Mathematics
New submissions
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New submissions for Thu, 28 Mar 24
- [1] arXiv:2403.17943 [pdf, ps, other]
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Title: A $(φ_n, φ)$-Poincaré inequality on John domainComments: arXiv admin note: substantial text overlap with arXiv:2305.04016Subjects: Functional Analysis (math.FA)
Given a bounded domain $\Omega \subset {\mathbb R}^{n}$ with $n\ge2$, let $\phi $ is a Young function satisfying the doubling condition with the constant $K_\phi<2^{n}$.
If $\Omega$ is a John domain, we show that $\Omega $ supports a $(\phi_{n}, \phi)$-Poincar\'e inequality.
Conversely, assume additionally that $\Omega$ is simply connected domain when $n=2$ or a bounded domain which is quasiconformally equivalent to some uniform domain when $n\ge3$. If $\Omega$ supports a $(\phi_n, \phi)$-Poincar\'e inequality, we show that it is a John domain. - [2] arXiv:2403.17944 [pdf, ps, other]
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Title: The sup-completion of a Dedekind complete vector lattice IISubjects: Functional Analysis (math.FA)
We persist in our investigation of the sup-completion of a Dedekind complete Riesz space, extending to the broader context of Riesz spaces. some results initially obtained by Feng, Li, Shen, and also by Erd\"os, and R\'enyi.
- [3] arXiv:2403.17945 [pdf, ps, other]
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Title: A proof of Sylvester's theoremAuthors: Saptak BhattacharyaComments: 3 pages, 0 figuresSubjects: Functional Analysis (math.FA)
We give a new elementary proof of existence and uniqueness of a solution to the Sylvester equation $AX-XB=Y$
- [4] arXiv:2403.17946 [pdf, ps, other]
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Title: Nonlinear Heisenberg-Robertson-Schrodinger Uncertainty PrincipleAuthors: K. Mahesh KrishnaComments: 4 Pages, 0 FiguresSubjects: Functional Analysis (math.FA); Information Theory (cs.IT); Mathematical Physics (math-ph)
We derive an uncertainty principle for Lipschitz maps acting on subsets of Banach spaces. We show that this nonlinear uncertainty principle reduces to the Heisenberg-Robertson-Schrodinger uncertainty principle for linear operators acting on Hilbert spaces.
- [5] arXiv:2403.17949 [pdf, other]
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Title: A prime number "Game of Life": can floor($y \cdot p\#$) be prime for all $p$>=2?Authors: Martin RaabComments: 26 pages, 6 figuresSubjects: General Mathematics (math.GM)
A new sequence in the spirit of the Mills primes is presented and its properties are investigated.
- [6] arXiv:2403.17950 [pdf, ps, other]
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Title: Networks and their degree distribution, leading to a new concept of small worldsAuthors: Leo EggheSubjects: General Mathematics (math.GM)
The degree distribution, referred to as the delta-sequence of a network is studied. Using the non-normalized Lorenz curve, we apply a generalized form of the classical majorization partial order. Next, we introduce a new class of small worlds, namely those based on degree centralities of networks. Similar to a previous study, small worlds are defined as sequences of networks with certain limiting properties. We distinguish between three types of small worlds: those based on the highest degree, those based on the average degree, and those based on the median degree. We show that these new classes of small worlds are different from those introduced previously based on the diameter of the network or the average and median distance between nodes. However, there exist sequences of networks that qualify as small worlds in both senses of the word, with stars being an example. Our approach enables the comparison of two networks with an equal number of nodes in terms of their small-worldliness. Finally, we introduced neighboring arrays based on the degrees of the zeroth and first-order neighbors and proved that for trees, equal neighboring arrays lead to equal delta-arrays.
- [7] arXiv:2403.17951 [pdf, ps, other]
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Title: Regular extreme semisimple Lie algebrasSubjects: Representation Theory (math.RT)
A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is regular extreme if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. Douglas and Repka previously showed that the simple Lie algebras of type $A_n$ are regular extreme. In this article, we show that, in fact, all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme.
- [8] arXiv:2403.17952 [pdf, ps, other]
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Title: Mneimneh-type Binomial Sums of Multiple Harmonic-type SumsComments: 15 pages. arXiv admin note: text overlap with arXiv:2403.04107Subjects: Number Theory (math.NT)
In this paper, we establish some expressions of Mneimneh-type binomial sums involving multiple harmonic-type sums in terms of finite sums of Stirling numbers, Bell numbers and some related variables. In particular, we present some new formulas of Mneimneh-type binomial sums involving generalized (alternating) harmonic numbers. Further, we establish a new identity relating the multiple zeta star values $\zeta^\star(m+2,\{1\}_{r-1})$ and specific multiple polylogarithms by applying the Toeplitz principle. Furthermore, we present some interesting consequences and illustrative examples.
- [9] arXiv:2403.17953 [pdf, ps, other]
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Title: Solutions to a Pillai-type equation involving Tribonacci numbers and S-unitsComments: 26 pagesSubjects: Number Theory (math.NT)
Let $ \{T_n\}_{n\geq 0} $ be the sequence of Tribonacci numbers. In this paper, we study the exponential Diophantine equation $T_n-2^x3^y=c$, for $n,x,y\in \mathbb{Z}_{\ge0}$. In particular, we show that there is no integer $c$ with at least six representations of the form $T_n-2^x3^y$.
- [10] arXiv:2403.17955 [pdf, ps, other]
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Title: An Effective Lower Bound on the Integer Cube SumAuthors: Saunak BhattacharjeeComments: 7 pagesSubjects: Number Theory (math.NT)
Let $f(x, y) \in \mathbb{Z}[x, y]$ be a cubic form with non-zero discriminant, and for each integer $m \in \mathbb{Z}$, let, $N_{f}(m)=\#\left\{(x, y) \in \mathbb{Z}^{2}: f(x, y)=m\right\} $. In 1983, Silverman proved that $N_{f}(m)>\Omega\left((\log |m|)^{3 / 5}\right)$ when $f(x, y)=x^{3}+y^{3}$. In this paper, we obtain an explicit bound for $N_f(m)$, namely, showing that $N_{f}(m)>13\times 10^{-6}(\log |m|)^{11/13}$ (holds for infinitely many integers m), when $f(x, y)=x^{3}+y^{3}$.
- [11] arXiv:2403.17956 [pdf, ps, other]
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Title: Automatic continuity of operator semigroups in the Calkin algebraAuthors: Tomasz KochanekComments: arXiv admin note: substantial text overlap with arXiv:2203.05635Subjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
We study operator semigroups in the Calkin algebra $\mathcal{Q}(\mathcal{H})$, represented as a subalgebra of the algebra of bounded linear operators on a Hilbert space via one of `canonical' Calkin's representations. Using the BDF theory, we associate with any normal $C_0$-semigroup $(q(t))_{t\geq 0}$ in $\mathcal{Q}(\mathcal{H})$ an extension $\Gamma\in\mathrm{Ext}(\Delta)$, where $\Delta$ is the inverse limit of certain compact metric spaces defined purely in terms of the spectrum $\sigma(A)$ of the generator of $(q(t))_{t\geq 0}$. Then we show that, in natural circumstances, if $(q(t))_{t\geq 0}$ is continuous in the strong operator topology, then it is actually uniformly continuous, although there are $C_0$-semigroups in $\mathcal{Q}(\mathcal{H})$ that are not uniformly continuous.
- [12] arXiv:2403.17957 [pdf, ps, other]
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Title: The Density of Borromean PrimesComments: 19 pages. To appear in Commentarii Mathematici Universitatis Sancti PauliSubjects: Number Theory (math.NT)
In this paper, we study an asymptotic distribution of sets of primes satisfying certain "linking conditions" in arithmetic topology, namely, conditions given by the Legendre and R\'edei symbols among sets of primes. As our Main Theorem, we prove an asymptotic density formula for Borromean primes among all primes. For the proof, we use the effective Chebotarev density formula under the Generalized Riemann Hypothesis and explicit computations of discriminants of the number fields involved in R\'edei's extension.
- [13] arXiv:2403.17959 [pdf, ps, other]
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Title: Rational Diophantine sextuples with strong pairComments: 9 pagesSubjects: Number Theory (math.NT)
A set of $m$ distinct nonzero rationals $\{a_1, a_2,\ldots, a_m\}$ such that $a_i a_j+1$ is a perfect square for all $1\le i <j \le m$, is called a rational Diophantine $m$-tuple. If in addition, $a_i^2+1$ is a perfect square for $1\le i\le m$, then we say the $m$-tuple is strong. In this paper, we construct infinite families of rational Diophantine sextuples containing a strong Diophantine pair.
- [14] arXiv:2403.17960 [pdf, ps, other]
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Title: On generalizations of Iwasawa's theoremSubjects: Group Theory (math.GR)
Iwasawa's theorem indicates that a finite group $G$ is supersolvable if and only if all maximal chains of the identity in $G$ have the same length. As generalizations of Iwasawa's theorem, we provide some characterizations of the structure of a finite group $G$ in which all maximal chains of every minimal subgroup have the same length. Moreover, let $\delta(G)$ be the number of subgroups of $G$ all of whose maximal chains in $G$ do not have the same length, we prove that $G$ is a non-solvable group with $\delta(G)\leq 16$ if and only if $G\cong A_5$.
- [15] arXiv:2403.17961 [pdf, ps, other]
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Title: A categorical formulation of Kraus' paradoxAuthors: Andrew W. SwanSubjects: Category Theory (math.CT); Logic in Computer Science (cs.LO); Logic (math.LO)
We give a categorical formulation of Kraus' "magic trick" for recovering information from truncated types. Rather than type theory, we work in Van den Berg-Moerdijk path categories with a univalent universe, and rather than propositional truncation we work with arbitrary cofibrations, which includes truncation as a special case. We show, using Kraus' argument that any cofibration with homogeneous domain is a monomorphism. We give some simple concrete examples in groupoids to illustrate the interaction between homogeneous types, cofibrations and univalent fibrations.
- [16] arXiv:2403.17962 [pdf, other]
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Title: X-matricesSubjects: Rings and Algebras (math.RA)
We evidence a family $\mathcal{X}$ of square matrices over a field $\mathbb{K}$, whose elements will be called X-matrices. We show that this family is shape invariant under multiplication as well as transposition. We show that $\mathcal{X}$ is a (in general non-commutative) subring of $GL(n,\mathbb{K})$. Moreover, we analyse the condition for a matrix $A \in \mathcal{X}$ to be invertible in $\mathcal{X}$. We also show that, if one adds a symmetry condition called here bi-symmetry, then the set $\mathcal{X}^b$ of bi-symmetric X-matrices is a commutative subring of $\mathcal{X}$. We propose results for eigenvalue inclusion, showing that for X-matrices eigenvalues lie exactly on the boundary of Cassini ovals. It is shown that any monic polynomial on $ \mathbb{R} $ can be associated with a companion matrix in $ \mathcal{X} $.
- [17] arXiv:2403.17963 [pdf, other]
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Title: A better compression driver? CutFEM 3D shape optimization taking viscothermal losses into accountSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
The compression driver, the standard sound source for midrange acoustic horns, contains a cylindrical compression chamber connected to the horn throat through a system of channels known as a phase plug. The main challenge in the design of the phase plug is to avoid resonance and interference phenomena. The complexity of these phenomena makes it difficult to carry out this design task manually, particularly when the phase-plug channels are radially oriented. Therefore, we employ an algorithmic technique that combines numerical solutions of the governing equations with a gradient-based optimization algorithm that can deform the walls of the phase plug. A particular modeling challenge here is that viscothermal losses cannot be ignored, due to narrow chambers and slits in the device. Fortunately, a recently developed, accurate, but computationally inexpensive boundary-layer model is applicable. We use this model, a level-set geometry description, and the Cut Finite Element technique to avoid mesh changes when the geometry is modified by the optimization algorithm. Moreover, the shape calculus needed to compute derivatives for the optimization algorithm is carried out in the fully discrete case. Applying these techniques, the algorithm was able to successfully design the shape of a set of radially-directed phase plugs so that the final frequency response surprisingly closely matches the ideal response, derived by a lumped circuit model where wave interference effects are not accounted for. This result may serve to resuscitate the radial phase plug design, rarely used in today's commercial compression drivers.
- [18] arXiv:2403.17964 [pdf, other]
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Title: Quantifying separability in RAAGs via representationsComments: arXiv admin note: substantial text overlap with arXiv:2303.03644Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
We answer the question asked by Louder, McReynolds and Patel, and prove the following statement. Let L be a RAAG, H a word quasiconvex subgroup of L, then there is a finite dimensional representation of L that separates the subgroup H in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate H in L. This implies the same statement for a virtually special group L and, in particular, a fundamental groups of a hyperbolic 3-manifold.
- [19] arXiv:2403.17965 [pdf, ps, other]
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Title: Linear Equation over Non-Commutative AlgebraAuthors: Aleks KleynComments: nglish text - 9 pages; Russian text - 9 pagesSubjects: General Mathematics (math.GM)
I considered solving of the system of linear equations $$a^1_{1s0}x^1a^1_{1s1}+...+a^1_{ns0}x^na^1_{ns1}=b^1$$ $$...$$ $$a^n_{1s0}x^1a^n_{1s1}+...+a^n_{ns0}x^na^n_{ns1}=b^n$$ over non-commutative associative algebra. I considered examples in quaternion algebra. I considered also Newton's method to solve the equation $$f(x)=a$$ over non-commutative associative algebra.
- [20] arXiv:2403.17966 [pdf, ps, other]
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Title: A review of Pythagorean triples from both classical and modern viewpointsAuthors: Ali TaghaviSubjects: History and Overview (math.HO)
In this note we present a survey on some classical and modern approaches on Pythagorean triples. Some questions are also posed in direction of some materials under review. In particular some non commutative and operator theoretical approaches of Pythagorean triples are discussed
- [21] arXiv:2403.17967 [pdf, other]
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Title: Algumas luminescências sobre o jogo Lights OutComments: in Spanish languageSubjects: History and Overview (math.HO)
The theory behind the Lights Out game has been developed by several authors. The aim of this work is to present some results related to this game using Linear Algebra. We establish a criterion for the solubility of this game in the case of an $m$ by $n$ grid, which depends on the invertibility of a matrix, and we present the conditions for this to occur, easily verifiable from $m$ and $n$. Furthermore, we explicitly determine the value of the determinant for a particular case.
- [22] arXiv:2403.17970 [pdf, ps, other]
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Title: Exploring Functional Identities: From Division Rings to Matrix AlgebrasComments: 10Subjects: Rings and Algebras (math.RA)
In this paper, we tackle unresolved inquiries by Ferreira et al. \cite{bruno} in their recent publication, ``Functional Identity on Division Algebras". We delve into the intricate behavior of additive functions on matrix algebras over division rings through rigorous analysis and theorem-proving. Our findings offer valuable insights into the nature of these functions and their implications for algebraic structures.
- [23] arXiv:2403.17971 [pdf, ps, other]
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Title: Unraveling Functional Equations in Composition Algebra: Resolving Conjectures and Examining ImplicationsComments: 13Subjects: Rings and Algebras (math.RA)
We address the conjectures left by the recent article by Ferreira et al. titled ``Commuting maps and identities with inverses on alternative division rings.'' We also present an example showing the necessity of the conditions of the results that answer the conjectures.
- [24] arXiv:2403.17972 [pdf, ps, other]
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Title: On the unit group and the $2$-class number of $\mathbb{Q}(\sqrt{2},\sqrt{p},\sqrt{q})$Comments: arXiv admin note: substantial text overlap with arXiv:2108.04171Subjects: Number Theory (math.NT)
Let $p\equiv 1\pmod{8}$ and $q\equiv7\pmod 8$ be two prime numbers. The purpose of this paper is to compute the unit group of the fields $\KK=\QQ(\sqrt 2, \sqrt{p}, \sqrt{q} )$ and give their $2$-class numbers.
- [25] arXiv:2403.17981 [pdf, ps, other]
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Title: On some classes of Saphar type operators IIComments: 20 pagesSubjects: Functional Analysis (math.FA)
The purpose of this paper is to explore additional properties of left Drazin invertible, essentially left Drazin invertible, right Drazin invertible, and essentially right Drazin invertible operators on Banach spaces, building upon the groundwork laid in [8] and [19]. Specifically, we propose alternative definitions for these operators and investigate their behavior in powers. Furthermore, we employ a specific operator decomposition to characterize these operators, providing a deeper understanding of their structure and properties. The operators we study are distinguished from other operators bearing the same name in existing literature, see [1], [9]. By employing more refined definitions, we uncover a broader range of properties for these operators, setting them apart from their counterparts in the literature.
- [26] arXiv:2403.17989 [pdf, ps, other]
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Title: Study guide for "On restriction projections to planes in $\mathbb R^3$"Subjects: Classical Analysis and ODEs (math.CA)
This article is a study guide for ``On restricted projections to planes in $\mathbb R^3$" [arXiv:2207.13844] by Gan, Guo, Guth, Harris, Maldague and Wang. We first present the main problems and preliminaries related to restricted projections in $\mathbb R^3$. Then we introduce the high-low method and decoupling, which are the two central and novel ideas in their proofs. We hope to provide as many details as possible so that this study guide is self-contained, with the only exception of the Bourgain-Demeter decoupling inequality for curves in the appendix.
- [27] arXiv:2403.17990 [pdf, ps, other]
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Title: On noncommutative Hölder inequality of Sukochev and Zanin for weak Schatten classSubjects: Functional Analysis (math.FA)
Sukochev and Zanin resolved an open problem due to B. Simon concerning optimal constants in H\"older inequality for the weak Schatten classes of compact operators. In this note we observe that these constants, by introducing the modified weak Schatten quasi-norms, can be renormalised so that the original Simon's conjecture (with optimal constant 1) does hold. We also provide an unexpectedly simple proof for the modified H\"older inequality and its sharpness.
- [28] arXiv:2403.17996 [pdf, ps, other]
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Title: Quantum fields on projective geometriesAuthors: Daniel SpitzComments: 46 pages, 1 figureSubjects: Mathematical Physics (math-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th)
Considering homogeneous four-dimensional space-time geometries within real projective geometry provides a mathematically well-defined framework to discuss their deformations and limits without the appearance of coordinate singularities. On Lie algebra level the related conjugacy limits act isomorphically to concatenations of contractions. We axiomatically introduce projective quantum fields on homogeneous space-time geometries, based on correspondingly generalized unitary transformation behavior and projectivization of the field operators. Projective correlators and their expectation values remain well-defined in all geometry limits, which includes their ultraviolet and infrared limits. They can degenerate with support on space-time boundaries and other lower-dimensional space-time subspaces. We explore fermionic and bosonic superselection sectors as well as the irreducibility of projective quantum fields. Dirac fermions appear, which obey spin-statistics as composite quantum fields. The framework might be of use for the consistent description of quantum fields in holographic correspondences and their flat limits.
- [29] arXiv:2403.17997 [pdf, ps, other]
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Title: La relation entre $ζ(4n-1)$, $ζ(2p)$ et $ζ(4n-1-2p)$Authors: Mundankulu KabongoComments: 12 pages and one figure for domaineSubjects: General Mathematics (math.GM)
The functional relation of the Riemann z\^eta function provides us with neither the nature nor the expression of z\^eta at positive odd numbers. From the function $F(z)=\frac{z^{-2n}}{e^z-1}$, we find a functional relation involving $\zeta(4n- 1)$, $\zeta(2p)$ and $\zeta(4n-1-2p)$. It is given by: \begin{equation} \zeta(4n-1)=\frac{1}{2n-1}\sum_{p=1}^{2n-2}\zeta(2p)\zeta(4n-1-2p). \end{equation} $n=2, 3, 4, 5, 6, ...$ From this formula we introduce a new approach to study the nature of $\zeta$ on these integers.
- [30] arXiv:2403.18014 [pdf, ps, other]
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Title: Generalized Chern-Simons-Schrodinger system with critical exponential growth: the zero mass caseComments: 20 pagesSubjects: Analysis of PDEs (math.AP)
We consider the existence of ground state solutions for a class of zero-mass Chern-Simons-Schr\"{o}dinger systems \[
\left\{
\begin{array}{ll} \displaystyle -\Delta u +A_0 u+\sum\limits_{j=1}^2A_j^2 u=f(u)-a(x)|u|^{p-2}u, \newline
\displaystyle \partial_1A_2-\partial_2A_1=-\frac{1}{2}|u|^2,~\partial_1A_1+\partial_2A_2=0, \newline
\displaystyle \partial_1A_0=A_2|u|^2,~ \partial_2A_0=-A_1|u|^2,
\end{array} \right. \] where $a:\mathbb R^2\to\mathbb R^+$ is an external potential, $p\in(1,2)$ and $f\in \mathcal{C}(\mathbb R)$ denotes a nonlinearity that fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. By introducing an improvement of the version of Trudinger-Moser inequality, we are able to investigate the existence of positive ground state solutions for the given system using variational method. - [31] arXiv:2403.18017 [pdf, ps, other]
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Title: Nonsingularity of unsymmetric Kansa matrices: random collocation by MultiQuadrics and Inverse MultiQuadricsSubjects: Numerical Analysis (math.NA)
Unisolvence of unsymmetric Kansa collocation is still a substantially open problem. We prove that Kansa matrices with MultiQuadrics and Inverse MultiQuadrics for the Dirichlet problem of the Poisson equation are almost surely nonsingular, when the collocation points are chosen by any continuous random distribution in the domain interior and arbitrarily on its boundary.
- [32] arXiv:2403.18027 [pdf, ps, other]
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Title: Chain Conditions in $C_p(X)$Subjects: General Topology (math.GN)
We present new results regarding calibers in function spaces $C_p(X)$. We calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $\lambda$-Lindel\"of extension of a discrete space of cardinality $\geq \lambda$. We give sufficient conditions to characterize the calibers of $C_p(X)$ when $X$ is a topological sum, and we calculate the calibers of $C_p(X)$ when $X = \prod_{\xi < \lambda}X_\xi$ is a product of non-trivial Tychonoff spaces with $i$-weight less or equal to $\lambda$. The main theorem is: If $\kappa$ and $\lambda$ are cardinals with $\omega \leq \lambda\leq \kappa$, then the set of calibers of the space of the real-valued continuous functions defined on the one-point $\lambda$-Lindel\"of extension of the discrete space of cardinality $\kappa$ with its pointwise convergence topology, $C_p(L(\lambda, \kappa))$, is $\{\mu\in CN : cf(\mu)>\omega \ \wedge \ \mu\not\in (\lambda ,\kappa] \ \wedge \ cf(\mu)\not\in (\lambda,\kappa]\}$.
- [33] arXiv:2403.18032 [pdf, ps, other]
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Title: A Note on Almost Everywhere Convergence Along Tangential Curves to the Schrödinger Equation Initial DatumAuthors: Javier MinguillónComments: 6 pagesSubjects: Classical Analysis and ODEs (math.CA)
In this short note, we give an easy proof of the following result: for $ n\geq 2, $ $\underset{t\to0}{\lim} \,e^{it\Delta }f\left(x+\gamma(t)\right) = f(x) $ almost everywhere whenever $ \gamma $ is an $ \alpha- $H\"older curve with $ \frac12\leq \alpha\leq 1 $ and $ f\in H^s(\mathbb{R}^n) $, with $ s > \frac{n}{2(n+1)} $. This is the optimal range of regularity up to the endpoint.
- [34] arXiv:2403.18034 [pdf, ps, other]
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Title: Rank distribution in cubic twist families of elliptic curvesComments: Version 1: 25 pagesSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
Let $a$ be an integer which is not of the form $n^2$ or $-3 n^2$ for $n\in \mathbb{Z}$. Let $E_a$ be the elliptic curve with rational $3$-isogeny defined by $E_a:y^2=x^3+a$, and $K:=\mathbb{Q}(\mu_3)$. Assume that the $3$-Selmer group of $E_a$ over $K$ vanishes. It is shown that there is an explicit infinite set of cubefree integers $m$ such that the $3$-Selmer groups over $K$ of $E_{m^2 a}$ and $E_{m^4 a}$ both vanish. In particular, the ranks of these cubic twists are seen to be $0$ over $K$. Our results are proven by studying stability properties of $3$-Selmer groups in cyclic cubic extensions of $K$, via local and global Galois cohomological techniques.
- [35] arXiv:2403.18037 [pdf, ps, other]
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Title: A remark on the distortion of twisted sumsAuthors: Jesús SuárezSubjects: Functional Analysis (math.FA)
Odell and Schlumprecht solved the distortion problem by proving that the classic sequence spaces $\ell_p$ for $1<p<\infty$ admit an inevitable biorthogonal system. In particular, these spaces are arbitrarily distortable. Later, Maurey extended the result to asymptotic $\ell_p$-spaces while Tomczak-Jaegermann did likewise for the Schatten classes. We observe that the Kalton-Peck spaces $Z_p$ for $1<p<\infty$ admit an inevitable biorthogonal system. Therefore, we may add this classic family to the list of arbitrarily distortable spaces.
- [36] arXiv:2403.18044 [pdf, other]
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Title: Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback designComments: 9 pages, 6 figures, 2 tablesSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Dynamical Systems (math.DS); Numerical Analysis (math.NA); Fluid Dynamics (physics.flu-dyn)
Polytopic autoencoders provide low-dimensional parametrizations of states in a polytope. For nonlinear PDEs, this is readily applied to low-dimensional linear parameter-varying (LPV) approximations as they have been exploited for efficient nonlinear controller design via series expansions of the solution to the state-dependent Riccati equation. In this work, we develop a polytopic autoencoder for control applications and show how it outperforms standard linear approaches in view of LPV approximations of nonlinear systems and how the particular architecture enables higher order series expansions at little extra computational effort. We illustrate the properties and potentials of this approach to computational nonlinear controller design for large-scale systems with a thorough numerical study.
- [37] arXiv:2403.18049 [pdf, ps, other]
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Title: Quillen (co)homology of divided power algebras over an operadSubjects: Rings and Algebras (math.RA); Algebraic Topology (math.AT)
Barr--Beck cohomology, put into the framework of model categories by Quillen, provides a cohomology theory for any algebraic structure, for example Andr\'e--Quillen cohomology of commutative rings. Quillen cohomology has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad $\mathcal P$: the commutative and Lie operad respectively. In this paper, we investigate the Quillen cohomology of divided power algebras over an operad $\mathcal P$, identifying Beck modules, derivations, and K\"ahler differentials in that setup. We also compare the cohomology of divided power algebras over $\mathcal P$ with that of $\mathcal P$-algebras, and work out some examples.
- [38] arXiv:2403.18053 [pdf, ps, other]
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Title: Shear banding and cracking in unsaturated porous media through a nonlocal THM meshfree paradigmSubjects: Numerical Analysis (math.NA)
The thermo-hydro-mechanical of unsaturated soils plays a significant role in dynamic shear banding and fracturing. In this article, we propose a thermo-hydro-mechanical material model in the periporomechanics paradigm to model shear banding and crack triggered by temperature. Periporomechanics is a nonlocal framework for the mechanics of unsaturated soil where a length scale dictates the nonlocal interaction between material points. Periporomechanics unites continuous and discontinuous deformation and fluid flow in porous media. As a new contribution, we incorporate the thermo-hydro-mechanical material model in the periporomechanics through the correspondence principle for modeling shear banding and cracking in unsaturated porous media. The stabilized PPM correspondence principle that mitigates the multiphase zero-energy mode instability is augmented. At the global level, we have numerically implemented the periporomechanics paradigm through an explicit Lagrangian meshfree algorithm in the global level. At the local level, we impose the return mapping algorithm to implement the thermo-hydro-mechanical constitutive model. We present numerical examples to demonstrate the efficacy and robustness of proposed periporomechanics for modeling the shear banding bifurcation and crack in unsaturated porous media triggered by temperature.
- [39] arXiv:2403.18060 [pdf, ps, other]
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Title: The Cordiality Game and the Game Cordiality NumberComments: 12 pagesSubjects: Combinatorics (math.CO)
The cordiality game is played on a graph $G$ by two players, Admirable (A) and Impish (I), who take turns selecting \track{unlabeled} vertices of $G$. Admirable labels the selected vertices by $0$ and Impish by $1$, and the resulting label on any edge is the sum modulo $2$ of the labels of the vertices incident to that edge. The two players have opposite goals: Admirable attempts to minimize the number of edges with different labels as much as possible while Impish attempts to maximize this number. When both Admirable and Impish play their optimal games, we define the \emph{game cordiality number}, $c_g(G)$, as the absolute difference between the number of edges labeled zero and one. Let $P_n$ be the path on $n$ vertices. We show $c_g(P_n)\le \frac{n-3}{3}$ when $n \equiv 0 \pmod 3$, $c_g(P_n)\le \frac{n-1}{3}$ when $n \equiv 1 \pmod 3$, and $c_g(P_n)\le \frac{n+1}{3}$ when $n \equiv 2\pmod 3$. Furthermore, we show a similar bound, $c_g(T) \leq \frac{|T|}{2}$ holds for any tree $T$.
- [40] arXiv:2403.18065 [pdf, ps, other]
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Title: Primitive elements in the Hall algebra of a cyclic quiverAuthors: Renda MaComments: PhD thesisSubjects: Representation Theory (math.RT); Quantum Algebra (math.QA)
We provide an explicit formula for primitive elements in the Hall algebras of nilpotent representations of cyclic quivers.
- [41] arXiv:2403.18068 [pdf, other]
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Title: On the boundedness of solutions of a forced discontinuous oscillatorSubjects: Dynamical Systems (math.DS)
We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators.
- [42] arXiv:2403.18075 [pdf, ps, other]
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Title: Finite and Symmetric Euler Sums and Finite and Symmetric (Alternating) Multiple $T$-ValuesAuthors: Jianqiang ZhaoComments: 17 pagesJournal-ref: Axioms 2024 Volume 13 Issue 4, 210Subjects: Number Theory (math.NT)
In this paper, we will study finite multiple $T$-values (MTVs) and their alternating versions, which are level two and level four variations of finite multiple zeta values, respectively. We will first provide some structural results for level two finite multiple zeta values (i.e., finite Euler sums) for small weights, guided by the author's previous conjecture that the finite Euler sum space of weight, $w$, is isomorphic to a quotient Euler sum space of weight, $w$. Then, by utilizing some well-known properties of the classical alternating MTVs, we will derive a few important $\Q$-linear relations among the finite alternating MTVs, including the reversal, linear shuffle, and sum relations. We then compute the upper bound for the dimension of the $\Q$-span of finite (alternating) MTVs for some small weights by rigorously using the newly discovered relations, numerically aided by computers.
- [43] arXiv:2403.18077 [pdf, ps, other]
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Title: The equivalence of smooth and synthetic notions of timelike sectional curvature boundsComments: 15 pagesSubjects: Differential Geometry (math.DG)
Timelike sectional curvature bounds play an important role in spacetime geometry, both for the understanding of classical smooth spacetimes and for the study of Lorentzian (pre-)length spaces introduced in \cite{kunzinger2018lorentzian}. In the smooth setting, a bound on the sectional curvature of timelike planes can be formulated via the Riemann curvature tensor. In the synthetic setting, bounds are formulated by comparing various geometric configurations to the corresponding ones in constant curvature spaces. The first link between these notions in the Lorentzian context was established in \cite{harris1982triangle}, which was instrumental in the proof of powerful results in spacetime geometry \cite{beem1985toponogov, beem1985decomposition, galloway2018existence}. For general semi-Riemannian manifolds, the equivalence between sectional curvature bounds and synthetic bounds was established in \cite{alexander2008triangle}, however in this approach the sectional curvatures of both timelike and spacelike planes have to be considered. In this article, we fill a gap in the literature by proving the full equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes. As an essential tool, we establish Hessian comparison for the time separation and signed distance functions.
- [44] arXiv:2403.18088 [pdf, other]
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Title: Discretize first, filter next: learning divergence-consistent closure models for large-eddy simulationComments: 40 pages, 16 figures, 5 tablesSubjects: Numerical Analysis (math.NA); Fluid Dynamics (physics.flu-dyn)
We propose a new neural network based large eddy simulation framework for the incompressible Navier-Stokes equations based on the paradigm "discretize first, filter and close next". This leads to full model-data consistency and allows for employing neural closure models in the same environment as where they have been trained. Since the LES discretization error is included in the learning process, the closure models can learn to account for the discretization.
Furthermore, we introduce a new divergence-consistent discrete filter defined through face-averaging. The new filter preserves the discrete divergence-free constraint by construction, unlike general discrete filters such as volume-averaging filters. We show that using a divergence-consistent LES formulation coupled with a convolutional neural closure model produces stable and accurate results for both a-priori and a-posteriori training, while a general (divergence-inconsistent) LES model requires a-posteriori training or other stability-enforcing measures. - [45] arXiv:2403.18091 [pdf, ps, other]
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Title: The Flying Sidekick Traveling Salesman Problem with Multiple Drops: A Simple and Effective Heuristic ApproachAuthors: Sarah K. Schaumann, Abhishake Kundu, Juan C. Pina-Pardo, Matthias Winkenbach, Ricardo A. Gatica, Stephan M. Wagner, Timothy I. MatisSubjects: Optimization and Control (math.OC)
We study the Flying Sidekick Traveling Salesman Problem with Multiple Drops (FSTSP-MD), a multi-modal last-mile delivery model where a single truck and a single drone cooperatively deliver customer packages. In the FSTSP-MD, the drone can be launched from the truck to deliver multiple packages before it returns to the truck for a new delivery operation. The FSTSP-MD aims to find the synchronized truck and drone delivery routes that minimize the completion time of the delivery process. We develop a simple and effective heuristic approach based on an order-first, split-second scheme. This heuristic combines standard local search and diversification techniques with a novel shortest-path problem that finds FSTSP-MD solutions in polynomial time. We show that our heuristic consistently outperforms state-of-the-art heuristics developed for the FSTSP-MD and the FSTSP (i.e., the single-drop case) through extensive numerical experiments. We also show that the FSTSP-MD substantially reduces completion times compared to a traditional truck-only delivery system. Several managerial insights are described regarding the effects of drone capacity, drone speed, drone flight endurance, and customer distribution.
- [46] arXiv:2403.18099 [pdf, other]
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Title: Nested Hilbert Schemes on Hirzebruch surfaces and quiver varietiesComments: 28 pagesSubjects: Algebraic Geometry (math.AG); Representation Theory (math.RT)
For $n\ge 1$ we show that the length 1 nested Hilbert scheme of the total space $X_n$ of the line bundle $\mathcal O_{\mathbb P^1}(-n)$, parameterizing pairs of nested 0-cycles in $X_n$, is a quiver variety associated with a suitable quiver with relations. This generalizes previous work about nested Hilbert schemes on $\mathbb C^2$ in one direction, and about the Hilbert schemes of points of $X_n$ in another direction.
- [47] arXiv:2403.18102 [pdf, ps, other]
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Title: The operadic theory of convexityComments: 42 pagesSubjects: Category Theory (math.CT); Information Theory (cs.IT); Quantum Physics (quant-ph)
In this article, we characterize convexity in terms of algebras over a PROP, and establish a tensor-product-like symmetric monoidal structure on the category of convex sets. Using these two structures, and the theory of $\scr{O}$-monoidal categories, we state and prove a Grothendieck construction for lax $\scr{O}$-monoidal functors into convex sets. We apply this construction to the categorical characterization of entropy of Baez, Fritz, and Leinster, and to the study of quantum contextuality in the framework of simplicial distributions.
- [48] arXiv:2403.18109 [pdf, ps, other]
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Title: Hölder continuity of core entropy for non-recurrent quadratic polynomialsComments: 24 pagesSubjects: Dynamical Systems (math.DS)
We prove that core entropy is H\"older continuous as a function of external angles for a large class of quadratic polynomials that are non-recurrent with respect to angle-doubling. The result follows from a symbolic analysis of the Mandelbrot set and the dynamics of Hubbard trees in terms of kneading sequences which has been established in previous work.
- [49] arXiv:2403.18110 [pdf, other]
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Title: Randomisation in the Josephus ProblemSubjects: Probability (math.PR); Combinatorics (math.CO); Number Theory (math.NT)
The Josephus problem is a well--studied elimination problem consisting in determining the position of the survivor after repeated applications of a deterministic rule removing one person at a time from a given group.
A natural probabilistic variant of this process is introduced in this paper. More precisely, in this variant, the survivor is determined after performing a succession of Bernouilli trials with parameter $p$ designating each time the person to remove. When the number of participants tends to infinity, the main result characterises the limit distribution of the position of the survivor with an increasing degree of precision as the parameter approaches the unbiaised case $p=1/2$. Then, the convergence rate to the position of the survivor is obtained in the form of a Central-Limit Theorem.
A number of other variants of the suggested probabilistic elimination process are also considered. They each admit a specific limit behavior which, in most cases, is stated in the form of an open problem. - [50] arXiv:2403.18113 [pdf, ps, other]
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Title: $\ell_1$ spreading models and the FPP for Cesàro mean nonexpansive mapsAuthors: C. S. BarrosoComments: All comments are welcome. arXiv admin note: substantial text overlap with arXiv:2302.04323Subjects: Functional Analysis (math.FA)
Let $K$ be a nonempty set in a Banach space $X$. A mapping $T\colon K\to K$ is called {\it $\mathfrak{cm}$-nonexpansive} if for any sequence $(x_i)_{i=1}^n$ and $y$ in $K$, one has $\|(1/n) \sum_{i=1}^n Tx_i -Ty\|\leq \|(1/n)\sum_{i=1}^n x_i - y\|$. As a subsclass of nonexpansive maps, the FPP for such maps is well-established in a great variety of spaces. The main result of this paper is a fixed point result relating $\mathfrak{cm}$-nonexpansiveness, $\ell_1$ spreading models and Schauder bases with not-so-large basis constants. As a result, we deduce that every Banach space with weak Banach-Saks property has the fixed point property for $\mathfrak{cm}$-nonexpansive maps.
- [51] arXiv:2403.18124 [pdf, other]
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Title: Stochastic Finite Volume Method for Uncertainty Management in Gas Pipeline Network FlowsSubjects: Optimization and Control (math.OC); Computation (stat.CO)
Natural gas consumption by users of pipeline networks is subject to increasing uncertainty that originates from the intermittent nature of electric power loads serviced by gas-fired generators. To enable computationally efficient optimization of gas network flows subject to uncertainty, we develop a finite volume representation of stochastic solutions of hyperbolic partial differential equation (PDE) systems on graph-connected domains with nodal coupling and boundary conditions. The representation is used to express the physical constraints in stochastic optimization problems for gas flow allocation subject to uncertain parameters. The method is based on the stochastic finite volume approach that was recently developed for uncertainty quantification in transient flows represented by hyperbolic PDEs on graphs. In this study, we develop optimization formulations for steady-state gas flow over actuated transport networks subject to probabilistic constraints. In addition to the distributions for the physical solutions, we examine the dual variables that are produced by way of the optimization, and interpret them as price distributions that quantify the financial volatility that arises through demand uncertainty modeled in an optimization-driven gas market mechanism. We demonstrate the computation and distributional analysis using a single-pipe example and a small test network.
- [52] arXiv:2403.18130 [pdf, other]
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Title: Generalized Maximum Entropy Differential Dynamic ProgrammingComments: 7 pages, 5 figures, This paper is for CDC 2024Subjects: Optimization and Control (math.OC); Information Theory (cs.IT)
We present a sampling-based trajectory optimization method derived from the maximum entropy formulation of Differential Dynamic Programming with Tsallis entropy. This method can be seen as a generalization of the legacy work with Shannon entropy, which leads to a Gaussian optimal control policy for exploration during optimization. With the Tsallis entropy, the optimal control policy takes the form of $q$-Gaussian, which further encourages exploration with its heavy-tailed shape. Moreover, in our formulation, the exploration variance, which was scaled by a fixed constant inverse temperature in the original formulation with Shannon entropy, is automatically scaled based on the value function of the trajectory. Due to this property, our algorithms can promote exploration when necessary, that is, the cost of the trajectory is high, rather than using the same scaling factor. The simulation results demonstrate the properties of the proposed algorithm described above.
- [53] arXiv:2403.18131 [pdf, other]
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Title: Convergence of Iterative Quadratic Programming for Robust Fixed-Endpoint Transfer of Bilinear SystemsSubjects: Optimization and Control (math.OC); Quantum Physics (quant-ph)
We present a computational method for open-loop minimum-norm control synthesis for fixed-endpoint transfer of bilinear ensemble systems that are indexed by two continuously varying parameters. We suppose that one ensemble parameter scales the homogeneous, linear part of the dynamics, and the second parameter scales the effect of the applied control inputs on the inhomogeneous, bilinear dynamics. This class of dynamical systems is motivated by robust quantum control pulse synthesis, where the ensemble parameters correspond to uncertainty in the free Hamiltonian and inhomogeneity in the control Hamiltonian, respectively. Our computational method is based on polynomial approximation of the ensemble state in parameter space and discretization of the evolution equations in the time domain using a product of matrix exponentials corresponding to zero-order hold controls over the time intervals. The dynamics are successively linearized about control and trajectory iterates to formulate a sequence of quadratic programs for computing perturbations to the control that successively improve the objective until the iteration converges. We use a two-stage computation to first ensure transfer to the desired terminal state, and then minimize the norm of the control function. The method is demonstrated for the canonical uniform transfer problem for the Bloch system that appears in nuclear magnetic resonance, as well as the matter-wave splitting problem for the Raman-Nath system that appears in ultra-cold atom interferometry.
- [54] arXiv:2403.18138 [pdf, other]
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Title: Constructing abelian varieties from rank 3 Galois representations with real trace fieldComments: 3 pages, comments welcome!Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 $\overline{\mathbb Q}_{\ell}$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$ extends to an integral model such that the Frobenius traces are contained in a fixed totally real number field. Then, after potentially shrinking $U$, there exists an abelian scheme $f\colon B_U\rightarrow U$ such that $L$ is a summand of $R^2f_*\overline{\mathbb Q}_{\ell}(1)$.
The key ingredients are: (1) the totally real assumption implies $L$ admits a square root $M$; (2) the trace field of $M$ is sufficiently bounded, allowing us to use recent work of Krishnamoorthy-Yang-Zuo to construct an abelian scheme over $U_{\bar K}$ geometrically realizing $L$; and (3) Deligne's weight-monodromy theorem and the Rapoport-Zink spectral sequence, which allow us to pin down the arithmetizations using the total degeneration. - [55] arXiv:2403.18141 [pdf, ps, other]
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Title: The 2D Toda lattice hierarchy for multiplicative functionals of Schur measuresAuthors: Pierre LazagSubjects: Mathematical Physics (math-ph); Combinatorics (math.CO); Probability (math.PR)
We prove Fredholm determinants build out from generalizations of Schur measures, or equivalently, arbitrary multiplicative functionals of the original Schur measures are tau-functions of the 2D Toda lattice hierarchy. Our result apply to finite temperature Schur measures, and extends both the result of Okounkov in \cite{okounkovschurmeasures} and of Cafasso-Ruzza in \cite{cafassoruzza} concerning the finite-temperature Plancherel measure. Our proof lies on the semi-infinite wedge formalism and the Boson-Fermion correspondance.
- [56] arXiv:2403.18146 [pdf, ps, other]
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Title: Adaptive TTD Configurations for Near-Field Communications: An Unsupervised Transformer ApproachSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
True-time delayers (TTDs) are popular analog devices for facilitating near-field wideband beamforming subject to the spatial-wideband effect. In this paper, an adaptive TTD configuration is proposed for short-range TTDs. Compared to the existing TTD configurations, the proposed one can effectively combat the spatial-widebandd effect for arbitrary user locations and array shapes with the aid of a switch network. A novel end-to-end deep neural network is proposed to optimize the hybrid beamforming with adaptive TTDs for maximizing spectral efficiency. 1) First, based on the U-Net architecture, a near-field channel learning module (NFC-LM) is proposed for adaptive beamformer design through extracting the latent channel response features of various users across different frequencies. In the NFC-LM, an improved cross attention (CA) is introduced to further optimize beamformer design by enhancing the latent feature connection between near-field channel and different beamformers. 2) Second, a switch multi-user transformer (S-MT) is proposed to adaptively control the connection between TTDs and phase shifters (PSs). In the S-MT, an improved multi-head attention, namely multi-user attention (MSA), is introduced to optimize the switch network through exploring the latent channel relations among various users. 3) Third, a multi feature cross attention (MCA) is introduced to simultaneously optimize the NFC-LM and S-MT by enhancing the latent feature correlation between beamformers and switch network. Numerical simulation results show that 1) the proposed adaptive TTD configuration effectively eliminates the spatial-wideband effect under uniform linear array (ULA) and uniform circular array (UCA) architectures, and 2) the proposed deep neural network can provide near optimal spectral efficiency, and solve the multi-user bemformer design and dynamical connection problem in real-time.
- [57] arXiv:2403.18153 [pdf, other]
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Title: Markov chains and mappings of distributions on compact spaces II: Numerics and ConjecturesSubjects: Probability (math.PR)
Consider a compact metric space $S$ and a pair $(j,k)$ with $k \ge 2$ and $1 \le j \le k$. For any probability distribution $\theta \in P(S)$, define a Markov chain on $S$ by: from state $s$, take $k$ i.i.d. ($\theta$) samples, and jump to the $j$'th closest. Such a chain converges in distribution to a unique stationary distribution, say $\pi_{j,k}(\theta)$. This defines a mapping $\pi_{j,k}: P(S) \to P(S)$. What happens when we iterate this mapping? In particular, what are the fixed points of this mapping? A few results are proved in a companion article; this article, not intended for formal publication, records numerical studies and conjectures.
- [58] arXiv:2403.18154 [pdf, ps, other]
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Title: Cohomology classes, periods, and special values of Rankin-Selberg $L$-functionsComments: 21 pagesSubjects: Number Theory (math.NT)
In this article, we give a cohomological interpretation of (a special case of) the integrals constructed by the second named author and Q. Zhang \cite{YanZhang2023} which represent the product of Rankin-Selberg $L$-functions of $\mathrm{GL}_n\times\mathrm{GL}_m$ and $\mathrm{GL}_n\times\mathrm{GL}_{n-m-1}$ for $m<n$. As an application, we prove an algebraicity result for the special values of certain $L$-functions. This work is a generalization of the algebraicity result of Raghuram for $\mathrm{GL}_n\times\mathrm{GL}_{n-1}$ \cite{Raghuram2010} in the special case $m=n-1$, and the results of Mahnkopf \cite{Mahnkopf1998, Mahnkopf2005} in the special case $m=n-2$.
- [59] arXiv:2403.18155 [pdf, other]
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Title: An inexact infeasible arc-search interior-point method for linear programming problemsComments: 25 pages, 3 figuresSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
Inexact interior-point methods (IPMs) are a type of interior-point methods that inexactly solve the linear equation system for obtaining the search direction. On the other hand,arc-search IPMs approximate the central path with an ellipsoidal arc obtained by solving two linear equation systems in each iteration, while conventional line-search IPMs solve one linear system, therefore, the improvement due to the inexact solutions of the linear equation systems can be more beneficial in arc-search IPMs than conventional IPMs. In this paper, we propose an inexact infeasible arc-search interior-point method.We establish that the proposed method is a polynomial-time algorithm through its convergence analysis. The numerical experiments with the conjugate gradient method show that the proposed method can reduce the number of iterations compared to an existing method for benchmark problems; the numbers of iterations are reduced to two-thirds for more than 70% of the problems.
- [60] arXiv:2403.18170 [pdf, ps, other]
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Title: Formal deformations, cohomology theory and $L_\infty[1]$-structures for differential Lie algebras of arbitrary weightSubjects: Rings and Algebras (math.RA)
Generalising a previous work of Jiang and Sheng, a cohomology theory for differential Lie algebras of arbitrary weight is introduced. The underlying $L_\infty[1]$-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between $L_\infty[1]$-structures for absolute and relative differential Lie algebras are established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras. In a forthcoming paper, we will show that the operad of homotopy (relative) differential Lie algebras is the minimal model of the operad of (relative) differential Lie algebras.
- [61] arXiv:2403.18171 [pdf, other]
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Title: Higher order multi-dimension reduction methods via Einstein-productSubjects: Numerical Analysis (math.NA)
This paper explores the extension of dimension reduction (DR) techniques to the multi-dimension case by using the Einstein product. Our focus lies on graph-based methods, encompassing both linear and nonlinear approaches, within both supervised and unsupervised learning paradigms. Additionally, we investigate variants such as repulsion graphs and kernel methods for linear approaches. Furthermore, we present two generalizations for each method, based on single or multiple weights. We demonstrate the straightforward nature of these generalizations and provide theoretical insights. Numerical experiments are conducted, and results are compared with original methods, highlighting the efficiency of our proposed methods, particularly in handling high-dimensional data such as color images.
- [62] arXiv:2403.18179 [pdf, ps, other]
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Title: Tagged particles and size-biased dynamics in mean-field interacting particle systemsComments: 12 pagesSubjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech)
We establish a connection between tagged particles and size-biased empirical processes in interacting particle systems, in analogy to classical results on the propagation of chaos. In a mean-field scaling limit, the evolution of the occupation number on the tagged particle site converges to a time-inhomogeneous Markov process with non-linear master equation given by the law of large numbers of size-biased empirical measures. The latter are important in recent efforts to understand the dynamics of condensation in interacting particle systems.
- [63] arXiv:2403.18190 [pdf, ps, other]
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Title: Green Functions in Small CharacteristicAuthors: Frank LübeckSubjects: Representation Theory (math.RT)
The values of the ordinary Green functions are known for almost all groups of Lie type, a long term achievement by various authors.
In this note we solve the last open cases, which are for exceptional groups of type $E_8(q)$ where $q$ is a power of $2$, $3$ or $5$. - [64] arXiv:2403.18213 [pdf, other]
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Title: Long-Term Mine Planning with Large Neighbourhood SearchSubjects: Optimization and Control (math.OC)
We present a Large Neighbourhood Search based approach for solving complex long-term open-pit mine planning problems. An initial feasible solution, generated by a sliding windows heuristic, is improved through repeated solves of a restricted mixed integer program. Each iteration leaves only a subset of the variables in our planning model free to take on new values. We form these subsets through the use of a novel path-based neighbourhood structure, and neighbourhood formation strategies that exploit the structure of the planning model. We show that our method is able to find near-optimal solutions to problems that cannot be solved by an off-the-shelf solver in a reasonable time, or with reasonable computational resources.
- [65] arXiv:2403.18217 [pdf, ps, other]
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Title: Mixed Variational Formulation of Coupled PlatesSubjects: Numerical Analysis (math.NA)
This paper proposes a mixed variational formulation for the problem of two coupled plates with a rigid {junction}. The proposed mixed {formulation} introduces {the union of} stresses and moments as {an auxiliary variable}, which {are} commonly of great interest in practical applications. The primary challenge lies in determining a suitable {space involving} both boundary and junction conditions of the auxiliary variable. The {theory} of densely defined operators in Hilbert spaces is employed to define {a nonstandard Sobolev space} without the use of trace operators. The well-posedness is established for the mixed formulation. Based on these conditions, this paper provides a framework {of} conforming {mixed} finite element methods. Numerical experiments are given to validate the theoretical results.
- [66] arXiv:2403.18231 [pdf, ps, other]
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Title: The Dimensions of the Hulls of Conorm Codes from Algebraic Geometry CodesSubjects: Information Theory (cs.IT)
Chara et al. introduced conorm codes defined over algebraic geometry codes, but the hulls of conorm codes were not determined yet. In this paper, we study the dimension of the hull of conorm codes using the method introduced by Camps et al. For an algebraic geometry code $\mathcal{C}:=C_\mathscr{L}(D, G)$, we consider the divisor $\gcd(G, H)$, where $H$ is the divisor satisfying \[C_\mathscr{L}(D, G)^\perp=C_\mathscr{L}(D, H).\] Given an extension $F'/\mathbb{F}_{q^t}$ of an algebraic function field $F/\mathbb{F}_q$, we assume that the divisor $\gcd(G, H)$ is non-special. If the degree of $\gcd(G, H)$ is greater than $2g-2+{t\over [F':F]}\deg\text{Diff}(F'/F)$, then we have determined the exact dimension of the hull of the conorm of $\mathcal{C}$. If not, we have determined the lower bound of the dimension of the hull of the conorm of $\mathcal{C}$. We provide some examples for the dimension of the hull of certain conorm codes of AG codes defined over a rational function field.
- [67] arXiv:2403.18237 [pdf, ps, other]
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Title: Analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problemComments: Our unified analytical framework provides a holistic view of the phase space structures near collinear libration points in the CRTBP. It successfully addresses analytical challenges related to quasihalo orbits and their invariant manifolds. The proposed coupling-induced bifurcation mechanism and the corresponding coupling coefficients can also be applied to general dynamical systemsSubjects: Mathematical Physics (math-ph); Earth and Planetary Astrophysics (astro-ph.EP)
A unified analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits and quasihalo orbits, their invariant manifolds, as well as transit and non-transit orbits. Traditional methods could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. By introducing a coupling coefficient {\eta} and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt-Poincar\'e method. Analyzing the third-order bifurcation equation reveals bifurcation conditions for halo orbits, quasihalo orbits, and their invariant manifolds. Furthermore, new families of periodic orbits similar to halo orbits are discovered, and non-periodic/quasi-periodic orbits, such as transit orbits and non-transit orbits, are found to undergo bifurcations. When {\eta} = 0, the series solution describes Lissajous orbits and their invariant manifolds, transit, and non-transit orbits. As {\eta} varies from zero to non-zero values, the solution seamlessly transitions to describe quasihalo orbits and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.
- [68] arXiv:2403.18255 [pdf, other]
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Title: Statistical inference for multi-regime threshold Ornstein-Uhlenbeck processesSubjects: Statistics Theory (math.ST)
In this paper, we investigate the parameter estimation for threshold Ornstein$\mathit{-}$Uhlenbeck processes. Least squares method is used to obtain continuous-type and discrete-type estimators for the drift parameters based on continuous and discrete observations, respectively. The strong consistency and asymptotic normality of the proposed least squares estimators are studied. We also propose a modified quadratic variation estimator based on the long-time observations for the diffusion parameters and prove its consistency. Our simulation results suggest that the performance of our proposed estimators for the drift parameters may show improvements compared to generalized moment estimators. Additionally, the proposed modified quadratic variation estimator exhibits potential advantages over the usual quadratic variation estimator with relatively small sample sizes. In particular, our method can be applied to the multi-regime cases ($m>2$), while the generalized moment method only deals with the two regime cases ($m=2$). The U.S. treasury rate data is used to illustrate the theoretical results.
- [69] arXiv:2403.18261 [pdf, ps, other]
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Title: Simplicity of the contactomorphism group of finite regularityAuthors: Yong-Geun OhComments: 61 pages, comments welcome!Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Geometric Topology (math.GT)
For a given coorientable contact manifold $(M^{2n+1},\xi)$, we consider the group $ \operatorname{Cont}_c^{(r,\delta)}(M,\alpha)$ consisting of $C^{r,\delta}$ contactomorphisms with compact support which is equipped with $C^{r,\delta}$-topology of H\"older regularity $(r,\delta)$ for $r \geq 1$ and $0 <\delta \leq 1$. We prove that for all H\"older class exponents with $r > n + 2$ or $r = n+1, \, \frac12 < \delta \leq 1$ (resp. $r < n+1$ or $r = n+1$ and $ 0< \delta <\frac12$), the group is a perfect (and so a simple) group. In particular, $\operatorname{Cont}_c^r(M,\xi)$ is simple for all integer $r \geq 1$. For the case of $\operatorname{Cont}_c^{(r,\delta)}(M,\alpha)$ of general H\"older regularity, we prove the simplicity for all pairs $(r,\delta)$ leaving only the case of $(r,\delta) = (n+1,\frac12)$ open.
- [70] arXiv:2403.18273 [pdf, ps, other]
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Title: Analysis of solution to an elliptic free boundary value problem equipped with a `bad' dataAuthors: Debajyoti ChoudhuriSubjects: Analysis of PDEs (math.AP)
We will study a free boundary value problem driven by a source term which is quite {\it irregular}. In the process, we will establish a monotonicity result, and regularity of the solution.
- [71] arXiv:2403.18279 [pdf, ps, other]
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Title: Forbidden complexes for the 3-sphereComments: 17 pages, 14 figuresSubjects: Geometric Topology (math.GT); Combinatorics (math.CO)
A simplicial complex is said to be {\em critical} (or {\em forbidden}) for the 3-sphere $S^3$ if it cannot be embedded in $S^3$ but after removing any one point, it can be embedded.
We show that if a multibranched surface cannot be embedded in $S^3$, it contains a critical complex which is a union of a multibranched surface and a (possibly empty) graph. We exhibit all critical complexes for $S^3$ which are contained in $K_5 \times S^1$ and $K_{3,3} \times S^1$ families. We also classify all critical complexes for $S^3$ which can be decomposed into $G\times S^1$ and $H$, where $G$ and $H$ are graphs.
In spite of the above property, there exist complexes which cannot be embedded in $S^3$, but they do not contain any critical complexes. From the property of those examples, we define an equivalence relation on all simplicial complexes $\mathcal{C}$ and a partially ordered set of complexes $(\mathcal{C}/\mathord\sim; \subseteqq)$, and refine the definition of critical. According to the refined definition of critical, we show that if a complex $X$ cannot be embedded in $S^3$, then there exists $[X']\subseteqq [X]$ such that $[X']$ is critical for $[S^3]$. - [72] arXiv:2403.18284 [pdf, other]
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Title: A new dual spectral projected gradient method for log-determinant semidefinite programming with hidden clustering structuresComments: 21 pages, 3 figuresSubjects: Optimization and Control (math.OC)
In this paper, we propose a new efficient method for a sparse Gaussian graphical model with hidden clustering structures by extending a dual spectral projected gradient (DSPG) method proposed by Nakagaki et al.~(2020). We establish the global convergence of the proposed method to an optimal solution, and we show that the projection onto the feasible region can be solved with a low computational complexity by the use of the pool-adjacent-violators algorithm. Numerical experiments on synthesis data and real data demonstrate the efficiency of the proposed method. The proposed method takes 0.91 seconds to achieve a similar solution to the direct application of the DSPG method which takes 4361 seconds.
- [73] arXiv:2403.18285 [pdf, other]
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Title: Stability and convergence of the penalty formulation for nonlinear magnetostaticsSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
The magnetostatic field distribution in a nonlinear medium amounts to the unique minimizer of the magnetic coenergy over all fields that can be generated by the same current. This is a nonlinear saddlepoint problem whose numerical solution can in principle be achieved by mixed finite element methods and appropriate nonlinear solvers. The saddlepoint structure, however, makes the solution cumbersome. A remedy is to split the magnetic field into a known source field and the gradient of a scalar potential which is governed by a convex minimization problem. The penalty approach avoids the use of artificial potentials and Lagrange multipliers and leads to an unconstrained convex minimization problem involving a large parameter. We provide a rigorous justification of the penalty approach by deriving error estimates for the approximation due to penalization. We further highlight the close connections to the Lagrange-multiplier and scalar potential approach. The theoretical results are illustrated by numerical tests for a typical benchmark problem
- [74] arXiv:2403.18287 [pdf, ps, other]
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Title: Frozen Gaussian approximation for the fractional Schrödinger equationSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
We develop the frozen Gaussian approximation (FGA) for the fractional Schr\"odinger equation in the semi-classical regime, where the solution is highly oscillatory when the scaled Planck constant $\varepsilon$ is small. This method approximates the solution to the Schr\"odinger equation by an integral representation based on asymptotic analysis and provides a highly efficient computational method for high-frequency wave function evolution. In particular, we revise the standard FGA formula to address the singularities arising in the higher-order derivatives of coefficients of the associated Hamiltonian flow that are second-order continuously differentiable or smooth in conventional FGA analysis. We then establish its convergence to the true solution. Additionally, we provide some numerical examples to verify the accuracy and convergence behavior of the frozen Gaussian approximation method.
- [75] arXiv:2403.18297 [pdf, other]
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Title: A Mean Field Game of Sequential TestingComments: 51 pages, 3 figuresSubjects: Optimization and Control (math.OC); Probability (math.PR)
We introduce a mean field game for a family of filtering problems related to the classic sequential testing of the drift of a Brownian motion. To the best of our knowledge this work presents the first treatment of mean field filtering games with stopping and an unobserved common noise in the literature. We show that the game is well-posed, characterize the solution, and establish the existence of an equilibrium under certain assumptions. We also perform numerical studies for several examples of interest.
- [76] arXiv:2403.18304 [pdf, ps, other]
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Title: Complete moment convergence of moving average processes for $m$-widely acceptable sequence under sub-linear expectationsComments: 16 pages,submitted to Journal of Inequalities and ApplicationsSubjects: Probability (math.PR)
In this article, the complete moment convergence for the partial sum of moving average processes $\{X_n=\sum_{i=-\infty}^{\infty}a_iY_{i+n},n\ge 1\}$ is estabished under some proper conditions, where $\{Y_i,-\infty<i<\infty\}$ is a sequence of $m$-widely acceptable ($m$-WA) random variables, which is stochastically dominated by a random variable $Y$ in sub-linear expectations space $(\Omega,\HH,\ee)$ and $\{a_i,-\infty<i<\infty\}$ is an absolutely summable sequence of real numbers. The results extend the relevant results in probability space to those under sub-linear expectations.
- [77] arXiv:2403.18307 [pdf, ps, other]
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Title: Mutual Information Optimization for SIM-Based Holographic MIMO SystemsComments: 5 pages, 2 figuresSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
In the context of emerging stacked intelligent metasurface (SIM)-based holographic MIMO (HMIMO) systems, a fundamental problem is to study the mutual information (MI) between transmitted and received signals to establish their capacity. However, direct optimization or analytical evaluation of the MI, particularly for discrete signaling, is often intractable. To address this challenge, we adopt the channel cutoff rate (CR) as an alternative optimization metric for the MI maximization. In this regard, we propose an alternating projected gradient method (APGM), which optimizes the CR of a SIM-based HMIMO system by adjusting signal precoding and the phase shifts across the transmit and receive SIMs in a layer-by-layer basis. Simulation results indicate that the proposed algorithm significantly enhances the CR, achieving substantial gains proportional to those observed for the corresponding MI. This justifies the effectiveness of using the channel CR for the MI optimization. Moreover, we demonstrate that the integration of digital precoding, even on a modest scale, has a significant impact on the ultimate performance of SIM-aided systems.
- [78] arXiv:2403.18315 [pdf, ps, other]
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Title: Topological Vector Spaces: a non-standard approach with monads and galaxiesSubjects: Logic (math.LO); Functional Analysis (math.FA)
By generalizing the overspill principle towards directed sets, a new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties associated with these maps can be extended not only to internal sets but to all monads and galaxies. An abstract theory of Intersections of Galaxies is introduced. These concepts are applied to basic topology as well (locally convex) topological vector spaces and their properties. Local properties and completeness can be defined and characterized effortlessly. Duality theory is studied in this framework, allowing in particular to formulate brief and insightful proofs for the theorems of Mackey-Arens and Grothendieck completeness without any technicalities.
- [79] arXiv:2403.18320 [pdf, other]
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Title: Online Prediction for Streaming Tensor Time SeriesSubjects: Optimization and Control (math.OC)
Real-time prediction plays a vital role in various control systems, such as traffic congestion control and wireless channel resource allocation. In these scenarios, the predictor usually needs to track the evolution of the latent statistical patterns in the modern high-dimensional streaming time series continuously and quickly, which presents new challenges for traditional prediction methods. This paper proposes a novel algorithm based on tensor factorization to predict streaming tensor time series online. The proposed algorithm updates the predictor in a low-complexity online manner to adapt to the time-evolving data. Additionally, an automatically adaptive version of the algorithm is presented to mitigate the negative impact of stale data. Simulation results demonstrate that our proposed methods achieve prediction accuracy similar to that of conventional offline tensor prediction methods, while being much faster than them during long-term online prediction. Therefore, our proposed algorithm provides an effective and efficient solution for the online prediction of streaming tensor time series.
- [80] arXiv:2403.18335 [pdf, ps, other]
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Title: Arc-transitive maps with coprime Euler characteristic and edge numberSubjects: Combinatorics (math.CO); Group Theory (math.GR)
This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one carries out the classification work for arc-transitive maps with nonsolvable automorphism groups, which illustrates how the edge number impacts on the Euler characteristic for maps. The classification is involved with the construction of some new and interesting arc-regular maps.
- [81] arXiv:2403.18345 [pdf, other]
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Title: Compactifications of the Eisenstein ancestral Deligne-Mostow varietyComments: 50 pages, comments welcomeSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
All arithmetic non-compact ball quotients by Deligne-Mostow's unitary monodromy group arise as sub-ball quotients of either of two spaces called ancestral cases, corresponding to Gaussian or Eisenstein Hermitian forms respectively. In our previous paper, we investigated the compactifications of the Gaussian Deligne-Mostow variety. Here we work on the remaining case, namely the ring of Eisenstein integers, which is related to the moduli space of unordered 12 points on $\mathbb P^1$. In particular, we show that Kirwan's partial resolution of the moduli space is not a semi-toroidal compactification and Deligne-Mostow's period map does not lift to the unique toroidal compactification. We give two interpretations of these phenomena in terms of the log minimal model program and automorphic forms. As an application, we prove that the above two compactifications are not (stacky) derived equivalent, as the $DK$-conjecture predicts. Furthermore, we construct an automorphic form on the moduli space of non-hyperelliptic curves of genus 4, which is isogenous to the Eisenstein Deligne-Mostow variety, giving another intrinsic proof, independent of lattice embeddings, of a result by Casalaina-Martin, Jensen and Laza.
- [82] arXiv:2403.18353 [pdf, other]
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Title: Early Stopping for Ensemble Kalman-Bucy InversionAuthors: Maia TienstraSubjects: Statistics Theory (math.ST)
Bayesian linear inverse problems aim to recover an unknown signal from noisy observations, incorporating prior knowledge. This paper analyses a data dependent method to choose the scale parameter of a Gaussian prior. The method we study arises from early stopping methods, which have been successfully applied to a range of problems for statistical inverse problems in the frequentist setting. These results are extended to the Bayesian setting. We study the use of a discrepancy based stopping rule in the setting of random noise. Our proposed stopping rule results in optimal rates under certain conditions on the prior covariance operator. We furthermore derive for which class of signals this method is adaptive. It is also shown that the associated posterior contracts at the optimal rate and provides a conservative measure of uncertainty. We implement the proposed stopping rule using the continuous-time ensemble Kalman--Bucy filter (EnKBF). The fictitious time parameter replaces the scale parameter, and the ensemble size is appropriately adjusted in order to not lose statistical optimality of the computed estimator. The EnKBF, then, gives a continuous process from the prior distribution to the posterior which is terminated using the proposed stopping rule.
- [83] arXiv:2403.18357 [pdf, ps, other]
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Title: Minimax density estimation in the adversarial framework under local differential privacyAuthors: Mélisande Albert (IMT, INSA Toulouse), Juliette Chevallier (IMT, INSA Toulouse), Béatrice Laurent (INSA Toulouse, IMT), Ousmane Sacko (UPN, MODAL'X)Subjects: Statistics Theory (math.ST)
We consider the problem of nonparametric density estimation under privacy constraints in an adversarial framework. To this end, we study minimax rates under local differential privacy over Sobolev spaces. We first obtain a lower bound which allows us to quantify the impact of privacy compared with the classical framework. Next, we introduce a new Coordinate block privacy mechanism that guarantees local differential privacy, which, coupled with a projection estimator, achieves the minimax optimal rates.
- [84] arXiv:2403.18362 [pdf, other]
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Title: Fractional variational integrators based on convolution quadratureSubjects: Numerical Analysis (math.NA)
Fractional dissipation is a powerful tool to study non-local physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. In [19], a new approach is proposed to deal with dissipative systems including fractionally damped systems in a variational way for both, the continuous and discrete setting. It is based on the doubling of variables and their fractional derivatives. The aim of this work is to derive higher-order fractional variational integrators by means of convolution quadrature (CQ) based on backward difference formulas. We then provide numerical methods that are of order 2 improving a previous result in [19]. The convergence properties of the fractional variational integrators and saturation effects due to the approximation of the fractional derivatives by CQ are studied numerically.
- [85] arXiv:2403.18363 [pdf, other]
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Title: Computing safe bicycle routes -- Berechnung sicherer FahrradwegeComments: 13 pages in GermanSubjects: Optimization and Control (math.OC)
The safety of streets is difficult to quantify numerically. However, it is possible to sort streets regarding their safety into ordered categories, like safe, neutral and unsafe. In this paper we model the computation of safe bicycle routes as an optimization problem with ordinal coefficients. We describe an appropriate optimality concept for ordinal optimization problems and introduce a solution strategy for ordinal routing problems. Furthermore, we introduce a concept to incorporate safety preferences by introducing weights such that longer path with a higher safety rating are preferred. We apply the concept of ordinal routing to compute safe bicycle routes in Stuttgart, Germany, based on dates from OpenStreetMaps. We show that the choice of the weights does not only represent the trade-off of safety vs. path length, but has also an impact on the number of alternative solutions and thus on the computation time.
--
Die Sicherheit von Wegen ist nur eingeschr\"ankt messbar und daher schwierig zu quantifizieren. Dahingegen ist es verh\"altnism\"a{\ss}ig leicht Wege bez\"uglich ihrer Sicherheit in geordnete Kategorien, wie beispielsweise sicher, neutral und gef\"ahrlich einzuordnen. In diesem Beitrag werden Optimierungsprobleme mit geordneten Kategorien formuliert und Optimalit\"at f\"ur diese definiert. Daraus wird eine L\"osungsstrategie f\"ur solche Probleme abgeleitet. Dar\"uber hinaus wird erkl\"art, wie die Abgrenzung zwischen den Kategorien erh\"oht werden kann, sodass l\"angere aber daf\"ur sicherere Wege mit Hilfe von Gewichten berechnet werden k\"onnen. Diese theoretischen Ergebnisse werden in der Praxis angewendet und es werden auf Grundlage von Daten von OpenStreetMaps sichere Fahrradwege in Stuttgart berechnet. Dabei zeigt sich, dass eine gute Wahl der Gewichte zu weniger L\"osungen und k\"urzeren Rechenzeiten f\"uhrt. - [86] arXiv:2403.18364 [pdf, other]
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Title: Intent-Aware DRL-Based Uplink Dynamic Scheduler for 5G-NRSubjects: Information Theory (cs.IT); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)
We investigate the problem of supporting Industrial Internet of Things user equipment (IIoT UEs) with intent (i.e., requested quality of service (QoS)) and random traffic arrival. A deep reinforcement learning (DRL) based centralized dynamic scheduler for time-frequency resources is proposed to learn how to schedule the available communication resources among the IIoT UEs. The proposed scheduler leverages an RL framework to adapt to the dynamic changes in the wireless communication system and traffic arrivals. Moreover, a graph-based reduction scheme is proposed to reduce the state and action space of the RL framework to allow fast convergence and a better learning strategy. Simulation results demonstrate the effectiveness of the proposed intelligent scheduler in guaranteeing the expressed intent of IIoT UEs compared to several traditional scheduling schemes, such as round-robin, semi-static, and heuristic approaches. The proposed scheduler also outperforms the contention-free and contention-based schemes in maximizing the number of successfully computed tasks.
- [87] arXiv:2403.18368 [pdf, other]
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Title: The Mercer-Young Theorem for Matrix-Valued Kernels on Separable Metric SpacesComments: 12 pages, 2 figuresSubjects: Functional Analysis (math.FA); Optimization and Control (math.OC)
We generalize the characterization theorem going back to Mercer and Young, which states that a symmetric and continuous kernel is positive definite if and only if it is integrally positive definite. More precisely, we extend the result from real-valued kernels on compact intervals to matrix-valued kernels on separable metric spaces. We also demonstrate the applications of the generalized theorem to the field of convex optimization.
- [88] arXiv:2403.18378 [pdf, ps, other]
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Title: Improvements to the theoretical estimates of the Schwarz preconditioner with $Δ$-GenEO coarse space for the indefinite Helmholtz problemSubjects: Numerical Analysis (math.NA)
The purpose of this work is to improve the estimates for the $\Delta$-GenEO method from the paper "Overlapping Schwarz methods with GenEO coarse spaces for indefinite and nonself-adjoint problems" by N. Bootland, V. Dolean, I. G Graham, C. Ma, R. Scheichl (https://doi.org/10.1093/imanum/drac036) when applied to the indefinite Helmholtz equation. We derive k-dependent estimates of quantities of interest ensuring the robustness of the method.
- [89] arXiv:2403.18382 [pdf, ps, other]
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Title: Joint distribution of $L$-values and orders of Sha groups of quadratic twists of elliptic curvesAuthors: Peng-Jie WongComments: 24 pages, comments welcomeSubjects: Number Theory (math.NT)
We study the joint distribution of central $L$-values and orders of Tate-Shafarevich groups of quadratic twists of elliptic curves. In particular, adapting Radziwill and Soundararajan's principles of establishing upper and lower bounds for the distribution of central values in families of $L$-functions, we obtain conditional upper and lower bounds for such a joint distribution for rank zero twists. These lead us to a conjecture on the full asymptotic for the joint distribution.
- [90] arXiv:2403.18384 [pdf, other]
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Title: Small planar hypohamiltonian graphsAuthors: Cheng-Chen TsaiComments: 16 pages, 16 figuresSubjects: Combinatorics (math.CO)
A graph is hypohamiltonian if it is non-Hamiltonian, but the deletion of every single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 40 vertices, a result due to Jooyandeh, McKay, \"Osterg{\aa}rd, Pettersson, and Zamfirescu. That result is here improved upon by two planar hypohamiltonian graphs on 34 vertices. We exploited a special subgraph contained in two graphs of Jooyandeh et al., and modified it to construct the two 34-vertex graphs and six planar hypohamiltonian graphs on 37 vertices. Each of the 34-vertex graphs has 26 cubic vertices, improving upon the result of Jooyandeh et al. that planar hypohamiltonian graphs have 30 cubic vertices. We use the 34-vertex graphs to construct hypohamiltonian graphs of order 34 with crossing number 1, improving the best-known bound of 36 due to Wiener. Whether there exists a planar hypohamiltonian graph on 41 vertices was an open question. We settled this question by applying an operation introduced by Thomassen to the 37-vertex graphs to obtain several planar hypohamiltonian graphs on 41 vertices. The 25 planar hypohamiltonian graphs on 40 vertices of Jooyandeh et al. have no nontrivial automorphisms. The result is here improved upon by six planar hypohamiltonian graphs on 40 vertices with nontrivial automorphisms. The smallest known cubic planar hypohamiltonian graph has 70 vertices, a graph due to Araya and Wiener. We present another cubic planar hypohamiltonian graph on 70 vertices.
- [91] arXiv:2403.18385 [pdf, ps, other]
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Title: Global solution of 2D hyperbolic liquid crystal system for small initial dataAuthors: Xuecheng WangComments: 24pages, comments are welcome!Subjects: Analysis of PDEs (math.AP)
We prove the global stability of small perturbation near the the constant equilibrium for the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction function of liquid crystal molecules satisfies a wave map equation with an acoustical metric. This improves the almost global existence result by Huang-Jiang-Zhao. As byproducts, we obtain the sharp (same as the linear solution) decay estimates for the nonlinear velocity and the nonlinear wave part. Moreover the nonlinear wave part scatters to a linear solution as time goes to infinity.
The main novelty of this paper is that we uncover a null structure inside the velocity equation on the Fourier side for the nonlinear interaction between nonlinear heat equation and nonlinear wave equation. - [92] arXiv:2403.18386 [pdf, other]
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Title: Distributed Feedback Optimization of Linear Multi-agent SystemsComments: 8 pages, 4 figuresSubjects: Optimization and Control (math.OC)
Feedback optimization is an increasingly popular control paradigm to optimize dynamical systems, accounting for control objectives that concern the system's operation at steady-state. Existing feedback optimization techniques heavily rely on centralized system and controller architectures, and thus suffer from scalability and privacy issues when systems become large-scale. In this paper, we propose and study a distributed architecture for feedback optimization, in which each agent updates its local control state by combining the average of its neighbors with a local negative-gradient step. Under convexity and smoothness assumptions, we establish convergence of the control method to a fixed point. By reinforcing the assumptions to restricted strong convexity of the cost, we show that our algorithm converges linearly to a neighborhood of the optimal point, where the size of the neighborhood depends on the choice of the stepsize. Simulations corroborate the theoretical results.
- [93] arXiv:2403.18387 [pdf, ps, other]
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Title: Morrey-Lorentz estimates for Hodge-type systemsSubjects: Analysis of PDEs (math.AP)
We prove up to the boundary regularity estimates in Morrey-Lorentz spaces for weak solutions of the linear system of differential forms with regular anisotropic coefficients
\begin{equation*}
d^{\ast} \left( A d\omega \right) + B^{\intercal}d d^{\ast} \left( B\omega \right) = \lambda B\omega + f \text{ in } \Omega,
\end{equation*}
with either $ \nu\wedge \omega$ and $\nu\wedge d^{\ast} \left( B\omega \right)$ or $\nu\lrcorner B\omega$ and
$\nu\lrcorner \left( A d\omega \right)$ prescribed on $\partial\Omega.$ We derive these estimates from the $L^{p}$ estimates obtained in \cite{Sil_linearregularity} in the spirit of Campanato's method. Unlike Lorentz spaces, Morrey spaces are neither interpolation spaces nor rearrangement invariant. So Morrey estimates can not be obtained directly from the $L^{p}$ estimates using interpolation. We instead adapt an idea of Lieberman \cite{Lieberman_morrey_from_Lp} to our setting to derive the estimates. Applications to Hodge decomposition in Morrey-Lorentz spaces, Gaffney type inequalities and estimates for related systems such as Hodge-Maxwell systems and `div-curl' systems are discussed. - [94] arXiv:2403.18389 [pdf, ps, other]
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Title: On the birational geometry of $\mathbf{Q}$-Fano threefolds of large Fano index, IAuthors: Yuri ProkhorovComments: 24 pages, LaTeX, submitted to Edge volume 2Subjects: Algebraic Geometry (math.AG)
We investigate the rationality problem for $\mathbf{Q}$-Fano threefolds of Fano index $\ge 2$.
- [95] arXiv:2403.18390 [pdf, ps, other]
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Title: Sails for universal quadratic formsComments: 21 pagesSubjects: Number Theory (math.NT)
We establish a new connection between sails, a key notion in the geometric theory of generalised continued fractions, and arithmetic of totally real number fields, specifically, universal quadratic forms and additively indecomposable integers. Our main application is to biquadratic fields, for which we show that if their signature rank is at least 3, then ranks of universal forms and numbers of indecomposables grow as a power of the discriminant. We also construct a family in which these numbers grow only logarithmically.
- [96] arXiv:2403.18399 [pdf, ps, other]
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Title: Cyclic operads through modulesAuthors: Thomas WillwacherSubjects: Algebraic Topology (math.AT); Quantum Algebra (math.QA)
We describe a way to compute mapping spaces of cyclic operads through modules. As an application we compute the homotopy automorphism space of the cyclic Batalin-Vilkovisky (Hopf co-)operad.
- [97] arXiv:2403.18400 [pdf, ps, other]
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Title: Reweighted Quasi Norm Regularized Low-Rank Factorization for Matrix Robust PCASubjects: Optimization and Control (math.OC)
Robust Principal Component Analysis (RPCA) and its associated non-convex relaxation methods constitute a significant component of matrix completion problems, wherein matrix factorization strategies effectively reduce dimensionality and enhance computational speed. However, some non-convex factorization forms lack theoretical guarantees. This paper proposes a novel strategy in non-convex quasi-norm representation, introducing a method to obtain weighted matrix quasi-norm factorization forms. Especially, explicit bilinear factor matrix factorization formulations for the weighted logarithmic norm and weighted Schatten-$q$ quasi norms with $q=1, 1/2, 2/3$ are provided, along with the establishment of corresponding matrix completion models. An Alternating Direction Method of Multipliers (ADMM) framework algorithm is employed for solving, and convergence results of the algorithm are presented.
- [98] arXiv:2403.18414 [pdf, ps, other]
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Title: Approximation of functions of many variables from the generalization Nikol'skii-Besov type classes in the uniform and integral metricsComments: 18 pages, in UkrainianSubjects: Classical Analysis and ODEs (math.CA)
We obtain the exact order estimates of the approximation of the functions of many variables from the generalized Nikol'skii-Besov classes $B^{\Omega}_{p,\theta}(\mathbb{R}^d)$ by sums of de la Vallee Poussin type in the metrics space $L_{\infty}(\mathbb{R}^d)$ and $L_{1}(\mathbb{R}^d)$. These classes of functions for some given $\Omega$ coincide with the well-known classical Nikol'skii-Besov isotropic classes.
- [99] arXiv:2403.18424 [pdf, ps, other]
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Title: On the existence of a second positive solution to mixed local-nonlocal concave-convex critical problemsComments: arXiv admin note: text overlap with arXiv:2308.09794Subjects: Analysis of PDEs (math.AP)
We prove the existence of a second positive weak solution for mixed local-nonlocal critical semilinear elliptic problems with a sublinear perturbation in the spirit of [Ambrosetti, Brezis, Cerami, 1994].
- [100] arXiv:2403.18428 [pdf, ps, other]
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Title: Fermion integrals for finite spectral triplesAuthors: John W. BarrettComments: Approx 9 pagesSubjects: Mathematical Physics (math-ph)
Fermion functional integrals are calculated for the Dirac operator of a finite real spectral triple. Complex, real and chiral functional integrals are considered for each KO-dimension where they are non-trivial, and phase ambiguities in the definition are noted.
- [101] arXiv:2403.18429 [pdf, other]
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Title: Reinforcement learning for graph theory, I. Reimplementation of Wagner's approachComments: 16 pages, 1 table, 8 figuresSubjects: Combinatorics (math.CO)
We reimplement here the recent approach of Adam Zsolt Wagner [arXiv:2104.14516], which applies reinforcement learning to construct (counter)examples in graph theory, in order to make it more readable, more stable and much faster. The presented concepts are illustrated by constructing counterexamples for a number of published conjectured bounds for the Laplacian spectral radius of graphs.
- [102] arXiv:2403.18431 [pdf, ps, other]
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Title: $l^2$ decoupling theorem for surfaces in $\mathbb{R}^3$Comments: 34 pagesSubjects: Classical Analysis and ODEs (math.CA)
We identify a new way to divide the $\delta$-neighborhood of surfaces $\mathcal{M}\subset\mathbb{R}^3$ into a finitely-overlapping collection of rectangular boxes $S$. We obtain a sharp $(l^2,L^p)$ decoupling estimate using this decomposition, for the sharp range of exponents $2\leq p\leq 4$. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line.
- [103] arXiv:2403.18432 [pdf, other]
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Title: Poisson Regression in one Covariate on Massive DataComments: 16 pages, 11 figuresSubjects: Statistics Theory (math.ST)
The goal of subsampling is to select an informative subset of all observations, when using the full data for statistical analysis is not viable. We construct locally $ D $-optimal subsampling designs under a Poisson regression model with a log link in one covariate. A Representation of the support of locally $ D $-optimal subsampling designs is established. We make statements on scale-location transformations of the covariate that require a simultaneous transformation of the regression parameter. The performance of the methods is demonstrated by illustrating examples. To show the advantage of the optimal subsampling designs, we examine the efficiency of uniform random subsampling as well as of two heuristic designs. Further, the efficiency of locally $ D $-optimal subsampling designs is studied when the parameter is misspecified.
- [104] arXiv:2403.18434 [pdf, ps, other]
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Title: On perspective Abelian groupsSubjects: Group Theory (math.GR)
As a special case of perspective R-modules, an Abelian goup is called perspective if isomorphic summands have a common complement. In this paper we describe many classes of such groups.
- [105] arXiv:2403.18440 [pdf, other]
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Title: Connections between metric differentiability and rectifiabilityAuthors: Iván Caamaño, Estíbalitz Durand-Cartagena, Jesús Á. Jaramillo, Ángeles Prieto, Elefterios SoultanisSubjects: Metric Geometry (math.MG)
We combine Kirchheim's metric differentials with Cheeger charts in order to establish a non-embeddability principle for any collection $\mathcal C$ of Banach (or metric) spaces: if a metric measure space $X$ bi-Lipschitz embeds in some element in $\mathcal C$, and if every Lipschitz map $X\to Y\in \mathcal C$ is differentiable, then $X$ is rectifiable. This gives a simple proof of the rectifiability of Lipschitz differentiability spaces that are bi-Lipschitz embeddable in Euclidean space, due to Kell--Mondino. Our principle also implies a converse to Kirchheim's theorem: if all Lipschitz maps from a domain space to arbitrary targets are metrically differentiable, the domain is rectifiable. We moreover establish the compatibility of metric and w$^*$-differentials of maps from metric spaces in the spirit of Ambrosio--Kirchheim.
- [106] arXiv:2403.18449 [pdf, other]
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Title: Generalizations of free monoidsSubjects: Category Theory (math.CT)
We generalize free monoids by defining $k$-monoids. These are nothing other than the one-vertex higher-rank graphs used in $C^{\ast}$-algebra theory with the cardinality requirement waived. The $1$-monoids are precisely the free monoids. We then take the next step and generalize $k$-monoids in such a way that self-similar group actions yield monoids of this type.
- [107] arXiv:2403.18450 [pdf, other]
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Title: Loop homology of moment-angle complexes in the flag caseAuthors: Fedor VylegzhaninComments: 32 pages, comments are welcome!Subjects: Algebraic Topology (math.AT); Combinatorics (math.CO); K-Theory and Homology (math.KT); Rings and Algebras (math.RA)
We develop a general homological approach to presentations of connected graded associative algebras, and apply it to loop homology of moment-angle complexes $Z_K$ that correspond to flag simplicial complexes $K$. For arbitrary coefficient ring, we describe generators of the Pontryagin algebra $H_*(\Omega Z_K)$ and defining relations between them. We prove that such moment-angle complexes are coformal over $\mathbb{Q},$ give a necessary condition for rational formality, and compute their homotopy groups in terms of homotopy groups of spheres.
- [108] arXiv:2403.18465 [pdf, ps, other]
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Title: Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal typeComments: 23 pagesSubjects: Representation Theory (math.RT); Quantum Algebra (math.QA); Rings and Algebras (math.RA)
We classify graded pre-Nichols algebras of diagonal type with finite Gelfand-Kirillov dimension. The characterization is made through an isomorphism of posets with the family of appropriate subsets of the set of positive roots coming from central extensions of Nichols algebras of diagonal type, generalizing the corresponding extensions for small quantum groups in de Concini-Kac-Procesi forms of quantum groups.
On the way to achieving this result, we also classify graded quotients of algebras of functions of unipotent algebraic groups attached to semisimple Lie algebras. - [109] arXiv:2403.18467 [pdf, other]
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Title: Variational principles and apllications to symmetric PDEsComments: 20 pagesSubjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP)
In this paper, we explore various equivalences of Ekeland's variational principle within the framework of group-invariant mappings. We introduce and analyze several key theorems, including the Drop theorem, the Petal theorem, Caristi-Kirk fxed-point theorem, and Takahashi's theorem, all of them within this context. Moreover, we extend the classical Drop theorem and Petal theorem to a more generalized setting. We also demonstrate the practical signifcance of these findings through numerous applications to diverse areas of mathematics. In particular, in the context of partial differential equations, we explore their implications on the solution of the Plateau problem, and in control theory. We also extend the classical Pontyargin maximum principle.
- [110] arXiv:2403.18472 [pdf, other]
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Title: Computational decomposition and composition technique for approximate solution of nonstationary problemsAuthors: P.N. VabishchevichComments: 22 pages, 3 figuresSubjects: Numerical Analysis (math.NA)
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit-implicit time approximations. Such a technique is currently applied in conditions where the decomposition step is uncomplicated. A general approach is proposed to construct decomposition-composition algorithms for evolution equations in finite-dimensional Hilbert spaces. It is based on two main variants of the decomposition of the unit operator in the corresponding spaces at the decomposition stage and the application of additive operator-difference schemes at the composition stage. The general results are illustrated on the boundary value problem for a second-order parabolic equation by constructing standard splitting schemes on spatial variables and region-additive schemes (domain decomposition schemes).
- [111] arXiv:2403.18478 [pdf, ps, other]
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Title: A consistent kinetic Fokker-Planck model for gas mixturesAuthors: Marlies PirnerComments: arXiv admin note: text overlap with arXiv:1806.09462Subjects: Analysis of PDEs (math.AP)
Rarefied gas dynamics is usually described by the Boltzmann equation. Unfortunately, the expense of evaluating this operator can be very prohibitive. This made it worthwhile to look for approximations that convey essentially an equivalent amount of physical information. One widely known approximative collision operator is the Bathnagar-Gross-Krook (BGK) operator. However, recently, the Foker-Planck approximation has become increasingly popular. Nevertheless, the modeling of gas mixtures in the context of the kinetic Fokker-Planck equation has so far only been addressed in a very few papers. In this paper, we propose a general multi-species Fokker-Planck model. We prove consistency of our model: conservation properties, positivity of all temperatures, H-Theorem and the shape of equilibrium as Maxwell distributions with the same mean velocity and temperature. Moreover, we derive the usual macroscopic equations.
- [112] arXiv:2403.18487 [pdf, other]
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Title: Singularity: a Seventh Memo?Subjects: History and Overview (math.HO)
In this paper we explore the relationships between Calvino's memos and Mathematics. In the first part, we discuss how Lightness, Quickness, Exactitude, Visibility, Multiplicity are present in the mathematical language, reasoning and in the work of the mathematician. In addiction, we follow a similar path for the topics of Calvino's lecture of which we only know the title or some notes. In the final part, we explain why `Singularity' could be chosen as topic for Calvino's seventh lecture.
- [113] arXiv:2403.18488 [pdf, ps, other]
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Title: The Guesswork of Ordered Statistics Decoding: Complexity and Practical DesignComments: Submitted for peer review;19 pages;15 figuresSubjects: Information Theory (cs.IT)
This paper investigates guesswork over ordered statistics and formulates the complexity of ordered statistics decoding (OSD) in binary additive white Gaussian noise (AWGN) channels. It first develops a new upper bound of guesswork for independent sequences, by applying the Holder's inequity to Hamming shell-based subspaces. This upper bound is then extended to the ordered statistics, by constructing the conditionally independent sequences within the ordered statistics sequences. We leverage the established bounds to formulate the best achievable decoding complexity of OSD that ensures no loss in error performance, where OSD stops immediately when the correct codeword estimate is found. We show that the average complexity of OSD at maximum decoding order can be accurately approximated by the modified Bessel function, which increases near-exponentially with code dimension. We also identify a complexity saturation threshold, where increasing the OSD decoding order beyond this threshold improves error performance without further raising decoding complexity. Finally, the paper presents insights on applying these findings to enhance the efficiency of practical decoder implementations.
- [114] arXiv:2403.18496 [pdf, ps, other]
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Title: Nijenhuis deformations of Poisson algebras and $F$-manifold algebrasComments: 24 pages; comments are welcomeSubjects: Rings and Algebras (math.RA); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
The notion of pre-Poisson algebras was introduced by Aguiar in his study of zinbiel algebras and pre-Lie algebras. In this paper, we first introduce NS-Poisson algebras as a generalization of both Poisson algebras and pre-Poisson algebras. An NS-Poisson algebra has an associated sub-adjacent Poisson algebra. We show that a Nijenhuis operator on a Poisson algebra deforms the structure into an NS-Poisson algebra. The semi-classical limit of an NS-algebra deformation and a suitable filtration of an NS-algebra produce NS-Poisson algebras. On the other hand, $F$-manifold algebras were introduced by Dotsenko as the underlying algebraic structure of $F$-manifolds. We also introduce NS-$F$-manifold algebras as a simultaneous generalization of NS-Poisson algebras, $F$-manifold algebras and pre-$F$-manifold algebras. In the end, we show that Nijenhuis deformations of $F$-manifold algebras and the semi-classical limits of NS-pre-Lie algebra deformations have NS-$F$-manifold algebra structures.
- [115] arXiv:2403.18498 [pdf, ps, other]
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Title: Global Representation Ring and Knutson IndexComments: 11 pages, 2 tablesSubjects: Rings and Algebras (math.RA); Group Theory (math.GR); Representation Theory (math.RT)
We follow on from the pioneering paper by the second author where he introduced the Knutson index for the character ring of a finite group. We introduce the Knutson index for general commutative rings. Then we study it for Burnside rings and global representation rings. We also introduce the global table of a finite group, that encompasses both the character table and the Burnside table of marks.
- [116] arXiv:2403.18500 [pdf, ps, other]
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Title: Global unique solutions to the planar inhomogeneous Navier--Stokes--Maxwell equationsSubjects: Analysis of PDEs (math.AP)
The evolution of an electrically conducting imcompressible fluid with nonconstant density can be described by a set of equations combining the continuity, momentum and Maxwell's equations; altogether known as the inhomogeneous Navier--Stokes--Maxwell system.
In this paper, we focus on the global well-posedness of these equations in two dimensions. Specifically, we are able to prove the existence of global energy solutions, provided that the initial velocity field belongs to the Besov space $\dot{B}^{r}_{p,1}(\mathbb{R}^2)$, with $r=-1+\frac{2}{p}$, for some $p\in (1,2)$, while the initial electromagnetic field enjoys some $H^s(\mathbb{R}^2)$ Sobolev regularity, for some $s \geq 2-\frac{2}{p} \in (0,1)$, and whenever the initial fluid density is bounded pointwise and close to a nonnegative constant. Moreover, if it is assumed that $s>\frac{1}{2}$, then the solution is shown to be unique in the class of all energy solutions.
It is to be emphasized that the solutions constructed here are global and uniformly bounded with respect to the speed of light $c\in (0,\infty)$. This important fact allows us to derive the inhomogeneous MHD system as the speed of light tends to infinity. - [117] arXiv:2403.18505 [pdf, ps, other]
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Title: Moduli spaces of untwisted wild Riemann surfacesComments: 15 pages: comments welcome!Subjects: Algebraic Geometry (math.AG)
We construct moduli stacks of wild Riemann surfaces in the (pure) untwisted case, for any complex reductive structure group, and we define the corresponding (pure) wild mapping class groups.
- [118] arXiv:2403.18507 [pdf, ps, other]
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Title: CI-sequences and almost complete intersectionsAuthors: Giuseppe ZappalàSubjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
We study the Hilbert function and the graded Betti numbers of almost complete intersection artinian algebras. We show that that every Hilbert function of a complete intersection artinian algebra is the Hilbert function of an almost complete intersection algebra. In codimension $3$ we focus on almost complete intersection artinian algebras whose Hilbert function coincides with that of a complete intersection defined by $3$ forms of the same degree. We classify all the possible graded Betti numbers of such algebras and we specify what cancellations are allowed in a minimal graded free resolution.
- [119] arXiv:2403.18511 [pdf, ps, other]
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Title: Bolzano's Conjecture: Measuring the Numerosity of Infinite SetsAuthors: Julian JackComments: 7 pagesSubjects: General Mathematics (math.GM)
Bolzano and Cantor were the first mathematicians to make significant attempts to measure the size (numerosity) of different infinite collections. They differed in their methodological approaches, with Cantor's prevailing. This led to the foundation of the theory of sets as well as Cantor's transfinite arithmetic. This paper argues that Bolzano's conjecture is correct and that Euclid's principle, 'that the whole is greater than a part', should be considered as a necessary condition for the quantification of infinite sets (rather than bijection). Cantor had concluded that the rational and the algebraic numbers were of the same size as the natural numbers, whilst, in contrast, the real numbers were a larger set. Using Cantor's methods it is shown in this paper that the rational numbers are of larger size than the natural numbers, thus showing that bijection is not a reliable measure of the size of infinite sets. It is also concluded, using mathematical induction, that different 'countably' infinite sets can have various different sizes. The implications for theorems using bijection as a measure of size is then briefly discussed. There already exist new methods of measuring numerosity, based on Euclid's principle, which may develop a consistent system of infinite arithmetic.
- [120] arXiv:2403.18520 [pdf, other]
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Title: Global convergence of iterative solvers for problems of nonlinear magnetostaticsSubjects: Numerical Analysis (math.NA)
We consider the convergence of iterative solvers for problems of nonlinear magnetostatics. Using the equivalence to an underlying minimization problem, we can establish global linear convergence of a large class of methods, including the damped Newton-method, fixed-point iteration, and the Kacanov iteration, which can all be interpreted as generalized gradient descent methods. Armijo backtracking isconsidered for an adaptive choice of the stepsize. The general assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, as well as permanent magnets. The main results are proven on the continuous level, but they carry over almost verbatim to various approximation schemes, including finite elements and isogeometric analysis, leading to bounds on the iteration numbers, which are independent of the particular discretization. The theoretical results are illustrated by numerical tests for a typical benchmark problem.
- [121] arXiv:2403.18522 [pdf, ps, other]
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Title: On the $A_α$-index of graphs with given order and dissociation numberComments: 16 pages; 6 figuresSubjects: Combinatorics (math.CO)
Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the dissociation number of $G$. The adjacency matrix and the degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G),$ respectively. In 2017, Nikiforov proposed the $A_\alpha$-matrix: $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$ where $\alpha\in[0,1].$ The largest eigenvalue of this novel matrix is called the $A_\alpha$-index of $G.$ In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest $A_\alpha$-index over all connected graphs (resp. bipartite graphs, trees) with fixed order and dissociation number. Secondly, we describe the structure of all the $n$-vertex graphs having the minimum $A_\alpha$-index with dissociation number $\tau$, where $\tau\geqslant\lceil\frac{2}{3}n\rceil.$ Finally, we identify all the connected $n$-vertex graphs with dissociation number $\tau\in\{2,\lceil\frac{2}{3}n\rceil,n-1,n-2\}$ having the minimum $A_\alpha$-index.
- [122] arXiv:2403.18527 [pdf, other]
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Title: Wirtinger gradient descent methods for low-dose Poisson phase retrievalSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
The problem of phase retrieval has many applications in the field of optical imaging. Motivated by imaging experiments with biological specimens, we primarily consider the setting of low-dose illumination where Poisson noise plays the dominant role. In this paper, we discuss gradient descent algorithms based on different loss functions adapted to data affected by Poisson noise, in particular in the low-dose regime. Starting from the maximum log-likelihood function for the Poisson distribution, we investigate different regularizations and approximations of the problem to design an algorithm that meets the requirements that are faced in applications. In the course of this, we focus on low-count measurements. For all suggested loss functions, we study the convergence of the respective gradient descent algorithms to stationary points and find constant step sizes that guarantee descent of the loss in each iteration. Numerical experiments in the low-dose regime are performed to corroborate the theoretical observations.
- [123] arXiv:2403.18528 [pdf, other]
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Title: Limited Attention Allocation in a Stochastic Linear Quadratic System with Multiplicative NoiseSubjects: Optimization and Control (math.OC); Mathematical Finance (q-fin.MF)
This study addresses limited attention allocation in a stochastic linear quadratic system with multiplicative noise. Our approach enables strategic resource allocation to enhance noise estimation and improve control decisions. We provide analytical optimal control and propose a numerical method for optimal attention allocation. Additionally, we apply our ffndings to dynamic mean-variance portfolio selection, showing effective resource allocation across time periods and factors, providing valuable insights for investors.
- [124] arXiv:2403.18532 [pdf, ps, other]
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Title: Scaling limits for random walks on long range percolation clustersComments: 32 pagesSubjects: Probability (math.PR)
We study limit laws for simple random walks on supercritical long-range percolation clusters on the integer lattice. For the long range percolation model, the probability that two vertices are connected behaves asymptotically as a negative power of distance between them. We prove that the scaling limit of simple random walk on the infinite component converges to an isotropic alpha-stable Levy process. This complements the work of Crawford and Sly, who proved the corresponding result for alpha between 0 and 1. The convergence holds in both the quenched and annealed senses.
- [125] arXiv:2403.18534 [pdf, other]
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Title: Creating spanning trees in Waiter-Client gamesAuthors: Grzegorz Adamski, Sylwia Antoniuk, Małgorzata Bednarska-Bzdęga, Dennis Clemens, Fabian Hamann, Yannick MoggeSubjects: Combinatorics (math.CO)
For a positive integer $n$ and a tree $T_n$ on $n$ vertices, we consider an unbiased Waiter-Client game $\textrm{WC}(n,T_n)$ played on the complete graph~$K_n$, in which Waiter's goal is to force Client to build a copy of $T_n$. We prove that for every constant $c<1/3$, if $\Delta(T_n)\le cn$ and $n$ is sufficiently large, then Waiter has a winning strategy in $\textrm{WC}(n,T_n)$. On the other hand, we show that there exist a positive constant $c'<1/2$ and a family of trees $T_{n}$ with $\Delta(T_n)\le c'n$ such that Client has a winning strategy in the $\textrm{WC}(n,T_n)$ game for every $n$ sufficiently large. We also consider the corresponding problem in the Client-Waiter version of the game.
- [126] arXiv:2403.18544 [pdf, ps, other]
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Title: On the distribution of the components of multicurves of given typeComments: 29 pagesSubjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)
We study the distribution of the individual components of a random multicurve under the action of the mapping class group.
- [127] arXiv:2403.18552 [pdf, other]
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Title: Generalized convergence of the deep BSDE method: a step towards fully-coupled FBSDEs and applications in stochastic controlComments: 25 pages, 3 figures, 1 tableSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to fully-coupled drift coefficients, and give sufficient conditions for convergence under standard assumptions. The resulting conditions are directly verifiable for any equation. Consequently, unlike in earlier theory, our convergence analysis enables the treatment of FBSDEs stemming from stochastic optimal control problems. In particular, we provide a theoretical justification for the non-convergence of the deep BSDE method observed in recent literature, and present direct guidelines for when convergence can be guaranteed in practice. Our theoretical findings are supported by several numerical experiments in high-dimensional settings.
- [128] arXiv:2403.18556 [pdf, other]
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Title: Numerical optimisation of Dirac eigenvaluesComments: 19 pages, 26 figuresSubjects: Optimization and Control (math.OC); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Spectral Theory (math.SP)
Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the optimality of the regular polygon (respectively, disk) among all polygons of a given number of sides (respectively, arbitrary sets), subject to area or perimeter constraints. We consider the three lowest positive eigenvalues and their ratios. Roughly, we find results analogous to known or expected for the Dirichlet Laplacian, except for the third eigenvalue which does not need to be minimised by the regular polygon (respectively, the disk) for all masses. In addition to the numerical results, a new, mass-dependent upper bound to the lowest eigenvalue in rectangles is proved and its extension to arbitrary quadrilaterals is conjectured.
- [129] arXiv:2403.18557 [pdf, other]
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Title: Stability Properties of the Impulsive Goodwin's Oscillator in 1-cycleComments: submitted to IEEE CDC 2024Subjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
The Impulsive Goodwin's Oscillator (IGO) is a mathematical model of a hybrid closed-loop system. It arises by closing a special kind of continuous linear positive time-invariant system with impulsive feedback, which employs both amplitude and frequency pulse modulation. The structure of IGO precludes the existence of equilibria, and all its solutions are oscillatory. With its origin in mathematical biology, the IGO also presents a control paradigm useful in a wide range of applications, in particular dosing of chemicals and medicines. Since the pulse modulation feedback mechanism introduces significant nonlinearity and non-smoothness in the closedloop dynamics, conventional controller design methods fail to apply. However, the hybrid dynamics of IGO reduce to a nonlinear, time-invariant discrete-time system, exhibiting a one-to-one correspondence between periodic solutions of the original IGO and those of the discrete-time system. The paper proposes a design approach that leverages the linearization of the equivalent discrete-time dynamics in the vicinity of a fixed point. A simple and efficient local stability condition of the 1-cycle in terms of the characteristics of the amplitude and frequency modulation functions is obtained.
- [130] arXiv:2403.18559 [pdf, ps, other]
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Title: Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie systemComments: 29 pagesSubjects: Analysis of PDEs (math.AP)
In dimension three, the existence of global weak solutions to the axisymmetric simplified Ericksen-Leslie system without swirl is established. This is achieved by analyzing weak convergence of solutions of the axisymmetric Ginzburg-Landau approximated solutions as the penalization parameter $\varepsilon$ tends to zero. The proof relies on the one hand on the use of a blow-up argument to rule out energy concentration off the $z$-axis, which exploits the topological restrictions of the axisymmetry. On the other hand, possible limiting energy concentrations on the $z$-axis can be dealt by a cancellation argument at the origin. Once more, the axisymmetry plays a substantial role. We will also show that the set of axisymmetric solutions without swirl $(u,d)$ to the simplified Ericksen-Leslie system is compact under weak convergence in $L^\infty_tL^2_x\times L^2_tH^1_x$.
- [131] arXiv:2403.18562 [pdf, ps, other]
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Title: Construction of Gross-Neveu model using Polchinski flow equationAuthors: Paweł DuchComments: 95 pages, comments welcomeSubjects: Mathematical Physics (math-ph); Probability (math.PR)
The Gross-Neveu model is a quantum field theory model of Dirac fermions in two dimensions with a quartic interaction term. Like Yang-Mills theory in four dimensions, the model is renormalizable (but not super-renormalizable) and asymptotically free (i.e. its short-distance behaviour is governed by the free theory). We give a new construction of the massive Euclidean Gross-Neveu model in infinite volume based on the renormalization group flow equation. The construction does not involve cluster expansion or discretization of phase-space. We express the Schwinger functions of the Gross-Neveu model in terms of the effective potential and construct the effective potential by solving the flow equation using the Banach fixed point theorem. Since we use crucially the fact that fermionic fields can be represented as bounded operators our construction does not extend to models including bosons. However, it is applicable to other asymptotically free purely fermionic theories such as the symplectic fermion model.
- [132] arXiv:2403.18566 [pdf, other]
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Title: Computer-Assisted Proofs of Existence of Invariant Tori in Quasi-periodic Systems via Fourier MethodsSubjects: Dynamical Systems (math.DS)
The goal of this paper is to provide a methodology to prove existence of (fiberwise hyperbolic) real-analytic invariant tori in real-analytic quasi-periodic skew-product dynamical systems that present nearly-invariant tori of the same characteristics. The methodology is based on the application of a Newton-Kantorovich theorem whose hypotheses are tested using Fourier analysis methods for a numerical approximation of the parameterization of an invariant torus.
- [133] arXiv:2403.18574 [pdf, other]
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Title: A Proof of the Box Conjecture for Commuting Pairs of MatricesSubjects: Combinatorics (math.CO); Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
We prove the Box Conjecture for pairs of commuting nilpotent matrices, as formulated by Iarrobino et al \cite{IKvSZ}. This describes the Jordan type of the dense orbit in the nilpotent commutator of a given nilpotent matrix. Our main tool is the Burge correspondence between the set of all partitions and a set of binary words \cite{Bur-1, Bur-2}. For connection with the algebraic and geometric setup of matrices and orbits we employ some of Shayman's results on invariant subspaces of a nilpotent matrix \cite{Sha-1,Sha-2}. Our proof is valid over an arbitrary field.
- [134] arXiv:2403.18576 [pdf, other]
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Title: Logarithmic singularity in the density four-point function of two-dimensional critical percolation in the bulkComments: 8 pages, 1 figureSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Probability (math.PR)
We provide definitive proof of the logarithmic nature of the percolation conformal field theory in the bulk by showing that the four-point function of the density operator has a logarithmic divergence as two points collide and that the same divergence appears in the operator product expansion (OPE) of two density operators. The right hand side of the OPE contains two operators with the same scaling dimension, one of them multiplied by a term with a logarithmic singularity. Our method involves a probabilistic analysis of the percolation events contributing to the four-point function. It does not require algebraic considerations, nor taking the $Q \to 1$ limit of the $Q$-state Potts model, and is amenable to a rigorous mathematical formulation. The logarithmic divergence appears as a consequence of scale invariance combined with independence.
- [135] arXiv:2403.18590 [pdf, ps, other]
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Title: $θ$-derivations on convolution algebrasSubjects: Functional Analysis (math.FA)
In this paper, we investigate $\theta$-derivations on Banach algebra $ L_0^{\infty} (w)^*$. First, we study the range of them and prove the Singer-Wermer conjucture. We also give a characterization of the space of all $\theta$-derivations on $ L_0^{\infty} (w)^*$. Then, we prove automatic continuity and Posner's theorems for $\theta$-derivations.
- [136] arXiv:2403.18592 [pdf, other]
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Title: Contact processes with quenched disorder on $\mathbb{Z}^d$ and on Erdos-Renyi graphsAuthors: Rick DurrettComments: 26 pages, 7 figuresSubjects: Probability (math.PR)
In real systems impurities and defects play an important role in determining their properties. Here we will consider what probabilists have called the contact process in a random environment and what physicists have more precisely named the contact process with quenched disorder. We will concentrate our efforts on the special case called the random dilution model, in which sites independently and with probability $p$ are active and particles on them give birth at rate $\lambda$, while the other sites are inert and particles on them do not give birth. We show that the resulting inhomogeniety can make dramatic changes in the behavior in the supercritical, subcritical, and critical behavior. In particular, the usual exponential decay of the desnity of particles in the subcritical phase becomes a power law (the Griffiths phase), and polynomial decay at the critical value becomes a power of $\log$.
- [137] arXiv:2403.18596 [pdf, ps, other]
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Title: Remarks on a theorem of Eells and SampsonComments: 8 pages. Comments are welcome!Subjects: Differential Geometry (math.DG)
We prove an extension of Eells and Sampson's rigidity theorem for harmonic maps from a closed manifold of non-negative Ricci curvature to a manifold of non-positive sectional curvature. We give an application of our result in the setting of harmonic-Einstein (or Ricci-harmonic) metrics and as a consequence we recover a classical rigidity result of Hamilton for the problem of prescribed positive definite Ricci curvature.
- [138] arXiv:2403.18611 [pdf, ps, other]
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Title: More on maximal line-free sets in $\mathbb{F}_p^n$Authors: Jakob FührerSubjects: Combinatorics (math.CO)
For a prime $p$ we construct a subset of $\mathbb{F}_p^{(k^2-k)/2}$ of size $p^{(k^2-k)/2-1}$ that does not contain progressions of length $k$. More generally, we show that for any prime power $q$ there is a subset of $\mathbb{F}_q^{(k^2-k)/2}$ of size $q^{(k^2-k)/2-1}$ that does not contain $k$ points on a line. This yields the first asympotic lower bounds $c^n$ for the size of $p$-progression-free sets in $\mathbb{F}_p^{n}$ with $c=p-o(1)$, as $p$ tends to infinity.
- [139] arXiv:2403.18612 [pdf, ps, other]
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Title: Bowen's formula for a rational graph-directed Markov systemSubjects: Dynamical Systems (math.DS)
We establish Bowen's formula for the Julia set of a non-elementary, expanding, irreducible and aperiodic rational graph-directed Markov system satisfying the backward separating condition. Towards this end, we shall prove that the associated skew product map is topologically exact on the skew product Julia set, and satisfies the density of repelling periodic points. Moreover, we give a criterion for expandingness in terms of hyperbolicity.
- [140] arXiv:2403.18617 [pdf, ps, other]
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Title: Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approachComments: 26 pagesSubjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)
We prove new concentration inequalities for quantum spin systems which apply to any local observable measured on any product state or on any state with exponentially decaying correlations. Our results do not require the spins to be arranged in a regular lattice, and cover the case of observables that contain terms acting on spins at arbitrary distance. Moreover, we introduce a local W1 distance, which quantifies the distinguishability of two states with respect to local observables. We prove a transportation-cost inequality stating that the local W1 distance between a generic state and a state with exponentially decaying correlations is upper bounded by a function of their relative entropy. Finally, we apply such inequality to prove the equivalence between the canonical and microcanonical ensembles of quantum statistical mechanics and the weak eigenstate thermalization hypothesis for the Hamiltonians whose Gibbs states have exponentially decaying correlations.
- [141] arXiv:2403.18618 [pdf, ps, other]
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Title: Accelerating preconditioned ADMM via degenerate proximal point mappingsSubjects: Optimization and Control (math.OC)
In this paper, we aim to accelerate a preconditioned alternating direction method of multipliers (pADMM), whose proximal terms are convex quadratic functions, for solving linearly constrained convex optimization problems. To achieve this, we first reformulate the pADMM into a form of proximal point method (PPM) with a positive semidefinite preconditioner which can be degenerate due to the lack of strong convexity of the proximal terms in the pADMM. Then we accelerate the pADMM by accelerating the reformulated degenerate PPM (dPPM). Specifically, we first propose an accelerated dPPM by integrating the Halpern iteration and the fast Krasnosel'ski\u{i}-Mann iteration into it, achieving asymptotic $o(1/k)$ and non-asymptotic $O(1/k)$ convergence rates. Subsequently, building upon the accelerated dPPM, we develop an accelerated pADMM algorithm that exhibits both asymptotic $o(1/k)$ and non-asymptotic $O(1/k)$ nonergodic convergence rates concerning the Karush-Kuhn-Tucker residual and the primal objective function value gap. Preliminary numerical experiments validate the theoretical findings, demonstrating that the accelerated pADMM outperforms the pADMM in solving convex quadratic programming problems.
- [142] arXiv:2403.18620 [pdf, ps, other]
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Title: Triple product $p$-adic $L$-functions for finite slope families and a $p$-adic Gross-Zagier formulaAuthors: Ting-Han HuangSubjects: Number Theory (math.NT)
In this paper, we generalize two results of H. Darmon and V. Rotger on triple product $p$-adic $L$-functions associated with Hida families to finite slope families. We first prove a $p$-adic Gross-Zagier formula, then demonstrate an application to a special case of the equivariant Birch and Swinnerton-Dyer conjecture for supersingular elliptic curves.
- [143] arXiv:2403.18621 [pdf, other]
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Title: Performance Analysis of Integrated Sensing and Communication Networks with Blockage EffectsSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Communication-sensing integration represents an up-and-coming area of research, enabling wireless networks to simultaneously perform communication and sensing tasks. However, in urban cellular networks, the blockage of buildings results in a complex signal propagation environment, affecting the performance analysis of integrated sensing and communication (ISAC) networks. To overcome this obstacle, this paper constructs a comprehensive framework considering building blockage and employs a distance-correlated blockage model to analyze interference from line of sight (LoS), non-line of sight (NLoS), and target reflection cascading (TRC) links. Using stochastic geometric theory, expressions for signal-to-interference-plus-noise ratio (SINR) and coverage probability for communication and sensing in the presence of blockage are derived, allowing for a comprehensive comparison under the same parameters. The research findings indicate that blockage can positively impact coverage, especially in enhancing communication performance. The analysis also suggests that there exists an optimal base station (BS) density when blockage is of the same order of magnitude as the BS density, maximizing communication or sensing coverage probability.
- [144] arXiv:2403.18626 [pdf, other]
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Title: Phase transition in the EM scheme of an SDE driven by $α$-stable noises with $α\in (0,2]$Subjects: Probability (math.PR)
We study in this paper the EM scheme for a family of well-posed critical SDEs with the drift $-x\log(1+|x|)$ and $\alpha$-stable noises. Specifically, we find that when the SDE is driven by a rotationally symmetric $\alpha$-stable processes with $\alpha=2$ (i.e. Brownian motion), the EM scheme is bounded in the $L^2$ sense uniformly w.r.t. the time. In contrast, if the SDE is driven by a rotationally symmetric $\alpha$-stable process with $\alpha \in (0,2)$, all the $\beta$-th moments, with $\beta \in (0,\alpha)$, of the EM scheme blow up. This demonstrates a phase transition phenomenon as $\alpha \uparrow 2$. We verify our results by simulations.
- [145] arXiv:2403.18634 [pdf, ps, other]
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Title: A Hilbert metric for bounded symmetric domainsComments: 26 pagesSubjects: Differential Geometry (math.DG); Complex Variables (math.CV); Metric Geometry (math.MG)
Bounded symmetric domains carry several natural invariant metrics, for example the Carath\'eodory, Kobayashi or the Bergman metric. We define another natural metric, from generalized Hilbert metric defined in [FGW20], by considering the Borel embedding of the domain as an open subset of its dual compact Hermitian symmetric space and then its Harish-Chandra realization in projective spaces. We describe this construction on the four classical families of bounded symmetric domains and compute both this metric and its associated Finsler metric. We compare it to Carath\'eodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.
- [146] arXiv:2403.18641 [pdf, other]
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Title: Improving Efficiency of Parallel Across the Method Spectral Deferred CorrectionsComments: 24 pagesSubjects: Numerical Analysis (math.NA); Distributed, Parallel, and Cluster Computing (cs.DC)
Parallel-across-the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDC) with a diagonal sweeper, which is closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence speed, efficiency and stability depends critically on the used coefficients. Previous approaches have used numerical optimization to find good parameters. Instead, we propose an ansatz that allows to find optimal parameters analytically. We show that the resulting parallel SDC methods provide stability domains and convergence order very similar to those of well established serial SDC variants. Using a model for computational cost that assumes 80% efficiency of an implementation of parallel SDC we show that our variants are competitive with serial SDC, previously published parallel SDC coefficients as well as Picard iteration, explicit RKM-4 and an implicit fourth-order diagonally implicit Runge-Kutta method.
- [147] arXiv:2403.18644 [pdf, other]
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Title: Analysis of the monotonicity method for an anisotropic scatterer with a conductive boundarySubjects: Analysis of PDEs (math.AP)
In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary. We will assume that the corresponding far-field pattern is known/measured and we consider two inverse problems. First, we show that the far-field data uniquely determines the boundary coefficient. Next, since it is known that anisotropic coefficients are not uniquely determined by this data we will develop a qualitative method to recover the scatterer. To this end, we study the so-called monotonicity method applied to this inverse shape problem. This method has recently been applied to some inverse scattering problems but this is the first time it has been applied to an anisotropic scatterer. This method allows one to recover the scatterer but considering the eigenvalues of an operator associated with the far--field operator. We present some simple numerical reconstructions to illustrate our theory in two dimensions. For our reconstructions, we need to compute the adjoint of the Herglotz wave function as an operator mapping into $H^1$ of a small ball.
- [148] arXiv:2403.18645 [pdf, ps, other]
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Title: Some remarks regarding special elements in algebras obtained by the Cayley-Dickson process over ZpSubjects: Rings and Algebras (math.RA)
In this paper we provide some properties of k-potent elements in algebras obtained by the Cayley-Dickson process over Zp. Moreover, we find a structure of nonunitary ring over Fibonacci quaternions over Z3 and we present a method to encrypt plain texts, by using invertible elements in such algebras.
- [149] arXiv:2403.18653 [pdf, other]
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Title: Indecomposable set-theoretical solutions to the Yang-Baxter equation of size $p^2$Comments: 21 Pages, Comments Welcome!Subjects: Quantum Algebra (math.QA); Group Theory (math.GR); Rings and Algebras (math.RA)
The quantum Yang-Baxter equation is a braiding condition on complex vector spaces which is of high relevance in several fields of mathematics, such as knot theory and quantum group theory. A combinatorial approach is the investigation of set-theoretic solutions to the Yang--Baxter equation and their associated algebraic structures. In this article, we focus on indecomposable set-theoretic solutions to the Yang--Baxter equation. More specifically, we give a full classification of those which are of size $p^2$.
- [150] arXiv:2403.18654 [pdf, ps, other]
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Title: A Dimca-Greuel type inequality for foliationsComments: 10 pagesSubjects: Complex Variables (math.CV); Differential Geometry (math.DG)
Let $\mathcal{F}$ be a holomorphic foliation at $p\in \mathbb{C}^2$, and $B$ be a separatrix of $\mathcal{F}$. We prove the following Dimca-Greuel type inequality $\frac{\mu_p(\mathcal{F},B)}{\tau_p(\mathcal{F},B)}<4/3$, where $\mu_p(\mathcal{F},B)$ is the multiplicity of $\mathcal{F}$ along $B$ and $\tau_p(\mathcal{F},B)$ is the dimension of the quotient of $\mathbb{C}[[x,y]]$ by the ideal generated by the components of any $1$-form defining $\mathcal{F}$ and any equation of $B$. As a consequence, we provide a new proof of the $\frac{4}{3}$-Dimca-Greuel's conjecture for singularities of irreducible plane curve germs, with foliations ingredients, that differs from those given by Alberich-Carrami\~nana, Almir\'on, Blanco, Melle-Hern\'andez and Genzmer-Hernandes but it is in line with the idea developed by Wang.
- [151] arXiv:2403.18658 [pdf, ps, other]
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Title: Theoretical Guarantees for the Subspace-Constrained Tyler's EstimatorSubjects: Statistics Theory (math.ST); Machine Learning (stat.ML)
This work analyzes the subspace-constrained Tyler's estimator (STE) designed for recovering a low-dimensional subspace within a dataset that may be highly corrupted with outliers. It assumes a weak inlier-outlier model and allows the fraction of inliers to be smaller than a fraction that leads to computational hardness of the robust subspace recovery problem. It shows that in this setting, if the initialization of STE, which is an iterative algorithm, satisfies a certain condition, then STE can effectively recover the underlying subspace. It further shows that under the generalized haystack model, STE initialized by the Tyler's M-estimator (TME), can recover the subspace when the fraction of iniliers is too small for TME to handle.
- [152] arXiv:2403.18663 [pdf, ps, other]
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Title: On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifoldsSubjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Spectral Theory (math.SP)
On a closed analytic manifold $(M,g)$, let $\phi_i$ be the eigenfunctions of $\Delta_g$ with eigenvalues $\lambda_i^2$ and let $f:=\prod \phi_{k_j}$ be a finite product of Laplace-Beltrami eigenfunctions. We show that $\left\langle f, \phi_i \right\rangle_{L^2(M)}$ decays exponentially as soon as $\lambda_i > C \sum \lambda_{k_j}$ for some constant $C$ depending only on $M$. Moreover, by using a lower bound on $\| f \|_{L^2(M)} $, we show that $99\%$ of the $L^2$-mass of $f$ can be recovered using only finitely many Fourier coefficients.
- [153] arXiv:2403.18665 [pdf, ps, other]
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Title: Lipschitz-type estimate for the frog model with Bernoulli initial configurationComments: 32 pagesSubjects: Probability (math.PR)
We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters $p,q \in (0,1]$ is bounded from above and below by multiples of $|p-q|$.
- [154] arXiv:2403.18669 [pdf, ps, other]
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Title: Orthogonal Polynomials with a Singularly Perturbed Airy WeightComments: 16 pagesSubjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph)
We study the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal polynomials. We find that the orthogonal polynomials satisfy a second-order linear ordinary differential equation, whose coefficients are all expressed in terms of the recurrence coefficients. By considering the time evolution, we obtain a system of differential-difference equations satisfied by the recurrence coefficients. Finally, we study the asymptotics of the recurrence coefficients when the degrees of the orthogonal polynomials tend to infinity.
- [155] arXiv:2403.18670 [pdf, other]
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Title: Globally integrable quantum systems and their perturbationsComments: 41 pages, 4 figuresSubjects: Analysis of PDEs (math.AP)
In this paper we present the notion of globally integrable quantum system that we introduced in [BL22]: we motivate it using the spectral theory of pseudodifferential operators and then we give some results on linear and nonlinear perturbations of a globally integrable quantum system. In particular, we give a spectral result ensuring stability of most of its eigenvalues under relatively bounded perturbations, and two results controlling the growth of Sobolev norms when it is subject either to linear unbounded time dependent perturbations or a small nonlinear Hamiltonian nonlinear perturbation.
- [156] arXiv:2403.18675 [pdf, ps, other]
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Title: Submanifolds with boundary and Stokes' Theorem in Heisenberg groupsComments: 30 pagesSubjects: Differential Geometry (math.DG); Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG)
We introduce and study the notion of $C^1_\mathbb{H}$-regular submanifold with boundary in sub-Riemannian Heisenberg groups. As an application, we prove a version of Stokes' Theorem for $C^1_\mathbb{H}$-regular submanifolds with boundary that takes into account Rumin's complex of differential forms in Heisenberg groups.
- [157] arXiv:2403.18678 [pdf, ps, other]
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Title: On the set of supercyclic operatorsComments: 11 pagesSubjects: Functional Analysis (math.FA)
In this article, we address a problem posed by F. Bayart regarding the existence of an infinite-dimensional closed vector subspace (excluding the null operator) within the set of supercyclic operators on Banach spaces. We resolve this problem by establishing the existence of the closed subspace. Furthermore, we prove that the set of supercyclic operators on $\ell_1$ contains, up to the null operator, an isometric copy of $\ell_1$.
- [158] arXiv:2403.18685 [pdf, other]
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Title: Representatividad Muestral en la Incertidumbre Simétrica Multivariada para la Selección de AtributosAuthors: Gustavo Sosa-CabreraComments: 52 pages, in Spanish. Advisors: Miguel Garc\'ia-Torres, Santiago G\'omez-Guerrero, Christian E. Schaerer SerraSubjects: Information Theory (cs.IT); Machine Learning (cs.LG); Statistics Theory (math.ST)
In this work, we analyze the behavior of the multivariate symmetric uncertainty (MSU) measure through the use of statistical simulation techniques under various mixes of informative and non-informative randomly generated features. Experiments show how the number of attributes, their cardinalities, and the sample size affect the MSU. In this thesis, through observation of results, it is proposed an heuristic condition that preserves good quality in the MSU under different combinations of these three factors, providing a new useful criterion to help drive the process of dimension reduction.
--
En el presente trabajo hemos analizado el comportamiento de una versi\'on multivariada de la incertidumbre sim\'etrica a trav\'es de t\'ecnicas de simulaci\'on estad\'isticas sobre varias combinaciones de atributos informativos y no-informativos generados de forma aleatoria. Los experimentos muestran como el n\'umero de atributos, sus cardinalidades y el tama\~no muestral afectan al MSU como medida. En esta tesis, mediante la observaci\'on de resultados hemos propuesto una condici\'on que preserva una buena calidad en el MSU bajo diferentes combinaciones de los tres factores mencionados, lo cual provee un nuevo y valioso criterio para llevar a cabo el proceso de reducci\'on de dimensionalidad. - [159] arXiv:2403.18686 [pdf, other]
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Title: Decision-Epoch Matters: Unveiling its Impact on the Stability of Scheduling with Randomly Varying ConnectivitySubjects: Probability (math.PR); Performance (cs.PF)
A classical queuing theory result states that in a parallel-queue single-server model, the maximum stability region does not depend on the scheduling decision epochs, and in particular is the same for preemptive and non-preemptive systems. We consider here the case in which each of the queues may be connected to the server or not, depending on an exogenous process. In our main result, we show that the maximum stability region now does strongly depend on how the decision epochs are defined. We compare the setting where decisions can be made at any moment in time (the unconstrained setting), to two other settings: decisions are taken either (i) at moments of a departure (non-preemptive scheduling), or (ii) when an exponentially clock rings with rate $\gamma$. We characterise the maximum stability region for the two constrained configurations, allowing us to observe a reduction compared to the unconstrained configuration. In the non-preemptive setting, the maximum stability region is drastically reduced compared to the unconstrained setting and we conclude that a non-preemptive scheduler cannot take opportunistically advantage (in terms of stability) of the random varying connectivity. Instead, for the $\gamma$ decision epochs, we observe that the maximum stability region is monotone in the rate of the decision moments $\gamma$, and that one can be arbitrarily close to the maximum stability region in the unconstrained setting if we choose $\gamma$ large enough. We further show that Serve Longest Connected (SLC) queue is maximum stable in both constrained settings, within the set of policies that select a queue among the connected ones. From a methodological viewpoint, we introduce a novel theoretical tool termed a ``test for fluid limits'' (TFL) that might be of independent interest. TFL is a simple test that, if satisfied by the fluid limit, allows us to conclude for stability.
- [160] arXiv:2403.18688 [pdf, ps, other]
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Title: The Gross--Kohnen--Zagier theorem via $p$-adic uniformizationSubjects: Number Theory (math.NT)
This article gives a new proof of the Gross--Kohnen--Zagier theorem for Shimura curves which exploits the $p$-adic uniformization of Cerednik--Drinfeld. The explicit description of CM points via this uniformization leads to an expression relating the Gross--Kohnen--Zagier generating series to the ordinary projection of the first derivative, with respect to a weight variable, of a $p$-adic family of positive definite ternary theta series.
- [161] arXiv:2403.18698 [pdf, ps, other]
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Title: Submanifold projections and hyperbolicity in ${\rm Out}(F_n)$Comments: 24 pagesSubjects: Geometric Topology (math.GT); Group Theory (math.GR)
The free splitting graph of a free group $F_n$ with $n\geq 2$ generators is a hyperbolic ${\rm Out}(F_n)$-graph which has a geometric realization as a sphere graph in the connected sum of $n$ copies of $S^1\times S^2$. We use this realization to construct submanifold projections of the free splitting graph into the free splitting graphs of proper free factors. This is used to construct for $n\geq 3$ a new hyperbolic ${\rm Out}(F_n)$-graph. If $n=3$, then every exponentially growing element acts on this graph with positive translation length.
- [162] arXiv:2403.18704 [pdf, ps, other]
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Title: Convergence rates under a range invariance condition with application to electrical impedance tomographyAuthors: Barbara KaltenbacherSubjects: Numerical Analysis (math.NA)
This paper is devoted to proving convergence rates of variational and iterative regularization methods under variational source conditions VSCs for inverse problems whose linearization satisfies a range invariance condition. In order to achieve this, often an appropriate relaxation of the problem needs to be found that is usually based on an augmentation of the set of unknowns and leads to a particularly structured reformulation of the inverse problem. We analyze three approaches that make use of this structure, namely a variational and a Newton type scheme, whose convergence without rates has already been established in \cite{rangeinvar}; additionally we propose a split minimization approach that can be show to satisfy the same rates results. \\ The range invariance condition has been verified for several coefficient identification problems for partial differential equations from boundary observations as relevant in a variety of tomographic imaging modalities. Our motivation particularly comes from the by now classical inverse problem of electrical impedance tomography EIT and we study both the original formulation by a diffusion type equation and its reformulation as a Schr\"odinger equation. For both of them we find relaxations that can be proven to satisfy the range invariance condition. Combining results on VSCs from \cite{Diss-Weidling} with the abstract framework for the three approaches mentioned above, we arrive at convergence rates results for the variational, split minimization and Newton type method in EIT.
- [163] arXiv:2403.18707 [pdf, other]
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Title: Connections between Reachability and Time OptimalityComments: Submitted to AutomaticaSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
This paper presents the concept of an equivalence relation between the set of optimal control problems. By leveraging this concept, we show that the boundary of the reachability set can be constructed by the solutions of time optimal problems. Alongside, a more generalized equivalence theorem is presented together. The findings facilitate the use of solution structures from a certain class of optimal control problems to address problems in corresponding equivalent classes. As a byproduct, we state and prove the construction methods of the reachability sets of three-dimensional curves with prescribed curvature bound. The findings are twofold: Firstly, we prove that any boundary point of the reachability set, with the terminal direction taken into account, can be accessed via curves of H, CSC, CCC, or their respective subsegments, where H denotes a helicoidal arc, C a circular arc with maximum curvature, and S a straight segment. Secondly, we show that any boundary point of the reachability set, without considering the terminal direction, can be accessed by curves of CC, CS, or their respective subsegments. These findings extend the developments presented in literature regarding planar curves, or Dubins car dynamics, into spatial curves in $\mathbb{R}^3$. For higher dimensions, we confirm that the problem of identifying the reachability set of curvature bounded paths subsumes the well-known Markov-Dubins problem. These advancements in understanding the reachability of curvature bounded paths in $\mathbb{R}^3$ hold significant practical implications, particularly in the contexts of mission planning problems and time optimal guidance.
- [164] arXiv:2403.18713 [pdf, other]
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Title: Characterization of Spatial-Temporal Channel Statistics from Indoor Measurement Data at D BandAuthors: Chathuri Weragama, Joonas Kokkoniemi, Mar Francis De Guzman, Katsuyuki Haneda, Pekka Kyosti, Markku JunttiComments: 6 pages, 22 figuresSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Millimeter-wave (mmWave) and D Band (110--170~GHz) frequencies are poised to play a pivotal role in the advancement of sixth-generation (6G) systems and beyond, owing to their ability to enhance performance metrics such as capacity, ultra-low latency, and spectral efficiency. This paper concentrates on deriving statistical insights into power, delay, and the number of paths based on measurements conducted across four distinct locations at a center frequency of 143.1 GHz. The findings underscore the suitability of various distributions in characterizing power behavior in line-of-sight (LOS) scenarios, including lognormal, Nakagami, gamma, and beta distributions, whereas the loglogistic distribution gives the optimal fit for power distribution in non-line-of-sight (NLOS) scenarios. Moreover, the exponential distribution shows to be the most appropriate model for the delay distribution in both LOS and NLOS scenarios. In terms of the number of paths, observations indicate a tendency for the highest concentration within the 10 m to 30 m distance range between the transmitter (Tx) and receiver (Rx). These insights shed light on the statistical nature of D band propagation characteristics, which are vital for informing the design and optimization of future 6G communication systems
- [165] arXiv:2403.18718 [pdf, ps, other]
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Title: Constructive proofs of existence and stability of solitary waves in the capillary-gravity Whitham equationAuthors: Matthieu CadiotSubjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS); Functional Analysis (math.FA)
In this manuscript, we present a method to prove constructively the existence and spectral stability of solitary waves (solitons) in the capillary-gravity Whitham equation. By employing Fourier series analysis and computer-aided techniques, we successfully approximate the Fourier multiplier operator in this equation, allowing the construction of an approximate inverse for the linearization around an approximate solution $u_0$. Then, using a Newton-Kantorovich approach, we provide a sufficient condition under which the existence of a unique solitary wave $\tilde{u}$ in a ball centered at $u_0$ is obtained. The verification of such a condition is established combining analytic techniques and rigorous numerical computations. Moreover, we derive a methodology to control the spectrum of the linearization around $\tilde{u}$, enabling the study of spectral stability of the solution. As an illustration, we provide a (constructive) computer-assisted proof of existence of a stable soliton in both the case with capillary effects ($T>0$) and without capillary effects ($T=0$). The methodology presented in this paper can be generalized and provides a new approach for addressing the existence and spectral stability of solitary waves in nonlocal nonlinear equations. All computer-assisted proofs, including the requisite codes, are accessible on GitHub at \cite{julia_cadiot}.
- [166] arXiv:2403.18719 [pdf, other]
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Title: On the scaling of random Tamari intervals and Schnyder woods of random triangulations (with an asymptotic D-finite trick)Authors: Guillaume ChapuyComments: 24 pagesSubjects: Combinatorics (math.CO); Symbolic Computation (cs.SC); Probability (math.PR)
We consider a Tamari interval of size $n$ (i.e., a pair of Dyck paths which are comparable for the Tamari relation) chosen uniformly at random. We show that the height of a uniformly chosen vertex on the upper or lower path scales as $n^{3/4}$, and has an explicit limit law. By the Bernardi-Bonichon bijection, this result also describes the height of points in the canonical Schnyder trees of a uniform random plane triangulation of size $n$.
The exact solution of the model is based on polynomial equations with one and two catalytic variables. To prove the convergence from the exact solution, we use a version of moment pumping based on D-finiteness, which is essentially automatic and should apply to many other models. We are not sure to have seen this simple trick used before.
It would be interesting to study the universality of this convergence for decomposition trees associated to positive Bousquet-M\'elou--Jehanne equations. - [167] arXiv:2403.18724 [pdf, other]
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Title: An exactly curl-free finite-volume scheme for a hyperbolic compressible barotropic two-phase modelSubjects: Numerical Analysis (math.NA)
We present a new second order accurate structure-preserving finite volume scheme for the solution of the compressible barotropic two-phase model of Romenski et. al in multiple space dimensions. The governing equations fall into the wider class of symmetric hyperbolic and thermodynamically compatible (SHTC) systems and consist of a set of first-order hyperbolic partial differential equations (PDE). In the absence of algebraic source terms, the model is subject to a curl-free constraint for the relative velocity between the two phases. The main objective of this paper is, therefore, to preserve this structural property exactly also at the discrete level. The new numerical method is based on a staggered grid arrangement where the relative velocity field is stored in the cell vertexes while all the remaining variables are stored in the cell centers. This allows the definition of discretely compatible gradient and curl operators, which ensure that the discrete curl errors of the relative velocity field remain zero up to machine precision. A set of numerical results confirms this property also experimentally.
- [168] arXiv:2403.18727 [pdf, ps, other]
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Title: Modular representations of the Yangian $Y_2$Comments: Preliminary versionSubjects: Representation Theory (math.RT)
Let $Y_2$ be the Yangian associated to the general linear Lie algebra $\mathfrak{gl}_2$, defined over an algebraically closed field $\mathbbm{k}$ of characteristic $p > 0$. In this paper, we study the representation theory of the restricted Yangian $Y^{[p]}_2$. This gives a description of the representations of $\mathfrak{gl}_{2n}$, whose $p$-character is a nilpotent whose Jordan type is the two-row partition $( n, n)$.
- [169] arXiv:2403.18736 [pdf, ps, other]
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Title: Characterization of genuine ramification using formal orbifoldsComments: Final versionSubjects: Algebraic Geometry (math.AG)
We give a characterization of genuinely ramified maps of formal orbifolds in the Tannakian framework. In particular we show that a morphism is genuinely ramified if and only if the pullback of every stable bundle remains stable in the orbifold category. We also give some other characterizations of genuine ramification. This generalizes the results of [BKP1] and [BP1]. In fact, it is a positive characteristic analogue of results in [BKP2].
- [170] arXiv:2403.18738 [pdf, other]
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Title: The extension of traces for Sobolev mappings between manifoldsAuthors: Jean Van SchaftingenComments: 56 pagesSubjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA)
The compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ for which the trace operator from the first-order Sobolev space of mappings $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$ to the fractional Sobolev-Slobodecki\u{\i} space $\smash{\smash{\dot{W}}^{1 - 1/p, p}} (\partial \mathcal{M}, \mathcal{N})$ is surjective when $1 < p < m$ are characterised. The traces are extended using a new construction which can be carried out assuming the absence of the known topological and analytical obstructions. When $p \ge m$ the same construction provides a Sobolev extension with linear estimates for maps that have a continuous extension, provided that there are no known analytical obstructions to such a control.
- [171] arXiv:2403.18744 [pdf, other]
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Title: A nonsmooth Frank-Wolfe algorithm through a dual cutting-plane approachSubjects: Optimization and Control (math.OC)
An extension of the Frank-Wolfe Algorithm (FWA), also known as Conditional Gradient algorithm, is proposed. In its standard form, the FWA allows to solve constrained optimization problems involving $\beta$-smooth cost functions, calling at each iteration a Linear Minimization Oracle. More specifically, the oracle solves a problem obtained by linearization of the original cost function. The algorithm designed and investigated in this article, named Dualized Level-Set (DLS) algorithm, extends the FWA and allows to address a class of nonsmooth costs, involving in particular support functions. The key idea behind the construction of the DLS method is a general interpretation of the FWA as a cutting-plane algorithm, from the dual point of view. The DLS algorithm essentially results from a dualization of a specific cutting-plane algorithm, based on projections on some level sets. The DLS algorithm generates a sequence of primal-dual candidates, and we prove that the corresponding primal-dual gap converges with a rate of $O(1/\sqrt{t})$.
- [172] arXiv:2403.18748 [pdf, ps, other]
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Title: Extended eigenvalues of composition operator on Bergman space and Fock spaceAuthors: Li JialeSubjects: Functional Analysis (math.FA)
A complex scalar k is said to be an extended eigenvalue of a bounded linear operator A on a complex Hilbert space if there is a nonzero operator X such that AX=kXA. There are some solutions to the problem of computing the extended eigenvalues for composition operators induced on the Fock space and Bergman space by linear fractional transformationsof the complex plane in this paper.
- [173] arXiv:2403.18749 [pdf, other]
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Title: Robust Numerical Algebraic GeometrySubjects: Numerical Analysis (math.NA)
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a member of a parameterized family of polynomial systems where the parameter values may be measured with imprecision or arise from prior numerical computations, uncertainty may arise in the structure of the solution set, including the number of isolated solutions, the existence of higher dimensional solution components, and the number of irreducible components along with their multiplicities. The loci where these structures change form a stratification of exceptional algebraic sets in the space of parameters. We describe methodologies for making the interpretation of numerical results more robust by searching for nearby parameter values on an exceptional set. We demonstrate these techniques on several illustrative examples and then treat several more substantial problems arising from the kinematics of mechanisms and robots.
- [174] arXiv:2403.18757 [pdf, ps, other]
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Title: The Fubini--Study metric on an `odd' Grassmannian is rigidAuthors: Stuart James HallComments: 27 pagesSubjects: Differential Geometry (math.DG)
Following the ideas of Gasqui and Goldschmidt, we give an explicit description of the infinitesimal Einstein deformations admitted by the Fubini--Study metric on complex Grassmannians $G_{m}(\mathbb{C}^{n+m})$ with $m,n\geq 2$. The deformations were first shown to exist by Koiso in the 1980s but it has remained an open question as to whether they can be integrated to give genuine deformations of the Fubini--Study metric. We show that when $n+m$ is odd, the answer is no.
- [175] arXiv:2403.18758 [pdf, ps, other]
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Title: On the cohomological dimension of kernels of maps to $\mathbb Z$Authors: Sam P FisherComments: 11 pagesSubjects: Group Theory (math.GR)
We prove that if $G$ is a finitely generated RFRS group of cohomological dimension $2$, then $G$ is virtually free-by-cyclic if and only if $b_2^{(2)}(G) = 0$. This answers a question of Wise and generalises and gives a new proof of a recent theorem of Kielak and Linton, where the same result is obtained under the additional hypotheses that $G$ is virtually compact special and hyperbolic. More generally, we show that if $G$ is a RFRS group of cohomological dimension $n$ and of type $\mathrm{FP}_{n-1}$, then $G$ admits a virtual map to $\mathbb Z$ with kernel of rational cohomological dimension $n-1$ if and only if $b_n^{(2)}(G) = 0$.
- [176] arXiv:2403.18763 [pdf, ps, other]
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Title: Duality for Hodge-Witt cohomology with modulusSubjects: Algebraic Geometry (math.AG)
Given an effective Cartier divisor D with simple normal crossing support on a smooth and proper scheme X over a perfect field of positive characteristic p, there is a natural notion of de Rham-Witt sheaves on X with zeros along D. We show that these sheaves correspond under Grothendieck duality for coherent sheaves to de Rham-Witt sheaves on X with modulus (X,D), as defined in the theory of cube invariant modulus sheaves with transfers developed by Kahn-Miyazaki-Saito-Yamazaki. From this we deduce refined versions of Ekedahl - and Poincar\'e duality for crystalline cohomology generalizing results of Mokrane and Nakkajima for reduced D, and a modulus version of Milne-Kato duality for \'etale motivic cohomology with p-primary torsion coefficients, which refines a result of Jannsen-Saito-Zhao. We furthermore get new integral models for rigid cohomology with compact supports on the complement of D and a modulus version of Milne's perfect Brauer group pairing for smooth projective surfaces over finite fields.
- [177] arXiv:2403.18767 [pdf, other]
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Title: The best approximation pair problem relative to two subsets in a normed spaceSubjects: Optimization and Control (math.OC); Functional Analysis (math.FA); Metric Geometry (math.MG)
In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. This problem, which has a long history, has found applications in science and technology. We discuss the problem in more general normed spaces and with possibly non-convex subsets, and focus our attention on the issues of uniqueness and existence of the solution to the problem. To the best of our knowledge these fundamental issues have not received much attention. In particular, we present several sufficient geometric conditions for the (at most) uniqueness of a BAP relative to these subsets. These conditions are related to the structure of the boundaries of the subsets, their relative orientation, and the structure of the unit sphere of the space. In addition, we present many sufficient conditions for the existence of a BAP, possibly without convexity . Our results allow us to significantly extend the horizon of the recent alternating simultaneous Halpern-Lions-Wittmann-Bauschke (A-S-HLWB) algorithm [Censor, Mansour and Reem, The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets, arXiv:2304.09600 (2023)] for solving the BAP problem.
- [178] arXiv:2403.18773 [pdf, ps, other]
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Title: On the equivalence of all notions of generalized derivations whose domain is a C$^{\ast}$-algebraSubjects: Operator Algebras (math.OA)
Let $\mathcal{M}$ be a Banach bimodule over an associative Banach algebra $\mathcal{A}$, and let $F: \mathcal{A}\to \mathcal{M}$ be a linear mapping. Three main uses of the term \emph{generalized derivation} are identified in the available literature, namely,
($\checkmark$) $F$ is a generalized derivation of the first type if there exists a derivation $ d : \mathcal{A}\to \mathcal{M}$ satisfying $F(a b ) = F(a) b + a d(b)$ for all $a,b\in \mathcal{A}$.
($\checkmark$) $F$ is a generalized derivation of the second type if there exists an element $\xi\in \mathcal{M}^{**}$ satisfying $F(a b ) = F(a) b + a F(b) - a \xi b$ for all $a,b\in \mathcal{A}$.
($\checkmark$) $F$ is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings $G,H : \mathcal{A}\to \mathcal{M}$ satisfying $F(a b ) = G(a) b + a H(b)$ for all $a,b\in \mathcal{A}$.
There are examples showing that these three definitions are not, in general, equivalent. Despite that the first two notions are well studied when $\mathcal{A}$ is a C$^*$-algebra, it is not known if the three notions are equivalent under these special assumptions. In this note we prove that every generalized derivation of the third type whose domain is a C$^*$-algebra is automatically continuous. We also prove that every (continuous) generalized derivation of the third type from a C$^*$-algebra $\mathcal{A}$ into a general Banach $\mathcal{A}$-bimodule is a generalized derivation of the first and second type. In particular, the three notions coincide in this case. We also explore the possible notions of generalized Jordan derivations on a C$^*$-algebra and establish some continuity properties for them. - [179] arXiv:2403.18780 [pdf, other]
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Title: Universal bounds on the entropy of toroidal attractorsSubjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)
A toroidal set is a compactum $K \subseteq \mathbb{R}^3$ which has a neighbourhood basis of solid tori. We study the topological entropy of toroidal attractors $K$, bounding it from below in terms of purely topological properties of $K$. In particular, we show that for a toroidal set $K$, either any smooth attracting dynamics on $K$ has an entropy at least $\log 2$, or (up to continuation) $K$ admits smooth attracting dynamics which are stationary (hence with a zero entropy).
- [180] arXiv:2403.18782 [pdf, ps, other]
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Title: Beyond boundaries: Gary Lorden's groundbreaking contributions to sequential analysisSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
Gary Lorden provided a number of fundamental and novel insights to sequential hypothesis testing and changepoint detection. In this article we provide an overview of Lorden's contributions in the context of existing results in those areas, and some extensions made possible by Lorden's work, mentioning also areas of application including threat detection in physical-computer systems, near-Earth space informatics, epidemiology, clinical trials, and finance.
- [181] arXiv:2403.18798 [pdf, ps, other]
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Title: Weak convergence of probability measures on hyperspaces with the upper Fell-topologyAuthors: Dietmar FergerSubjects: Probability (math.PR)
Let E be a locally compact second countable Hausdorff space and F the pertaining family of all closed sets. We endow F respectively with the Fell-topology, the upper Fell topology or the upper Vietoris-topology and investigate weak convergence of probability measures on the corresponding hyperspaces with a focus on the upper Fell topology. The results can be transferred to distributional convergence of random closed sets in E with applications to the asymptotic behavior of measurable selection.
- [182] arXiv:2403.18799 [pdf, ps, other]
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Title: Dimension-independent functional inequalities by tensorization and projection argumentsComments: 28 pagesSubjects: Probability (math.PR); Analysis of PDEs (math.AP); Functional Analysis (math.FA)
We study stability under tensorization and projection-type operations of gradient-type estimates and other functional inequalities for Markov semigroups on metric spaces. Using transportation-type inequalities obtained by F. Baudoin and N. Eldredge in 2021, we prove that constants in the gradient estimates can be chosen to be independent of the dimension. Our results are applicable to hypoelliptic diffusions on sub-Riemannian manifolds and some hypocoercive diffusions. As a byproduct, we obtain dimension-independent reverse Poincar\'{e}, reverse logarithmic Sobolev, and gradient bounds for Lie groups with a transverse symmetry and for non-isotropic Heisenberg groups.
- [183] arXiv:2403.18800 [pdf, other]
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Title: On two algebras of token graphsSubjects: Combinatorics (math.CO)
The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$.
In this article, we describe some properties of the Laplacian matrix $\L_k$ of $F_k(G)$ and the Laplacian matrix $\overline{\L}_k$ of the $k$-token graph $F_k(\overline{G})$ of its complement $\overline{G}$.
In this context, a result about the commutativity of the matrices $\L_k$ and $\overline{\L}_k$ was given in [C. Dalf\'o, F. Duque, R. Fabila-Monroy, M. A. Fiol, C. Huemer, A. L. Trujillo-Negrete, and F. J. Zaragoza Mart\'{\i}nez,
On the Laplacian spectra of token graphs,
{\em Linear Algebra Appl.} {\bf 625} (2021) 322--348], but the proof was incomplete, and there were some typos. Here, we give the correct proof.
Based on this result, and fixed the pair $(n,k)$ and the graph $G$, we first introduce a `local' algebra ${\cal L}(G)$, generated by the pair $(\L_k, \overline{\L}_k)$, showing its closed relationship with the Bose-Mesner algebra of the Johnson graphs $J(n,k)$.
Finally, fixed only $(n,k)$, we present a `global' algebra ${\cal A}(n,k)$ that contains ${\cal L}(G)$ together with the Laplacian and adjacency matrices of the $k$-token graph of any graph $G$ on $n$ vertices. - [184] arXiv:2403.18801 [pdf, other]
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Title: Nonstandard Lagrangians and branched Hamiltonians: A brief reviewComments: Preliminary version, comments are welcomeSubjects: Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI); Classical Physics (physics.class-ph); Quantum Physics (quant-ph)
Time and again, non-conventional forms of Lagrangians have found attention in the literature. For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics including specifically the types of the Li\'{e}nard class; for another, very often the problem of their quantization opens up multiple branches of the corresponding Hamiltonians, ending up with the presence of singularities in the associated eigenfunctions. In this article, we furnish a brief review of the classical theory of such Lagrangians and the associated branched Hamiltonians, starting with the example of Li\'{e}nard-type systems. We then take up other cases where the Lagrangians depend upon the velocity with powers greater than two while still having a tractable mathematical structure, while also describing the associated branched Hamiltonians for such systems. For various examples, we emphasize upon the emergence of the notion of momentum-dependent mass in the theory of branched Hamiltonians.
- [185] arXiv:2403.18805 [pdf, ps, other]
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Title: Using an invariant knot of a flow to find additional invariant structureAuthors: J.J. Sánchez-GabitesSubjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)
Consider a continuous flow in $\mathbb{R}^3$ or any orientable $3$-manifold. Let $(Q_1, Q_0)$ be an index pair in the sense of Conley and consider the region $N := \overline{Q_1 - Q_0}$. (An example of this is a compact $3$-manifold $N$ such that trajectories of the flow cross $\partial N$ inwards or outwards transversally, or bounce off it from the outside). Suppose we know there is an invariant knot or link $K$ in the interior of $N$. We prove the following: if $K$ is contractible and nontrivial (in the sense of knot theory) in $N$, then every neighbourhood $U$ of $K$ contains a point $p \in N - K$ such that the whole trajectory of $p$ is contained in $N$. In other words, the presence of $K$ forces the existence of additional invariant structure in $N$ (besides $K$), and the latter can actually be found arbitrarily close to $K$.
To prove this result we develop a ``coloured'' handle theory which may be of independent interest to study flows in $3$-manifolds. - [186] arXiv:2403.18806 [pdf, ps, other]
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Title: The socle of subshift algebras, with applications to subshift conjugacySubjects: Rings and Algebras (math.RA); Operator Algebras (math.OA)
We introduce the concept of "irrational paths" for a given subshift and use it to characterize all minimal left ideals in the associated unital subshift algebra. Consequently, we characterize the socle as the sum of the ideals generated by irrational paths. Proceeding, we construct a graph such that the Leavitt path algebra of this graph is graded isomorphic to the socle. This realization allows us to show that the graded structure of the socle serves as an invariant for the conjugacy of Ott-Tomforde-Willis subshifts and for the isometric conjugacy of subshifts constructed with the product topology. Additionally, we establish that the socle of the unital subshift algebra is contained in the socle of the corresponding unital subshift C*-algebra.
- [187] arXiv:2403.18808 [pdf, ps, other]
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Title: Solid lines in axial algebras of Jordan type half and Jordan algebrasAuthors: Jari DesmetComments: 17 pages. Comments welcomeSubjects: Rings and Algebras (math.RA); Group Theory (math.GR)
Primitive axial algebras of Jordan type $\eta$ were introduced in 2015 by Hall,
Rehren and Shpectorov. These algebras are not characterized by polynomial identities. Rather, they are required to be generated by primitive idempotents whose multiplication is diagonalizable with eigenvalues $0,1,\eta$, and its eigenvectors multiply following specific fusion rules.
Based on the concept of solid subalgebras, introduced very recently by Gorshkov, Shpectorov and Staroletov, we provide a criterion for when a primitive axial algebra of Jordan type with parameter $\eta = \frac{1}{2}$ is a Jordan algebra. We also prove that this is equivalent to associators being derivations of the algebra. - [188] arXiv:2403.18809 [pdf, other]
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Title: $L^\infty$-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decompositionComments: 21 pages, 2 figures, 2 tablesSubjects: Dynamical Systems (math.DS); Numerical Analysis (math.NA)
Extended dynamic mode decomposition (EDMD) is a well-established method to generate a data-driven approximation of the Koopman operator for analysis and prediction of nonlinear dynamical systems. Recently, kernel EDMD (kEDMD) has gained popularity due to its ability to resolve the challenging task of choosing a suitable dictionary by defining data-based observables. In this paper, we provide the first pointwise bounds on the approximation error of kEDMD. The main idea consists of two steps. First, we show that the reproducing kernel Hilbert spaces of Wendland functions are invariant under the Koopman operator. Second, exploiting that the learning problem given by regression in the native norm can be recast as an interpolation problem, we prove our novel error bounds by using interpolation estimates. Finally, we validate our findings with numerical experiments.
- [189] arXiv:2403.18815 [pdf, ps, other]
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Title: A dynamical interpretation of the connection map of an attractor-repeller decompositionAuthors: J.J. Sánchez-GabitesSubjects: Dynamical Systems (math.DS)
In Conley index theory one may study an invariant set $S$ by decomposing it into an attractor $A$, a repeller $R$, and the orbits connecting the two. The Conley indices of $S$, $A$ and $R$ fit into an exact sequence where a certain connection homomorphism $\Gamma$ plays an important role. In this paper we provide a dynamical interpretation of this map. Roughly, $R$ "emits" an element of its Conley index as a "wavefront", part of which intersects the connecting orbits in $S$. This subset of the wavefront evolves towards $A$ and is then "received" by it to produce an element in its Conley index.
Cross-lists for Thu, 28 Mar 24
- [190] arXiv:2403.17969 (cross-list from cs.DM) [pdf, other]
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Title: Antimagic Labeling of Graphs Using Prime NumbersComments: 11 pages, 15 figuresSubjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO)
Graph labeling is a technique that assigns unique labels or weights to the vertices or edges of a graph, often used to analyze and solve various graph-related problems. There are few methods with certain limitations conducted by researchers previously on this topic. This research paper focuses on antimagic labeling of different types of graphs and trees. It entails the assignment of distinct prime values to edges in a manner that ensures the cumulative sum of edge labels at each vertex remains unique. This research proposes a conjecture on antimagic labeling of any graphs and proves two theories. Firstly, we tried to give weights to the edges randomly, as some exceptions are faced in particular phases in this way, we followed a whole new way to mitigate this problem. This research paper demonstrates computational and mathematical verification to prove that antimagic labeling of any perfect binary tree and complete graph is possible.
- [191] arXiv:2403.17982 (cross-list from stat.ME) [pdf, ps, other]
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Title: Markov chain models for inspecting response dynamics in psychological testingAuthors: Andrea BoscoComments: 20 pages, 1 figure, 3 tables, 25 equations/matrices. Part of this paper was presented to the XXIX AIP Congress, Experimental Psychology Section. September 18th-20th 2023, Lucca, Italy. Title of the talk: "Differentiating students with signs of ADHD or OCD based on hysteresis in responses to a mind-wandering test. A Study of Markov Chain Test Response Sequences"Subjects: Methodology (stat.ME); Machine Learning (cs.LG); Probability (math.PR)
The importance of considering contextual probabilities in shaping response patterns within psychological testing is underscored, despite the ubiquitous nature of order effects discussed extensively in methodological literature. Drawing from concepts such as path-dependency, first-order autocorrelation, state-dependency, and hysteresis, the present study is an attempt to address how earlier responses serve as an anchor for subsequent answers in tests, surveys, and questionnaires. Introducing the notion of non-commuting observables derived from quantum physics, I highlight their role in characterizing psychological processes and the impact of measurement instruments on participants' responses. We advocate for the utilization of first-order Markov chain modeling to capture and forecast sequential dependencies in survey and test responses. The employment of the first-order Markov chain model lies in individuals' propensity to exhibit partial focus to preceding responses, with recent items most likely exerting a substantial influence on subsequent response selection. This study contributes to advancing our understanding of the dynamics inherent in sequential data within psychological research and provides a methodological framework for conducting longitudinal analyses of response patterns of test and questionnaire.
- [192] arXiv:2403.18007 (cross-list from quant-ph) [pdf, other]
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Title: Typical thermalization of low-entanglement statesComments: 6+14 pages, one figureSubjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
Proving thermalization from the unitary evolution of a closed quantum system is one of the oldest questions that is still nowadays only partially resolved. Several efforts have led to various formulations of what is called the eigenstate thermalization hypothesis. The latter, however, assume initial states which are highly concentrated around a specific energy window and, as such, cannot account for a large class of states that are of paramount importance and that are operationally accessible in natural physical settings, including many experimental schemes for testing thermalization and for quantum simulation: low-entanglement states. In this work, we prove thermalization of these states under precise conditions that have operational significance. More specifically, we define a random energy smoothing - motivated by arguments of unavoidable finite resolution - on local Hamiltonians that lead to local thermalization when the initial state has low entanglement. Finally we show that such transformation affects neither the Gibbs state locally nor, under a mild condition, the short time dynamics.
- [193] arXiv:2403.18011 (cross-list from hep-th) [pdf, other]
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Title: On AdS$_4$ deformations of celestial symmetriesComments: 13 pagesSubjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
Celestial holography has led to the discovery of new symmetry algebras arising from the study of collinear limits of perturbative gravity amplitudes in flat space. We explain from the twistor perspective how a non-vanishing cosmological constant $\Lambda$ naturally modifies the celestial chiral algebra. The cosmological constant deforms the Poisson bracket on twistor space, so the corresponding deformed algebra of Hamiltonians under the new bracket is automatically consistent. This algebra is equivalent to that recently found by Taylor and Zhu. We find a number of variations of the deformed algebra. We give the Noether charges arising from the expression of this algebra as a symmetry of the twistor action for self-dual gravity with cosmological constant.
- [194] arXiv:2403.18055 (cross-list from eess.SY) [pdf, other]
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Title: Adaptive Boundary Control of the Kuramoto-Sivashinsky Equation Under Intermittent SensingComments: Submitted to AutomaticaSubjects: Systems and Control (eess.SY); Analysis of PDEs (math.AP)
We study in this paper boundary stabilization, in the L2 sense, of the one-dimensional Kuramoto-Sivashinsky equation subject to intermittent sensing. We assume that we measure the state of this spatio-temporal equation on a given spatial subdomain during certain intervals of time, while we measure the state on the remaining spatial subdomain during the remaining intervals of time. As a result, we assign a feedback law at the boundary of the spatial domain and force to zero the value of the state at the junction of the two subdomains. Throughout the study, the destabilizing coefficient is assumed to be space-dependent and bounded but unknown. Adaptive boundary controllers are designed under different assumptions on the forcing term. In particular, when the forcing term is null, we guarantee global exponential stability of the origin. Furthermore, when the forcing term is bounded and admits a known upper bound, we guarantee input-to-state stability, and only global uniform ultimate boundedness is guaranteed when the upper bound is unknown. Numerical simulations are performed to illustrate our results
- [195] arXiv:2403.18071 (cross-list from eess.SY) [pdf, other]
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Title: From Sontag s to Cardano-Lyapunov Formula for Systems Not Affine in the Control: Convection-Enabled PDE StabilizationComments: To be presented at the 2024 American Control ConferenceSubjects: Systems and Control (eess.SY); Analysis of PDEs (math.AP)
We propose the first generalization of Sontag s universal controller to systems not affine in the control, particularly, to PDEs with boundary actuation. We assume that the system admits a control Lyapunov function (CLF) whose derivative, rather than being affine in the control, has either a depressed cubic, quadratic, or depressed quartic dependence on the control. For each case, a continuous universal controller that vanishes at the origin and achieves global exponential stability is derived. We prove our result in the context of convectionreaction-diffusion PDEs with Dirichlet actuation. We show that if the convection has a certain structure, then the L2 norm of the state is a CLF. In addition to generalizing Sontag s formula to some non-affine systems, we present the first general Lyapunov approach for boundary control of nonlinear PDEs. We illustrate our results via a numerical example.
- [196] arXiv:2403.18084 (cross-list from physics.ao-ph) [pdf, other]
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Title: Properties and baroclinic instability of stratified thermal upper-ocean flowComments: Submitted to Proceeding of the Royal Society ASubjects: Atmospheric and Oceanic Physics (physics.ao-ph); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn)
We study the properties of, and perform a stability analysis of a baroclinic zonal current in, a thermal rotating shallow-water model, sometimes called \emph{Ripa's model}, featuring stratification for quasigeostrophic upper-ocean dynamics. The model has Lie--Poisson Hamiltonian structure. In addition to Casimirs, the model supports integrals of motion that neither form the kernel of the bracket nor are related to explicit symmetries. The model sustains Rossby waves and a neutral model, whose spurious growth is prevented by a positive-definite integral, quadratic on the deviation from the motionless state. A baroclinic zonal jet with vertical curvature is found to be spectrally stable for specific configurations of the gradients of layer thickness, vertically averaged buoyancy, and buoyancy frequency. Only a subset of such states was found Lyapunov stable using the available integrals, except the newly reported ones, whose role in constraining stratified thermal flow remains elusive. The existence of Lyapunov-stable states enabled bounding the nonlinear growth of perturbations to spectrally unstable states. Our results do not support the generality of earlier numerical evidence on the suppression of submesoscale wave activity as a result of the inclusion of stratification in thermal shallow-water theory, which we supported with direct numerical simulations.
- [197] arXiv:2403.18087 (cross-list from eess.SP) [pdf, other]
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Title: Channel Estimation and Beamforming for Beyond Diagonal Reconfigurable Intelligent SurfacesComments: 12 pages, 10 figures, submitted to IEEE journalSubjects: Signal Processing (eess.SP); Information Theory (cs.IT)
Beyond diagonal reconfigurable intelligent surface (BD-RIS) is a new advance and generalization of the RIS technique. BD-RIS breaks through the isolation between RIS elements by creatively introducing inter-element connections, thereby enabling smarter wave manipulation and enlarging coverage. However, exploring proper channel estimation schemes suitable for BD-RIS aided communication systems still remains an open problem. In this paper, we study channel estimation and beamforming design for BD-RIS aided multi-antenna systems. We first describe the channel estimation strategy based on the least square (LS) method, derive the mean square error (MSE) of the LS estimation, and formulate the joint pilot sequence and BD-RIS design problem with unique constraints induced by BD-RIS architectures. Specifically, we propose an efficient pilot sequence and BD-RIS design which theoretically guarantees to achieve the minimum MSE. With the estimated channel, we then consider two BD-RIS scenarios and propose beamforming design algorithms. Finally, we provide simulation results to verify the effectiveness of the proposed channel estimation scheme and beamforming design algorithms. We also show that more interelement connections in BD-RIS improves the performance while increasing the training overhead for channel estimation.
- [198] arXiv:2403.18127 (cross-list from cs.LG) [pdf, ps, other]
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Title: A Correction of Pseudo Log-Likelihood MethodComments: 7 pagesSubjects: Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
Pseudo log-likelihood is a type of maximum likelihood estimation (MLE) method used in various fields including contextual bandits, influence maximization of social networks, and causal bandits. However, in previous literature \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}, the log-likelihood function may not be bounded, which may result in the algorithm they proposed not well-defined. In this paper, we give a counterexample that the maximum pseudo log-likelihood estimation fails and then provide a solution to correct the algorithms in \citep{li2017provably, zhang2022online, xiong2022combinatorial, feng2023combinatorial1, feng2023combinatorial2}.
- [199] arXiv:2403.18149 (cross-list from cs.RO) [pdf, other]
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Title: Code Generation for Conic Model-Predictive Control on Microcontrollers with TinyMPCComments: Submitted to CDC, 2024. First two authors contributed equallySubjects: Robotics (cs.RO); Systems and Control (eess.SY); Optimization and Control (math.OC)
Conic constraints appear in many important control applications like legged locomotion, robotic manipulation, and autonomous rocket landing. However, current solvers for conic optimization problems have relatively heavy computational demands in terms of both floating-point operations and memory footprint, making them impractical for use on small embedded devices. We extend TinyMPC, an open-source, high-speed solver targeting low-power embedded control applications, to handle second-order cone constraints. We also present code-generation software to enable deployment of TinyMPC on a variety of microcontrollers. We benchmark our generated code against state-of-the-art embedded QP and SOCP solvers, demonstrating a two-order-of-magnitude speed increase over ECOS while consuming less memory. Finally, we demonstrate TinyMPC's efficacy on the Crazyflie, a lightweight, resource-constrained quadrotor with fast dynamics. TinyMPC and its code-generation tools are publicly available at https://tinympc.org.
- [200] arXiv:2403.18164 (cross-list from eess.SY) [pdf, other]
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Title: Incentive Designs for Learning Agents to Stabilize Coupled Exogenous SystemsComments: 8 pages, 3 figuresSubjects: Systems and Control (eess.SY); Dynamical Systems (math.DS); Optimization and Control (math.OC)
We consider a large population of learning agents noncooperatively selecting strategies from a common set, influencing the dynamics of an exogenous system (ES) we seek to stabilize at a desired equilibrium. Our approach is to design a dynamic payoff mechanism capable of shaping the population's strategy profile, thus affecting the ES's state, by offering incentives for specific strategies within budget limits. Employing system-theoretic passivity concepts, we establish conditions under which a payoff mechanism can be systematically constructed to ensure the global asymptotic stabilization of the ES's equilibrium. In comparison to previous approaches originally studied in the context of the so-called epidemic population games, the method proposed here allows for more realistic epidemic models and other types of ES, such as predator-prey dynamics. Stabilization is established with the support of a Lyapunov function, which provides useful bounds on the transients.
- [201] arXiv:2403.18166 (cross-list from eess.SY) [pdf, other]
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Title: Incentive-Compatible Vertiport Reservation in Advanced Air Mobility: An Auction-Based ApproachComments: 26 pages, 2 figures, 1 tableSubjects: Systems and Control (eess.SY); Multiagent Systems (cs.MA); Theoretical Economics (econ.TH); Optimization and Control (math.OC)
The rise of advanced air mobility (AAM) is expected to become a multibillion-dollar industry in the near future. Market-based mechanisms are touted to be an integral part of AAM operations, which comprise heterogeneous operators with private valuations. In this work, we study the problem of designing a mechanism to coordinate the movement of electric vertical take-off and landing (eVTOL) aircraft, operated by multiple operators each having heterogeneous valuations associated with their fleet, between vertiports, while enforcing the arrival, departure, and parking constraints at vertiports. Particularly, we propose an incentive-compatible and individually rational vertiport reservation mechanism that maximizes a social welfare metric, which encapsulates the objective of maximizing the overall valuations of all operators while minimizing the congestion at vertiports. Additionally, we improve the computational tractability of designing the reservation mechanism by proposing a mixed binary linear programming approach that is based on constructing network flow graph corresponding to the underlying problem.
- [202] arXiv:2403.18176 (cross-list from cs.LG) [pdf, other]
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Title: Mistake, Manipulation and Margin Guarantees in Online Strategic ClassificationSubjects: Machine Learning (cs.LG); Computer Science and Game Theory (cs.GT); Optimization and Control (math.OC)
We consider an online strategic classification problem where each arriving agent can manipulate their true feature vector to obtain a positive predicted label, while incurring a cost that depends on the amount of manipulation. The learner seeks to predict the agent's true label given access to only the manipulated features. After the learner releases their prediction, the agent's true label is revealed. Previous algorithms such as the strategic perceptron guarantee finitely many mistakes under a margin assumption on agents' true feature vectors. However, these are not guaranteed to encourage agents to be truthful. Promoting truthfulness is intimately linked to obtaining adequate margin on the predictions, thus we provide two new algorithms aimed at recovering the maximum margin classifier in the presence of strategic agent behavior. We prove convergence, finite mistake and finite manipulation guarantees for a variety of agent cost structures. We also provide generalized versions of the strategic perceptron with mistake guarantees for different costs. Our numerical study on real and synthetic data demonstrates that the new algorithms outperform previous ones in terms of margin, number of manipulation and number of mistakes.
- [203] arXiv:2403.18181 (cross-list from cs.LG) [pdf, ps, other]
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Title: Compression of the Koopman matrix for nonlinear physical models via hierarchical clusteringComments: 9 pages, 10 figuresSubjects: Machine Learning (cs.LG); Dynamical Systems (math.DS)
Machine learning methods allow the prediction of nonlinear dynamical systems from data alone. The Koopman operator is one of them, which enables us to employ linear analysis for nonlinear dynamical systems. The linear characteristics of the Koopman operator are hopeful to understand the nonlinear dynamics and perform rapid predictions. The extended dynamic mode decomposition (EDMD) is one of the methods to approximate the Koopman operator as a finite-dimensional matrix. In this work, we propose a method to compress the Koopman matrix using hierarchical clustering. Numerical demonstrations for the cart-pole model and comparisons with the conventional singular value decomposition (SVD) are shown; the results indicate that the hierarchical clustering performs better than the naive SVD compressions.
- [204] arXiv:2403.18184 (cross-list from physics.app-ph) [pdf, other]
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Title: Topology Optimization for the Full-Cell Design of Porous Electrodes in Electrochemical Energy Storage DevicesAuthors: Hanyu Li, Giovanna Bucci, Nicholas W. Brady, Nicholas R. Cross, Victoria M. Ehlinger, Tiras Y. Lin, Miguel Salazar de Troya, Daniel Tortorelli, Marcus A. Worsley, Thomas RoySubjects: Applied Physics (physics.app-ph); Optimization and Control (math.OC)
In this paper, we introduce a density-based topology optimization framework to design porous electrodes for maximum energy storage. We simulate the full cell with a model that incorporates electronic potential, ionic potential, and electrolyte concentration. The system consists of three materials, namely pure liquid electrolyte and the porous solids of the anode and cathode, for which we determine the optimal placement. We use separate electronic potentials to model each electrode, which allow interdigitated designs. As the result, a penalization is required to ensure that the anode and cathode do not touch, i.e. causing a short circuit. We compare multiple 2D designs generated for different fixed conditions, e.g. material properties. A 3D design with complex channel and interlocking structure is also created. All optimized designs are far superior to the traditional monolithic electrode design with respect to energy storage metrics. We observe up to 750% increase in energy storage for cases with slow effective ionic diffusion within the porous electrode.
- [205] arXiv:2403.18216 (cross-list from stat.ML) [pdf, other]
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Title: Minimax Optimal Fair Classification with Bounded Demographic DisparitySubjects: Machine Learning (stat.ML); Computers and Society (cs.CY); Machine Learning (cs.LG); Statistics Theory (math.ST)
Mitigating the disparate impact of statistical machine learning methods is crucial for ensuring fairness. While extensive research aims to reduce disparity, the effect of using a \emph{finite dataset} -- as opposed to the entire population -- remains unclear. This paper explores the statistical foundations of fair binary classification with two protected groups, focusing on controlling demographic disparity, defined as the difference in acceptance rates between the groups. Although fairness may come at the cost of accuracy even with infinite data, we show that using a finite sample incurs additional costs due to the need to estimate group-specific acceptance thresholds. We study the minimax optimal classification error while constraining demographic disparity to a user-specified threshold. To quantify the impact of fairness constraints, we introduce a novel measure called \emph{fairness-aware excess risk} and derive a minimax lower bound on this measure that all classifiers must satisfy. Furthermore, we propose FairBayes-DDP+, a group-wise thresholding method with an offset that we show attains the minimax lower bound. Our lower bound proofs involve several innovations. Experiments support that FairBayes-DDP+ controls disparity at the user-specified level, while being faster and having a more favorable fairness-accuracy tradeoff than several baselines.
- [206] arXiv:2403.18235 (cross-list from eess.SY) [pdf, other]
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Title: An Execution-time-certified QP Algorithm for $\ell_1$ penalty-based Soft-constrained MPCComments: 6 pagesSubjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
Providing an execution time certificate and handling possible infeasibility in closed-loop are two pressing requirements of Model Predictive Control (MPC). To simultaneously meet these two requirements, this paper uses $\ell_1$ penalty-based soft-constrained MPC formulation and innovatively transforms the resulting non-smooth QP into a box-constrained QP, which is solved by our previously proposed direct and execution-time certified algorithm with only dimension-dependent (data-independent) and exact number of iterations [1]. This approach not only overcomes the limitation of our previously proposed algorithm [1], only applicable to input-constrained MPC, but also enjoys exact recovery feature (exactly recover the same solution when the original problem is feasible) of $\ell_1$ penalty-based soft-constrained MPC formulation without suffering numerical difficulty of the resulting non-smoothness. Other various real-time QP applications, not limited to MPC, will also benefit from our QP algorithm with execution-time certificate and global feasibility.
- [207] arXiv:2403.18242 (cross-list from hep-th) [pdf, other]
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Title: Flows in the Space of Interacting Chiral Boson TheoriesComments: 106 pages; LaTeXSubjects: High Energy Physics - Theory (hep-th); Strongly Correlated Electrons (cond-mat.str-el); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
We study interacting theories of $N$ left-moving and $\overline{N}$ right-moving Floreanini-Jackiw bosons in two dimensions. A parameterized family of such theories is shown to enjoy (non-manifest) Lorentz invariance if and only if its Lagrangian obeys a flow equation driven by a function of the energy-momentum tensor. We discuss the canonical quantization of such theories along classical stress tensor flows, focusing on the case of the root-$T \overline{T}$ deformation, where we obtain perturbative results for the deformed spectrum in a certain large-momentum limit. In the special case $N = \overline{N}$, we consider the quantum effective action for the root-$T \overline{T}$-deformed theory by expanding around a general classical background, and we find that the one-loop contribution vanishes for backgrounds with constant scalar gradients. Our analysis can also be interpreted via dual $U(1)$ Chern-Simons theories in three dimensions, which might be used to describe deformations of charged $\mathrm{AdS}_3$ black holes or quantum Hall systems.
- [208] arXiv:2403.18247 (cross-list from cs.CR) [pdf, other]
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Title: An Experimentally Validated Feasible Quantum Protocol for Identity-Based Signature with Application to Secure Email CommunicationAuthors: Tapaswini Mohanty, Vikas Srivastava, Sumit Kumar Debnath, Debasish Roy, Kouichi Sakurai, Sourav MukhopadhyaySubjects: Cryptography and Security (cs.CR); Information Theory (cs.IT)
Digital signatures are one of the simplest cryptographic building blocks that provide appealing security characteristics such as authenticity, unforgeability, and undeniability. In 1984, Shamir developed the first Identity-based signature (IBS) to simplify public key infrastructure and circumvent the need for certificates. It makes the process uncomplicated by enabling users to verify digital signatures using only the identifiers of signers, such as email, phone number, etc. Nearly all existing IBS protocols rely on several theoretical assumption-based hard problems. Unfortunately, these hard problems are unsafe and pose a hazard in the quantum realm. Thus, designing IBS algorithms that can withstand quantum attacks and ensure long-term security is an important direction for future research. Quantum cryptography (QC) is one such approach. In this paper, we propose an IBS based on QC. Our scheme's security is based on the laws of quantum mechanics. It thereby achieves long-term security and provides resistance against quantum attacks. We verify the proposed design's correctness and feasibility by simulating it in a prototype quantum device and the IBM Qiskit quantum simulator. The implementation code in qiskit with Jupyternotebook is provided in the Annexure. Moreover, we discuss the application of our design in secure email communication.
- [209] arXiv:2403.18264 (cross-list from hep-th) [pdf, other]
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Title: Analytic Approach for Computation of Topological Number of Integrable Vortex on TorusComments: 29 pages, 9 figuresSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
An analytic method to calculate the vortex number on a torus is constructed, focusing on analytic vortex solutions to the Chern-Simons-Higgs theory, whose governing equation is the so-called Jackiw-Pi equation. The equation is one of the integrable vortex equations and is reduced to Liouville's equation. The requirement of continuity of the Higgs field strongly restricts the characteristics and the fundamental domain of the vortices. Also considered are the decompactification limits of the vortices on a torus, in which "flux loss" phenomena occasionally occur.
- [210] arXiv:2403.18269 (cross-list from stat.ML) [pdf, other]
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Title: Clustering Change Sign Detection by Fusing Mixture ComplexityComments: 23 pagesSubjects: Machine Learning (stat.ML); Information Theory (cs.IT); Machine Learning (cs.LG)
This paper proposes an early detection method for cluster structural changes. Cluster structure refers to discrete structural characteristics, such as the number of clusters, when data are represented using finite mixture models, such as Gaussian mixture models. We focused on scenarios in which the cluster structure gradually changed over time. For finite mixture models, the concept of mixture complexity (MC) measures the continuous cluster size by considering the cluster proportion bias and overlap between clusters. In this paper, we propose MC fusion as an extension of MC to handle situations in which multiple mixture numbers are possible in a finite mixture model. By incorporating the fusion of multiple models, our approach accurately captured the cluster structure during transitional periods of gradual change. Moreover, we introduce a method for detecting changes in the cluster structure by examining the transition of MC fusion. We demonstrate the effectiveness of our method through empirical analysis using both artificial and real-world datasets.
- [211] arXiv:2403.18310 (cross-list from cs.LG) [pdf, ps, other]
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Title: A thermodynamically consistent physics-informed deep learning material model for short fiber/polymer nanocompositesComments: arXiv admin note: text overlap with arXiv:2305.08102Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
This work proposes a physics-informed deep learning (PIDL)-based constitutive model for investigating the viscoelastic-viscoplastic behavior of short fiber-reinforced nanoparticle-filled epoxies under various ambient conditions. The deep-learning model is trained to enforce thermodynamic principles, leading to a thermodynamically consistent constitutive model. To accomplish this, a long short-term memory network is combined with a feed-forward neural network to predict internal variables required for characterizing the internal dissipation of the nanocomposite materials. In addition, another feed-forward neural network is used to indicate the free-energy function, which enables defining the thermodynamic state of the entire system. The PIDL model is initially developed for the three-dimensional case by generating synthetic data from a classical constitutive model. The model is then trained by extracting the data directly from cyclic loading-unloading experimental tests. Numerical examples show that the PIDL model can accurately predict the mechanical behavior of epoxy-based nanocomposites for different volume fractions of fibers and nanoparticles under various hygrothermal conditions.
- [212] arXiv:2403.18398 (cross-list from eess.SY) [pdf, ps, other]
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Title: Adaptive Economic Model Predictive Control for linear systems with performance guaranteesComments: 8 pages, 3 figures, submitted to IEEE CDC 2024Subjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
We present a model predictive control (MPC) formulation to directly optimize economic criteria for linear constrained systems subject to disturbances and uncertain model parameters. The proposed formulation combines a certainty equivalent economic MPC with a simple least-squares parameter adaptation. For the resulting adaptive economic MPC scheme, we derive strong asymptotic and transient performance guarantees. We provide a numerical example involving building temperature control and demonstrate performance benefits of online parameter adaptation.
- [213] arXiv:2403.18415 (cross-list from cs.LG) [pdf, ps, other]
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Title: The Topos of Transformer NetworksSubjects: Machine Learning (cs.LG); Category Theory (math.CT)
The transformer neural network has significantly out-shined all other neural network architectures as the engine behind large language models. We provide a theoretical analysis of the expressivity of the transformer architecture through the lens of topos theory. From this viewpoint, we show that many common neural network architectures, such as the convolutional, recurrent and graph convolutional networks, can be embedded in a pretopos of piecewise-linear functions, but that the transformer necessarily lives in its topos completion. In particular, this suggests that the two network families instantiate different fragments of logic: the former are first order, whereas transformers are higher-order reasoners. Furthermore, we draw parallels with architecture search and gradient descent, integrating our analysis in the framework of cybernetic agents.
- [214] arXiv:2403.18457 (cross-list from cond-mat.stat-mech) [pdf, ps, other]
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Title: Specificity of $τ$ -- approximation for chaotic electron trajectories on complex Fermi surfacesAuthors: A.Ya. MaltsevComments: 11 pages, 5 figures, revtexSubjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
The work examines a special behavior of the magnetic conductivity of metals that arises when chaotic electron trajectories appear on the Fermi surface. This behavior is due to the scattering of electrons at singular points of the dynamic system describing the dynamics of electrons in $\, {\bf p}$-space, and caused by small-angle scattering of electrons on phonons. In this situation, the electronic system is described by a "non-standard" relaxation time, which plays the main role in a certain range of temperature and magnetic field values.
- [215] arXiv:2403.18466 (cross-list from physics.flu-dyn) [pdf, other]
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Title: Kinetic data-driven approach to turbulence subgrid modelingSubjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph)
Numerical simulations of turbulent flows are well known to pose extreme computational challenges due to the huge number of dynamical degrees of freedom required to correctly describe the complex multi-scale statistical correlations of the velocity. On the other hand, kinetic mesoscale approaches based on the Boltzmann equation, have the potential to describe a broad range of flows, stretching well beyond the special case of gases close to equilibrium, which results in the ordinary Navier-Stokes dynamics. Here we demonstrate that, by properly tuning, a kinetic approach can statistically reproduce the quantitative dynamics of the larger scales in turbulence, thereby providing an alternative, computationally efficient and physically rooted approach towards subgrid scale (SGS) modeling in turbulence. More specifically we show that by leveraging on data from fully resolved Direct Numerical Simulation (DNS) data we can learn a collision operator for the discretized Boltzmann equation solver (the lattice Boltzmann method), which effectively implies a turbulence subgrid closure model. The mesoscopic nature of our formulation makes the learning problem fully local in both space and time, leading to reduced computational costs and enhanced generalization capabilities. We show that the model offers superior performance compared to traditional methods, such as the Smagorinsky model, being less dissipative and, therefore, being able to more closely capture the intermittency of higher-order velocity correlations.
- [216] arXiv:2403.18517 (cross-list from cs.LG) [pdf, other]
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Title: Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation ModelsSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA); Optimization and Control (math.OC)
Regularized nonnegative low-rank approximations such as sparse Nonnegative Matrix Factorization or sparse Nonnegative Tucker Decomposition are an important branch of dimensionality reduction models with enhanced interpretability. However, from a practical perspective, the choice of regularizers and regularization coefficients, as well as the design of efficient algorithms, is challenging because of the multifactor nature of these models and the lack of theory to back these choices. This paper aims at improving upon these issues. By studying a more general model called the Homogeneous Regularized Scale-Invariant, we prove that the scale-invariance inherent to low-rank approximation models causes an implicit regularization with both unexpected beneficial and detrimental effects. This observation allows to better understand the effect of regularization functions in low-rank approximation models, to guide the choice of the regularization hyperparameters, and to design balancing strategies to enhance the convergence speed of dedicated optimization algorithms. Some of these results were already known but restricted to specific instances of regularized low-rank approximations. We also derive a generic Majorization Minimization algorithm that handles many regularized nonnegative low-rank approximations, with convergence guarantees. We showcase our contributions on sparse Nonnegative Matrix Factorization, ridge-regularized Canonical Polyadic decomposition and sparse Nonnegative Tucker Decomposition.
- [217] arXiv:2403.18571 (cross-list from eess.SY) [pdf, ps, other]
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Title: Bootstrapping Guarantees: Stability and Performance Analysis for Dynamic Encrypted ControlSubjects: Systems and Control (eess.SY); Cryptography and Security (cs.CR); Optimization and Control (math.OC)
Encrypted dynamic controllers that operate for an unlimited time have been a challenging subject of research. The fundamental difficulty is the accumulation of errors and scaling factors in the internal state during operation. Bootstrapping, a technique commonly employed in fully homomorphic cryptosystems, can be used to avoid overflows in the controller state but can potentially introduce significant numerical errors. In this paper, we analyze dynamic encrypted control with explicit consideration of bootstrapping. By recognizing the bootstrapping errors occurring in the controller's state as an uncertainty in the robust control framework, we can provide stability and performance guarantees for the whole encrypted control system. Further, the conservatism of the stability and performance test is reduced by using a lifted version of the control system.
- [218] arXiv:2403.18573 (cross-list from quant-ph) [pdf, ps, other]
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Title: Classifying symmetric and symmetry-broken spin chain phases with anomalous group actionsSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
We consider the classification problem of quantum spin chains invariant under local decomposable group actions, covering matrix product unitaries (MPUs), using an operator algebraic approach. We focus on finite group symmetries hosting both symmetric and symmetry broken phases. The local-decomposable group actions we consider have a 3-cocycle class of the symmetry group associated to them. We derive invariants for our classification that naturally cover one-dimensional symmetry protected topological (SPT) phases. We prove that these invariants coincide with the ones of [J. Garre Rubio et al, Quantum 7, 927 (2023)] using matrix product states (MPSs) techniques, by explicitly working out the GNS representation of MPSs and MPUs, resulting in a useful dictionary between both approaches that could be of independent interest.
- [219] arXiv:2403.18705 (cross-list from cs.LG) [pdf, other]
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Title: Conditional Wasserstein Distances with Applications in Bayesian OT Flow MatchingComments: This paper supersedes arXiv:2310.13433Subjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
In inverse problems, many conditional generative models approximate the posterior measure by minimizing a distance between the joint measure and its learned approximation. While this approach also controls the distance between the posterior measures in the case of the Kullback--Leibler divergence, this is in general not hold true for the Wasserstein distance. In this paper, we introduce a conditional Wasserstein distance via a set of restricted couplings that equals the expected Wasserstein distance of the posteriors. Interestingly, the dual formulation of the conditional Wasserstein-1 flow resembles losses in the conditional Wasserstein GAN literature in a quite natural way. We derive theoretical properties of the conditional Wasserstein distance, characterize the corresponding geodesics and velocity fields as well as the flow ODEs. Subsequently, we propose to approximate the velocity fields by relaxing the conditional Wasserstein distance. Based on this, we propose an extension of OT Flow Matching for solving Bayesian inverse problems and demonstrate its numerical advantages on an inverse problem and class-conditional image generation.
- [220] arXiv:2403.18735 (cross-list from cs.LG) [pdf, other]
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Title: Nonlinear model reduction for operator learningComments: Published as a Tiny Paper at ICLR 2024 (Notable)Subjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
Operator learning provides methods to approximate mappings between infinite-dimensional function spaces. Deep operator networks (DeepONets) are a notable architecture in this field. Recently, an extension of DeepONet based on model reduction and neural networks, proper orthogonal decomposition (POD)-DeepONet, has been able to outperform other architectures in terms of accuracy for several benchmark tests. We extend this idea towards nonlinear model order reduction by proposing an efficient framework that combines neural networks with kernel principal component analysis (KPCA) for operator learning. Our results demonstrate the superior performance of KPCA-DeepONet over POD-DeepONet.
- [221] arXiv:2403.18750 (cross-list from cond-mat.stat-mech) [pdf, other]
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Title: Full counting statistics of 1d short-range Riesz gases in confinementAuthors: Jitendra Kethepalli, Manas Kulkarni, Anupam Kundu, Satya N. Majumdar, David Mukamel, Grégory SchehrComments: 36 pages, 7 figuresSubjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
We investigate the full counting statistics (FCS) of a harmonically confined 1d short-range Riesz gas consisting of $N$ particles in equilibrium at finite temperature. The particles interact with each other through a repulsive power-law interaction with an exponent $k>1$ which includes the Calogero-Moser model for $k=2$. We examine the probability distribution of the number of particles in a finite domain $[-W, W]$ called number distribution, denoted by $\mathcal{N}(W, N)$. We analyze the probability distribution of $\mathcal{N}(W, N)$ and show that it exhibits a large deviation form for large $N$ characterised by a speed $N^{\frac{3k+2}{k+2}}$ and by a large deviation function of the fraction $c = \mathcal{N}(W, N)/N$ of the particles inside the domain and $W$. We show that the density profiles that create the large deviations display interesting shape transitions as one varies $c$ and $W$. This is manifested by a third-order phase transition exhibited by the large deviation function that has discontinuous third derivatives. Monte-Carlo (MC) simulations show good agreement with our analytical expressions for the corresponding density profiles. We find that the typical fluctuations of $\mathcal{N}(W, N)$, obtained from our field theoretic calculations are Gaussian distributed with a variance that scales as $N^{\nu_k}$, with $\nu_k = (2-k)/(2+k)$. We also present some numerical findings on the mean and the variance. Furthermore, we adapt our formalism to study the index distribution (where the domain is semi-infinite $(-\infty, W])$, linear statistics (the variance), thermodynamic pressure and bulk modulus.
- [222] arXiv:2403.18787 (cross-list from cond-mat.stat-mech) [pdf, other]
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Title: Universality classes for percolation models with long-range correlationsComments: 6 pages, 4 figuresSubjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
We consider a class of percolation models where the local occupation variables have long-range correlations decaying as a power law $\sim r^{-a}$ at large distances $r$, for some $0< a< d$ where $d$ is the underlying spatial dimension. For several of these models, we present both, rigorous analytical results and matching simulations that determine the critical exponents characterizing the fixed point associated to their phase transition, which is of second order. The exact values we obtain are rational functions of the two parameters $a$ and $d$ alone, and do not depend on the specifics of the model.
Replacements for Thu, 28 Mar 24
- [223] arXiv:0805.2295 (replaced) [pdf, ps, other]
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Title: On the length of lemniscatesJournal-ref: Michigan Math. J., 46 (1999), 409-415Subjects: Complex Variables (math.CV)
- [224] arXiv:1212.6092 (replaced) [src]
- [225] arXiv:1709.05504 (replaced) [pdf, other]
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Title: A priori bounds for geodesic diameter. Part III. A Sobolev-Poincaré inequality and applications to a variety of geometric variational problemsComments: 46 pages, no figures. This is the final paper of a series of three papers which supersedes version 1 (35 pages). In version 2, we added applications to a variety of geometric variational problems (including several formulations of Plateau's problem)Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)
- [226] arXiv:1904.07184 (replaced) [pdf, ps, other]
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Title: A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theoremComments: 33 pagesSubjects: Probability (math.PR); Numerical Analysis (math.NA)
- [227] arXiv:2005.11533 (replaced) [pdf, ps, other]
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Title: Arakelov class groups of random number fieldsComments: 22 pages; minor expository changes; to appear in Math. AnnSubjects: Number Theory (math.NT)
- [228] arXiv:2008.09973 (replaced) [pdf, ps, other]
- [229] arXiv:2010.00335 (replaced) [pdf, ps, other]
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Title: Cohomology and Deformations of left-symmetric Rinehart AlgebrasComments: To appear in Communications in MathematicsSubjects: Rings and Algebras (math.RA); Representation Theory (math.RT)
- [230] arXiv:2103.11039 (replaced) [pdf, ps, other]
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Title: A priori bounds for quasi-linear SPDEs in the full sub-critical regimeComments: 38 pages, submitted. Substantially revised: improved main result, simplified proofsSubjects: Analysis of PDEs (math.AP); Probability (math.PR)
- [231] arXiv:2104.14413 (replaced) [pdf, ps, other]
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Title: The Chern class for $K_3$ and the cyclic quantum dilogarithmAuthors: Kevin HutchinsonComments: 8 pages. Extensively rewritten for greater clarity of the main argument Now revised in line with referee comments. To appear in Journal of AlgebraSubjects: K-Theory and Homology (math.KT)
- [232] arXiv:2109.00970 (replaced) [pdf, ps, other]
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Title: A Construction of 2-D Z-Complementary Array Code Sets with Flexible Even Row Lengths and Applications in Massive MIMOSubjects: Information Theory (cs.IT)
- [233] arXiv:2112.01352 (replaced) [pdf, ps, other]
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Title: Boundedness of elliptic Calabi-Yau threefoldsComments: Final version, to appear in J. Eur. Math. Soc. (JEMS)Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th)
- [234] arXiv:2201.06180 (replaced) [pdf, other]
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Title: Nonlinear Control Allocation: A Learning Based ApproachComments: submitted to IEEE Conference on Decision and Control (CDC), 2024Subjects: Systems and Control (eess.SY); Artificial Intelligence (cs.AI); Optimization and Control (math.OC)
- [235] arXiv:2203.07084 (replaced) [pdf, ps, other]
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Title: Projective dimension and Castelnuovo-Mumford regularity of t-spread idealsComments: 26 pagesSubjects: Commutative Algebra (math.AC)
- [236] arXiv:2206.03975 (replaced) [pdf, other]
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Title: Functional linear and single-index models: A unified approach via Gaussian Stein identityComments: To appear in Bernoulli JournalSubjects: Statistics Theory (math.ST)
- [237] arXiv:2206.14159 (replaced) [pdf, ps, other]
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Title: Thinness of some hypergeometric groups in Sp(6)Comments: Final version; accepted for publication in Glasgow Mathematical JournalSubjects: Group Theory (math.GR)
- [238] arXiv:2208.02767 (replaced) [pdf, other]
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Title: Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integrationSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
- [239] arXiv:2209.08865 (replaced) [pdf, ps, other]
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Title: Subregular nilpotent orbits and explicit character formulas for modules over affine Lie algebrasComments: 42 pages, v2: few typos corrected, v3: extended the proof of lemma 4.7Journal-ref: Pure and Applied Mathematics Quarterly Volume 20, No. 1 (2024) Pure and Applied Mathematics Quarterly, Volume 20, No. 1 (2024) (Special Issue dedicated to Corrado De Concini)Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG)
- [240] arXiv:2210.11634 (replaced) [pdf, other]
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Title: A Polynomial-time Algorithm for the Large Scale of Airplane Refueling ProblemComments: 18 pages, 2 figuresSubjects: Data Structures and Algorithms (cs.DS); Optimization and Control (math.OC)
- [241] arXiv:2210.14726 (replaced) [pdf, ps, other]
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Title: Hofer geometry via toric degenerationAuthors: Yusuke KawamotoComments: v2: added results for del Pezzo surfaces, 36 pages; v3: final version, to appear in Mathematische AnnalenSubjects: Symplectic Geometry (math.SG)
- [242] arXiv:2212.09838 (replaced) [pdf, ps, other]
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Title: Two-species chemotaxis-competition system with singular sensitivity: Global existence, boundedness, and persistenceSubjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS)
- [243] arXiv:2301.04262 (replaced) [pdf, ps, other]
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Title: Rational singularities and $q$-birational morphismAuthors: Donghyeon KimComments: Comments Welcome!Subjects: Algebraic Geometry (math.AG)
- [244] arXiv:2301.05561 (replaced) [pdf, ps, other]
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Title: Lacunary sequences in analysis, probability and number theoryComments: 58 pages. This is mostly a survey paper. The final section contains new results (with proofs). Version 2: minor corrections and updates. Version 3: corrected an error on p.8 (thanks to Jean-Francois Burnol for pointing out the error)Subjects: Number Theory (math.NT); Classical Analysis and ODEs (math.CA); Probability (math.PR)
- [245] arXiv:2301.13168 (replaced) [pdf, other]
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Title: The noncommutative minimal model programAuthors: Daniel Halpern-LeistnerComments: 33 pages, 2 figuresSubjects: Algebraic Geometry (math.AG)
- [246] arXiv:2302.01421 (replaced) [pdf, other]
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Title: Follower Agnostic Methods for Stackelberg GamesComments: 31 pagesSubjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Computer Science and Game Theory (cs.GT); Dynamical Systems (math.DS)
- [247] arXiv:2302.08124 (replaced) [pdf, other]
- [248] arXiv:2302.08193 (replaced) [pdf, ps, other]
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Title: Compatible E-differential forms on Lie algebroids over (pre-)multisymplectic manifoldsAuthors: Noriaki IkedaComments: 31 pagesSubjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Symplectic Geometry (math.SG)
- [249] arXiv:2302.14492 (replaced) [pdf, ps, other]
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Title: On covering dimension and sections of vector bundlesAuthors: M. C. CrabbSubjects: Algebraic Topology (math.AT)
- [250] arXiv:2303.03489 (replaced) [pdf, ps, other]
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Title: Large time behavior for the 3D Navier-Stokes with Navier boundary conditionsAuthors: James P. Kelliher, Christophe Lacave, Milton C. Lopes Filho, Helena J. Nussenzveig Lopes, Edriss S. TitiSubjects: Analysis of PDEs (math.AP)
- [251] arXiv:2303.12213 (replaced) [pdf, other]
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Title: Alpha shapes in kernel density estimationComments: 28 pagesSubjects: Algebraic Topology (math.AT)
- [252] arXiv:2303.15997 (replaced) [pdf, ps, other]
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Title: The Burnside problem for odd exponentsComments: Small editorial changes, 58ppSubjects: Group Theory (math.GR)
- [253] arXiv:2303.16335 (replaced) [pdf, other]
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Title: Boundary current fluctuations for the half space ASEP and six vertex modelAuthors: Jimmy HeComments: v3: final version. 60 pages, many figures. Comments are welcome!Subjects: Probability (math.PR); Mathematical Physics (math-ph); Combinatorics (math.CO)
- [254] arXiv:2304.01847 (replaced) [pdf, ps, other]
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Title: Isolated hypersurface singularities, spectral invariants, and quantum cohomologyAuthors: Yusuke KawamotoComments: 43 pages; v2: final version, to appear in Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal)Subjects: Symplectic Geometry (math.SG); Algebraic Geometry (math.AG)
- [255] arXiv:2304.09155 (replaced) [pdf, other]
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Title: Rainbow Hamiltonicity in uniformly coloured perturbed digraphsComments: Incorporated referee's comments. Accepted for publication in Combinatorics, Probability and ComputingSubjects: Combinatorics (math.CO)
- [256] arXiv:2305.01716 (replaced) [pdf, other]
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Title: The Pseudoinverse of $A=CR$ is $A^+=R^+C^+$ (?)Comments: 10 pages, 5 figures, matlab code, new paragraphs introduce general formulas for the pseudoinverse of CR, new Figures and the randomized pseudoinverse algorithmSubjects: Numerical Analysis (math.NA)
- [257] arXiv:2305.12523 (replaced) [pdf, ps, other]
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Title: Multi-Static Target Detection and Power Allocation for Integrated Sensing and Communication in Cell-Free Massive MIMOComments: 16 pages, 7 figuresSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
- [258] arXiv:2305.13265 (replaced) [pdf, ps, other]
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Title: Semi-galois Categories IV: A deformed reciprocity law for Siegel modular functionsAuthors: Takeo UramotoComments: made minor correction in Remark 4.2.7; preprint; 24 pagesSubjects: Number Theory (math.NT)
- [259] arXiv:2305.17294 (replaced) [pdf, other]
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Title: A boundary integral equation method for the complete electrode model in electrical impedance tomography with tests on experimental dataComments: 27 pages. The published version linked to below is substantially expandedJournal-ref: SIAM Journal on Imaging Sciences, volume 17, number 1, 672--705, 2024Subjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
- [260] arXiv:2306.07357 (replaced) [pdf, ps, other]
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Title: Noise Sensitivity of the Minimum Spanning Tree of the Complete GraphSubjects: Probability (math.PR); Combinatorics (math.CO)
- [261] arXiv:2306.10594 (replaced) [pdf, other]
- [262] arXiv:2306.13829 (replaced) [pdf, other]
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Title: Selective inference using randomized group lasso estimators for general modelsComments: 64pages, 4 figures, 3 tablesSubjects: Methodology (stat.ME); Statistics Theory (math.ST); Machine Learning (stat.ML)
- [263] arXiv:2307.00975 (replaced) [pdf, other]
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Title: Fast Convergence of Inertial Multiobjective Gradient-like Systems with Asymptotic Vanishing DampingComments: 25 pages, 3 FiguresSubjects: Optimization and Control (math.OC)
- [264] arXiv:2307.01315 (replaced) [pdf, other]
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Title: A log-linear model for non-stationary time series of countsSubjects: Statistics Theory (math.ST)
- [265] arXiv:2307.06445 (replaced) [pdf, ps, other]
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Title: Small cap decoupling for the paraboloid in $\mathbb{R}^n$Comments: 17 pages, small corrections following referee reportSubjects: Classical Analysis and ODEs (math.CA)
- [266] arXiv:2307.16075 (replaced) [pdf, ps, other]
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Title: Redesigning Large-Scale Multimodal Transit Networks with Shared Autonomous Mobility ServicesComments: 48 pages, 18 figures, accepted for publication in Transportation Research Part C: Emerging Technologies, and presentation in the 25th International Symposium on Transportation and Traffic Theory (ISTTT25)Subjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
- [267] arXiv:2308.01642 (replaced) [pdf, ps, other]
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Title: Weak uniqueness by noise for singular stochastic PDEsComments: 39 pagesSubjects: Probability (math.PR); Analysis of PDEs (math.AP)
- [268] arXiv:2308.06822 (replaced) [pdf, other]
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Title: Approximate and Weighted Data Reconstruction Attack in Federated LearningSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Cryptography and Security (cs.CR); Optimization and Control (math.OC)
- [269] arXiv:2308.08080 (replaced) [pdf, ps, other]
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Title: Descent conditions for generation in derived categoriesAuthors: Pat LankComments: Current version: Pre-final for publication. Previous: Corrected typos and filled gaps, add addendum added for recent development, improved presentationJournal-ref: J. Pure Appl. Algebra (2024), 107671Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)
- [270] arXiv:2308.11637 (replaced) [pdf, other]
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Title: Classical values of Zeta, as simple as possible but not simplerAuthors: Olga HoltzComments: 10 page, 1 figureSubjects: History and Overview (math.HO); Combinatorics (math.CO); Complex Variables (math.CV); Number Theory (math.NT)
- [271] arXiv:2308.12407 (replaced) [pdf, other]
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Title: The secular equation for elastic surface waves under non standard boundary conditions of impedance type: A perspective from linear algebraAuthors: Fabio VallejoComments: In preparation for journal submissionSubjects: Mathematical Physics (math-ph)
- [272] arXiv:2309.02032 (replaced) [pdf, other]
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Title: A novel strong duality-based reformulation for trilevel infrastructure models in energy systems developmentComments: 22 pages, 4 figuresSubjects: Optimization and Control (math.OC)
- [273] arXiv:2309.04381 (replaced) [pdf, other]
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Title: Generalization Bounds: Perspectives from Information Theory and PAC-BayesComments: 228 pagesSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Information Theory (cs.IT); Statistics Theory (math.ST); Machine Learning (stat.ML)
- [274] arXiv:2309.09104 (replaced) [pdf, ps, other]
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Title: Characterization of Solubilizers of Elements in Minimal Simple GroupsSubjects: Group Theory (math.GR)
- [275] arXiv:2309.12198 (replaced) [pdf, ps, other]
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Title: On aspherical configuration Lie groupoidsAuthors: S K RoushonComments: 14p. arXiv admin note: substantial text overlap with arXiv:2106.08110Subjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
- [276] arXiv:2309.13771 (replaced) [pdf, ps, other]
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Title: Matching powers of monomial ideals and edge ideals of weighted oriented graphsComments: New version following the referees suggestionsSubjects: Commutative Algebra (math.AC); Combinatorics (math.CO)
- [277] arXiv:2310.03366 (replaced) [pdf, ps, other]
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Title: Elementary Properties of Free LatticesComments: 13 pagesSubjects: Logic (math.LO)
- [278] arXiv:2310.10350 (replaced) [pdf, ps, other]
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Title: On evolution PDEs on co-evolving graphsSubjects: Analysis of PDEs (math.AP)
- [279] arXiv:2310.10900 (replaced) [pdf, other]
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Title: Stability of Sequential Lateration and of Stress Minimization in the Presence of NoiseComments: arXiv admin note: substantial text overlap with arXiv:2207.07218Subjects: Statistics Theory (math.ST); Networking and Internet Architecture (cs.NI); Probability (math.PR)
- [280] arXiv:2310.11471 (replaced) [pdf, other]
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Title: Modeling lower-truncated and right-censored insurance claims with an extension of the MBBEFD classComments: 36 pagesSubjects: Methodology (stat.ME); Statistics Theory (math.ST); Applications (stat.AP)
- [281] arXiv:2310.14647 (replaced) [pdf, ps, other]
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Title: Indicated domination gameComments: 19 pagesSubjects: Combinatorics (math.CO)
- [282] arXiv:2310.17038 (replaced) [pdf, ps, other]
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Title: Weak reservoirs are superexponentially irrelevant for misanthrope processesAuthors: Julian KernSubjects: Probability (math.PR)
- [283] arXiv:2311.01346 (replaced) [pdf, other]
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Title: On the Proportion of Coprime Fractions in Number FieldsComments: 12 pages, 1 figure, 1 tableSubjects: Number Theory (math.NT)
- [284] arXiv:2311.06373 (replaced) [pdf, other]
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Title: Partial Information Decomposition for Continuous Variables based on Shared Exclusions: Analytical Formulation and EstimationAuthors: David A. Ehrlich, Kyle Schick-Poland, Abdullah Makkeh, Felix Lanfermann, Patricia Wollstadt, Michael WibralComments: 32 pages, 15 figuresSubjects: Information Theory (cs.IT); Probability (math.PR); Statistics Theory (math.ST); Computation (stat.CO)
- [285] arXiv:2311.08214 (replaced) [pdf, ps, other]
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Title: Frequentist Guarantees of Distributed (Non)-Bayesian InferenceSubjects: Statistics Theory (math.ST); Machine Learning (stat.ML)
- [286] arXiv:2311.13888 (replaced) [pdf, other]
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Title: On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation lawsAuthors: Hendrik Ranocha, Andrew R. Winters, Michael Schlottke-Lakemper, Philipp Öffner, Jan Glaubitz, Gregor J. GassnerSubjects: Numerical Analysis (math.NA)
- [287] arXiv:2312.07394 (replaced) [pdf, other]
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Title: On model predictive control with sampled-data input for output tracking with prescribed performanceSubjects: Optimization and Control (math.OC)
- [288] arXiv:2312.09605 (replaced) [pdf, ps, other]
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Title: Rigid Lid limit in shallow water over a flat bottomAuthors: Benjamin Melinand (CEREMADE)Subjects: Analysis of PDEs (math.AP)
- [289] arXiv:2312.12019 (replaced) [pdf, ps, other]
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Title: Separable algebras in multitensor C$^*$-categories are unitarizableComments: 14 pagesJournal-ref: AIMS Mathematics 9 (2024) 11320-11334. Special Issue Operator Theory: Advances and ApplicationsSubjects: Operator Algebras (math.OA); Category Theory (math.CT); Quantum Algebra (math.QA)
- [290] arXiv:2312.12558 (replaced) [pdf, other]
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Title: Sample Efficient Reinforcement Learning with Partial Dynamics KnowledgeComments: Published in the 38th Annual AAAI Conference on Artificial IntelligenceSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
- [291] arXiv:2312.14722 (replaced) [pdf, ps, other]
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Title: On strictly elliptic K3 surfaces and del Pezzo surfacesComments: v2: 21 pages. Corrections in Table 1. Comments are very welcome!Subjects: Algebraic Geometry (math.AG)
- [292] arXiv:2401.00079 (replaced) [pdf, ps, other]
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Title: Computable Scott sentences and the weak Whitehead problem for finitely presented groupsAuthors: Gianluca PaoliniSubjects: Logic (math.LO)
- [293] arXiv:2401.00600 (replaced) [pdf, ps, other]
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Title: Stability conditions and semiorthogonal decompositions I: quasi-convergenceComments: 41 pages, 2 figures, several typos correctedSubjects: Algebraic Geometry (math.AG)
- [294] arXiv:2401.02792 (replaced) [pdf, other]
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Title: The Origin of Calabi-Yau Crystals in BPS States CountingComments: 53 pages; v3 minor corrections, references addedJournal-ref: J. High Energ. Phys. 2024, 140 (2024)Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Combinatorics (math.CO)
- [295] arXiv:2401.03244 (replaced) [pdf, other]
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Title: Artificial Intelligence for Operations Research: Revolutionizing the Operations Research ProcessSubjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI)
- [296] arXiv:2401.04072 (replaced) [pdf, ps, other]
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Title: K3 surfaces with real or complex multiplicationComments: 32 pages; v2: major extension covering also higher-dimensional hyperk\"ahler manifolds and possible Picard latticesSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
- [297] arXiv:2401.06956 (replaced) [pdf, ps, other]
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Title: A note on Rational Functions with three branched points on the Riemann sphereComments: 8 pages. Comments are welcomeSubjects: Complex Variables (math.CV); Differential Geometry (math.DG)
- [298] arXiv:2401.13387 (replaced) [pdf, other]
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Title: A Mathematical Theory of Semantic CommunicationComments: (version 2.0 updated) 96 pages, 18 figures. This paper is submitted to IEEE Transactions on Information Theory (TIT)Subjects: Information Theory (cs.IT)
- [299] arXiv:2401.16063 (replaced) [pdf, ps, other]
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Title: Shannon Capacity of Channels with Markov Insertions, Deletions and SubstitutionsComments: 15 pages, 1 figureSubjects: Information Theory (cs.IT)
- [300] arXiv:2401.16927 (replaced) [pdf, ps, other]
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Title: $G$-complete reducibility and saturationComments: 15 pages; v2 minor changes; v3 18 pages, various changes; new is Proposition 4.8 which shows that saturation is compatible with standard Frobenius endomorphismsSubjects: Representation Theory (math.RT); Group Theory (math.GR)
- [301] arXiv:2401.17908 (replaced) [pdf, ps, other]
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Title: Duality of quantum geometriesAuthors: Jan NaudtsComments: v2: Real-valued metric tensor and connection coefficients, added section on curvature, minor corrections v3: added sections on the quantum exponential family and on the commutative limitSubjects: Mathematical Physics (math-ph); Differential Geometry (math.DG)
- [302] arXiv:2402.09829 (replaced) [pdf, ps, other]
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Title: On a conjecture on shifted primes with large prime factors, IIAuthors: Yuchen DingComments: The expositions on the history of this topic is updated in the new versionSubjects: Number Theory (math.NT)
- [303] arXiv:2402.11800 (replaced) [pdf, other]
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Title: Stochastic Approximation with Delayed Updates: Finite-Time Rates under Markovian SamplingAuthors: Arman Adibi, Nicolo Dal Fabbro, Luca Schenato, Sanjeev Kulkarni, H. Vincent Poor, George J. Pappas, Hamed Hassani, Aritra MitraComments: Accepted to the 27th International Conference on Artificial Intelligence and Statistics (AISTATS) 2024!Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Multiagent Systems (cs.MA); Systems and Control (eess.SY); Optimization and Control (math.OC)
- [304] arXiv:2402.13376 (replaced) [pdf, ps, other]
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Title: Probabilistic automatic complexity of finite stringsAuthors: Kenneth GillComments: 41 pages, 5 figures. This work extends Chapter 2 of the author's PhD dissertation at Penn State. V2: fix statement of Proposition 3.4Subjects: Formal Languages and Automata Theory (cs.FL); Logic (math.LO)
- [305] arXiv:2402.18501 (replaced) [pdf, ps, other]
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Title: Yangian of the periplectic Lie superalgebraAuthors: Maxim NazarovComments: Section 26 addedSubjects: Quantum Algebra (math.QA); Representation Theory (math.RT); Exactly Solvable and Integrable Systems (nlin.SI)
- [306] arXiv:2402.18983 (replaced) [pdf, other]
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Title: Free energy expansions of a conditional GinUE and large deviations of the smallest eigenvalue of the LUEComments: 40 pages, 7 figures; v2 41 pages, 7 figures, references and associated text addedSubjects: Mathematical Physics (math-ph); Complex Variables (math.CV); Probability (math.PR)
- [307] arXiv:2403.00465 (replaced) [pdf, other]
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Title: Polyamorous SchedulingComments: v2: stronger and simplified hardness-of-approximation results, corrected constant in layering approximation algorithmSubjects: Data Structures and Algorithms (cs.DS); Social and Information Networks (cs.SI); Optimization and Control (math.OC)
- [308] arXiv:2403.01280 (replaced) [pdf, ps, other]
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Title: Rigidity results for group von Neumann algebras with diffuse centerComments: After the first version was posted, Stefaan Vaes pointed out one of the equivalences in Theorem 4.5 does not hold as stated. In the current version we fixed this along with the proofs where it was used. This does not affect the main result or the substance of the arguments used. The only difference is that now Corollary D is slightly more restrictive. We also corrected a number of typosSubjects: Operator Algebras (math.OA)
- [309] arXiv:2403.01650 (replaced) [pdf, ps, other]
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Title: On the maximum principle for general linear elliptic equationsAuthors: Neil S. TrudingerComments: 12 pagesSubjects: Analysis of PDEs (math.AP)
- [310] arXiv:2403.03271 (replaced) [pdf, ps, other]
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Title: Low-Complexity Linear Decoupling of Users for Uplink Massive MU-MIMO DetectionSubjects: Signal Processing (eess.SP); Information Theory (cs.IT)
- [311] arXiv:2403.06629 (replaced) [pdf, other]
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Title: Assembly Theory is an approximation to algorithmic complexity based on LZ compression that does not explain selection or evolutionComments: 15 pages + appendix, 2 figuresSubjects: Information Theory (cs.IT); Biomolecules (q-bio.BM)
- [312] arXiv:2403.06633 (replaced) [pdf, other]
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Title: Fractal spatio-temporal scale-free messaging: amplitude modulation of self-executable carriers given by the Weierstrass function's componentsComments: 15 pages + appendix (21 pages total)Subjects: Information Theory (cs.IT)
- [313] arXiv:2403.06646 (replaced) [pdf, ps, other]
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Title: Unisolvence of random Kansa collocation by Thin-Plate Splines for the Poisson equationSubjects: Numerical Analysis (math.NA)
- [314] arXiv:2403.06708 (replaced) [pdf, ps, other]
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Title: Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert SpacesComments: 34 pages, 2 tables. arXiv admin note: text overlap with arXiv:2207.02750Subjects: Optimization and Control (math.OC)
- [315] arXiv:2403.07961 (replaced) [pdf, other]
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Title: The $L_p$-discrepancy for finite $p>1$ suffers from the curse of dimensionalityComments: arXiv admin note: substantial text overlap with arXiv:2303.01787Subjects: Numerical Analysis (math.NA)
- [316] arXiv:2403.08070 (replaced) [pdf, ps, other]
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Title: On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-LaplacianComments: 19 pages. Comments are welcome. Several typos have been corrected to v1. A new reference has been addedSubjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
- [317] arXiv:2403.08075 (replaced) [pdf, ps, other]
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Title: Several isoperimetric inequalities of Dirichlet and Neumann eigenvalues of the Witten-LaplacianComments: 31 pages. Comments are welcome. Several typos have been corrected to v1Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
- [318] arXiv:2403.08296 (replaced) [pdf, ps, other]
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Title: Quasinormability and property $(Ω)$ for spaces of smooth and ultradifferentiable vectors associated with Lie group representationsComments: 33 pagesSubjects: Functional Analysis (math.FA)
- [319] arXiv:2403.12452 (replaced) [src]
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Title: On the Salient Limitations of `On the Salient Limitations of the Methods of Assembly Theory and their Classification of Molecular Biosignatures'Authors: Leroy CroninComments: the arguments here are being discussed in more detail and will be ready laterSubjects: Information Theory (cs.IT); Adaptation and Self-Organizing Systems (nlin.AO); Biological Physics (physics.bio-ph)
- [320] arXiv:2403.14396 (replaced) [pdf, ps, other]
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Title: Infinite horizon McKean-Vlasov FBSDEs and applications to mean field control problemsSubjects: Optimization and Control (math.OC); Probability (math.PR)
- [321] arXiv:2403.14585 (replaced) [pdf, ps, other]
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Title: Correlations and ergodic averages for strongly and mildly mixing automorphismsAuthors: Valery V. RyzhikovComments: in Russian languageSubjects: Dynamical Systems (math.DS)
- [322] arXiv:2403.14761 (replaced) [pdf, ps, other]
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Title: Quantitative Steinitz theorem and polarityAuthors: Grigory IvanovSubjects: Metric Geometry (math.MG)
- [323] arXiv:2403.15201 (replaced) [pdf, other]
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Title: Flip-Breakability: A Combinatorial Dichotomy for Monadically Dependent Graph ClassesComments: v2: added section "Conclusions and Future Work"Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Logic in Computer Science (cs.LO); Logic (math.LO)
- [324] arXiv:2403.15578 (replaced) [pdf, ps, other]
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Title: Exact distance Kneser graphsSubjects: Combinatorics (math.CO)
- [325] arXiv:2403.15719 (replaced) [pdf, ps, other]
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Title: Harmonic Bundles with Symplectic StructuresAuthors: Takashi OnoComments: Comments are welcome! 23 pagesSubjects: Algebraic Geometry (math.AG)
- [326] arXiv:2403.16121 (replaced) [pdf, other]
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Title: Log-rank test with coarsened exact matchingSubjects: Statistics Theory (math.ST)
- [327] arXiv:2403.16255 (replaced) [pdf, other]
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Title: Phase retrieval on circles and linesComments: 13 pages, 1 figureSubjects: Complex Variables (math.CV); Functional Analysis (math.FA)
- [328] arXiv:2403.16290 (replaced) [pdf, other]
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Title: An Information Theoretic Treatment of Animal Movement TracksAuthors: Wayne M GetzComments: 20 pages, 2 tables, 1 figureSubjects: Populations and Evolution (q-bio.PE); Information Theory (cs.IT)
- [329] arXiv:2403.16828 (replaced) [pdf, other]
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Title: Asymptotics of predictive distributions driven by sample means and variancesSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
- [330] arXiv:2403.16975 (replaced) [pdf, other]
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Title: Unconditionally positivity-preserving approximations of the Ait-Sahalia type model: Explicit Milstein-type schemesComments: 19 pages, 3 figuresSubjects: Numerical Analysis (math.NA)
- [331] arXiv:2403.17490 (replaced) [pdf, ps, other]
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Title: Covariant reconstruction of forms from their invariantsAuthors: Thomas Bouchet (LJAD)Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
- [332] arXiv:2403.17579 (replaced) [pdf, ps, other]
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Title: Kurokawa-Mizumoto congruence and differential operators on automorphic formsAuthors: Nobuki TakedaSubjects: Number Theory (math.NT)
- [333] arXiv:2403.17850 (replaced) [pdf, ps, other]
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Title: A Mixed-Integer Linear Program to create the shifts in a supermarketSubjects: Optimization and Control (math.OC)
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