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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_POSTSUPERS... | π−1(γ(0))superscript𝜋1𝛾0\pi^{-1}(\gamma(0))italic_π start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_γ ( 0 ) ) | 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 ... | 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. | 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_POSTSUB... | C |
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 ... | 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. | We will need the following C0superscript𝐶0C^{0}italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT-close version of Theorem 2.4. | 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 ... | We will also need the following statement that allows us to approximate a formal Legendrian by a loose Legendrian. | B |
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^{*... | 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)}% | 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^{% | 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% | 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}(% | A |
\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 ) . | The following theorem is a consequence of the results in [Co16, BZ15, HLMXZ19] and our investigations on little Lipschitz functions. | 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)... | 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. | Next, we record the following strengthened version of [BZ15, Lemma 2], which follows from Lemma 2 and its proof in [BZ15]. | D |
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_POSTSUBSCRI... | 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 b... | 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: | 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, th... | 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. | B |
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... | 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... | 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... | 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. | 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])... | C |
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. Moreove... | 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. | 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. | 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 a... | 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-iso... | B |
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... | 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... | An interesting byproduct of Theorem 1.3 is a new geometric interpretation of the exponent of modulus of continuity β𝛽\operatorname{\beta}italic_β. | 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}-\over... | 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_POSTS... | B |
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_... | 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⊂ℝ... | 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. | 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 ... | 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_{\... | C |
|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|supe... | 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 | Case k+ℓ≡0modv(q)𝑘ℓmodulo0𝑣𝑞k+\ell\equiv 0\mod v(q)italic_k + roman_ℓ ≡ 0 roman_mod italic_v ( italic_q ). | 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 | 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... | C |
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 | 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 | 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. | 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 | We have shown that the conclusion of Theorem 4.1 can go either way when dropping the two hypotheses. | D |
_{\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... | 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. | 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∈�... | 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]. | 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 supercuspida... | C |
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. | 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 betw... | 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 su... | 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]. | 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... | D |
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. | 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. | 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. | 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. | 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). | D |
Several factors can significantly impact the efficiency and accuracy of the approximate tensor network contraction process. | 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]. | 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. | 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 ac... | 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. | B |
Noetherian frames}\}roman_Fr bold_Grz bold_.3 = { transitive reflexive non-branching Noetherian frames } | 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 GL𝐚𝐬superscriptG... | 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. | 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-[... | Fr𝐊𝐃𝟓={serial Euclidean frames};Fr𝐊𝐃𝟓serial Euclidean frames\operatorname{Fr}\mathbf{KD5}=\{\text{serial Euclidean frames}\};roman_Fr bold_KD5 = { serial Euclidean frames } ; | B |
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... | 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... | 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 paramet... | 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 poin... | 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 proced... | B |
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],... | 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 ra... | We wish to thank N. Andruskiewitsch for useful discussions and I. Heckenberger for pointing out the content of Remark 4.5. | 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 ... | 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... | B |
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 ... | 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... | 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)... | 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 ... | As discussed in [23], one can obtain the results obtained in the previous Theorem in the following way also. Let us define | B |
‖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 | 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 genera... | 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]. | 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. | 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 | B |
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_POSTSU... | 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}(% | 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_% | 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% | 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,% | D |
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... | 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 ... | 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 ( itali... | 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 ... | 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_... | A |
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... | 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 | 1. Since the number of iterations M𝑀Mitalic_M is a shifted geometric random variable, it is finite with probability 1111. | 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 ... | 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). | A |
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 ). | 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. | 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 ). | for some r≥1𝑟1r\geq 1italic_r ≥ 1 and H𝐻Hitalic_H a finite subgroup inside Spin(5)Spin5\operatorname{Spin}(5)roman_Spin ( 5 ). | In turn Sp(2)Sp2\operatorname{Sp}(2)roman_Sp ( 2 ) is isomorphic to Spin(5)Spin5\operatorname{Spin}(5)roman_Spin ( 5 ). | D |
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 lin... | 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... | \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 ) , ⋯ , ita... | -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_POSTSUP... | (d−k)𝑑𝑘(d-k)( italic_d - italic_k ) trailing eigenvectors of Hessian of g𝑔gitalic_g | D |
[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 itali... | The class of ICtautq⋆(ICtautq)∗⋆subscriptsuperscriptIC𝑞tautsuperscriptsubscriptsuperscriptIC𝑞taut{\operatorname{IC}}^{q}_{\operatorname{taut}}\star({\operatorname{IC}}^{q}_{% | It remains to check that ICtautq⋆(ICtautq)∗⋆subscriptsuperscriptIC𝑞tautsuperscriptsubscriptsuperscriptIC𝑞taut{\operatorname{IC}}^{q}_{\operatorname{taut}}\star({\operatorname{IC}}^{q}_{% | generated by the sheaves ICtautq,(ICtautq)∗subscriptsuperscriptIC𝑞tautsuperscriptsubscriptsuperscriptIC𝑞taut{\operatorname{IC}}^{q}_{\operatorname{taut}},({\operatorname{IC}}^{q}_{% | ICtautq,(ICtautq)∗,IC0,0qsubscriptsuperscriptIC𝑞tautsuperscriptsubscriptsuperscriptIC𝑞tautsubscriptsuperscriptIC𝑞00{\operatorname{IC}}^{q}_{\operatorname{taut}},({\operatorname{IC}}^{q}_{% | B |
ψ:ℝ→ℝ:𝜓→ℝℝ\psi:\mathbb{R}\to\mathbb{R}italic_ψ : blackboard_R → blackboard_R is the derivative of ρ𝜌\rhoitalic_ρ, | and its derivative ψ=ρ′𝜓superscript𝜌′\psi=\rho^{\prime}italic_ψ = italic_ρ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT is 1-Lipschitz. | 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 ... | ψ′superscript𝜓′\psi^{\prime}italic_ψ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT the derivative of ψ𝜓\psiitalic_ψ and | , 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. | C |
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_marka... | 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... | 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: | 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_POSTSUBSCR... | 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... | B |
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)=0Homsubscript𝐸𝑗subscript𝐸�... | It is easy to convert a rooted labeled forest into a complete exceptional sequence. Start with the following. | 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: | 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: | 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_POSTSUBSCR... | A |
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_POSTSUBSCRI... | 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 t... | 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 nonsingul... | 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 classica... | 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 c... | A |
Despite their effectiveness in computer applications, error-correcting codes are quite useless if we want to model consensus in biological systems. | 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... | 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.... | 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 f... | 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. | D |
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_POST... | 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: | 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 expansi... | 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... | 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: | D |
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). | 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 ar... | 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... | 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 t... | 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 ... | C |
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 Xtsubscript𝑋�... | 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: | 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 ... | 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... | 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 questio... | A |
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 h... | 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... | 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\rho... | 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... | 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 ... | C |
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 | If (M,S,η)𝑀𝑆𝜂\left(M,S,\eta\right)( italic_M , italic_S , italic_η ) is a factorable monoid, then the restriction | Assume that (M,S,η)𝑀𝑆𝜂\left(M,S,\eta\right)( italic_M , italic_S , italic_η ) is a factorable monoid. The | 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 ) | (M,S,η)𝑀𝑆𝜂\left(M,S,\eta\right)( italic_M , italic_S , italic_η ) be a factorable monoid. If Condition (4.3) | D |
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). | 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... | 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. | 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 perf... | 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 asympt... | A |
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)subscriptsuperscrip... | We will use the same notation to denote the corresponding normal vector fields along ΣΣ\Sigmaroman_Σ. | 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. | where we use the fact that ΣΣ\Sigmaroman_Σ is totally geodesic and Ricci-flat. We conclude that, at b𝑏bitalic_b, | The restriction to normal vector fields corresponds to the fact that ℳℳ\mathcal{M}caligraphic_M is defined modulo reparametrizations, | A |
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 s... | 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 P... | 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 stac... | 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 fr... | 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)... | D |
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 integr... | Every nontrivial integral solution (a,b,c)𝑎𝑏𝑐(a,b,c)( italic_a , italic_b , italic_c ) of (⋆italic-⋆\star⋆ ‣ 1) is given by | 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 | . | 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). | 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 | A |
\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_POSTSUBSCRI... | Additionally, we are going to require some control on the Taylor approximation of the gradients around the solution, as established next: | 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}divi... | 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. Fo... | 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. | A |
_{*}(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 = ∑ sta... | 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 ... | 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 fun... | 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. | 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\mathca... | C |
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. | We finish the introduction by pointing out that our results also give improvements for multidimensional lexicographic-monotone array. | 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... | 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 | A multidimensional array is said to be monotone if for each direction all the 1111-dimensional subarrays in that direction are increasing or decreasing. | A |
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-uni... | 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). | Similarly, there exists a general procedure to construct AME(5,d)AME5𝑑\text{AME}(5,d)AME ( 5 , italic_d ) | such a state is then denoted by AME(n,d)AME𝑛𝑑\text{AME}(n,d)AME ( italic_n , italic_d ). Interestingly, not for all | Finally, the recently found AME(4,6)AME46\text{AME}(4,6)AME ( 4 , 6 ) Rather et al. (2022) is not maximally entangled | D |
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)<1superscripts... | 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 ... | 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 t... | 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. | 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]. | C |
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 im... | 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 wer... | 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 im... | 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 hig... | 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 ca... | C |
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 local... | The focus of the present article is on locally cyclic graphs, that is, graphs for which the neighbourhood of each vertex induces a cycle. | 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. | 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). | 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? | A |
\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_POSTSUPERS... | 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{... | 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 alg... | The MP pseudoinverse of a tensor can also be computed in the Fourier domain and this is shown in Algorithm 2. | 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. | C |
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? | 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? | The bisector of one of the angles and that of the supplementary angle are perpendicular to each other. | Given two straight lines, we consider the two angles formed by them, regardless of whether they are acute or not. | 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 . | C |
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}... | 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}_{% | (α1⊗IdEndA0(X),(X⊗αB)⊗ψEndA0(X))≅(i⊗IdEndA0(X),X⊗ϕEndA0(X))tensor-productsubscript𝛼1subscriptIdsuperscriptsubscriptEnd𝐴0𝑋subscripttensor-product𝜓subscripttensor-product𝛼𝑋𝐵superscriptsubscriptEnd𝐴0𝑋tensor-product𝑖subscriptIdsuperscriptsubscriptEnd𝐴0𝑋subscripttensor-productitalic-ϕ𝑋superscriptsubscriptEn... | 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... | Then (i⊗IdEndA0(X),X⊗EndA0(X))tensor-product𝑖subscriptIdsuperscriptsubscriptEnd𝐴0𝑋tensor-product𝑋superscriptsubscriptEnd𝐴0𝑋(i\otimes\operatorname{Id}_{\operatorname{End}_{A}^{0}(X)},X\otimes% | D |
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_{\v... | 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. | 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_{\v... | 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. | 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. | A |
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... | For this class of Prob-solvable loops, conditions (A)–(C) in Section 3.1 hold, but (D) and/or (E) may not be fulfilled. | 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. | 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. | 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. | D |
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 Wi... | 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^{% | 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{% | 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{% | 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{% | A |
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 ... | 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)}/\sim... | 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 ... | 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_{\math... | 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 | B |
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. Th... | 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. | By Corollary 3.4 and an argument similar to the proof of Corollary 2.4 we have the following result. | 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. | 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. | A |
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. ... | 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: | Our next theorem shows that we should not expect κ𝜅\kappaitalic_κ to scale polynomially with λ𝜆\lambdaitalic_λ, | 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. | In Section 4, we develop a perturbative expansion in λ𝜆\lambdaitalic_λ for the observables appearing in our theorems. | A |
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{... | 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): | 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_... | Using these matrices, we define the following spaces which will capture the dissipation produced by the various collision operators. | 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]. | C |
\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_P... | 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 ), | 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 ). | 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_C... | 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, | D |
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 sc... | 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; | 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 frakt... | 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... | We will write these four matrices relative to the bases we used in the previous section and that we recall here. | A |
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 e... | 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 th... | 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 e... | 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 trun... | 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... | C |
Let L=K(a)𝐿𝐾𝑎L=K(\sqrt{a})italic_L = italic_K ( square-root start_ARG italic_a end_ARG ). Then (25) gives | 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}({% | (Br(A)/Br1(A))3∞=Br(A)3∞/Br1(A)3∞.(\operatorname{Br}(A)/\operatorname{Br}_{1}(A))_{3^{\infty}}=\operatorname{Br}% | Br(Y)/Br1(Y)=Br(Y)ℓ/Br1(Y)ℓ.\operatorname{Br}(Y)/\operatorname{Br}_{1}(Y)=\operatorname{Br}(Y)_{\ell}/% | 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}(% | D |
In this article, we have provided an affirmative answer to this above question by establishing Theorem 1111. | 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 ... | 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})ital... | 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: | 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 ... | C |
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 ) - ital... | 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... | 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]. | 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]. | 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. | C |
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, mult... | These advancements indicate that the spectral valleys observed in this work are promising for applications to optical | 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, mult... | 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. | 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 cu... | A |
(\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... | 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... | 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]. | and also Poggesi’s work [Pog22], where a corresponding integral identity for the overdetermined problem in [PT20] can be found. | Alexandrov-type theorem for capillary CMC hypersurfaces in the half-space has been proved by Wente [Wente80] via the moving plane method. | B |
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). | 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... | 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 ). | 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 f... | 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. | D |
{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 ) . | 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 | 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. | 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 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 ana... | A |
\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 roma... | In the second line we have used the connection formulas reported in [OLBC10, Eqs. 10.27.6 and 10.27.8]. | 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 | To begin with, from (2.21), (2.22) and (2.23) (see also [OLBC10, §10.27 and §10.29(ii)]), we deduce that | 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 | → + ∞: | C |
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)subscriptGL𝑛�... | 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. | 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 ). | 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 multipl... | We use our results and the purity results of twisted Kloosterman sheaves to obtain good bounds for twisted matrix Kloosterman sums (Corollary 4.8). | C |
}(\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_POSTSUBSCR... | We prove this proposition by applying the Gärtner Ellis Theorem. More precisely we have to show that | We prove this proposition by applying the Gärtner Ellis Theorem. More precisely we have to show that | 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 | The desired LDP can be derived by applying the Gärtner Ellis Theorem (i.e. Theorem 2.1). In fact we have | A |
(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 i... | 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 ... | 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]. | Therefore, one may interpret (1.22) as a continuum limit as k→+∞→𝑘k\to+\inftyitalic_k → + ∞ of the system (1.33). | The equation might be better interpreted, following [11], as an infinite system of coupled Painlevé V equations. | C |
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. | 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]). | (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 principa... | 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). | 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. | D |
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