktur_E norm here controls a small amount of v𝑣vitalic_v regularity through the ⟨∇v⟩rsuperscriptdelimited-⟨⟩subscript∇𝑣𝑟\langle\nabla_{v}\rangle^{r}⟨ ∇ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ⟩ start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT term. Such control on velocity derivatives was not present in the previous paper, and requires additional analysis. This additional control is necessary to handle the non-perturbative setting of this paper.
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Using these matrices, we define the following spaces which will capture the dissipation produced by the various collision operators.
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Before we can state our main theorem, we must define a number of function spaces in which the solutions shall be constructed. The framework used here builds on the techniques developed in [6].
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C
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\operatorname{des}(t_{0},\mathbf{t}_{1},t_{p})}Y_{t_{1}}\ldots Y_{t_{p-1}}.italic_λ start_POSTSUPERSCRIPT roman_asc ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( italic_λ - 1 ) start_POSTSUPERSCRIPT roman_des ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , bold_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT italic_Y start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT … italic_Y start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT italic_p - 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT .
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In spirit, it sends the formal variable Ytsubscript𝑌𝑡Y_{t}italic_Y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT into Yt+νsubscript𝑌𝑡𝜈Y_{t+\nu}italic_Y start_POSTSUBSCRIPT italic_t + italic_ν end_POSTSUBSCRIPT if t∈[0,1−ν)𝑡01𝜈t\in[0,1-\nu)italic_t ∈ [ 0 , 1 - italic_ν ),
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and it sends the formal variable Ytsubscript𝑌𝑡Y_{t}italic_Y start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT into Yt+ν−1subscript𝑌𝑡𝜈1Y_{t+\nu-1}italic_Y start_POSTSUBSCRIPT italic_t + italic_ν - 1 end_POSTSUBSCRIPT if t∈[1−ν,1)𝑡1𝜈1t\in[1-\nu,1)italic_t ∈ [ 1 - italic_ν , 1 ).
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Z_{[a+\nu-1,b+\nu-1)}&\text{if }[a,b)\subset[1-\nu,1).\end{cases}roman_Tns start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ( italic_Z start_POSTSUBSCRIPT [ italic_a , italic_b ) end_POSTSUBSCRIPT ) = { start_ROW start_CELL italic_Z start_POSTSUBSCRIPT [ italic_a + italic_ν , italic_b + italic_ν ) end_POSTSUBSCRIPT end_CELL start_CELL if [ italic_a , italic_b ) ⊂ [ 0 , 1 - italic_ν ) , end_CELL end_ROW start_ROW start_CELL italic_Z start_POSTSUBSCRIPT [ italic_a + italic_ν - 1 , italic_b + italic_ν - 1 ) end_POSTSUBSCRIPT end_CELL start_CELL if [ italic_a , italic_b ) ⊂ [ 1 - italic_ν , 1 ) . end_CELL end_ROW
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Whenever 𝐭𝐭\mathbf{t}bold_t makes an excursion into [1−ν,1)1𝜈1[1-\nu,1)[ 1 - italic_ν , 1 ) or [0,ν)0𝜈[0,\nu)[ 0 , italic_ν ), respectively,
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D
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This comes entirely from observing the coefficients of matrices Atsubscript𝐴𝑡A_{t}italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT and At~~subscript𝐴𝑡\tilde{A_{t}}over~ start_ARG italic_A start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG. In this case, we can check that the t𝑡titalic_t-degree of the scalar only depends on the value of the pair (j,k)𝑗𝑘(j,k)( italic_j , italic_k ), which encourages us to interpret vectors e1subscript𝑒1e_{1}italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, e2subscript𝑒2e_{2}italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, e3subscript𝑒3e_{3}italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and e4subscript𝑒4e_{4}italic_e start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT as states that evolve as the computation progresses, with the t𝑡titalic_t-degree that will evolve in parallel. We refer to the degree of a specific scalar as its weight like in the proof of Proposition 3.13 and we use the same graph notation to track these weights. We write all possible such graphs below. Specifically, the left most edge in the top left graph should be understood as stating: “for any i𝑖iitalic_i such that b(vi⊗e1)𝑏tensor-productsubscript𝑣𝑖subscript𝑒1b(v_{i}\otimes e_{1})italic_b ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) is non zero, b(vi⊗e1)=λi(q,t)e1𝑏tensor-productsubscript𝑣𝑖subscript𝑒1subscript𝜆𝑖𝑞𝑡subscript𝑒1b(v_{i}\otimes e_{1})=\lambda_{i}(q,t)\leavevmode\nobreak\ e_{1}italic_b ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_q , italic_t ) italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with the weight of λi(q,t)subscript𝜆𝑖𝑞𝑡\lambda_{i}(q,t)italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_q , italic_t ) being 1111.”
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On each of the four previous rows, the left graph and the right one are the same. This means that with respect to what we are interested in, going through b𝑏bitalic_b or through c𝑐citalic_c is equivalent. This means that boxes b𝑏bitalic_b and c𝑐citalic_c can be identified here;
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Like in the proof of Proposition 3.13, we compute QUq𝔤𝔩(2|1)σ,Vα(T)(e1)superscript𝑄subscript𝑈𝑞𝔤𝔩superscriptconditional21𝜎subscript𝑉𝛼𝑇subscript𝑒1Q^{U_{q}\mathfrak{gl}(2|1)^{\sigma},V_{\alpha}}(T)(e_{1})italic_Q start_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT fraktur_g fraktur_l ( 2 | 1 ) start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT , italic_V start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_T ) ( italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and we want to keep track of the t𝑡titalic_t-degree of the quantity we have while we advance the computation towards finding the result. We just have to understand what the cost of entering a box b𝑏bitalic_b or c𝑐citalic_c is in terms of degree in t𝑡titalic_t for a vector e1subscript𝑒1e_{1}italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, e2subscript𝑒2e_{2}italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, e3subscript𝑒3e_{3}italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT or e4subscript𝑒4e_{4}italic_e start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT (and what the outcoming vector is). Like in the previous proof, this amounts to computing b(vi⊗ej)𝑏tensor-productsubscript𝑣𝑖subscript𝑒𝑗b(v_{i}\otimes e_{j})italic_b ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) and c(vi∗⊗ej)𝑐tensor-productsuperscriptsubscript𝑣𝑖subscript𝑒𝑗c(v_{i}^{*}\otimes e_{j})italic_c ( italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ⊗ italic_e start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) for all possible values of i𝑖iitalic_i and j𝑗jitalic_j. For each of these quantities that is non zero, it is a scalar multiple of one of the basis vectors eksubscript𝑒𝑘e_{k}italic_e start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. We are interested in the t𝑡titalic_t-degree of that scalar, and in the value of k𝑘kitalic_k as a function of (i,j)𝑖𝑗(i,j)( italic_i , italic_j ).
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Now as we advance the computation, the initial vector entering box b𝑏bitalic_b will not necessarily be e1subscript𝑒1e_{1}italic_e start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT anymore. However, drawing similar graphs from matrix Azsubscript𝐴𝑧A_{z}italic_A start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT for initial vectors e2subscript𝑒2e_{2}italic_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, e3subscript𝑒3e_{3}italic_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and e4subscript𝑒4e_{4}italic_e start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT, one can see that this z4superscript𝑧4z^{4}italic_z start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT upper bound is true regardless of the basis vector entering b𝑏bitalic_b. Analog considerations on matrix A~zsubscript~𝐴𝑧\tilde{A}_{z}over~ start_ARG italic_A end_ARG start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT prove that in the worst case scenario, the contribution of c𝑐citalic_c to the degree in z𝑧zitalic_z is z0=1superscript𝑧01z^{0}=1italic_z start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT = 1.
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We will write these four matrices relative to the bases we used in the previous section and that we recall here.
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A
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Because the bound on the truncated Red’s function is uniform for all x∈ℤd𝑥superscriptℤ𝑑x\in\mathbb{Z}^{d}italic_x ∈ blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, we may sum over all x𝑥xitalic_x on each side of the hyperplane {x⋅u→=0}⋅𝑥→𝑢0\{x\cdot\vec{u}=0\}{ italic_x ⋅ over→ start_ARG italic_u end_ARG = 0 } and get the same bounds. This argument essentially shows that conditioning on the walk’s being transient in one direction or the other does not affect the bound on the expected proportion of time it spends at red sites, and therefore does not affect the bounds on the expected displacement after a large number of steps. Since there is at most one possible limiting velocity on each side of the hyperplane, these bounds on conditional expectations can then be used to obtain almost-sure bounds on the limiting velocity. The result of this work is Proposition 1, which is used in the proof of Theorems 1 and 3, reducing each of them to a matter of proving bounds, uniform in x∈ℤd𝑥superscriptℤ𝑑x\in\mathbb{Z}^{d}italic_x ∈ blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, on the truncated annealed Red’s functions in terms of the truncated annealed Blue’s functions.
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is more difficult, and for this we use a coupling technique. Red sites are typically surrounded by blue sites. To show that a walk does not spend too much time on such red sites, we can couple two walks in environments that are exactly the same except at site x𝑥xitalic_x, which is blue in one environment and red in the other. The two walks decouple whenever they hit x𝑥xitalic_x. Then the particle in the environment for which x𝑥xitalic_x is blue freezes, while the particle for which x𝑥xitalic_x is red will hit x𝑥xitalic_x at most a geometric number of times before it uses the blue sites surrounding x𝑥xitalic_x to navigate to the point adjacent to x𝑥xitalic_x where the first particle is waiting. At this time, they recouple and continue as before. This allows us to bound the expected amount of time at x𝑥xitalic_x in the environment where x𝑥xitalic_x is red and surrounded by blue sites in terms of the expected amount of time at x𝑥xitalic_x in the environment where x𝑥xitalic_x is blue, irrespective of the rest of the environment. Then, by taking p𝑝pitalic_p large enough, we can make the environments where x𝑥xitalic_x is red rare enough that under the annealed measure, the walk is expected to spend much more time in blue sites than in red sites surrounded by blue sites.
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Because the bound on the truncated Red’s function is uniform for all x∈ℤd𝑥superscriptℤ𝑑x\in\mathbb{Z}^{d}italic_x ∈ blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, we may sum over all x𝑥xitalic_x on each side of the hyperplane {x⋅u→=0}⋅𝑥→𝑢0\{x\cdot\vec{u}=0\}{ italic_x ⋅ over→ start_ARG italic_u end_ARG = 0 } and get the same bounds. This argument essentially shows that conditioning on the walk’s being transient in one direction or the other does not affect the bound on the expected proportion of time it spends at red sites, and therefore does not affect the bounds on the expected displacement after a large number of steps. Since there is at most one possible limiting velocity on each side of the hyperplane, these bounds on conditional expectations can then be used to obtain almost-sure bounds on the limiting velocity. The result of this work is Proposition 1, which is used in the proof of Theorems 1 and 3, reducing each of them to a matter of proving bounds, uniform in x∈ℤd𝑥superscriptℤ𝑑x\in\mathbb{Z}^{d}italic_x ∈ blackboard_Z start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, on the truncated annealed Red’s functions in terms of the truncated annealed Blue’s functions.
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For the proof of Theorem 1, our comparison of the Red’s and Blue’s functions comes from exact expressions of ratios of these functions that involve probabilities of hitting and then returning to a site x𝑥xitalic_x, conditioned on the first step away from x𝑥xitalic_x. The main calculation is brief, but we need to truncate the walk at a random (geometric) time τ𝜏\tauitalic_τ, rather than a fixed T𝑇Titalic_T, in order to make it work. This geometric truncation is inspired by the methods of [13].
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In this section, we give a criterion for directional transience and ballisticity in terms of truncated Blue’s and Red’s functions. The criterion will be used in the proofs of both Theorem 3 and Theorem 1. Because the latter proof requires truncating at a random time, we give the criterion in terms of a geometric random variable τ𝜏\tauitalic_τ rather than a fixed T𝑇Titalic_T.
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C
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Let L=K(a)𝐿𝐾𝑎L=K(\sqrt{a})italic_L = italic_K ( square-root start_ARG italic_a end_ARG ). Then (25) gives
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Br1(Y)/Br(ℚ)=H1(ℚ,Hom(E¯,E¯′))=H1(Gal(L/ℚ),Hom(E¯,E¯′)).subscriptBr1𝑌BrℚsuperscriptH1ℚHom¯𝐸superscript¯𝐸′superscriptH1Gal𝐿ℚHom¯𝐸superscript¯𝐸′\operatorname{Br}_{1}(Y)/\operatorname{Br}({\mathbb{Q}})=\mathrm{H}^{1}({%
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(Br(A)/Br1(A))3∞=Br(A)3∞/Br1(A)3∞.(\operatorname{Br}(A)/\operatorname{Br}_{1}(A))_{3^{\infty}}=\operatorname{Br}%
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Br(Y)/Br1(Y)=Br(Y)ℓ/Br1(Y)ℓ.\operatorname{Br}(Y)/\operatorname{Br}_{1}(Y)=\operatorname{Br}(Y)_{\ell}/%
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Br1(Y)/Br(ℚ)=H1(Gal(L/ℚ),Hom(E¯,E¯′))subscriptBr1𝑌BrℚsuperscriptH1Gal𝐿ℚHom¯𝐸superscript¯𝐸′\operatorname{Br}_{1}(Y)/\operatorname{Br}({\mathbb{Q}})=\mathrm{H}^{1}(%
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D
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In this article, we have provided an affirmative answer to this above question by establishing Theorem 1111.
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In this paper, our aim is to prove the following Theorem 1 which indicates the structure of rank 2222 Fuchsian Schottky groups with generating sets that are non-classical on the hyperbolic plane. After that, as a consequence of Theorem 1, we deduce Corollary 1.0.1 and 1.0.2 that provide the two non-trivial examples of such groups in the upper-half plane with the circle at infinity as the boundary.
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It is well known that a Fuchsian group is a discrete subgroup of PSL(2,ℝ)𝑃𝑆𝐿2ℝPSL(2,\mathbb{R})italic_P italic_S italic_L ( 2 , blackboard_R ). On the other hand, a Schottky group is a special type of Kleinian group whereas the Kleinian groups are the discrete subgroups of PSL(2,ℂ)𝑃𝑆𝐿2ℂPSL(2,\mathbb{C})italic_P italic_S italic_L ( 2 , blackboard_C ) (see, [7] and [3] for details). In this paper, our goal is to set up the structure of the Fuchsian Schottky groups with non-classical generating sets by using the above two conditions (a)𝑎(a)( italic_a ) and (b)𝑏(b)( italic_b ) in the upper-half plane model with the circle at infinity as the boundary (see, Figure 2). In particular, in this article, after looking at the strategy performed by Yamamoto in [11], we explicitly present the structure of the rank 2222 Fuchsian Schottky groups with non-classical generating sets by imposing some new ideas in the technique operated by Yamamoto in the manuscript [11]. Firstly, for the rank 2222 Schottky group, Yamamoto [11] pointed out a suitable positive real number (viz., 22\sqrt{2}square-root start_ARG 2 end_ARG) in his used hyperbolic Möbius transformations (one of the generators out of two) so that the group became non-classical in the extended complex plane. But in this investigation, for rank 2222 Fuchsian Schottky groups we have observed that there doesn’t exist an analogous positive real number such that the groups become non-classical in the upper-half plane. Although the presence of negative real numbers is identified. For the lacuna of the existence of such a positive real number, in this study, we have taken the ‘distance𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒distanceitalic_d italic_i italic_s italic_t italic_a italic_n italic_c italic_e’ (in the Euclidean sense) between the origin and the particular number (to specify the position of that number in the hyperbolic plane, i.e., the modulus values of the numbers, without loss of generality of the construction) in the used Möbius transformations (count as the rank of that group) to become these types of Schottky groups non-classical. Secondly, Yamamoto’s construction [11] was fabricated by two circles and two rectangles in the Riemann sphere where circles were occupied by the reflection of the imaginary axis and rectangles were built up with some kind of dilation (with rotation). On the other hand, here we have approached the construction of non-classical Fuchsian Schottky groups on the upper-half plane with four semi-circles centered on the circle at infinity, and all the four semi-circles are situated by the reflections of the upper imaginary axis. Thirdly, in Yamamoto’s paper [11], the centers of the circles are lying on the real axis whereas the midpoints of the rectangles are the same and it is the origin of the axis. However, the centers of all the semi-circles in our construction belong to ℝ−[−1,+1]ℝ11\mathbb{R}-[-1,+1]blackboard_R - [ - 1 , + 1 ]. Further, the defining curves for Yamamoto’s non-classical Schottky group form the Jordan curves, whereas, in non-classical Fuchsian Schottky construction the semi-circles together with the diameters lying on the real axis create the Jordan loops look like the hollow half-moons (see, the red color curves in Figure 2222). In fact, in this manuscript, we have utilized these four half-moons as the defining curves to create the non-classical Fuchsian Schottky groups in the hyperbolic plane. Fourthly, Yamamoto [11] has applied the sufficiently small positive number in one generator out of two, but the nature of the construction of Fuchsian Schottky groups demands to use of the sufficiently small positive number in both the generators to become the group non-classical which we have provided in this work. These are the four new notions that we have imposed in this paper. In this way, we have developed the literature by explicitly setting up the hollow half-moons as Jordan curves in the Schottky structure for Fuchsian Schottky groups and produced two non-trivial examples of such groups with non-classical generating sets in the upper-half plane with boundary.
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In [9], we have provided the structure of arbitrary finite rank classical Fuchsian Schottky groups in the hyperbolic plane with a boundary on the harmony of the real Schottky groups with two subsequent additional conditions:
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In 1974, Marden ([5], [6]) introduced the concept of non-classical Schottky groups with a non-constructive proof. In 1975, Zarrow [12] claimed that he had found an example of a rank 2222 non-classical Schottky group, but later it was proved to be classical by Sato [8]. More precisely, the Schottky group constructed by Zarrow [12] was, in fact, a classical Schottky group, but on a different set of generators, and the demonstration of this was the main result in Sato’s paper [8]. After that, in 1991 Yamamoto [11] explicitly presented an example of the rank 2222 non-classical Schottky group in the Riemann sphere. Then, in 2009 Williams [10] also delivered an example of the rank 2 non-classical Schottky group. So, in the literature, the works related to the non-classical structure that have been studied in Schottky groups are all in Kleinian flavor. In this article, we aim to investigate the Schottky structure in the Fuchsian Schottky groups with non-classical generating sets in Fuchsian flavor. In [2], Button proved that all Fuchsian Schottky groups are classical Schottky groups, but not necessarily on the same set of generators. Furthermore, Button in [2] gave an example (with figure) of a Fuchsian group which is Schottky on a particular generating set, but non-classical on those generators. On the other hand, Marden ([5], [6]) also established that, all Fuchsian Schottky groups are classical. Therefore, in this direction, it is quite a difficult and fascinating problem to examine the following question.
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C
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As n→∞→𝑛n\rightarrow\inftyitalic_n → ∞ with |un(yn)−un(y1)|≤d(yn,y1)subscript𝑢𝑛subscript𝑦𝑛subscript𝑢𝑛subscript𝑦1𝑑subscript𝑦𝑛subscript𝑦1|u_{n}(y_{n})-u_{n}(y_{1})|\leq d(y_{n},y_{1})| italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) - italic_u start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) | ≤ italic_d ( italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ), it follows that u(y0)−u(y1)=r=d(y0,y1)𝑢subscript𝑦0𝑢subscript𝑦1𝑟𝑑subscript𝑦0subscript𝑦1u(y_{0})-u(y_{1})=r=d(y_{0},y_{1})italic_u ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) - italic_u ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_r = italic_d ( italic_y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ). Thus u𝑢uitalic_u is a metric viscosity solution.
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However, the classical theory of viscosity solution relies heavily on the differential structure, hindering its extension to general metric spaces. Fortunately, various kinds of metric viscosity solutions have been provided in [2, 14, 15, 29, 30, 33] on metric spaces, and these metric viscosity solutions only depend on the metric structure. For their concrete definitions and relationships among them, refer to [16].
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From Corollary 3.9, generally speaking, the concept of the metric viscosity solution in our study differs from the definition of the curve-based solution of (1.1) in [14], where a classical viscosity solution (classical solution) of (1.1) may not necessarily be a curve-based solution of (1.1), see [14, Example 2.4].
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It is known that, in infinite-dimensional Banach spaces, such stability may fail for various types of viscosity solution. For details, we refer to [31, Subsection 3.1], [32, Example 2.1] and [14, Example 5.5].
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Next, we will discuss the stability of the metric viscosity solutions. Meanwhile, we will present an example that dlC-function is not a metric viscosity solution.
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C
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The multiple lobes of the spectral valleys realized in the current study highlight two primary advancements. First, due to the beneficial interplay between the FGVD and SPM, these lobes can be observed even in the case of weaker nonlinearity. In contrast, for purely nonlinear systems, where regular SPM dominates, multiple lobes typically require a much higher degree of nonlinearity to be noticeable [39, 9]. Second, the lobes of the spectral valleys can be controlled on demand via the phase modulation, which offers a high-efficiency nonlinear shaping regime, in comparison with the linear pulse shaping methods by employing spectral amplitude modulations [63, 28].
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These advancements indicate that the spectral valleys observed in this work are promising for applications to optical
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The multiple lobes of the spectral valleys realized in the current study highlight two primary advancements. First, due to the beneficial interplay between the FGVD and SPM, these lobes can be observed even in the case of weaker nonlinearity. In contrast, for purely nonlinear systems, where regular SPM dominates, multiple lobes typically require a much higher degree of nonlinearity to be noticeable [39, 9]. Second, the lobes of the spectral valleys can be controlled on demand via the phase modulation, which offers a high-efficiency nonlinear shaping regime, in comparison with the linear pulse shaping methods by employing spectral amplitude modulations [63, 28].
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spectral valley can be found. This suggests a promising application for dense data encoding by using these spectral valley states, as shown in the next subsection.
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Although the concept of the fractional derivatives and dispersion has a long history, their realization in physical systems, such as nonlinear fiber optics, is a relatively new and emerging topic. This inspires several promising perspectives: 1) The immediate goal is to explore additional solutions of the FNLSE. The current results focus primarily on positive fractional dispersion lengths, corresponding to the “repulsion” case in the force model, which explains the observed spectral valleys. However, equally intriguing results are expected in the “attraction” case, which would illustrate spectral squeezing. Recently two works have demonstrated an “attraction” effect (narrowed spectrum) in similar contexts [65, 66]. By involving additional parameters, such as higher-order dispersion or stronger nonlinearity, even greater diversity in the pulse dynamics could be uncovered. 2) The second perspective is the realization of the spatial-temporal light synthesis by incorporating both fractional dispersion and fractional diffraction. This approach has significant potential for applications to optical encoding and spatiotemporal mode-locked laser architectures [67, 68]. 3) Finally, due to the mathematical similarity between the FNLSE in nonlinear optics and its quantum counterpart, this regime may serve as an effective model for emulating fractional quantum mechanics [1, 69].
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A
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(\hat{g})_{\nu}-\hat{g}=c&\text{on }T.\end{cases}{ start_ROW start_CELL over¯ start_ARG roman_Δ end_ARG over^ start_ARG italic_g end_ARG = 1 end_CELL start_CELL in roman_Ω , end_CELL end_ROW start_ROW start_CELL ( over^ start_ARG italic_g end_ARG ) start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT - over^ start_ARG italic_g end_ARG = italic_c end_CELL start_CELL on italic_T . end_CELL end_ROW
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Both of these works give a characterization of free boundary spherical caps, that is with θ=π2𝜃𝜋2\theta=\frac{\pi}{2}italic_θ = divide start_ARG italic_π end_ARG start_ARG 2 end_ARG. See also Magnanini-Poggesi [MP22], where they establish an integral identity to give a new proof of Guo-Xia’s rigidity result [GX19] as well as a quantitative stability analysis;
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Theorem 1.1 is a direct consequence of (LABEL:iden-integral). In the case c=0𝑐0c=0italic_c = 0, this identity has been proved by Magnanini-Poggesi [MP22].
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and also Poggesi’s work [Pog22], where a corresponding integral identity for the overdetermined problem in [PT20] can be found.
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Alexandrov-type theorem for capillary CMC hypersurfaces in the half-space has been proved by Wente [Wente80] via the moving plane method.
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B
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We denote by F(x,y,z)𝐹𝑥𝑦𝑧F(x,y,z)italic_F ( italic_x , italic_y , italic_z ) the right hand side of the unperturbed system with γ=0𝛾0\gamma=0italic_γ = 0 in (2.1).
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and}\quad\sin s=\varepsilon+\mathcal{O}\left((s-s^{*})^{2}\right),roman_cos italic_s = - italic_ε ( italic_s - italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) + caligraphic_O ( ( italic_s - italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ) and roman_sin italic_s = italic_ε + caligraphic_O ( ( italic_s - italic_s start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ,
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and meet S2superscript𝑆2S^{2}italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT in two flow-invariant circles connecting the equilibria (0,0,±1)00plus-or-minus1(0,0,\pm 1)( 0 , 0 , ± 1 ).
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Since β<0<α𝛽0𝛼\beta<0<\alphaitalic_β < 0 < italic_α and β2<8α2superscript𝛽28superscript𝛼2\beta^{2}<8\alpha^{2}italic_β start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < 8 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, then these two equilibria are saddles and there is a pair of heteroclinic trajectories going from each equilibrium to the other.
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The unit sphere S2superscript𝑆2S^{2}italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is flow-invariant for (2.1) and attracts all trajectories except the origin which is a repelling equilibrium.
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D
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{q}\right).italic_d ≥ 3 , 2 < italic_q < 2 start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = divide start_ARG 2 italic_d end_ARG start_ARG italic_d - 2 end_ARG , italic_θ = italic_d ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG - divide start_ARG 1 end_ARG start_ARG italic_q end_ARG ) .
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The inequality (2) can be deduced from the results of Brezis and Vázquez [3, Theorem 4.1 and Extension 4.3]; see [23] for related results. The bound (2) can be also derived from Sobolev’s embedding theorem and the kinetic estimate
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We acknowledge the support from the Deutsche Forschungsgemeinschaft through the DFG project Nr. 426365943. CD also acknowledges the support from the Jean-Paul Gimon Fund and from the Erasmus+ programme.
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which was proved by Palatucci–Pisante [17, Theorem 1], using subtle weighted Lpsuperscript𝐿𝑝L^{p}italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT-estimates for Riesz potentials in [19] and Calderón-Zygmund type techniques in the spirit of the Fefferman–Phong argument [5]. The bound (6) is helpful to obtain the compactness of minimizing sequences of the critical Sobolev inequality; see [17, Theorem 3] for details. In contrast, our inequality (4) is stronger than (5) and it only holds for the special case d=1/θ=3𝑑1𝜃3d=1/\theta=3italic_d = 1 / italic_θ = 3.
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The optimal constant of the one-dimensional inequality (17) was already obtained by Nagy in 1941 [16], with 2∗=2d/(d−2)superscript22𝑑𝑑22^{*}=2d/(d-2)2 start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT = 2 italic_d / ( italic_d - 2 ) replaced by a general positive power. The existence and uniqueness of optmizers of the analogue of (17) in higher dimensions are also well-known; we refer to the classical works of Weinstein [24] and Kwong [11] for instance. The uniqueness of optimizers of (17) can be translated straightforwardly to the classification of optmizers of (9) as stated in Theorem 3 (i); we refer to [21, Eq. (4.6)] for a similar analysis.
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A
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\prime}}\,\delta_{\ell\,\ell^{\prime}}\,.roman_Λ start_POSTSUBSCRIPT italic_s italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( roman_ℓ roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_POSTSUPERSCRIPT ( italic_z ) = divide start_ARG italic_π end_ARG start_ARG 2 roman_sin ( italic_π italic_α ) end_ARG [ ( - italic_i square-root start_ARG italic_z end_ARG ) start_POSTSUPERSCRIPT 2 | roman_ℓ + italic_α | end_POSTSUPERSCRIPT - 1 ] italic_δ start_POSTSUBSCRIPT italic_s italic_s start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_δ start_POSTSUBSCRIPT roman_ℓ roman_ℓ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT .
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In the second line we have used the connection formulas reported in [OLBC10, Eqs. 10.27.6 and 10.27.8].
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To say more, using (2.8), (2.9) and (3.8), together with the Bessel connection formula [OLBC10, Eq. 10.27.8], for any 𝐪∈ℂ4𝐪superscriptℂ4\mathbf{q}\in\mathbb{C}^{4}bold_q ∈ blackboard_C start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, we infer
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To begin with, from (2.21), (2.22) and (2.23) (see also [OLBC10, §10.27 and §10.29(ii)]), we deduce that
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To say more, in view of (3.13) and (3.14), by means of [OLBC10, Eq. 10.17.5] we deduce the following, for |𝐱|→+∞→𝐱|\mathbf{x}|\to+\infty| bold_x | → + ∞:
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C
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The idea for the proofs of Theorem 1.1 and Theorem 1.2 comes from a deeper idea in representation theory of GLn(𝔽)subscriptGL𝑛𝔽\mathrm{GL}_{n}\left(\mathbb{F}\right)roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F ). Twisted matrix Kloosterman sums are class functions of GLn(𝔽)subscriptGL𝑛𝔽\mathrm{GL}_{n}\left(\mathbb{F}\right)roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F ), so they can be written as a linear combination of characters of irreducible representations of GLn(𝔽)subscriptGL𝑛𝔽\mathrm{GL}_{n}\left(\mathbb{F}\right)roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F ). What is the coefficient in this linear combination of trπtr𝜋\operatorname{tr}\piroman_tr italic_π, for an irreducible representation π𝜋\piitalic_π? It turns out that the answer is closely related to the tensor product representation gamma factor associated to π𝜋\piitalic_π and the principal series representation associated with the character α:(𝔽×)k→ℂ×:𝛼→superscriptsuperscript𝔽𝑘superscriptℂ\alpha\colon\left(\mathbb{F}^{\times}\right)^{k}\to\mathbb{C}^{\times}italic_α : ( blackboard_F start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT → blackboard_C start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT.
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Finally, as a corollary of Theorem 4.4, and of the purity result of twisted Kloosterman sheaves, we obtain upper bounds for twisted matrix Kloosterman sums.
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In this section, we write an identity that expresses twisted matrix Kloosterman sums in terms of the non-abelian Gauss sums discussed in Section 2.4 and irreducible characters of GLn(𝔽)subscriptGL𝑛𝔽\mathrm{GL}_{n}\left(\mathbb{F}\right)roman_GL start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( blackboard_F ).
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More generally, in a recent work with Oded Carmon [3], we encountered exotic matrix Kloosterman sums while studying a finite field analog of Ginzburg–Kaplan gamma factors [1, Appendix A]. Our results in [3] translate between properties of gamma factors and properties of matrix Kloosterman sums. For example, the multiplicativity property of gamma factors is closely related to the multiplicativity property of matrix Kloosterman sums discussed above. In the appendix, we explain one of our results from [3], that relates matrix Kloosterman sums to Speh representations.
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We use our results and the purity results of twisted Kloosterman sheaves to obtain good bounds for twisted matrix Kloosterman sums (Corollary 4.8).
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C
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}(\theta)\}.italic_J start_POSTSUBSCRIPT roman_LD end_POSTSUBSCRIPT ( italic_x ) := roman_sup start_POSTSUBSCRIPT italic_θ ∈ blackboard_R end_POSTSUBSCRIPT { italic_θ italic_x - roman_Ψ start_POSTSUBSCRIPT italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_θ ) } .
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We prove this proposition by applying the Gärtner Ellis Theorem. More precisely we have to show that
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We prove this proposition by applying the Gärtner Ellis Theorem. More precisely we have to show that
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For every m∈ℝ𝑚ℝm\in\mathbb{R}italic_m ∈ blackboard_R we apply the Gärtner Ellis Theorem (Theorem 2.1). So we have to show that
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The desired LDP can be derived by applying the Gärtner Ellis Theorem (i.e. Theorem 2.1). In fact we have
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A
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(for all j=1,…,k𝑗1…𝑘j=1,\dots,kitalic_j = 1 , … , italic_k) in terms of qj(x)=sj+1−sjg(x;rj)subscript𝑞𝑗𝑥subscript𝑠𝑗1subscript𝑠𝑗𝑔𝑥subscript𝑟𝑗q_{j}(x)=\sqrt{s_{j+1}-s_{j}}\,g(x;r_{j})italic_q start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x ) = square-root start_ARG italic_s start_POSTSUBSCRIPT italic_j + 1 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG italic_g ( italic_x ; italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ).
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Here, the residues are meant as formal residues: namely, the formal residue at t,rj𝑡subscript𝑟𝑗t,r_{j}italic_t , italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is 1/(2πi)12𝜋i1/(2\pi\mathrm{i})1 / ( 2 italic_π roman_i ) times the limit of the contour integral over a positively oriented circle centered at t,rj𝑡subscript𝑟𝑗t,r_{j}italic_t , italic_r start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT as the radius tends to zero, and the formal residue at ∞\infty∞ is 1/(2πi)12𝜋i1/(2\pi\mathrm{i})1 / ( 2 italic_π roman_i ) times the limit of the contour integral over a negatively oriented circle as the radius diverges to +∞+\infty+ ∞.
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The latter is connected with the Airy process and the (cylindrical) Korteweg–de Vries equation, in the same way as (1.22) is connected with the Bessel process and to the nonlinear partial differential equation (1.5), as well as to the narrow-wedge solution to Kardar–Parisi–Zhang equation, cf. [2, 7].
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Therefore, one may interpret (1.22) as a continuum limit as k→+∞→𝑘k\to+\inftyitalic_k → + ∞ of the system (1.33).
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The equation might be better interpreted, following [11], as an infinite system of coupled Painlevé V equations.
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C
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The inequalities in Theorem 1.1 are optimal. Moreover, Theorem 1.1 is global in the sense that the completeness assumption on Mnsuperscript𝑀𝑛M^{n}italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT cannot be dropped. See §3§3\S{3}§ 3 for details.
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It may be checked that the arc-length parametrization condition cannot be continued for all s∈ℝ.𝑠ℝs\in\mathbb{R}.italic_s ∈ blackboard_R . Similar examples can be constructed in any dimension and any space form by solving a certain ODE (See Leite [13]).
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(b) Assume that Mnsuperscript𝑀𝑛M^{n}italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is conformally flat as before and k≥2.𝑘2k\geq 2.italic_k ≥ 2 . Although f𝑓fitalic_f is not necessarily of type (1,n−1),1𝑛1(1,n-1),( 1 , italic_n - 1 ) , by combining a result of Moore [15] on the existence of a principal normal vector field of high multiplicity with that of Dajczer, Onti and Vlachos ([6], Th. 1111), we infer that the possible types of f𝑓fitalic_f are limited when the codimension k𝑘kitalic_k is small. In particular, when k=2𝑘2k=2italic_k = 2 and n≥5𝑛5n\geq 5italic_n ≥ 5 the only possible type of f𝑓fitalic_f other than the type (1,n−1)1𝑛1(1,n-1)( 1 , italic_n - 1 ) is (1,1,n−2).11𝑛2(1,1,n-2).( 1 , 1 , italic_n - 2 ) .
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Remark. We note that the flat normal bundle assumption in Theorem 1.2 is redundant when the codimension k=2𝑘2k=2italic_k = 2 and Mnsuperscript𝑀𝑛M^{n}italic_M start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT is not minimal ([9], Th. 1′superscript1′1^{\prime}1 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT).
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Remark. Parts (ii) and (iii) of Theorem 1.1 extend some results of Cheng ([2], Th. 3.13.13.13.1), Hu and Zhai ([12], Th. 5.15.15.15.1) (also see Leite [13]) on hypersurfaces to submanifolds with flat normal bundle.
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D
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