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Showing new listings for Thursday, 14 November 2024
- [1] arXiv:2411.08075 [pdf, html, other]
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Title: Input-to-State Stability of time-varying infinite-dimensional control systemsComments: 87 pages. arXiv admin note: substantial text overlap with arXiv:2302.00535 by other authorsSubjects: Optimization and Control (math.OC)
The concept of input-to-state stability (ISS) proposed in the late 1980s is one of the central notions in robust nonlinear control. ISS has become indispensable for various branches of nonlinear systems theory, such as robust stabilization of nonlinear systems, design of nonlinear observers, analysis of large-scale networks, etc. The success of the ISS theory of ODEs and the need for robust stability analysis of partial differential equations (PDEs) motivated the development of ISS theory in the infinite-dimensional setting. For instance, the Lyapunov method for analysis of iISS of nonlinear parabolic equations. For an overview of the ISS theory for distributed parameter systems. ISS of control systems with application to robust global stabilization of the chemostat has been studied with the help of vector Lyapunov functions.
- [2] arXiv:2411.08080 [pdf, html, other]
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Title: Jacobi convolution polynomial for Petrov-Galerkin scheme and general fractional calculus of arbitrary order over finite intervalSubjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
Recently, general fractional calculus was introduced by Kochubei (2011) and Luchko (2021) as a further generalisation of fractional calculus, where the derivative and integral operator admits arbitrary kernel. Such a formalism will have many applications in physics and engineering, since the kernel is no longer restricted. We first extend the work of Al-Refai and Luchko (2023) on finite interval to arbitrary orders. Followed by, developing an efficient Petrov-Galerkin scheme by introducing Jacobi convolution polynomials as basis functions. A notable property of this basis function, the general fractional derivative of Jacobi convolution polynomial is a shifted Jacobi polynomial. Thus, with a suitable test function it results in diagonal stiffness matrix, hence, the efficiency in implementation. Furthermore, our method is constructed for any arbitrary kernel including that of fractional operator, since, its a special case of general fractional operator.
- [3] arXiv:2411.08084 [pdf, html, other]
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Title: Application of Operator Theory for the Collatz ConjectureComments: 27 pagesSubjects: Operator Algebras (math.OA); Dynamical Systems (math.DS); Number Theory (math.NT)
In this paper, we formulate the Collatz conjecture (or the $3n{+}1$-problem) as some operator theoretic problems and prove that each of those statements is equivalent to the conjecture.
- [4] arXiv:2411.08086 [pdf, html, other]
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Title: Absolutely dilatable bimodule mapsComments: 30 pagesSubjects: Operator Algebras (math.OA); Functional Analysis (math.FA)
We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.
- [5] arXiv:2411.08137 [pdf, html, other]
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Title: New matrices for spectral hypergraph theory, IIComments: 41 pages, 7 figuresSubjects: Combinatorics (math.CO); Spectral Theory (math.SP)
The properties of a hypergraph explored through the spectrum of its unified matrix was made by the authors in [26]. In this paper, we introduce three different hypergraph matrices: unified Laplacian matrix, unified signless Laplacian matrix, and unified normalized Laplacian matrix, all defined using the unified matrix. We show that these three matrices of a hypergraph are respectively identical to the Laplacian matrix, signless Laplacian matrix, and normalized Laplacian matrix of the associated graph. This allows us to use the spectra of these hypergraph matrices as a means to connect the structural properties of the hypergraph with those of the associated graph. Additionally, we introduce certain hypergraph structures and invariants during this process, and relate them to the eigenvalues of these three matrices.
- [6] arXiv:2411.08139 [pdf, html, other]
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Title: Visualizing the Sum-Product ConjectureComments: 27 pages, many color imagesSubjects: Number Theory (math.NT); Combinatorics (math.CO)
Let $SPP(n)$ be the set $\{\big(|A+A|,|A\cdot A|\big) : A\subseteq {\mathbb N}, |A|=n \}$, where $A+A$ is the sumset $\{a+b : a,b\in A\}$ and $AA$ is the product set $\{ab : a,b\in A\}$. We prove the value of $SPP(n)$ exactly for $n\le 6$, and compute a large subset of $SPP(n)$ for $n\le 32$. This is a dataset of 1,158,717 sets of positive integers that are addtiively and multiplicatively diverse. We do {\bf not} see evidence in favor of Erdős's Sum-Product Conjecture in our dataset. We include a number of conjectures, open problems, and observations motivated by this dataset, and a large number of color visualizations.
- [7] arXiv:2411.08141 [pdf, other]
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Title: Probably approximately correct high-dimensional causal effect estimation given a valid adjustment setSubjects: Statistics Theory (math.ST); Machine Learning (stat.ML)
Accurate estimates of causal effects play a key role in decision-making across applications such as healthcare, economics, and operations. In the absence of randomized experiments, a common approach to estimating causal effects uses \textit{covariate adjustment}. In this paper, we study covariate adjustment for discrete distributions from the PAC learning perspective, assuming knowledge of a valid adjustment set $\bZ$, which might be high-dimensional. Our first main result PAC-bounds the estimation error of covariate adjustment by a term that is exponential in the size of the adjustment set; it is known that such a dependency is unavoidable even if one only aims to minimize the mean squared error. Motivated by this result, we introduce the notion of an \emph{$\eps$-Markov blanket}, give bounds on the misspecification error of using such a set for covariate adjustment, and provide an algorithm for $\eps$-Markov blanket discovery; our second main result upper bounds the sample complexity of this algorithm. Furthermore, we provide a misspecification error bound and a constraint-based algorithm that allow us to go beyond $\eps$-Markov blankets to even smaller adjustment sets. Our third main result upper bounds the sample complexity of this algorithm, and our final result combines the first three into an overall PAC bound. Altogether, our results highlight that one does not need to perfectly recover causal structure in order to ensure accurate estimates of causal effects.
- [8] arXiv:2411.08146 [pdf, html, other]
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Title: Semiclassical measure of the spherical harmonics by Bourgain on $\mathbb{S}^3$Comments: 11 pagesSubjects: Classical Analysis and ODEs (math.CA); Analysis of PDEs (math.AP); Spectral Theory (math.SP)
Bourgain used the Rudin-Shapiro sequences to construct a basis of uniformly bounded holomorphic functions on the unit sphere in $\mathbb{C}^2$. They are also spherical harmonics (i.e., Laplacian eigenfunctions) on $\mathbb{S}^3 \subset \mathbb{R}^4$. In this paper, we prove that these functions tend to be equidistributed on $\mathbb{S}^3$, based on an estimate of the auto-correlation of the Rudin-Shapiro sequences. Moreover, we identify the semiclassical measure associated to these spherical harmonics by the singular measure supported on the family of Clifford tori in $\mathbb{S}^3$. In particular, this demonstrates a new localization pattern in the study of Laplacian eigenfunctions.
- [9] arXiv:2411.08175 [pdf, html, other]
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Title: Well-posedness of a Variable-Exponent Telegraph Equation Applied to Image DespecklingComments: 33 pages, 19 figures, 3 tablesSubjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
In this paper, we present a telegraph diffusion model with variable exponents for image despeckling. Moving beyond the traditional assumption of a constant exponent in the telegraph diffusion framework, we explore three distinct variable exponents for edge detection. All of these depend on the gray level of the image or its gradient. We rigorously prove the existence and uniqueness of weak solutions of our model in a functional setting and perform numerical experiments to assess how well it can despeckle noisy gray-level images. We consider both a range of natural images contaminated by varying degrees of artificial speckle noise and synthetic aperture radar (SAR) images. We finally compare our method with the nonlocal speckle removal technique and find that our model outperforms the latter at speckle elimination and edge preservation.
- [10] arXiv:2411.08177 [pdf, other]
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Title: Erasure Decoding for Quantum LDPC Codes via Belief Propagation with Guided DecimationComments: Published in 2024 60th Annual Allerton Conference ProceedingsSubjects: Information Theory (cs.IT)
Quantum low-density parity-check (LDPC) codes are a promising family of quantum error-correcting codes for fault tolerant quantum computing with low overhead. Decoding quantum LDPC codes on quantum erasure channels has received more attention recently due to advances in erasure conversion for various types of qubits including neutral atoms, trapped ions, and superconducting qubits. Belief propagation with guided decimation (BPGD) decoding of quantum LDPC codes has demonstrated good performance in bit-flip and depolarizing noise. In this work, we apply BPGD decoding to quantum erasure channels. Using a natural modification, we show that BPGD offers competitive performance on quantum erasure channels for multiple families of quantum LDPC codes. Furthermore, we show that the performance of BPGD decoding on erasure channels can sometimes be improved significantly by either adding damping or adjusting the initial channel log-likelihood ratio for bits that are not erased. More generally, our results demonstrate BPGD is an effective general-purpose solution for erasure decoding across the quantum LDPC landscape.
- [11] arXiv:2411.08180 [pdf, html, other]
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Title: A rank one mild mixing system without minimal self joiningsComments: 46 pages, 6 figuresSubjects: Dynamical Systems (math.DS)
We show that there is a rank 1 transformation that is mildly mixing but does not have minimal self-joinings, answering a question of Thouvenot.
- [12] arXiv:2411.08184 [pdf, html, other]
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Title: Conic programming to understand sums of squares of eigenvalues of graphsComments: 33 pages. Our collaboration started after arXiv:2308.04475 was posted by two of the authors. We decided that the results in that submission fit well here, so we are keeping them in Section 2, with a significant improvement in the presentationSubjects: Combinatorics (math.CO); Optimization and Control (math.OC); Spectral Theory (math.SP)
In this paper we prove a conjecture by Wocjan, Elphick and Anekstein (2018) which upper bounds the sum of the squares of the positive (or negative) eigenvalues of the adjacency matrix of a graph by an expression that behaves monotonically in terms of the vector chromatic number. One of our lemmas is a strengthening of the Cauchy-Schwarz inequality for Hermitian matrices when one of the matrices is positive semidefinite.
A related conjecture due to Bollobás and Nikiforov (2007) replaces the vector chromatic number by the clique number and sums over the first two eigenvalues only. We prove a version of this conjecture with weaker constants. An important consequence of our work is a proof that for any fixed $r$, computing a rank $r$ optimum solution to the vector chromatic number semidefinite programming is NP-hard.
We also present a vertex weighted version of some of our results, and we show how it leads quite naturally to the known vertex-weighted version of the Motzkin-Straus quadratic optimization formulation for the clique number. - [13] arXiv:2411.08198 [pdf, html, other]
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Title: Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product SpacesComments: 20 pagesSubjects: Differential Geometry (math.DG)
In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree $-1$ (including the inverse mean curvature flow) in warped product spaces $I \times_{\phi} M^n$, where $M^n$ is a compact homogeneous manifold and $\phi'' \geq 0$, must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than $-1$ and with an extra assumption $\phi' \geq 0$.
Furthermore, we also show that any complete non-compact star-shaped, asymptotically concial expanding self-similar solutions to the flow by positive power of mean curvature in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric. - [14] arXiv:2411.08201 [pdf, html, other]
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Title: Fibers of point cloud persistenceComments: 33 pages, 8 figuresSubjects: Algebraic Topology (math.AT)
Persistent homology (PH) studies the topology of data across multiple scales by building nested collections of topological spaces called filtrations, computing homology and returning an algebraic object that can be vizualised as a barcode--a multiset of intervals. The barcode is stable and interpretable, leading to applications within mathematics and data science. We study the spaces of point clouds with the same barcode by connecting persistence with real algebraic geometry and rigidity theory. Utilizing a semi-algebraic setup of point cloud persistence, we give lower and upper bounds on its dimension and provide combinatorial conditions in terms of the local and global rigidity properties of graphs associated with point clouds and filtrations. We prove that for generic point clouds in $\mathbb{R}^d$ ($d \geq 2$), a point cloud is identifiable up to isometry from its VR persistence if the associated graph is globally rigid, and locally identifiable up to isometry from its Čech persistence if the associated hypergraph is rigid.
- [15] arXiv:2411.08208 [pdf, html, other]
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Title: On homomorphisms between Weyl modules: The case of a column transpostionComments: Number of pages 39Subjects: Representation Theory (math.RT)
Let $G=GL_n(K)$ be the general linear group defined over an infinite field $K$ of positive characteristic $p$ and let $\Delta(\lambda)$ be the Weyl module of $G$ which corresponds to a partition $\lambda$. In this paper we classify all homomorphisms $\Delta(\lambda) \to \Delta(\mu)$ when $\lambda=(a,b,1^d)$ and $\mu=(a+d,b)$, $d>1$. In particular, we show that $Hom_G(\Delta(\lambda),\Delta(\mu))$ is nonzero if and only if $p=2$ and $a$ is even. In this case, we show that the dimension of the homomorphism space is equal to 1 and we provide an explicit generator whose description depends on binary expansions of various integers. We also show that these generators in general are not compositions of Carter-Payne homomorphisms.
- [16] arXiv:2411.08215 [pdf, html, other]
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Title: Equidistribution of Stark-Heegner and ATR cyclesSubjects: Number Theory (math.NT)
We prove the equidistribution of some cycles of S-arithmetic nature that are related to RM points and Stark-Heegner points. We also prove the equidistribution of Picard orbits of ATR cycles as defined by Darmon, Rotger and Zhao.
- [17] arXiv:2411.08220 [pdf, other]
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Title: Nonlocal operators with Neumann conditionsComments: 85 pagesSubjects: Probability (math.PR)
We construct a strong Markov process corresponding to the Dirichlet form of Servadei and Valdinoci and use the process to solve the corresponding Neumann boundary problem for the fractional Laplacian and the half-line.
- [18] arXiv:2411.08233 [pdf, html, other]
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Title: Improved Constructions of Skew-Tolerant Gray CodesComments: 13 pagesSubjects: Information Theory (cs.IT); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
We study skew-tolerant Gray codes, which are Gray codes in which changes in consecutive codewords occur in adjacent positions. We present the first construction of asymptotically non-vanishing skew-tolerant Gray codes, offering an exponential improvement over the known construction. We also provide linear-time encoding and decoding algorithms for our codes. Finally, we extend the definition to non-binary alphabets, and provide constructions of complete $m$-ary skew-tolerant Gray codes for every base $m\geq 3$.
- [19] arXiv:2411.08239 [pdf, html, other]
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Title: Exact Eigenvalues and Eigenvectors for Some n-Dimensional MatricesSubjects: Spectral Theory (math.SP); Numerical Analysis (math.NA)
Building on previous work that provided analytical solutions to generalised matrix eigenvalue problems arising from numerical discretisations, this paper develops exact eigenvalues and eigenvectors for a broader class of $n$-dimensional matrices, focusing on non-symmetric and non-persymmetric matrices. These matrices arise in one-dimensional Laplacian eigenvalue problems with mixed boundary conditions and in a few quantum mechanics applications where standard Toeplitz-plus-Hankel matrix forms do not suffice. By extending analytical methodologies to these broader matrix categories, the study not only widens the scope of applicable matrices but also enhances computational methodologies, leading to potentially more accurate and efficient solutions in physics and engineering simulations.
- [20] arXiv:2411.08245 [pdf, html, other]
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Title: Vertex orders in higher dimensionsSubjects: Combinatorics (math.CO)
Unit interval and interval complexes are higher-dimensional generalizations of unit interval and interval graphs, respectively. We show that strongly connected unit interval complexes are shellable with shellings induced by their unit interval orders. We also show that these complexes are vertex decomposable and hence shelling completable. On the other hand, we give simple examples of strongly connected interval complexes that are not shellable in dimensions two and higher.
- [21] arXiv:2411.08247 [pdf, other]
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Title: On the Nature and Complexity of an Impartial Two-Player Variant of the Game Lights-OutSubjects: Combinatorics (math.CO); Computational Complexity (cs.CC)
In this paper we study a variant of the solitaire game Lights-Out, where the player's goal is to turn off a grid of lights. This variant is a two-player impartial game where the goal is to make the final valid move. This version is playable on any simple graph where each node is given an assignment of either a 0 (representing a light that is off) or 1 (representing a light that is on). We focus on finding the Nimbers of this game on grid graphs and generalized Petersen graphs. We utilize a recursive algorithm to compute the Nimbers for 2 x n grid graphs and for some generalized Petersen graphs.
- [22] arXiv:2411.08252 [pdf, html, other]
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Title: Inheritance of local topological properties of schemesComments: 16 pages, 0 figuresSubjects: Algebraic Geometry (math.AG)
This paper shows that for certain local topological properties, given a locally quasi-finite, flat and locally finitely presented map of schemes $f\colon X\to Y$, if $Y$ has the property, then so does $X$. We also show that being locally topologically noetherian or locally connected or having locally finitely many irreducible components are examples for these local topological properties.
- [23] arXiv:2411.08258 [pdf, html, other]
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Title: Efficient encoding and decoding algorithm for a class of perfect single-deletion-correcting permutation codesComments: 27 pages, 4 figuresSubjects: Information Theory (cs.IT); Combinatorics (math.CO)
A permutation code is a nonlinear code whose codewords are permutation of a set of symbols. We consider the use of permutation code in the deletion channel, and consider the symbol-invariant error model, meaning that the values of the symbols that are not removed are not affected by the deletion. In 1992, Levenshtein gave a construction of perfect single-deletion-correcting permutation codes that attain the maximum code size. Furthermore, he showed in the same paper that the set of all permutations of a given length can be partitioned into permutation codes so constructed. This construction relies on the binary Varshamov-Tenengolts codes. In this paper we give an independent and more direct proof of Levenshtein's result that does not depend on the Varshamov-Tenengolts code. Using the new approach, we devise efficient encoding and decoding algorithms that correct one deletion.
- [24] arXiv:2411.08259 [pdf, html, other]
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Title: Transport of Zariski density in compatible collections of $G$-representationsComments: 18 pages. comments welcome!Subjects: Number Theory (math.NT)
Let $X$ be a connected normal scheme of finite type over $\mathbf{Z}$, let $G$ be a connected reductive group over $\mathbf{Q}$, and let $\{\rho_\ell\colon\pi_1(X[1/\ell])\to G(\mathbf{Q}_\ell)\}_\ell$ be a Frobenius-compatible collection of continuous homomorphisms indexed by the primes. Assume $\mathrm{Img}(\rho_\ell)$ is Zariski-dense in $G_{\mathbf{Q}_\ell}$ for all $\ell$ in a nonempty finite set $\mathcal{R}$. We prove that, under certain hypotheses on $\mathcal{R}$ (depending only on $G$), $\mathrm{Img}(\rho_\ell)$ is Zariski-dense in $G_{\mathbf{Q}_\ell}$ for all $\ell$ in a set of Dirichlet density $1$. As an application, we combine this result with a version of Hilbert's irreducibility theorem and recent work of Klevdal--Patrikis to obtain new information about the "canonical" local systems attached to Shimura varieties not of Abelian type.
- [25] arXiv:2411.08265 [pdf, html, other]
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Title: Sharp character bounds for symmetric groups in terms of partition lengthComments: 7 pagesSubjects: Representation Theory (math.RT)
Let $S_n$ denote a symmetric group, $\chi$ an irreducible character of $S_n$, and $g\in S_n$ an element which decomposes into $k$ disjoint cycles, where $1$-cycles are included. Then $|\chi(g)|\le k!$, and this upper bound is sharp for fixed $k$ and varying $n$, $\chi$, and $g$. This implies a sharp upper bound of $k!$ for unipotent character values of $SL_n(q)$ at regular semisimple elements with characteristic polynomial $P(t)=P_1(t)\cdots P_k(t)$, where the $P_i$ are irreducible over $F_q[t]$.
- [26] arXiv:2411.08268 [pdf, html, other]
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Title: Modified Dirichlet character sums over the $k$-free integersSubjects: Number Theory (math.NT)
The main question of this paper is the following: how much cancellation can the partial sums restricted to the $k$-free integers up to $x$ of a $\pm 1$ multiplicative function $f$ be in terms of $x$? Building upon the recent paper by Q. Liu, Acta Math. Sin. (Engl. Ser.) 39 (2023), no. 12, 2316-2328, we prove that under the Riemann Hypothesis for quadratic Dirichlet $L$-functions, we can get $x^{1/(k+1)}$ cancellation when $f$ is a modified quadratic Dirichlet character, i.e., $f$ is completely multiplicative and for some quadratic Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite subset of prime numbers. This improves the conditional results by Aymone, Medeiros and the author cf. Ramanujan J. 59 (2022), no. 3, 713-728.
- [27] arXiv:2411.08269 [pdf, html, other]
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Title: Rings of Hilbert modular forms, computations on Hilbert modular surfaces, and the Oda-Hamahata conjectureComments: 37 pagesSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
The modularity of an elliptic curve $E/\mathbb Q$ can be expressed either as an analytic statement that the $L$-function is the Mellin transform of a modular form, or as a geometric statement that $E$ is a quotient of a modular curve $X_0(N)$. For elliptic curves over number fields these notions diverge; a conjecture of Hamahata asserts that for every elliptic curve $E$ over a totally real number field there is a correspondence between a Hilbert modular variety and the product of the conjugates of $E$. In this paper we prove the conjecture by explicit computation for many cases where $E$ is defined over a real quadratic field and the geometric genus of the Hilbert modular variety is $1$.
- [28] arXiv:2411.08270 [pdf, html, other]
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Title: Absolutely irreducible quasisimple linear groups containing elements of order a specified Zsigmondy primeComments: 34 pages, 9 tables, dedicated to the memory of Richard A. ParkerSubjects: Representation Theory (math.RT); Group Theory (math.GR)
This paper is concerned with absolutely irreducible quasisimple subgroups $G$ of a finite general linear group $GL_d(\mathbb{F}_q)$ for which some element $g\in G$ of prime order $r$, in its action on the natural module $V=(\mathbb{F}_q)^d$, is irreducible on a subspace of the form $V(1-g)$ of dimension $d/2$. We classify $G,d,r$, the characteristic $p$ of the field $\mathbb{F}_q$, and we identify those examples where the element $g$ has a fixed point subspace of dimension $d/2$. Our proof relies on representation theory, in particular, the multiplicities of eigenvalues of $g$, and builds on earlier results of DiMuro.
- [29] arXiv:2411.08271 [pdf, html, other]
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Title: High-order and Mass-conservative Regularized Implicit-explicit relaxation Runge-Kutta methods for the logarithmic Schr\"{o}dinger equationSubjects: Numerical Analysis (math.NA)
The non-differentiability of the singular nonlinearity (such as $f=\ln|u|^2$) at $u=0$ presents significant challenges in devising accurate and efficient numerical schemes for the logarithmic Schrödinger equation (LogSE). To address this singularity, we propose an energy regularization technique for the LogSE. For the regularized model, we utilize Implicit-Explicit Relaxation Runge-Kutta methods, which are linearly implicit, high-order, and mass-conserving for temporal discretization, in conjunction with the Fourier pseudo-spectral method in space. Ultimately, numerical results are presented to validate the efficiency of the proposed methods.
- [30] arXiv:2411.08273 [pdf, html, other]
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Title: On the inadequacy of nudging data assimilation algorithms for non-dissipative systems: A computational examination of the Korteweg de-Vries and Lorenz equationsComments: [1] Michael S. Jolly, Tural Sadigov, and Edriss S. Titi. Determining form and data assimilation algorithm for weakly damped and driven Korteweg-de Vries equation-Fourier modes case. Nonlinear Analysis: Real World Applications, 36:287-317, 2017Subjects: Optimization and Control (math.OC); Mathematical Physics (math-ph)
In this work, we study the applicability of the Azouani-Olson-Titi (AOT) nudging algorithm for continuous data assimilation to evolutionary dynamical systems that are not dissipative. Specifically, we apply the AOT algorithm to the Korteweg de-Vries (KdV) equation and a partially dissipative variant of the Lorenz 1963 system. Our analysis reveals that the KdV equation lacks the finitely many determining modes property, leading to the construction of infinitely many solutions with exactly the same sparse observational data, which data assimilation methods cannot distinguish between. We numerically verify that the AOT algorithm successfully recovers these counterexamples for the damped and driven KdV equation, as studied in [1], which is dissipative. Additionally, we demonstrate numerically that the AOT algorithm is not effective in accurately recovering solutions for a partially dissipative variant of the Lorenz 1963 system.
- [31] arXiv:2411.08284 [pdf, html, other]
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Title: Dynamic Thresholding Algorithm with Memory for Linear Inverse ProblemsSubjects: Information Theory (cs.IT); Numerical Analysis (math.NA)
The relaxed optimal $k$-thresholding pursuit (ROTP) is a recent algorithm for linear inverse problems. This algorithm is based on the optimal $k$-thresholding technique which performs vector thresholding and error metric reduction simultaneously. Although ROTP can be used to solve small to medium-sized linear inverse problems, the computational cost of this algorithm is high when solving large-scale problems. By merging the optimal $k$-thresholding technique and iterative method with memory as well as optimization with sparse search directions, we propose the so-called dynamic thresholding algorithm with memory (DTAM), which iteratively and dynamically selects vector bases to construct the problem solution. At every step, the algorithm uses more than one or all iterates generated so far to construct a new search direction, and solves only the small-sized quadratic subproblems at every iteration. Thus the computational complexity of DTAM is remarkably lower than that of ROTP-type methods. It turns out that DTAM can locate the solution of linear inverse problems if the matrix involved satisfies the restricted isometry property. Experiments on synthetic data, audio signal reconstruction and image denoising demonstrate that the proposed algorithm performs comparably to several mainstream thresholding and greedy algorithms, and it works much faster than the ROTP-type algorithms especially when the sparsity level of signal is relatively low.
- [32] arXiv:2411.08295 [pdf, html, other]
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Title: Improving the convergence of Markov chains via permutations and projectionsComments: 42 pages, 2 figuresSubjects: Probability (math.PR); Optimization and Control (math.OC); Computation (stat.CO)
This paper aims at improving the convergence to equilibrium of finite ergodic Markov chains via permutations and projections. First, we prove that a specific mixture of permuted Markov chains arises naturally as a projection under the KL divergence or the squared-Frobenius norm. We then compare various mixing properties of the mixture with other competing Markov chain samplers and demonstrate that it enjoys improved convergence. This geometric perspective motivates us to propose samplers based on alternating projections to combine different permutations and to analyze their rate of convergence. We give necessary, and under some additional assumptions also sufficient, conditions for the projection to achieve stationarity in the limit in terms of the trace of the transition matrix. We proceed to discuss tuning strategies of the projection samplers when these permutations are viewed as parameters. Along the way, we reveal connections between the mixture and a Markov chain Sylvester's equation as well as assignment problems, and highlight how these can be used to understand and improve Markov chain mixing. We provide two examples as illustrations. In the first example, the projection sampler (with a suitable choice of the permutation) improves upon Metropolis-Hastings in a discrete bimodal distribution with a reduced relaxation time from exponential to polynomial in the system size, while in the second example, the mixture of permuted Markov chain yields a mixing time that is logarithmic in system size (with high probability under random permutation), compared to a linear mixing time in the Diaconis-Holmes-Neal sampler.
- [33] arXiv:2411.08296 [pdf, html, other]
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Title: On the computation of the arcsin function in the Kerala school of astronomy and mathematicsComments: 25 pages, 5 Sanskrit verses, 3 figures, numerical examplesSubjects: History and Overview (math.HO)
This paper examines how the mathematicians and astronomers of the Kerala school tackled the problem of computing the values of the arcsin function. Four different approaches are discussed all of which are found in Nilakantha Somayaji's (1444 - 1545 CE) Tantrasangraha and the roots of all of which can be traced to ideas originally articulated by Sangamagrama Madhava (c. 1340 - 1425 CE): (i) a simple method when the argument is small; (ii) an iterative method when the argument is small; (iii) a method based on a lookup table; (iv) a method when the argument is large. The paper also contains the original Sanskrit verses describing the various methods and English translations thereof. Moreover, there is a presentation of a novel method for computing the circumference of a circle found in Jyeshthadeva's (c. 1500 - 1575 CE) Yuktibhasha which is based on method (i) for computing the arcsin function. All methods have been illustrated with numerical examples. A surprising by-product of the investigation is a totally unexpected appearance of a core integer sequence, namely, the entry A001764 in the Online Encyclopedia of Integer Sequence, while studying the iterative method for computing the arcsin function.
- [34] arXiv:2411.08303 [pdf, html, other]
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Title: The discrepancy in min-max statistics between two random matrices with finite third momentsComments: 11 pagesSubjects: Probability (math.PR)
We propose a novel coupling inequality of the min-max type for two random matrices with finite absolute third moments, which generalizes the quantitative versions of the well-known inequalities by Gordon. Previous results have calculated the quantitative bounds for pairs of Gaussian random matrices. Through integrating the methods utilized by Chatterjee-Meckes and Reinert-Röllin in adapting Stein's method of exchangeable pairs for multivariate normal approximation, this study eliminates the Gaussian restriction on random matrices, enabling us to achieve more extensive results.
- [35] arXiv:2411.08317 [pdf, html, other]
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Title: On Regular H\'enon-like RenormalizationComments: 75 pages, 9 figuresSubjects: Dynamical Systems (math.DS)
We develop a renormalization theory of non-perturbative dissipative Hénon-like maps with combinatorics of bounded type. The main novelty of our approach is the incorporation of Pesin theoretic ideas to the renormalization method, which enables us to control the small-scale geometry of dynamics in the higher-dimensional setting. In a prequel to this paper, it is shown that, under certain regularity conditions on the return maps, renormalizations of Hénon-like maps have $\textit{a priori}$ bounds. The current paper is devoted to the applications of this critical estimate. First, we prove that Hénon-like maps converge under renormalization to the same renormalization attractor as for 1D unimodal maps. Second, we show that the necessary and sufficient conditions for renormalization convergence are finite-time checkable. Lastly, we show that every infinitely renormalizable Hénon-like map is $\textit{regularly unicritical}$: there exists a unique orbit of tangencies between strong-stable and center manifolds, and outside a slow-exponentially shrinking neighborhood of this orbit, the dynamics behaves as a uniformly partially hyperbolic system.
- [36] arXiv:2411.08319 [pdf, html, other]
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Title: On the Euler characteristics for quandlesSubjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Group Theory (math.GR)
A quandle is an algebraic system whose axioms generalize the algebraic structure of the point symmetries of symmetric spaces. In this paper, we give a definition of Euler characteristics for quandles. In particular, the quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic. Additionally, we calculate the Euler characteristics of some finite quandles, including generalized Alexander quandles, core quandles, discrete spheres, and discrete tori. Furthermore, we prove several properties of quandle Euler characteristics, which suggest that they share similar properties with topological Euler characteristics.
- [37] arXiv:2411.08321 [pdf, html, other]
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Title: Elliptic curves of conductor $2^m p$, quadratic twists, and Watkins's conjectureComments: comments welcomeSubjects: Number Theory (math.NT)
Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve. By the modularity theorem, it admits a surjection from a modular curve $X_0(N) \to \mathsf{E}$, and the minimal degree among such maps is called the modular degree of $\mathsf{E}$. By the Mordell--Weil Theorem, $\mathsf{E}(\mathbb{Q})\simeq \mathbb{Z}^r \oplus T$ for some nonnegative integer $r$ and some finite group $T$. Watkins's Conjecture predicts that $2^r$ divides the modular degree, thus suggesting an intriguing link between these geometrically- and algebraically-defined invariants. We offer some new cases of Watkins's Conjecture, specifically for elliptic curves with additive reduction at $2$, good reduction outside of at most two odd primes, and a rational point of order two.
- [38] arXiv:2411.08325 [pdf, html, other]
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Title: Stabilities of Kleitman diameter theoremComments: 37pagesSubjects: Combinatorics (math.CO)
Let $\mathcal{F}$ be a family of subsets of $[n]$. The diameter of $\mathcal{F}$ is the maximum size of symmetric differences among pairs of its members. Solving a conjecture of Erdős, in 1966, Kleitman determined the maximum size of a family with fixed diameter, which states that a family with diameter $s$ has cardinality at most that of a Hamming ball of radius $s/2$. More precisely, suppose that $\mathcal{F} \subseteq 2^{[n]}$ is a family with diameter $s$, if $s=2d$ for an integer $d\ge 1$, then $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i}$; if $s=2d+1$ for an integer $d\ge 1$, then $|\mathcal{F}|\le \sum_{i=0}^d {n \choose i} + {n-1 \choose d}$. This result extends the well-known Katona intersection theorem and the Erdős--Ko--Rado theorem, and it is now known as the Kleitman diameter theorem. In 2017, Frankl characterized the extremal families of Kleitman's theorem and presented a stability result. In this paper, we solve a recent problem due to Li and Wu by determining the extremal families of Frankl's theorem, and establishing a further stability result. These theorems can be viewed as the first and second stability results for the Kleitman diameter theorem. Combining our framework with some appropriate structural analysis, it is feasible to further extend our method to establish higher-layer stability results of the Kleitman diameter theorem.
- [39] arXiv:2411.08336 [pdf, html, other]
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Title: Branched covers between 2-spheresComments: 16 pagesSubjects: Geometric Topology (math.GT)
By employing methods from complex analysis and topology, we have established a criterion for the existence of specific types of branched covers between 2-spheres. This criterion generalizes the results previously obtained by Jiang (2004), Pervova-Petronio (2006), Zhu (2019), and Wei-Wu-Xu (2024). As an application of our findings, we introduce several new types of exceptional branching datum.
- [40] arXiv:2411.08338 [pdf, other]
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Title: Quantifying uncertainty in the numerical integration of evolution equations based on Bayesian isotonic regressionSubjects: Numerical Analysis (math.NA); Data Analysis, Statistics and Probability (physics.data-an); Methodology (stat.ME)
This paper presents a new Bayesian framework for quantifying discretization errors in numerical solutions of ordinary differential equations. By modelling the errors as random variables, we impose a monotonicity constraint on the variances, referred to as discretization error variances. The key to our approach is the use of a shrinkage prior for the variances coupled with variable transformations. This methodology extends existing Bayesian isotonic regression techniques to tackle the challenge of estimating the variances of a normal distribution. An additional key feature is the use of a Gaussian mixture model for the $\log$-$\chi^2_1$ distribution, enabling the development of an efficient Gibbs sampling algorithm for the corresponding posterior.
- [41] arXiv:2411.08339 [pdf, html, other]
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Title: Expected degrees in random plane graphsSubjects: Combinatorics (math.CO)
We prove that, for every set of $n$ points $\mathcal{P}$ in $\mathbb{R}^2$, a random plane graph drawn on $\mathcal{P}$ is expected to contain less than $n/10.18$ isolated vertices. In the other direction, we construct a point set where the expected number of isolated vertices in a random plane graph is about $n/23.32$. For $i\ge 1$, we prove that the expected number of vertices of degree $i$ is always less than $n/\sqrt{\pi i}$
Our analysis is based on cross-graph charging schemes. That is, we move charge between vertices from different plane graphs of the same point set. This leads to information about the expected behavior of a random plane graph. - [42] arXiv:2411.08342 [pdf, html, other]
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Title: Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensionsComments: 33 pages, 8 figuresSubjects: Numerical Analysis (math.NA)
A high-order quadrature rule is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density {\it on each patch}, which allows for a natural harmonic polynomial extension in a {\it volumetric neighborhood} of the patch. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin-centered 1-forms that often require expensive adaptive integration. These singularity-preserving line integrals can be semi-analytically evaluated using singularity-swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations on the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high-order quadratures for singular and nearly singular surface integrals, leading to improved efficiency and robustness. The algorithmic steps are discussed in detail. The accuracy and efficiency of the recursive reduction quadrature are illustrated via several numerical examples.
- [43] arXiv:2411.08345 [pdf, html, other]
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Title: Brualdi-Hoffman-Tur\'{a}n problem of the gemSubjects: Combinatorics (math.CO)
A graph is said to be $F$-free if it does not contain $F$ as a subgraph. Brualdi-Hoffman-Turán problem seeks to determine the maximum spectral radius of an $F$-free graph with given size. The gem consists of a path on $4$ vertices, along with an additional vertex that is adjacent to every vertex of the path. Concerning Brualdi-Hoffman-Turán problem of the gem, when the size is odd, Zhang and Wang [Discrete Math. 347 (2024) 114171] and Yu, Li and Peng [arXiv:2404. 03423] solved it. In this paper, we completely solve the Brualdi-Hoffman-Turán problem type problem of the gem.
- [44] arXiv:2411.08351 [pdf, html, other]
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Title: Alphabet-affine 2-neighbour-transitive codesComments: 18 pagesSubjects: Combinatorics (math.CO)
A code ${\mathcal C}$ is a subset of the vertex set of a Hamming graph $H(n,q)$, and ${\mathcal C}$ is $2$-neighbour-transitive if the automorphism group $G={\rm Aut}({\mathcal C})$ acts transitively on each of the sets ${\mathcal C}$, ${\mathcal C}_1$ and ${\mathcal C}_2$, where ${\mathcal C}_1$ and ${\mathcal C}_2$ are the (non-empty) sets of vertices that are distances $1$ and $2$, respectively, (but no closer) to some element of ${\mathcal C}$.
Suppose that ${\mathcal C}$ is a $2$-neighbour-transitive code with minimum distance at least $5$. For $q=2$, all `minimal' such ${\mathcal C}$ have been classified. Moreover, it has previously been shown that a subgroup of the automorphism group of the code induces an affine $2$-transitive group action on the alphabet of the Hamming graph. The main results of this paper are to show that this affine $2$-transitive group must be a subgroup of ${\rm A}\Gamma{\rm L}_1(q)$ and to provide a number of infinite families of examples of such codes. These examples are described via polynomial algebras related to representations of certain classical groups. - [45] arXiv:2411.08355 [pdf, html, other]
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Title: Communication Efficient Decentralization for Smoothed Online Convex OptimizationComments: 39 pagesSubjects: Optimization and Control (math.OC); Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG)
We study the multi-agent Smoothed Online Convex Optimization (SOCO) problem, where $N$ agents interact through a communication graph. In each round, each agent $i$ receives a strongly convex hitting cost function $f^i_t$ in an online fashion and selects an action $x^i_t \in \mathbb{R}^d$. The objective is to minimize the global cumulative cost, which includes the sum of individual hitting costs $f^i_t(x^i_t)$, a temporal "switching cost" for changing decisions, and a spatial "dissimilarity cost" that penalizes deviations in decisions among neighboring agents. We propose the first decentralized algorithm for multi-agent SOCO and prove its asymptotic optimality. Our approach allows each agent to operate using only local information from its immediate neighbors in the graph. For finite-time performance, we establish that the optimality gap in competitive ratio decreases with the time horizon $T$ and can be conveniently tuned based on the per-round computation available to each agent. Moreover, our results hold even when the communication graph changes arbitrarily and adaptively over time. Finally, we establish that the computational complexity per round depends only logarithmically on the number of agents and almost linearly on their degree within the graph, ensuring scalability for large-system implementations.
- [46] arXiv:2411.08361 [pdf, html, other]
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Title: Auto-tuned Primal-dual Successive Convexification for Hypersonic Reentry GuidanceSkye Mceowen, Daniel J. Calderone, Aman Tiwary, Jason S. K. Zhou, Taewan Kim, Purnanand Elango, Behcet AcikmeseComments: 38 pages, 27 figures; submitted to the AIAA Journal of Guidance, Control, and Dynamics (JGCD)Subjects: Optimization and Control (math.OC)
This paper presents auto-tuned primal-dual successive convexification (Auto-SCvx), an algorithm designed to reliably achieve dynamically-feasible trajectory solutions for constrained hypersonic reentry optimal control problems across a large mission parameter space. In Auto-SCvx, we solve a sequence of convex subproblems until convergence to a solution of the original nonconvex problem. This method iteratively optimizes dual variables in closed-form in order to update the penalty hyperparameters used in the primal variable updates. A benefit of this method is that it is auto-tuning, and requires no hand-tuning by the user with respect to the constraint penalty weights. Several example hypersonic reentry problems are posed and solved using this method, and comparative studies are conducted against current methods. In these numerical studies, our algorithm demonstrates equal and often improved performance while not requiring hand-tuning of penalty hyperparameters.
- [47] arXiv:2411.08362 [pdf, html, other]
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Title: Large-scale boundary estimates of parabolic homogenization over rough boundariesSubjects: Analysis of PDEs (math.AP)
In this paper, for a family of second-order parabolic system or equation with rapidly oscillating and time-dependent periodic coefficients over rough boundaries, we obtain the large-scale boundary estimates, by a quantitative approach. The quantitative approach relies on approximating twice: we first approximate the original parabolic problem over rough boundary by the same equation over a non-oscillating boundary and then approximate the oscillating equation over a non-oscillating boundary by its homogenized equation over the same non-oscillating boundary.
- [48] arXiv:2411.08364 [pdf, html, other]
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Title: Non-zero values of a family of approximations of a class of L-functionsSubjects: Number Theory (math.NT)
Motivated by the approximate functional equation, consider the approximation $\zeta_N(s) = \sum_{n=1}^N n^{-s} + \chi(s) \sum_{n=1}^N n^{1-s}$ of the Riemann zeta function.
It has been previously shown that $\zeta_N(s)$ has 100\% of its zeros lie on the critical line, and that $a$-values of $\zeta_N(s)$ cluster arbitrarily close to the critical line.
In this paper, we show that, despite the above, 0\% of a-values actually lie on the critical line itself.
We also extend our results to a wider class of $L$-functions. - [49] arXiv:2411.08366 [pdf, other]
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Title: Stability of the catenoid for the hyperbolic vanishing mean curvature equation in 4 spatial dimensionsComments: 105 pages, 1 figureSubjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
We establish the asymptotic stability of the catenoid, as a nonflat stationary solution to the hyperbolic vanishing mean curvature (HVMC) equation in Minkowski space $\mathbb{R}^{1 + (n + 1)}$ for $n = 4$. Our main result is under a ``codimension-$1$'' assumption on initial perturbation, modulo suitable translation and boost (i.e. modulation), without any symmetry assumptions. In comparison to the $n \geq 5$ case addressed by Lührmann-Oh-Shahshahani arXiv:2212.05620, proving catenoid stability in $4$ dimensions shares additional difficulties with its $3$ dimensional analog, namely the slower spatial decay of the catenoid and slower temporal decay of waves. To overcome these difficulties in the $n = 3$ case, the strong Huygens principle, as well as a miraculous cancellation in the source term, plays an important role in arXiv:2409.05968 to obtain strong late time tails. In $n = 4$ dimensions, without these special structural advantages, our novelty is to introduce an appropriate commutator vector field to derive a new hierarchy of estimates with higher $r^p$-weights so that an improved pointwise decay can be established. We expect this to be applicable for proving improved late time tails of other quasilinear wave equations in even dimensions or wave equations with inverse square potential.
- [50] arXiv:2411.08369 [pdf, html, other]
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Title: On curve-flat Lipschitz functions and their linearizationsComments: 37 pagesSubjects: Functional Analysis (math.FA); Metric Geometry (math.MG)
We show that several operator ideals coincide when intersected with the class of linearizations of Lipschitz maps. In particular, we show that the linearization $\hat{f}$ of a Lipschitz map $f:M\to N$ is Dunford-Pettis if and only if it is Radon-Nikodým if and only if it does not fix any copy of $L_1$. We also identify and study the corresponding metric property of $f$, which is a natural extension of the curve-flatness introduced in [arXiv:2103.09370]. Further, we show that $\hat{f}$ is compact if and only if it does not fix any copy of $\ell_1$.
- [51] arXiv:2411.08372 [pdf, html, other]
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Title: Equitable list coloring of sparse graphsSubjects: Combinatorics (math.CO)
A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/k\rceil$ times. Call a graph equitably $k$-choosable if it has an equitable $L$-coloring for every $k$-list assignment $L$. A graph $G$ is $(a,b)$-sparse if for every $A\subseteq V(G)$, the number of edges in the subgraph $G[A]$ of $G$ induced by $A$ is at most $a|A|+b$.
Our first main result is that every $(\frac{7}{6},\frac{1}{3})$-sparse graph with minimum degree at least $2$ is equitably $3$-colorable and equitably $3$-choosable. This is sharp. Our second main result is that every $(\frac{5}{4},\frac{1}{2})$-sparse graph with minimum degree at least $2$ is equitably $4$-colorable and equitably $4$-choosable. This is also sharp.
One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring. - [52] arXiv:2411.08377 [pdf, html, other]
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Title: Dual-Valued Functions of Dual Matrices with Applications in Causal EmergenceSubjects: Numerical Analysis (math.NA)
Dual continuation, an innovative insight into extending the real-valued functions of real matrices to the dual-valued functions of dual matrices with a foundation of the Gâteaux derivative, is proposed. Theoretically, the general forms of dual-valued vector and matrix norms, the remaining properties in the real field, are provided. In particular, we focus on the dual-valued vector $p$-norm $(1\!\leq\! p\!\leq\!\infty)$ and the unitarily invariant dual-valued Ky Fan $p$-$k$-norm $(1\!\leq\! p\!\leq\!\infty)$. The equivalence between the dual-valued Ky Fan $p$-$k$-norm and the dual-valued vector $p$-norm of the first $k$ singular values of the dual matrix is then demonstrated. Practically, we define the dual transitional probability matrix (DTPM), as well as its dual-valued effective information (${\rm{EI_d}}$). Additionally, we elucidate the correlation between the ${\rm{EI_d}}$, the dual-valued Schatten $p$-norm, and the dynamical reversibility of a DTPM. Through numerical experiments on a dumbbell Markov chain, our findings indicate that the value of $k$, corresponding to the maximum value of the infinitesimal part of the dual-valued Ky Fan $p$-$k$-norm by adjusting $p$ in the interval $[1,2)$, characterizes the optimal classification number of the system for the occurrence of the causal emergence.
- [53] arXiv:2411.08398 [pdf, html, other]
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Title: Arithmetic Polygons and Sums of Consecutive SquaresComments: 22 pages, 6 figures, 2 tablesSubjects: Number Theory (math.NT)
We introduce and study arithmetic polygons. We show that these arithmetic polygons are connected to triples of square pyramidal numbers. For every odd $N\geq3$, we prove that there is at least one arithmetic polygon with $N$ sides. We also show that there are infinitely many arithmetic polygons with an even number of sides.
- [54] arXiv:2411.08403 [pdf, html, other]
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Title: Purity of the anisotropic affine Springer fibers for $\mathbf{GL}_{n}$Subjects: Algebraic Geometry (math.AG); Representation Theory (math.RT)
For the group $\mathrm{GL}_{n}$ and the anisotropic elements, we confirm the purity hypothesis of Goresky, Kottwitz and MacPherson, which states that the affine Springer fibers are cohomologically pure in the sense of Grothendieck-Deligne.
- [55] arXiv:2411.08406 [pdf, html, other]
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Title: On Kazama-Suzuki duality between $\mathcal W_k(sl_4, f_{sub})$ and $N= 2$ superconformal vertex algebraComments: 23 pagesSubjects: Quantum Algebra (math.QA)
We classify all possible occurrences of Kazama-Suzuki duality between the $N=2$ superconformal algebra $L^{N=2}_c$ and the subregular algebra $\mathcal{W}$-algebra $\mathcal{W}_{k}(sl_4, f_{sub})$. We establish a new Kazama-Suzuki duality between the subregular $\mathcal{W}$-algebra $\mathcal{W}^k(sl_4, f_{\text{sub}})$ and the the $N = 2$ superconformal algebra $L^{N=2}_{c}$ for $c=-15$. As a consequence of duality, we classify the irreducible $\mathcal{W}_{k=-1}(sl_4, f_{\text{sub}})$-modules.
- [56] arXiv:2411.08411 [pdf, html, other]
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Title: Modeling and Optimization for Rotatable Antenna Enabled Wireless CommunicationComments: 7 pages, 6 figuresSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
In this paper, we propose a new rotatable antenna (RA) model to improve the performance of wireless communication systems. Different from conventional fixed-position antenna (FPA), the proposed RA system can independently and flexibly change the three-dimensional (3D) orientation of each antenna by adjusting its declination angles to achieve desired channel realizations. Specifically, we study an RA-enabled uplink communication system, where the receive beamforming and the declination angles of all RAs are jointly optimized to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among all the users. In the special single-user and free-space propagation setup, the optimal declination angles are derived in closed form with the maximum-ratio combining (MRC) beamformer applied at the base station (BS). In the general multi-user and multi-path setup, we propose an alternating optimization (AO) algorithm to alternately optimize the receive beamforming and the declination angles in an iterative manner. Simulation results are provided to demonstrate that the proposed RA-enabled system can significantly outperform other benchmark schemes.
- [57] arXiv:2411.08412 [pdf, other]
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Title: Enumeration of crossings in two-step puzzlesQuentin François (CEREMADE, DMA)Subjects: Combinatorics (math.CO)
We prove a formula which gives the number of occurrences of certain labels and local configurations inside two-step puzzles introduced by Buch, Kresch, Purbhoo and Tamvakis from the work of Knutson. Puzzles are tilings of the triangular lattice by edge labeled tiles and are known to compute the Schubert structure constants of the cohomology of two-step flag varieties. The formula that we obtain depends only on the boundary conditions of the puzzle. The proof is based on the study of color maps which are tilings of the triangular lattice by edge labeled tiles obtained from puzzles.
- [58] arXiv:2411.08416 [pdf, html, other]
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Title: On wavelet coorbit spaces associated to different dilation groupsComments: To appear in a special volume dedicated to K. GröchenigSubjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA)
This paper develops methods based on coarse geometry for the comparison of wavelet coorbit spaces defined by different dilation groups, with emphasis on establishing a unified approach to both irreducible and reducible quasi-regular representations. We show that the use of reducible representations is essential to include a variety of examples, such as anisotropic Besov spaces defined by general expansive matrices, in a common framework. The obtained criteria yield, among others, a simple characterization of subgroups of a dilation group yielding the same coorbit spaces. They also allow to clarify which anisotropic Besov spaces have an alternative description as coorbit spaces associated to irreducible quasi-regular representations.
- [59] arXiv:2411.08421 [pdf, html, other]
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Title: Modest Sets are Equivalent to PERsComments: 14 pages; Expository articleSubjects: Category Theory (math.CT); Logic in Computer Science (cs.LO)
The aim of this article is to give an expository account of the equivalence between modest sets and partial equivalence relations. Our proof is entirely self-contained in that we do not assume any knowledge of categorical realizability. At the heart of the equivalence lies the subquotient construction on a partial equivalence relation. The subquotient construction embeds the category of partial equivalence relations into the category of modest sets. We show that this embedding is a split essentially surjective functor, and thereby, an equivalence of categories. Our development is both constructive and predicative, and employs the language of homotopy type theory. All the mathematics presented in this article has been mechanised in Cubical Agda.
- [60] arXiv:2411.08426 [pdf, html, other]
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Title: Enumerative aspects of Caylerian polynomialsComments: 23 pages, 3 tablesSubjects: Combinatorics (math.CO)
Eulerian polynomials record the distribution of descents over permutations. Caylerian polynomials likewise record the distribution of descents over Cayley permutations. Using combinatorial species and sign-reversing involutions we derive counting formulas and generating functions for the Caylerian polynomials as well as for related refined polynomials.
- [61] arXiv:2411.08428 [pdf, html, other]
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Title: Partially concentrating solutions for systems with Lotka-Volterra type interactionsSubjects: Analysis of PDEs (math.AP)
In this paper we consider the existence of standing waves for a coupled system of $k$ equations with Lotka-Volterra type interaction. We prove the existence of a standing wave solution with all nontrivial components satisfying a prescribed asymptotic profile. In particular, the $k-1$-last components of such solution exhibits a concentrating behavior, while the first one keeps a quantum nature. We analyze first in detail the result with three equations since this is the first case in which the coupling has a role contrary to what happens when only two densities appear. We also discuss the existence of solutions of this form for systems with other kind of couplings making a comparison with Lotka-Volterra type systems.
- [62] arXiv:2411.08430 [pdf, html, other]
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Title: The Restricted Isometry Property of Block Diagonal Matrices Generated by $\varphi$-Sub-Gaussian VariablesComments: 13 pagesSubjects: Probability (math.PR)
In this paper, we prove the restricted isometry property of block diagonal random matrices with elements from $\varphi$-sub-Gaussian variables, which extends the previously known results for the sub-Gaussian case. A crucial ingredient of our proof is an improved uniform Hanson-Wright deviation inequality, which should be of independent interest.
- [63] arXiv:2411.08435 [pdf, html, other]
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Title: Tractable Robust Markov Decision ProcessesSubjects: Optimization and Control (math.OC)
In this paper we investigate the tractability of robust Markov Decision Processes (RMDPs) under various structural assumptions on the uncertainty set. Surprisingly, we show that in all generality (i.e. without any assumption on the instantaneous rewards), s-rectangular and sa-rectangular uncertainty sets are the only models of uncertainty that are tractable. Our analysis also shows that existing non-rectangular models, including r-rectangular uncertainty and new generalizations, are only weakly tractable in that they require an additional structural assumption that the instantaneous rewards do not depend on the next state, and in this case they are equivalent to rectangular models, which severely undermines their significance and usefulness. Interestingly, our proof techniques rely on identifying a novel simultaneous solvability property, which we show is at the heart of several important properties of RMDPs, including the existence of stationary optimal policies and dynamic programming-based formulations. The simultaneous solvability property enables a unified approach to studying the tractability of all existing models of uncertainty, rectangular and non-rectangular alike.
- [64] arXiv:2411.08445 [pdf, html, other]
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Title: Coexistence of Hilbert space effects and orthogonalityComments: 17 pagesSubjects: Functional Analysis (math.FA)
In this paper, we show that every pair of absolutely compatible Hilbert space effects are coexistent and exhibit a partial orthogonality property. We introduce the notion of partially ortho-coexistence. We generalize absolute compatibility to obtain more examples of partially ortho-coexistent pairs and introduce the notion of generalized compatibility. In the case of $\mathbb{M}_2$, we discuss a geometric behaviour of the generalized compatibility.
- [65] arXiv:2411.08456 [pdf, html, other]
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Title: The Sylvester question in $\mathbb{R}^d$: convex sets with a flat floorComments: 28 pages, 8 figuresSubjects: Combinatorics (math.CO); Probability (math.PR)
Pick $n$ independent and uniform random points $U_1,\ldots,U_n$ in a compact convex set $K$ of $\mathbb{R}^d$ with volume 1, and let $P^{(d)}_K(n)$ be the probability that these points are in convex position. The Sylvester conjecture in $\mathbb{R}^d$ is that $\min_K P^{(d)}_K(d+2)$ is achieved by the $d$-dimensional simplices $K$ (only).
In this paper, we focus on a companion model, already studied in the $2d$ case, which we define in any dimension $d$: we say that $K$ has $F$ as a flat floor, if $F$ is a subset of $K$, contained in a hyperplan $P$, such that $K$ lies in one of the half-spaces defined by $P$.
We define $Q_K^F(n)$ as the probability that $U_1,\cdots,U_n$ together with $F$ are in convex position (i.e., the $U_i$ are on the boundary of the convex hull ${\sf CH}(\{U_1,\cdots,U_n\}\cup F\})$). We prove that, for all fixed $F$,
$K\mapsto Q_K^F(2)$ reaches its minimum on the "mountains" with floor $F$ (mountains are convex hull of $F$ union an additional vertex), while the maximum is not reached, but $K\mapsto Q_K^F(2)$ has values arbitrary close to 1. If the optimisation is done on the set of $K$ contained in $F\times[0,d]$ (the "subprism case"), then the minimum is also reached by the mountains, and the maximum by the "prism" $F\times[0,1]$. Since again, $Q_K^F{(2)}$ relies on the expected volume (of ${\sf CH}(\{V_1,V_2\}\cup F\})$), this result can be seen as a proof of the Sylvester problem in the floor case.
In $2d$, where $F$ can essentially be the segment $[0,1],$ we give a general decomposition formula for $Q_K^F(n)$ so to compute several formulas and bounds for different $K$. In 3D, we give some bounds for $Q_K^F(n)$ for various floors $F$ and special cases of $K$. - [66] arXiv:2411.08458 [pdf, html, other]
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Title: Cellular sheaf Laplacians on the set of simplices of symmetric simplicial set induced by hypergraphComments: 20 pages, 1 figureSubjects: Algebraic Topology (math.AT)
We generalize cellular sheaf Laplacians on an ordered finite abstract simplicial complex to the set of simplices of a symmetric simplicial set. We construct a functor from the category of hypergraphs to the category of finite symmetric simplicial sets and define cellular sheaf Laplacians on the set of simplices of finite symmetric simplicial set induced by hypergraph. We provide formulas for cellular sheaf Laplacians and show that cellular sheaf Laplacian on an ordered finite abstract simplicial complex is exactly the ordered cellular sheaf Laplacian on the set of simplices induced by abstract simplicial complex.
- [67] arXiv:2411.08459 [pdf, html, other]
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Title: Revisiting Atomic Norm Minimization: A Sequential Approach for Atom Identification and RefinementSubjects: Optimization and Control (math.OC)
Atomic norm minimization (ANM) is a key approach for line spectral estimation (LSE). Most related algorithms formulate ANM as a semidefinite programming (SDP), which incurs high computational cost. In this letter, we revisit the ANM problem and present a novel limit-based formulation, which dissects the essential components of the semidefinite characterization of ANM. Our new formulation does not depend on SDP and can be extended to handle more general atomic sets beyond mixture of complex sinusoids. Furthermore, we reveal the connection between ANM and Bayesian LSE approaches, bridging the gap between these two methodologies. Based on this new formulation, we propose a low-complexity algorithm called Sequential Atom Identification and Refinement (SAIR) for ANM. Simulation results demonstrate that SAIR achieves superior estimation accuracy and computational efficiency compared to other state-of-the-art methods.
- [68] arXiv:2411.08465 [pdf, html, other]
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Title: Generating Series of Key Polynomials and Bounded Ascending Sequences of IntegersComments: 48 pagesSubjects: Combinatorics (math.CO)
The fact that Schubert polynomials are the weighted counting functions for reduced RC-graphs, also known as reduced pipe dreams, was established using their generating functions inside an appropriate Demazure algebra. Here we investigate the generating functions of another family of polynomials, the key polynomials, also known as Demazure characters. Each component in that function is a rational function, whose denominator is an explicit product whose definition is based on bounded ascending sequences of integers. We determine the first terms of the polynomial numerator, and pose conjectures about these terms in general as well as some of the next ones. The form of our generating functions suggests relations between the coefficients in key polynomials and signed sums of numbers of integral points on polytopes.
- [69] arXiv:2411.08468 [pdf, html, other]
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Title: $\ell_0$ factor analysisSubjects: Optimization and Control (math.OC); Machine Learning (stat.ML)
Factor Analysis is about finding a low-rank plus sparse additive decomposition from a noisy estimate of the signal covariance matrix. In order to get such a decomposition, we formulate an optimization problem using the nuclear norm for the low-rank component, the $\ell_0$ norm for the sparse component, and the Kullback-Leibler divergence to control the residual in the sample covariance matrix. An alternating minimization algorithm is designed for the solution of the optimization problem. The effectiveness of the algorithm is verified via simulations on synthetic and real datasets.
- [70] arXiv:2411.08473 [pdf, html, other]
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Title: Fractional Fourier Domain PAPR ReductionSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
High peak-to-average power ratio (PAPR) has long posed a challenge for multi-carrier systems, impacting amplifier efficiency and overall system performance. This paper introduces dynamic angle fractional Fourier division multiplexing (DA-FrFDM), an innovative multi-carrier system that effectively reduces PAPR for both QAM and Gaussian signals with minimal signaling overhead. DA-FrFDM leverages the fractional Fourier domain to balance PAPR characteristics between the time and frequency domains, achieving significant PAPR reduction while preserving signal quality. Furthermore, DA-FrFDM refines signal processing and enables one-tap equalization in the fractional Fourier domain through the simple multiplication of time-domain signals by a quadratic phase sequence. Our results show that DA-FrFDM not only outperforms existing PAPR reduction techniques but also retains efficient inter-carrier interference (ICI) mitigation capabilities in doubly dispersive channels.
- [71] arXiv:2411.08475 [pdf, html, other]
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Title: Anti-Ramsey Number of Friendship GraphsSubjects: Combinatorics (math.CO)
An edge-colored graph is called \textit{rainbow graph} if all the colors on its edges are distinct. For a given positive integer $n$ and a family of graphs $\mathcal{G}$, the anti-Ramsey number $ar(n, \mathcal{G})$ is the smallest number of colors $r$ required to ensure that, no matter how the edges of the complete graph $K_n$ are colored using exactly $r$ colors, there will always be a rainbow copy of some graph $G$ from the family $\mathcal{G}$. A friendship graph $F_k$ is the graph obtained by combining $k$ triangles that share a common vertex. In this paper, we determine the anti-Ramsey number $ar(n, \{F_k\})$ for large values of $n$. Additionally, we also determine the $ar(n, \{K_{1,k}, M_k\}$, where $K_{1,k}$ is a star graph with $ k+1$ vertices and $M_k$ is a matching of size $k$.
- [72] arXiv:2411.08479 [pdf, html, other]
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Title: An elementary proof of Hilbert's theorem on ternary quartics: Some complementsSubjects: Algebraic Geometry (math.AG)
In 1888, Hilbert proved that every nonnegative quartic form $f=f(x,y,z)$ with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. In a 2012 paper we presented a new approach that used only elementary techniques. In this note we add some further simplifications to this proof.
- [73] arXiv:2411.08481 [pdf, html, other]
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Title: Variable-Length Feedback Codes via Deep LearningSubjects: Information Theory (cs.IT)
Variable-length feedback coding has the potential to significantly enhance communication reliability in finite block length scenarios by adapting coding strategies based on real-time receiver feedback. Designing such codes, however, is challenging. While deep learning (DL) has been employed to design sophisticated feedback codes, existing DL-aided feedback codes are predominantly fixed-length and suffer performance degradation in the high code rate regime, limiting their adaptability and efficiency. This paper introduces deep variable-length feedback (DeepVLF) code, a novel DL-aided variable-length feedback coding scheme. By segmenting messages into multiple bit groups and employing a threshold-based decoding mechanism for independent decoding of each bit group across successive communication rounds, DeepVLF outperforms existing DL-based feedback codes and establishes a new benchmark in feedback channel coding.
- [74] arXiv:2411.08484 [pdf, html, other]
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Title: Some integrals which are not in table 129 of Bierens de HaanSubjects: Classical Analysis and ODEs (math.CA)
In the standard work of Bierens de Haan about integrals we look at table 129. This table lists a number of integrals of a certain kind. In this paper the table is expanded with a number of similar integrals. These are determined by a number of different methods. Some of these methods are not treated in the freshman standard works on integral calculus. As a bonus, we treat an integral using a complete different method which is not so well-known.
- [75] arXiv:2411.08497 [pdf, html, other]
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Title: General Order Virtual Element Methods for Neumann Boundary Optimal Control Problems in Saddle Point FormulationSubjects: Numerical Analysis (math.NA)
In this work, we explore the application of the Virtual Element Methods for Neumann boundary Optimal Control Problems in saddle point formulation. The method is proposed for arbitrarily polynomial order of accuracy and general polygonal meshes. Our contribution includes a rigorous a priori error estimate that holds for general polynomial degree. On the numerical side, we present (i) an initial convergence test that reflects our theoretical findings, and (ii) a second test case based on a more application-oriented experiment. For the latter test, we focus on the role of VEM stabilization, conducting a detailed experimental analysis, and proposing an alternative structure-preserving strategy to circumvent issues related to the choice of the stabilization parameter.
- [76] arXiv:2411.08500 [pdf, html, other]
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Title: On linear equations over split octonionsComments: 19 ppSubjects: Rings and Algebras (math.RA)
Over an algebraically closed field, affine varieties of solutions of the linear equations a(xb)=c and a(bx)=c over split octonions are described. The set of solutions of a general linear monomial equation over split octonions is considered. We also study how the dimension of the set of solutions is changed under the degeneration of a linear equation.
- [77] arXiv:2411.08501 [pdf, html, other]
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Title: Categorical characterisations of quasi-isometric embeddingsComments: 19 pages, 1 appendixSubjects: Metric Geometry (math.MG); Category Theory (math.CT); Group Theory (math.GR); Geometric Topology (math.GT)
We characterise the (closeness classes of) quasi-isometric embeddings as the regular monomorphisms in the coarsely Lipschitz category, formalising the notion that they are isomorphisms onto their image. Furthermore, we prove that the coarsely Lipschitz category is coregular, and hence admits an (Epi, RegMono)--orthogonal factorisation system. Consequently, quasi-isometric embeddings are equivalently characterised as the effective, strong, or extremal monomorphisms. Finally, we prove that the coarsely Lipschitz category is not coexact in the sense of Barr.
- [78] arXiv:2411.08509 [pdf, html, other]
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Title: Sum Rate Maximization for Movable Antenna-Aided Downlink RSMA SystemsSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Rate splitting multiple access (RSMA) is regarded as an essential and powerful physical-layer (PHY) paradigm for next generation communication systems. Under such a system, users employ successive interference cancellation (SIC), allowing them to decode a portion of the interference and treat the remainder as noise. However, a problem is that current RSMA systems rely on fixed-position antenna arrays, limiting their capacity to fully exploit spatial freedom. This constraint restricts beamforming gain, which substantially degrades RSMA performance. To address this problem, we propose an movable antenna (MA)-aided RSMA scheme that allows the antennas at the base station (BS) to adjust their positions dynamically. Our target is to maximize the system's sum rate of both common and private messages by jointly optimizing the MA positions, beamforming matrix, and common rate allocation. To tackle the formulated non-convex problem, we employ fractional programming (FP) and develop a two-stage, coarse-to-fine-grained search algorithm to obtain suboptimal solutions. Numerical results demonstrate that, with appropriate antenna adjustments, the MA-enabled system significantly enhances the overall performance and reliability of RSMA when employing the proposed algorithm compared to fixed-position antenna configurations.
- [79] arXiv:2411.08518 [pdf, html, other]
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Title: On the numerical integration of the Fokker-Planck equation driven by a mechanical force and the Bismut-Elworthy-Li formulaComments: 30 pages, 5 figuresSubjects: Optimization and Control (math.OC); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
Optimal control theory aims to find an optimal protocol to steer a system between assigned boundary conditions while minimizing a given cost functional in finite time. Equations arising from these types of problems are often non-linear and difficult to solve numerically. In this note, we describe numerical methods of integration for two partial differential equations that commonly arise in optimal control theory: the Fokker-Planck equation driven by a mechanical potential for which we use Girsanov theorem; and the Hamilton-Jacobi-Bellman, or dynamic programming, equation for which we find the gradient of its solution using the Bismut-Elworthy-Li formula. The computation of the gradient is necessary to specify the optimal protocol. Finally, we give an example application in solving an optimal control problem without spacial discretization by machine learning.
- [80] arXiv:2411.08520 [pdf, html, other]
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Title: On the Design of Variable Modulation and Adaptive Modulation for Uplink Sparse Code Multiple AccessSubjects: Information Theory (cs.IT)
Sparse code multiple access (SCMA) is a promising non-orthogonal multiple access scheme for enabling massive connectivity in next generation wireless networks. However, current SCMA codebooks are designed with the same size, leading to inflexibility of user grouping and supporting diverse data rates. To address this issue, we propose a variable modulation SCMA (VM-SCMA) that allows users to employ codebooks with different modulation orders. To guide the VM-SCMA design, a VM matrix (VMM) that assigns modulation orders based on the SCMA factor graph is first introduced. We formulate the VM-SCMA design using the proposed average inverse product distance and the asymptotic upper bound of sum-rate, and jointly optimize the VMM, VM codebooks, power and codebook allocations. The proposed VM-SCMA not only enables diverse date rates but also supports different modulation order combinations for each rate. Leveraging these distinct advantages, we further propose an adaptive VM-SCMA (AVM-SCMA) scheme which adaptively selects the rate and the corresponding VM codebooks to adapt to the users' channel conditions by maximizing the proposed effective throughput. Simulation results show that the overall designs are able to simultaneously achieve a high-level system flexibility, enhanced error rate results, and significantly improved throughput performance, when compared to conventional SCMA schemes.
- [81] arXiv:2411.08522 [pdf, html, other]
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Title: Digital Euler Characteristic TransformSubjects: Algebraic Topology (math.AT)
The Euler Characteristic Transform (ECT) of Turner et al. provides a way to statistically analyze non-diffeomorphic shapes without relying on landmarks. In applications, this transform is typically approximated by a discrete set of directions and heights, which results in potential loss of information as well as problems in inverting the transform. In this work we present a fully digital algorithm for computing the ECT exactly, up to computer precision; we introduce the Ectoplasm package that implements this algorithm, and we demonstrate this is fast and convenient enough to compute distances in real life data sets. We also discuss the implications of this algorithm to related problems in shape analysis, such as shape inversion and sub-shape selection. We also show a proof-of-concept application for solving the shape alignment problem with gradient descent and adaptive grid search, which are two powerful methods neither of which is possible using the discretized transform.
- [82] arXiv:2411.08536 [pdf, html, other]
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Title: Extended Shuffle Product for Multiple Zeta ValuesSubjects: Number Theory (math.NT); Rings and Algebras (math.RA)
The shuffle product on positive integer points, which corresponds to the shuffle algebra for multiple zeta values, is extended uniquely to all integer points, by making the linear operator which decreases the first entry by one a differential operator. We then show that all convergent integer points form a subalgebra under this extended shuffle product. By lifting the extended shuffle product to the locality algebra of Chen symbols, we prove that the multiple zeta series defines an algebra homomorphism from the subalgebra of convergent points to real numbers, which shows that the extended shuffle product is a structure for convergent integer points.
- [83] arXiv:2411.08546 [pdf, html, other]
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Title: A result for hemi-bundled cross-intersecting familiesComments: 17pages. arXiv admin note: text overlap with arXiv:2411.03674Subjects: Combinatorics (math.CO)
Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. It is significant to determine the maximum sum of sizes of cross-intersecting families under the additional assumption that one of the two families is intersecting. Such a pair of families is said to be hemi-bundled. In particular, Frankl (2016) proved that for $k \geq 1, t\ge 0$ and $n \geq 2 k+t$, if $\mathcal{F} \subseteq\binom{[n]}{k+t}$ and $\mathcal{G} \subseteq\binom{[n]}{k}$ are cross-intersecting families, in which $\mathcal{F}$ is non-empty and $(t+1)$-intersecting, then $|\mathcal{F}|+|\mathcal{G}| \leq\binom{n}{k}-\binom{n-k-t}{k}+1$. This bound can be attained when $\mathcal{F}$ consists of a single set. In this paper, we generalize this result under the constraint $|\mathcal{F}| \geq r$ for every $r\leq n-k-t+1$. Moreover, we investigate the stability results of Katona's theorem for non-uniform families with the $s$-union property. Our result extends the stabilities established by Frankl (2017) and Li and Wu (2024). As applications, we revisit a recent result of Frankl and Wang (2024) as well as a result of Kupavskii (2018). Furthermore, we determine the extremal families in these two results.
- [84] arXiv:2411.08549 [pdf, html, other]
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Title: A Framework for Robust Lossy Compression of Heavy-Tailed SourcesComments: 28 pages, 6 figuresSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
We study the rate-distortion problem for both scalar and vector memoryless heavy-tailed $\alpha$-stable sources ($0 < \alpha < 2$). Using a recently defined notion of ``strength" as a power measure, we derive the rate-distortion function for $\alpha$-stable sources subject to a constraint on the strength of the error, and show it to be logarithmic in the strength-to-distortion ratio. We showcase how our framework paves the way to finding optimal quantizers for $\alpha$-stable sources and more generally to heavy-tailed ones. In addition, we study high-rate scalar quantizers and show that uniform ones are asymptotically optimal under the strength measure. We compare uniform Gaussian and Cauchy quantizers and show that more representation points for the Cauchy source are required to guarantee the same quantization quality. Our findings generalize the well-known rate-distortion and quantization results of Gaussian sources ($\alpha = 2$) under a quadratic distortion measure.
- [85] arXiv:2411.08568 [pdf, html, other]
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Title: Enumerating orders in number fieldsSubjects: Number Theory (math.NT)
We arrange the orders in an algebraic number field in a tree. This tree can be used to enumerate all orders of bounded index in the maximal order as well as the orders over some given order.
- [86] arXiv:2411.08571 [pdf, html, other]
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Title: Essential dynamics in chaotic attractorsSubjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)
We prove that if a smooth vector field $F$ of $S^3$ generates a sufficiently complicated heteroclinic knot, the flow also generates infinitely many periodic orbits, which persist under smooth perturbations which preserve the heteroclinic knot. Consequentially, we then associate a Template with the flow dynamics - regardless of whether $F$ satisfies any hyperbolicity condition or not. In addition, inspired by the Thurston-Nielsen Classification Theorem, we also conclude topological criteria for the existence of chaotic dynamics for three-dimensional flows - which we apply to study both the Rössler and Lorenz attractors.
- [87] arXiv:2411.08581 [pdf, html, other]
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Title: Groups with a Fixed Character DegreeSubjects: Group Theory (math.GR)
Let $G$ be a finite group, and let $d$ be the degree of an irreducible character of $G$ such that $|G|=d(d+e)$ for some $e>1$. Consider the case when $G$ is solvable, $d$ is square-free, and $(d,d+e)=1$. We wish to explore an equivalent condition on $G$ when $d\in\text{cd}(G)$. We show that if $d\in\text{cd}(G)$ then there is a sequence of congruences relating the prime power factors of $d+e$ to the product of prime factors of $d$ such that the product of the moduli in this sequence of congruences is $d$. Moreover, the argument will hold in both directions.
- [88] arXiv:2411.08584 [pdf, html, other]
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Title: Separated Variables on Plane Algebraic CurvesComments: 13 pagesSubjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)
We present a semi-algorithm which for any rational function $r\in\mathbb{K}(x,y)$ and any irreducible polynomial $p\in\mathbb{K}[x,y]$ decides whether the restriction of $r$ to the curve defined by $p$ is the restriction of an element of $\mathbb{K}(x)+\mathbb{K}(y)$. In case it is, it finds all such elements.
- [89] arXiv:2411.08585 [pdf, html, other]
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Title: Hardy type inequalities with mixed cylindrical-spherical weights: the general caseComments: 30 pagesSubjects: Analysis of PDEs (math.AP)
We continue our investigation of Hardy-type inequalities involving combinations of cylindrical and spherical weights. Compared to [Cora-Musina-Nazarov, Ann. Sc. Norm. Sup., 2024], where the quasi-spherical case was considered, we handle the full range of allowed parameters. This has led to the observation of new phenomena related to lack of compactness.
- [90] arXiv:2411.08591 [pdf, html, other]
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Title: Non-affine fractal hypersurfaces: construction and dimensionsComments: Submitted to a reputable journal for potential publicationSubjects: Dynamical Systems (math.DS)
This article presents the construction of a non-affine hypersurface on an $n$-simplex in $\mathbb{R}^n$. Additionally, fractal dimension of the graph of a non-affine multivariate real-valued fractal function is estimated under certain conditions. Furthermore, the upper bound of the Hausdorff dimension of the invariant probability measure supported on the graph of such fractal function is estimated.
- [91] arXiv:2411.08593 [pdf, html, other]
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Title: Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundleComments: Final version to appear in Journal of Differential Geometry and its ApplicationsSubjects: Complex Variables (math.CV); Algebraic Geometry (math.AG); Differential Geometry (math.DG)
Let $M$ be a compact complex manifold, and $D\, \subset\, M$ a reduced normal crossing divisor on it, such that the logarithmic tangent bundle $TM(-\log D)$ is holomorphically trivial. Let ${\mathbb A}$ denote the maximal connected subgroup of the group of all holomorphic automorphisms of $M$ that preserve the divisor $D$. Take a holomorphic Cartan geometry $(E_H,\,\Theta)$ of type $(G,\, H)$ on $M$, where $H\, \subset\, G$ are complex Lie groups. We prove that $(E_H,\,\Theta)$ is isomorphic to $(\rho^* E_H,\,\rho^* \Theta)$ for every $\rho\, \in\, \mathbb A$ if and only if the principal $H$--bundle $E_H$ admits a logarithmic connection $\Delta$ singular on $D$ such that $\Theta$ is preserved by the connection $\Delta$.
- [92] arXiv:2411.08595 [pdf, html, other]
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Title: Convergence Rate of Payoff-based Generalized Nash Equilibrium LearningSubjects: Optimization and Control (math.OC)
We consider generalized Nash equilibrium (GNE) problems in games with strongly monotone pseudo-gradients and jointly linear coupling constraints. We establish the convergence rate of a payoff-based approach intended to learn a variational GNE (v-GNE) in such games. While convergent algorithms have recently been proposed in this setting given full or partial information of the gradients, rate of convergence in the payoff-based information setting has been an open problem. Leveraging properties of a game extended from the original one by a dual player, we establish a convergence rate of $O(\frac{1}{t^{4/7}})$ to a v-GNE of the game.
- [93] arXiv:2411.08602 [pdf, html, other]
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Title: Projective Banach Lie bialgebras, the projective Yang-Baxter equation and projective Poisson Lie groupsSubjects: Rings and Algebras (math.RA); Differential Geometry (math.DG)
In this paper, we first introduce the notion of projective Banach Lie bialgebras as the projective tensor product analogue of Banach Lie bialgebras. Then we consider the completion of the classical Yang-Baxter equation and classical r-matrices, and propose the notions of the projective Yang-Baxter equation and projective r-matrices. As in the finite-dimensional case, we prove that every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra. Next adapting Poisson Banach-Lie groups to the projective tensor product setting, we propose the notion of projective Banach Poisson-Lie groups and show that the differentiation of a projective Banach Poisson-Lie group has the projective Banach Lie bialgebra structure. Finally considering bounded $\mathcal{O}$-operators on Banach Lie algebras, we give an equivalent description of triangular projective r-matrices.
- [94] arXiv:2411.08611 [pdf, html, other]
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Title: The Game Value of Sequential Compounds of Integers and StarsSubjects: Combinatorics (math.CO)
A combinatorial game is a two-player game without hidden information or chance elements. One of the major approaches to analyzing games in combinatorial game theory is to break down a given game position into a disjunctive sum of multiple sub-positions, then evaluate the game value of each component of the sum, and finally integrate these game values to find which player has a winning strategy in the whole position. Accordingly, finding the game value of a given position is a major topic in combinatorial game theory. The sequential compound proposed by Stromquist and Ullman is a combinatorial game consisting of two combinatorial games. In the sequential compound of games $G$ and $H$, the players make moves on $G$ until $G$ is over, and then they play on $H$. In this paper, we investigate the general properties of sequential compounds. As the main result, we give the game values of sequential compounds of a finite number of integers and stars, which are basic and typical games in combinatorial game theory.
- [95] arXiv:2411.08614 [pdf, html, other]
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Title: Longtime and chaotic dynamics in microscopic systems with singular interactionsSubjects: Analysis of PDEs (math.AP)
This paper investigates the long time dynamics of interacting particle systems subject to singular interactions. We consider a microscopic system of $N$ interacting point particles, where the time evolution of the joint distribution $f_N(t)$ is governed by the Liouville equation. Our primary objective is to analyze the system's behavior over extended time intervals, focusing on stability, potential chaotic dynamics and the impact of singularities. In particular, we aim to derive reduced models in the regime where $N \gg 1$, exploring both the mean-field approximation and configurations far from chaos, where the mean-field approximation no longer holds. These reduced models do not always emerge but in these cases it is possible to derive uniform bounds in $ L^2 $, both over time and with respect to the number of particles, on the marginals $ \left(f_{k,N}\right)_{1\leq k \leq N}$, irrespective of the initial state's chaotic nature. Furthermore, we extend previous results by considering a wide range of singular interaction kernels surpassing the traditional $L^d$ regularity barriers, $K \in W^{\frac{-2}{d+2},d+2}(\mathbb{T}^d)$, where $\mathbb{T}$ denotes the $1$-torus and $d\geq2$ is the dimension. Finally, we address the highly singular case of $K \in H^{-1}(\mathbb{T}^d)$ within high-temperature regimes, offering new insights into the behavior of such systems.
- [96] arXiv:2411.08620 [pdf, html, other]
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Title: A finitary Kronecker's lemma and large deviations in the Strong Law of Large numbers on Banach spacesComments: 29 pagesSubjects: Logic (math.LO); Probability (math.PR)
We explore the computational content of Kronecker's lemma via the proof-theoretic perspective of proof mining and utilise the resulting finitary variant of this fundamental result to provide new rates for the Strong Law of Large Numbers for random variables taking values in type $p$ Banach spaces, which in particular are very uniform in the sense that they do not depend on the distribution of the random variables. Furthermore, we provide computability-theoretic arguments to demonstrate the ineffectiveness of Kronecker's lemma and investigate the result from the perspective of Reverse Mathematics. In addition, we demonstrate how this ineffectiveness from Kronecker's lemma trickles down to the Strong Law of Large Numbers by providing a construction that shows that computable rates of convergence are not always possible. Lastly, we demonstrate how Kronecker's lemma falls under a class of deterministic formulas whose solution to their Dialectica interpretation satisfies a continuity property and how, for such formulas, one obtains an upgrade principle that allows one to lift computational interpretations of deterministic results to quantitative results for their probabilistic analogue. This result generalises the previous work of the author and Pischke.
- [97] arXiv:2411.08623 [pdf, html, other]
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Title: Non-local homogenization limits of discrete elastic spring network models with random coefficientsSubjects: Analysis of PDEs (math.AP)
This work examines a discrete elastic energy system with local interactions described by a discrete second-order functional in the symmetric gradient and additional non-local random long-range interactions. We analyze the asymptotic behavior of this model as the grid size tends to zero. Assuming that the occurrence of long-range interactions is Bernoulli distributed and depends only on the distance between the considered grid points, we derive - in an appropriate scaling regime - a fractional p-Laplace-type term as the long-range interactions' homogenized limit. A specific feature of the presented homogenization process is that the random weights of the p-Laplace-type term are non-stationary, thus making the use of standard ergodic theorems impossible. For the entire discrete energy system, we derive a non-local fractional p-Laplace-type term and a local second-order functional in the symmetric gradient. Our model can be used to describe the elastic energy of standard, homogeneous, materials that are reinforced with long-range stiff fibers.
- [98] arXiv:2411.08624 [pdf, html, other]
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Title: Gram Matrices for Isotropic VectorsComments: 23 pagesSubjects: Commutative Algebra (math.AC); High Energy Physics - Theory (hep-th)
We investigate determinantal varieties for symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories. We study the ideals of relations among functions in the matrix entries that serve as building blocks for conformal correlators.
- [99] arXiv:2411.08643 [pdf, html, other]
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Title: David regularity of the Yoccoz extensionComments: 32 pages, 8 figuresSubjects: Dynamical Systems (math.DS)
It is a central problem in the study of critical circle dynamics to understand the regularity of Yoccoz conjugators: the circle homeomorphisms that conjugate critical circle maps of irrational rotation numbers with their corresponding rigid rotations. One can shift the perspective on this problem and look at the regularity of extensions of such maps to the unit disk. Of particular interest to us, one can ask when this conjugator has David extensions. Following a conjecture of Petersen and Zakeri, we classify the David regularity of one extension process, the Yoccoz extension, and discuss its optimality.
- [100] arXiv:2411.08653 [pdf, html, other]
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Title: Hilbert space embeddings of independence tests of several variablesSubjects: Functional Analysis (math.FA); Probability (math.PR)
In this paper, we present the general theory of embedding independence tests on Hilbert spaces that generalizes the concepts of distance covariance, distance multivariance and HSIC. This is done by defining new types of kernel on an $n$ Cartesian product called positive definite independent of order $k$. An emphasis is given on the continuous case in order to obtain a version of the Kernel Mean Embedding for this new classes of kernels. We also provide $2$ explicit methods to construct examples for this new type of kernel on a general space by using Bernstein functions of several variables and completely monotone functions of higher order.
- [101] arXiv:2411.08657 [pdf, html, other]
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Title: Well-posedness and inverse problems for the nonlocal third-order acoustic equation with time-dependent nonlinearitySubjects: Analysis of PDEs (math.AP)
In this paper, we study the inverse problems of simultaneously determining a potential term and a time-dependent nonlinearity for the nonlinear Moore-Gibson-Thompson equation with a fractional Laplacian. This nonlocal equation arises in the field of peridynamics describing the ultrasound waves of high amplitude in viscous thermally fluids. We first show the well-posedness for the considered nonlinear equations in general dimensions with small exterior data. Then, by applying the well-known linearization approach and the unique continuation property for the fractional Laplacian, the potential and nonlinear terms are uniquely determined by the Dirichlet-to-Neumann (DtN) map taking on arbitrary subsets of the exterior in the space-time domain. The uniqueness results for general nonlinearities to some extent extend the existing works for nonlocal wave equations. Particularly, we also show a uniqueness result of determining a time-dependent nonlinear coefficient for a 1-dimensional fractional Jordan-Moore-Gibson-Thompson equation of Westervelt type.
- [102] arXiv:2411.08675 [pdf, html, other]
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Title: Twisted Kitaev quantum double model as local topological orderComments: 29 pagesSubjects: Quantum Algebra (math.QA)
We study the twisted Kitaev quantum double model within the framework of Local Topological Order (LTO). We extend its definition to arbitrary 2D lattices, enabling an explicit characterization of the ground state space through invariant spaces of monomial representations. We reformulate the LTO conditions for including general lattices and prove that the twisted model satisfies all four LTO axioms on any 2D lattice. As a corollary, we show that its ground state space is a quantum error-correcting code.
- [103] arXiv:2411.08679 [pdf, html, other]
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Title: Connected components of the space of triples of transverse partial flags in $\mathrm{SO}_0(p,q)$ and Anosov representationsComments: 35 pages, 7 figuresSubjects: Differential Geometry (math.DG); Group Theory (math.GR); Representation Theory (math.RT)
We count the number of connected components in the space of triples of transverse flags in any flag manifold of $\mathrm{SO}_0(p,q)$. We compute the effect the involution of the unipotent radical has on those components and deduce that for certain parabolic subgroups $P_{\Theta}$, any $P_{\Theta}$-Anosov subgroup is virtually isomorphic to either a surface group of a free group. We give examples of Anosov subgroups which are neither free nor surface groups for some sets of roots which do not fall under the previous results. As a consequence of the methods developed here, we get an explicit algorithm based on computation of minors to check if a unipotent matrix in $\mathrm{SO}_0(p,q)$ belong to the $\Theta$-positive semigroup $U_\Theta^{>0}$ when $p\neq q$.
- [104] arXiv:2411.08680 [pdf, html, other]
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Title: Integrated Precoder and Trajectory Design for MIMO UAV-Assisted Relay System With Finite-Alphabet InputsSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Unmanned aerial vehicles (UAVs) are gaining widespread use in wireless relay systems due to their exceptional flexibility and cost-effectiveness. This paper focuses on the integrated design of UAV trajectories and the precoders at both the transmitter and UAV in a UAV-assisted relay communication system, accounting for transmit power constraints and UAV flight limitations. Unlike previous works that primarily address multiple-input single-output (MISO) systems with Gaussian inputs, we investigate a more realistic scenario involving multiple-input multiple-output (MIMO) systems with finite-alphabet inputs. To tackle the challenging and inherently non-convex problem, we propose an efficient solution algorithm that leverages successive convex approximation and alternating optimization techniques. Simulation results validate the effectiveness of the proposed algorithm, demonstrating its capability to optimize system performance.
- [105] arXiv:2411.08685 [pdf, html, other]
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Title: Long induced paths in sparse graphs and graphs with forbidden patternsComments: 26 pages, 8 figures. Comments welcome !Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete bipartite graph $K_{t,t}$ as a subgraph. In this case we show that $G$ has an induced path of order $(\log \log n)^{1/5-o(1)}$. This is an exponential improvement over a result of Galvin, Rival, and Sands (1982) and comes close to a recent upper bound of order $O((\log \log n)^2)$.
Another way to approach this problem is by viewing $G$ as an ordered graph (where the vertices are ordered according to their position on the path $P$). From this point of view it is most natural to consider which ordered subgraphs need to be forbidden in order to force the existence of a long induced path. Focusing on the exclusion of ordered matchings, we improve or recover a number of existing results with much simpler proofs, in a unified way. We also show that if some forbidden ordered subgraph forces the existence of a long induced path in $G$, then this induced path has size at least $\Omega((\log \log \log n)^{1/3})$, and can be chosen to be increasing with respect to $P$. - [106] arXiv:2411.08690 [pdf, html, other]
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Title: Eisenstein classes and generating series of modular symbols in $\mathrm{SL}_N$Subjects: Number Theory (math.NT)
We define a theta lift between the homology in degree $N-1$ of a locally symmetric space associated to $\mathrm{SL}_N(\mathbb{R})$ and the space of modular forms of weight $N$. We show that the Fourier coefficients of this lift are Poincaré duals to modular symbols associated to maximal parabolic subgroups. The constant term is a canonical cohomology classes obtained by transgressing the Euler class of a torus bundle. This Eisenstein lift realizes a geometric theta correspondence for the pair $\mathrm{SL}_N \times \mathrm{SL}_2$, in the spirit of Kudla-Millson. When $N=2$, we show that the lift surjects on the space of weight $2$ modular forms spanned by an Eisenstein series and the eigenforms with non-vanishing $L$-function.
- [107] arXiv:2411.08695 [pdf, html, other]
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Title: Derived categories of Quot schemes on smooth curves and tautological bundlesSubjects: Algebraic Geometry (math.AG); Representation Theory (math.RT)
We define a categorical action of the shifted quantum loop group of $\mathfrak{sl}_2$ on the derived categories of Quot schemes of finite length quotient sheaves on a smooth projective curve. As an application, we obtain a semi-orthogonal decomposition of the derived categories of Quot schemes, of representation theoretic origin. We use this decomposition to calculate the cohomology of interesting tautological vector bundles over the Quot scheme.
- [108] arXiv:2411.08697 [pdf, html, other]
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Title: Local generation of tilingsSubjects: Dynamical Systems (math.DS)
In this article, we investigate the possibility of generating all the configurations of a subshift in a local way. We propose two definitions of local generation, explore their properties and develop techniques to determine whether a subshift satisfies these definitions. We illustrate the results with several examples.
- [109] arXiv:2411.08702 [pdf, html, other]
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Title: A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz MethodsComments: 19 pages, 13 figuresSubjects: Numerical Analysis (math.NA)
We introduce a deep learning-based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics-informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary conditions, even for singularly perturbed problems.
We provide a comprehensive mathematical analysis demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm through numerical experiments, including high-dimensional, singularly perturbed problems and those posed over non-convex domains. - [110] arXiv:2411.08711 [pdf, html, other]
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Title: Duality on symmetric multiple polylogarithmsSubjects: Number Theory (math.NT)
There are three kinds of multiple polylogarithms; complex, finite and symmetric. The dualities for the complex and finite cases are known. In this paper, we present proofs of them via iterated integrals and its symmetric counterpart by a similar method.
- [111] arXiv:2411.08713 [pdf, html, other]
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Title: An existence result in annular regions times conical shells and its application to nonlinear Poisson systemsComments: 18 pages, 2 figuresSubjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA)
We provide a new existence result for abstract nonlinear operator systems in normed spaces, by means of topological methods. The solution is located within the product of annular regions and conical shells. The theoretical result possesses a wide range of applicability, which, for concreteness, we illustrate in the context of systems of nonlinear Poisson equations subject to homogeneous Dirichlet boundary conditions. For the latter problem we obtain existence and localization of solutions having all components nontrivial. This is also illustrated with an explicit example in which we also furnish a numerically approximated solution, consistent with the theoretical results.
- [112] arXiv:2411.08716 [pdf, html, other]
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Title: An infinite family of hyperbolic 3-manifolds without tight projectively Anosov flowsComments: 18 pages, 3 figures. Comments welcome!Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)
In this paper we find the first infinite family of hyperbolic 3-manifolds which admit tight contact structures but do not have any tight projectively Anosov flow. These manifolds are obtained as rational surgeries on the figure eight knot.
- [113] arXiv:2411.08722 [pdf, html, other]
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Title: Shadow systems, decomposability and isotropic constantsComments: 25 pages, 2 figuresSubjects: Metric Geometry (math.MG)
We study necessary conditions for local maximizers of the isotropic constant that are related to notions of decomposability. Our main result asserts that the polar body of a local maximizer of the isotropic constant can only have few Minkowski summands; more precisely, its dimension of decomposability is at most $\frac12(n^2+3n)$. Using a similar proof strategy, a result by Campi, Colesanti and Gronchi concerning RS-decomposability is extended to a larger class of shadow systems. We discuss the polytopal case, which turns out to have connections to (affine) rigidity theory, and investigate how the bound on the maximal number of irredundant summands can be improved if we restrict our attention to convex bodies with certain symmetries.
- [114] arXiv:2411.08725 [pdf, html, other]
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Title: Berry-Esseen bounds for large-time asymptotics of one-dimensional diffusion processes via Malliavin-Stein methodComments: 41 pagesSubjects: Probability (math.PR)
We consider solutions of stochastic differential equations which diverge to infinity as the time parameter goes to infinity. If the coefficients converge as the spacial variable goes to infinity, then the solutions will get close to some Gaussian processes with positive drifts as the time parameter goes to infinity. In this paper, we prove Berry-Esseen type bounds for the solutions in this setting. In particular, we obtain bounds of the total variation distance between the law of the centered and scaled solutions of the stochastic differential equations and the standard normal distribution with an optimal rate of convergence in the time parameter. In the proof we apply the Malliavin-Stein method to estimate the total variation distance.
- [115] arXiv:2411.08746 [pdf, other]
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Title: Higher K-theory of forms II. From exact categories to chain complexesSubjects: K-Theory and Homology (math.KT)
We prove basic statements about the Hermitian K-theory of exact form categories with weak equivalences. Notably, we extend a quadratic functor with values in abelian groups from an exact category to its category of bounded chain complexes in a way that does not change Grothendieck-Witt spaces. This is used in joint work with Marlowe for the comparison of the classical 1-categorical version of the Hermitian K-theory of exact categories with the infinity-categorical version of Calmes-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle.
- [116] arXiv:2411.08748 [pdf, html, other]
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Title: Continuity of matings of Kleinian groups and polynomialsComments: 29 pages, 10 figuresSubjects: Dynamical Systems (math.DS)
In recent years, the study of holomorphic correspondences as dynamical systems that can display behaviors of both rational maps and Kleinian groups has gained a good amount of attention. This phenomenon is related to the Sullivan dictionary, a list of parallels between the theories of these two systems. We build upon a surgical construction of such matings, due to Bullett and Harvey, increasing the degree of maps we consider, and proving regularity properties of the mating map on parameter spaces: namely, analyticity on the interior of its domain of definition, and continuity under quasiconformal rigidity on the boundary.
- [117] arXiv:2411.08749 [pdf, html, other]
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Title: Maximal root subsystems of affine reflection systems and dualitySubjects: Rings and Algebras (math.RA)
Any maximal root subsystem of a finite crystallographic reduced root system is either a closed root subsystem or its dual is a closed root subsystem in the dual root system. In this article, we classify the maximal root subsystems of an affine reflection system (reduced and non-reduced) and prove that this result holds in much more generality for reduced affine reflection systems. Moreover, we explicitly determine when a maximal root subsystem is a maximal closed root subsystem. Using our classification, at the end, we characterize the maximal root systems of affine reflection systems with nullity less than or equal to $2$ using Hermite normal forms; especially for Saito's EARS of nullity $2.$ This in turn classifies the maximal subgroups of the Weyl group of an affine reflection system that are generated by reflections.
- [118] arXiv:2411.08750 [pdf, html, other]
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Title: Optimal Transport-Based Displacement Interpolation with Data Augmentation for Reduced Order Modeling of Nonlinear Dynamical SystemsSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG)
We present a novel reduced-order Model (ROM) that leverages optimal transport (OT) theory and displacement interpolation to enhance the representation of nonlinear dynamics in complex systems. While traditional ROM techniques face challenges in this scenario, especially when data (i.e., observational snapshots) is limited, our method addresses these issues by introducing a data augmentation strategy based on OT principles. The proposed framework generates interpolated solutions tracing geodesic paths in the space of probability distributions, enriching the training dataset for the ROM. A key feature of our approach is its ability to provide a continuous representation of the solution's dynamics by exploiting a virtual-to-real time mapping. This enables the reconstruction of solutions at finer temporal scales than those provided by the original data. To further improve prediction accuracy, we employ Gaussian Process Regression to learn the residual and correct the representation between the interpolated snapshots and the physical solution. We demonstrate the effectiveness of our methodology with atmospheric mesoscale benchmarks characterized by highly nonlinear, advection-dominated dynamics. Our results show improved accuracy and efficiency in predicting complex system behaviors, indicating the potential of this approach for a wide range of applications in computational physics and engineering.
- [119] arXiv:2411.08757 [pdf, other]
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Title: The Non-Commutative Brillouin Torus, a Non-Commutative Geometry perspectiveSubjects: Mathematical Physics (math-ph)
Non-commutative geometry has significantly contributed to quantum mechanics by providing mathematical tools to extract topological and geometrical information from these systems. This thesis explores the methods used by Jean Bellissard and collaborators for analyzing homogeneous materials. The focus is on two topological algebras that extend Fourier analysis over $\mathbb{T}^d$ to study tight-binding models for homogeneous materials. These algebras, generalizations of $C(\mathbb{T}^d)$ and $C^{\infty}(\mathbb{T}^d)$, are considered anon-commutative smooth manifold, referred to as the Non-Commutative Brillouin Torus. Techniques from Fourier analysis, such as Fourier coefficients and Fejér summation, are adapted to this context, capturing the topological and smooth structure of the non-commutative manifold. The topological structure is represented by a C* algebra, while the smooth structure is captured by a smooth sub-algebra, forming the basis for constructing topological invariants of Hamiltonians through continuous cyclic cohomology.
Additionally, the K theory of C* algebras and smooth sub-algebras is a crucial tool for studying topological invariants of Hamiltonians. Smooth sub-algebras share a functional calculus with C* algebras, providing sufficient information to study their K theory. This relationship is vital for identifying topological invariants of Hamiltonians, particularly under conditions like low temperature and electron density, which contribute to the quantization of transversal conductivity in homogeneous materials. This thesis aims to be a valuable resource for newcomers to the field. - [120] arXiv:2411.08760 [pdf, html, other]
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Title: Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn EquationsSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Computational Physics (physics.comp-ph)
This paper investigates a numerical solution of Allen-Cahn equation with constant and degenerate mobility, with polynomial and logarithmic energy functionals, with deterministic and random initial functions, and with advective term in one, two, and three spatial dimensions, based on the physics-informed neural network (PINN). To improve the learning capacity of the PINN, we incorporate the energy dissipation property of the Allen-Cahn equation as a penalty term into the loss function of the network. To facilitate the learning process of random initials, we employ a continuous analogue of the initial random condition by utilizing the Fourier series expansion. Adaptive methods from traditional numerical analysis are also integrated to enhance the effectiveness of the proposed PINN. Numerical results indicate a consistent decrease in the discrete energy, while also revealing phenomena such as phase separation and metastability.
- [121] arXiv:2411.08772 [pdf, html, other]
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Title: On an indivisibility version of Iizuka's conjectureSubjects: Number Theory (math.NT)
In this paper, we establish the existence of positive density collection of $d\in\mathbb{N}$ such that class numbers of $\mathbb{Q}(\sqrt{d}), \ \mathbb{Q}(\sqrt{d+1})\dots\mathbb{Q}(\sqrt{d+n})$ are not divisible by $3^k$ for $n=3^{k+1}-5$ for any $k\in\mathbb{N}$. This result constitutes the indivisibility counterpart of Iizuka's conjecture. For the same choice of $n$, we prove the existence of positive density collection of $d$, in the set of negative integers, such that the class numbers of $\mathbb{Q}(\sqrt{d}), \ \mathbb{Q}(\sqrt{d+1}),\dots,\mathbb{Q}(\sqrt{d+n})$ are not divisible by $3^{k+1}$. Further, we write the set of all square-free natural numbers as an increasing sequence $(d_n)$ and prove the existence of positive density collection of $i$ in the set of natural numbers such that the class numbers of the number fields $\mathbb{Q}(\sqrt{d_i}), \ \mathbb{Q}(\sqrt{d_{i+1}}),\ \mathbb{Q}(\sqrt{d_{i+2}}),$ $ \mathbb{Q}(\sqrt{d_{i+3}}),\dots, \mathbb{Q}(\sqrt{d_{i+n}})$ are not divisible by $3^k$ for $n=3^{k+1}-5$. For higher degree, we show that certain limit of the collection of imaginary bi-quadratic fields whose class number is not divisible by $3$ over all the imaginary biquadratic fields is positive.
- [122] arXiv:2411.08775 [pdf, html, other]
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Title: Algorithms in 4-manifold topologyStefan Bastl, Rhuaidi Burke, Rima Chatterjee, Subhankar Dey, Alison Durst, Stefan Friedl, Daniel Galvin, Alejandro García Rivas, Tobias Hirsch, Cara Hobohm, Chun-Sheng Hsueh, Marc Kegel, Frieda Kern, Shun Ming Samuel Lee, Clara Löh, Naageswaran Manikandan, Léo Mousseau, Lars Munser, Mark Pencovitch, Patrick Perras, Mark Powell, José Pedro Quintanilha, Lisa Schambeck, David Suchodoll, Martin Tancer, Annika Thiele, Paula Truöl, Matthias Uschold, Simona Veselá, Melvin Weiß, Magdalina von Wunsch-RolshovenComments: 24 pages, 1 FigureSubjects: Geometric Topology (math.GT)
We show that there exists an algorithm that takes as input two closed, simply connected, topological 4-manifolds and decides whether or not these 4-manifolds are homeomorphic. In particular, we explain in detail how closed, simply connected, topological 4-manifolds can be naturally represented by a Kirby diagram consisting only of 2-handles. This representation is used as input for our algorithm. Along the way, we develop an algorithm to compute the Kirby-Siebenmann invariant of a closed, simply connected, topological 4-manifold from any of its Kirby diagrams and describe an algorithm that decides whether or not two intersection forms are isometric.
In a slightly different direction, we discuss the decidability of the stable classification of smooth manifolds with more general fundamental groups. Here we show that there exists an algorithm that takes as input two closed, oriented, smooth 4-manifolds with fundamental groups isomorphic to a finite group with cyclic Sylow 2-subgroup, an infinite cyclic group, or a group of geometric dimension at most 3 (in the latter case we additionally assume that the universal covers of both 4-manifolds are not spin), and decides whether or not these two 4-manifolds are orientation-preserving stably diffeomorphic. - [123] arXiv:2411.08782 [pdf, html, other]
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Title: Restoring Accessibility During Urban Rail Disruptions via Bus Network RedesignSubjects: Optimization and Control (math.OC)
In broad terms, accessibility measures opportunities reachable (such as shops, residents, etc.) within a given time frame. Urban Rail Transit (URT) plays a crucial role in providing accessibility, but it is susceptible to disruptions. In city centers with dense public transport (PT) networks, travelers can often find alternative lines. However, in suburbs where PT is sparse, disruptions have a more significant impact on accessibility. The traditional approach consists in deploying bridge and replacement buses to mitigate URT disruptions without specific care to accessibility. Yet, the question arises: is this approach the most effective way to restore accessibility? To the best of our knowledge, our paper is the first to propose a bus re-routing method with the objective of restoring accessibility during URT disruptions. We formulate an integer program and develop a two-stage heuristic algorithm to maximize restored accessibility. The efficacy of our method is always the present assessed in Évry-Courcouronnes and Choisy-le-Roi, France. The results show that, compared to conventional replacement methods, our strategy improves accessibility in particular in the areas most affected by the disruption. Such results are observed even when no additional vehicles are deployed, and at the same time, achieving a reduction in the kilometers traveled. Despite it is well understood that accessibility is the most relevant benefit a transportation system can produce, this aspect is reflected by the traditional approaches in remediation to disruption. With this work, we show instead how to make accessibility the main guiding principle in remediation.
- [124] arXiv:2411.08786 [pdf, html, other]
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Title: Conditional reasoning and the shadows it casts onto the first-order logic: the Nelsonian caseComments: 53 pages, 1 figureSubjects: Logic (math.LO)
We define a natural notion of standard translation for the formulas of conditional logic which is analogous to the standard translation of modal formulas into the first-order logic. We briefly show that this translation works (modulo a lightweight first-order encoding of the conditional models) for the minimal classical conditional logic $\mathsf{CK}$ introduced by Brian Chellas; however, the main result of the paper is that a classically equivalent reformulation of these notions (i.e. of standard translation plus theory of conditional models) also faithfully embeds the basic Nelsonian conditional logic $\mathsf{N4CK}$, introduced in arXiv:2311.02361 into $\mathsf{QN4}$, the paraconsistent variant of Nelson's first-order logic of strong negation. Thus $\mathsf{N4CK}$ is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of $\mathsf{CK}$ on the non-classical basis provided by the propositional fragment of $\mathsf{QN4}$. Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which allows to improve an older result by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic $\mathsf{FSK}^d$ into $\mathsf{QN4}$.
- [125] arXiv:2411.08796 [pdf, html, other]
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Title: Optimal stopping for Markov processes with positive jumpsComments: 20 pages, 2 figuresSubjects: Probability (math.PR)
Consider the discounted optimal stopping problem for a real valued Markov process with only positive jumps. We provide a theorem to verify that the optimal stopping region has the form {x >= x^*} for some critical threshold x^*, and a representation formula for the value function of the problem in terms of the Green kernel of the process, based on Dynkin's characterization of the value function as the least excessive majorant. As an application of our results, using the Fourier transform to compute the Green kernel of the process, we solve a new example: the optimal stopping for a Levy-driven Ornstein-Uhlenbeck process used to model prices in electricity markets.
- [126] arXiv:2411.08797 [pdf, html, other]
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Title: Complexity of Finite Borel Asymptotic DimensionComments: 15 pagesSubjects: Logic (math.LO); Combinatorics (math.CO)
We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\mathbf{\Sigma}^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated by a single Borel function. As an application of this characterization, we classify the complexities of digraph homomorphism problems for this class of graphs.
- [127] arXiv:2411.08799 [pdf, html, other]
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Title: On limiting distributions of arithmetic functionsComments: 18 pages, Comments welcomeSubjects: Number Theory (math.NT); Probability (math.PR)
For a natural number n, let M(n) denote the maximum exponent of any prime power dividing n, and let m(n) denote the minimum exponent of any prime power dividing n. We study the second moments of these arithmetic functions and establish their limiting distributions. We introduce a new discrete probabilistic distribution dependent on a function f taking values in [0,1], study its first two moments, and provide examples of several arithmetic functions satisfying such distribution as their limiting behavior.
- [128] arXiv:2411.08803 [pdf, html, other]
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Title: On the Terwilliger algebra of the group association scheme of the symmetric group $\operatorname {sym}(7)$Subjects: Combinatorics (math.CO); Rings and Algebras (math.RA)
Terwilliger algebras are finite-dimensional semisimple algebras that were first introduced by Paul Terwilliger in 1992 in studies of association schemes and distance-regular graphs. The Terwilliger algebras of the conjugacy class association schemes of the symmetric groups $\operatorname {sym}(n)$, for $3\leq n \leq 6$, have been studied and completely determined. The case for $\operatorname {sym}(7)$ is computationally much more difficult and has a potential application to find the size of the largest permutation codes of $\operatorname {sym}(7)$ with a minimal distance of at least $4$. In this paper, the dimension, the Wedderburn decomposition, and the block dimension decomposition of the Terwilliger algebra of the conjugacy class scheme of the group $\operatorname {sym}(7)$ are determined.
- [129] arXiv:2411.08806 [pdf, html, other]
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Title: Numerically flat foliations and holomorphic Poisson geometryComments: To Jean-Pierre Demailly, in memoriam. 27 pages, comments welcomeSubjects: Algebraic Geometry (math.AG); Differential Geometry (math.DG); Symplectic Geometry (math.SG)
We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact Kähler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that such foliations induce a decomposition of the tangent bundle of the ambient manifold, have leaves uniformized by Euclidean spaces, and have torsion canonical bundle. Additionally, we prove that smooth two-dimensional foliations with numerically trivial canonical bundle on projective manifolds are either isotrivial fibrations or have numerically flat tangent bundles. This in turn implies a global Weinstein splitting theorem for rank-two Poisson structures on projective manifolds. We also derive new Hodge-theoretic conditions for the existence of zeros of Poisson structures on compact Kähler manifolds.
- [130] arXiv:2411.08808 [pdf, html, other]
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Title: Products of pseudofinite structuresSubjects: Logic (math.LO)
We prove that any product of a family of pseudofinite structures is pseudofinite. The main tools are the fundamental results on products of first order structures due to Feferman and Vaught.
- [131] arXiv:2411.08809 [pdf, html, other]
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Title: The Impact of Social Value Orientation on Nash Equilibria of Two Player Quadratic GamesSubjects: Optimization and Control (math.OC)
We consider two player quadratic games in a cooperative framework known as social value orientation, motivated by the need to account for complex interactions between humans and autonomous agents in dynamical systems. Social value orientation is a framework from psychology, that posits that each player incorporates the other player's cost into their own objective function, based on an individually pre-determined degree of cooperation. The degree of cooperation determines the weighting that a player puts on their own cost relative to the other player's cost. We characterize the Nash equilibria of two player quadratic games under social value orientation by creating expansions that elucidate the relative difference between this new equilibria (which we term the SVO-Nash equilibria) and more typical equilibria, such as the competitive Nash equilibria, individually optimal solutions, and the fully cooperative solution. Specifically, each expansion parametrizes the space of cooperative Nash equilibria as a family of one-dimensional curves where each curve is computed by solving an eigenvalue problem. We show that both bounded and unbounded equilibria may exist. For equilibria that are bounded, we can identify bounds as the intersection of various ellipses; for equilibria that are unbounded, we characterize conditions under which unboundedness will occur, and also compute the asymptotes that the unbounded solutions follow. We demonstrate these results in trajectory coordination scenario modeled as a linear time varying quadratic game.
- [132] arXiv:2411.08828 [pdf, html, other]
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Title: Study of dynamical symmetrietry algebra of $\Psi_2$-Humbert functionSubjects: Classical Analysis and ODEs (math.CA); Operator Algebras (math.OA)
The study is devoted to the construction of dynamical symmetry algebra of confluent hypergeometric function $_1F_1$ and $\Psi_2$-Humbert function and to derive some generating relations and reduction formulas for $_1F_1$ and $\Psi_2$ functions.
- [133] arXiv:2411.08829 [pdf, html, other]
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Title: Embeddings of anisotropic Sobolev spaces into spaces of anisotropic H\"{o}lder-continuous functionsJournal-ref: Demonstr. Math. 57:1 (2024), art. 20240079, 10 ppSubjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP)
We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent Hölder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of Hölder continuity to anisotropic settings with variable exponents, providing deeper insight into the regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.
- [134] arXiv:2411.08830 [pdf, html, other]
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Title: Homogeneous quadratic Lie super algebrasSubjects: Rings and Algebras (math.RA)
In this work we state a version of the double extension for homogeneous quadratic Lie super algebras that includes even and odd cases. We prove that any indecomposable, non-simple and homogeneous quadratic Lie super algebra is obtained by means of this type of double extension. We also show that with this construction we can recover previously studied cases as well as some other which can not be recover with former versions of double extensions.
- [135] arXiv:2411.08839 [pdf, html, other]
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Title: Sparser Abelian High Dimensional ExpandersSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
We present two new explicit constructions of Cayley high dimensional expanders (HDXs) over the abelian group $\mathbb{F}_2^n$. Our expansion proofs use only linear algebra and combinatorial arguments.
The first construction gives local spectral HDXs of any constant dimension and subpolynomial degree $\exp(n^\epsilon)$ for every $\epsilon >0$, improving on a construction by Golowich [Gol23] which achieves $\epsilon =1/2$. [Gol23] derives these HDXs by sparsifying the complete Grassmann poset of subspaces. The novelty in our construction is the ability to sparsify any expanding Grassmannian posets, leading to iterated sparsification and much smaller degrees. The sparse Grassmannian (which is of independent interest in the theory of HDXs) serves as the generating set of the Cayley graph.
Our second construction gives a 2-dimensional HDXs of any polynomial degree $\exp(\epsilon n$) for any constant $\epsilon > 0$, which is simultaneously a spectral expander and a coboundary expander. To the best of our knowledge, this is the first such non-trivial construction. We name it the Johnson complex, as it is derived from the classical Johnson scheme, whose vertices serve as the generating set of this Cayley graph. This construction may be viewed as a derandomization of the recent random geometric complexes of [LMSY23]. Establishing coboundary expansion through Gromov's "cone method" and the associated isoperimetric inequalities is the most intricate aspect of this construction.
While these two constructions are quite different, we show that they both share a common structure, resembling the intersection patterns of vectors in the Hadamard code. We propose a general framework of such "Hadamard-like" constructions in the hope that it will yield new HDXs. - [136] arXiv:2411.08841 [pdf, other]
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Title: Spectral equivalence of nearby LagrangiansComments: 99 pages, no figures. Comments welcomeSubjects: Symplectic Geometry (math.SG); Algebraic Topology (math.AT)
Let $R$ be a commutative ring spectrum. We construct the wrapped Donaldson--Fukaya category with coefficients in $R$ of any stably polarized Liouville sector. We show that any two $R$-orientable and isomorphic objects admit $R$-orientations so that their $R$-fundamental classes coincide. Our main result is that any closed exact Lagrangian $R$-brane in the cotangent bundle of a closed manifold is isomorphic to an $R$-brane structure on the zero section in the wrapped Donaldson--Fukaya category, generalizing a well-known result over the integers. To achieve this, we prove that the Floer homotopy type of the cotangent fiber is given by the stable homotopy type of the based loop space of the zero section.
- [137] arXiv:2411.08846 [pdf, html, other]
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Title: On the number of crossings and bouncings of a diffusion at a sticky thresholdSubjects: Probability (math.PR)
We distinguish between three types of crossing behaviors at a sticky threshold for a one-dimensional sticky diffusion process and analyze the asymptotic properties of the three corresponding number of crossings statistics in case the process is sticky Brownian motion. To each type of crossings corresponds a distinct asymptotic regime of the statistic. Additionally, we consider three types of bouncing behavior as symmetric counterparts to crossing behaviors. The number of bouncings behaves similarly to its number of crossings counterpart when the process is sticky Brownian motion. From this we deduce the behavior of the number of bouncings of sticky-reflected Brownian motion. The results on crossings and bouncings are extended to sticky diffusions. As an application, we propose consistent estimators for the stickiness parameter of sticky diffusions and sticky-reflected Brownian motion.
- [138] arXiv:2411.08848 [pdf, html, other]
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Title: Stationary random measures : Covariance asymptotics, variance bounds and central limit theoremsComments: 40 pagesSubjects: Probability (math.PR)
We consider covariance asymptotics for linear statistics of general stationary random measures in terms of their truncated pair correlation measure. We give exact infinite series-expansion formulas for covariance of smooth statistics of random measures involving higher-order integrals of the truncated correlation measures and higher-order derivatives of the test functions and also equivalently in terms of their Fourier transforms. Exploiting this, we describe possible covariance and variance asymptotics for Sobolev and indicator statistics. In the smooth case, we show that that order of variance asymptotics drops by even powers and give a simple example of random measure exhibiting such a variance reduction. In the case of indicator statistics of C1-smooth sets, we derive covariance asymptotics at surface-order scale with the limiting constant depending on intersection of the boundaries of the two sets. We complement this by a lower bound for random measures with a non-trivial atomic part. Restricting to simple point processes, we prove a central limit theorem for Sobolev and Holder continuous statistics of simple point processes satifying a certain integral identity for higher-order truncated correlation functions.
- [139] arXiv:2411.08863 [pdf, html, other]
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Title: Probability Laws Concerning Zeta IntegralsComments: 9 pagesSubjects: Number Theory (math.NT); Probability (math.PR)
We give a probabilistic interpretation of the Dedekind zeta functions of $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-2})$ using zeta integrals and use this to show that the first two Li coefficients of these zeta functions are positive. This extends a result of Biane, Pitman, and Yor (2001) which considered the case of the Riemann zeta function.
- [140] arXiv:2411.08871 [pdf, html, other]
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Title: Restriction estimates using decoupling theorems and two-ends Furstenberg inequalitiesComments: This paper supersedes arXiv:2210.03878Subjects: Classical Analysis and ODEs (math.CA); Combinatorics (math.CO); Metric Geometry (math.MG)
We propose to study the restriction conjecture using decoupling theorems and two-ends Furstenberg inequalities. Specifically, we pose a two-ends Furstenberg conjecture, which implies the restriction conjecture. As evidence, we prove this conjecture in the plane by using the Furstenberg set estimate. Moreover, we use this planar result to prove a restriction estimate for $p>22/7$ in three dimensions, which implies Wolff's $5/2$-hairbrush bound for Kakeya sets in $\mathbb{R}^3$. Our approach also makes improvements for the restriction conjecture in higher dimensions.
- [141] arXiv:2411.08872 [pdf, html, other]
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Title: Large Wireless Model (LWM): A Foundation Model for Wireless ChannelsComments: The LWM model and relevant scripts are available on the LWM website: this https URLSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
This paper presents the Large Wireless Model (LWM) -- the world's first foundation model for wireless channels. Designed as a task-agnostic model, LWM generates universal, rich, contextualized channel embeddings (features) that potentially enhance performance across a wide range of downstream tasks in wireless communication and sensing systems. Towards this objective, LWM, which has a transformer-based architecture, was pre-trained in a self-supervised manner on large-scale wireless channel datasets. Our results show consistent improvements in classification and regression tasks when using the LWM embeddings compared to raw channel representations, especially in scenarios with high-complexity machine learning tasks and limited training datasets. This LWM's ability to learn from large-scale wireless data opens a promising direction for intelligent systems that can efficiently adapt to diverse tasks with limited data, paving the way for addressing key challenges in wireless communication and sensing systems.
New submissions (showing 141 of 141 entries)
- [142] arXiv:2411.08064 (cross-list from physics.flu-dyn) [pdf, html, other]
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Title: Energy and entropy conserving compatible finite elements with upwinding for the thermal shallow water equationsSubjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph); Geophysics (physics.geo-ph)
In this work, we develop a new compatible finite element formulation of the thermal shallow water equations that conserves energy and mathematical entropies given by buoyancy-related quadratic tracer variances. Our approach relies on restating the governing equations to enable discontinuous approximations of thermodynamic variables and a variational continuous time integration. A key novelty is the inclusion of centred and upwinded fluxes. The proposed semi-discrete system conserves discrete entropy for centred fluxes, monotonically damps entropy for upwinded fluxes, and conserves energy. The fully discrete scheme reflects entropy conservation at the continuous level. The ability of a new linearised Jacobian, which accounts for both centred and upwinded fluxes, to capture large variations in buoyancy and simulate thermally unstable flows for long periods of time is demonstrated for two different transient case studies. The first involves a thermogeostrophic instability where including upwinded fluxes is shown to suppress spurious oscillations while successfully conserving energy and monotonically damping entropy. The second is a double vortex where a constrained fully discrete formulation is shown to achieve exact entropy conservation in time.
- [143] arXiv:2411.08083 (cross-list from q-bio.PE) [pdf, html, other]
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Title: An age-structured diffusive model for epidemic modelling: Lie symmetries and exact solutionsSubjects: Populations and Evolution (q-bio.PE); Mathematical Physics (math-ph)
A new age-structured diffusive model for the mathematical modelling of epidemics is suggested. The model can be considered as a generalization of two models suggested earlier for the same purposes. The Lie symmetry classification of the model is derived. It is shown that the model admits an infinite-dimensional Lie algebra of invariance. Using the Lie symmetries, exact solutions, in particular those of the travelling wave types and in terms of special functions, are constructed. An example of application of the correctly-specified exact solution for calculation of total numbers of infected individuals during an epidemic is presented.
- [144] arXiv:2411.08085 (cross-list from cs.LG) [pdf, other]
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Title: Deep Learning 2.0: Artificial Neurons That Matter -- Reject Correlation, Embrace OrthogonalityComments: Submitted to CVPR 2025Subjects: Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV); General Topology (math.GN)
We introduce a yat-product-powered neural network, the Neural Matter Network (NMN), a breakthrough in deep learning that achieves non-linear pattern recognition without activation functions. Our key innovation relies on the yat-product and yat-product, which naturally induces non-linearity by projecting inputs into a pseudo-metric space, eliminating the need for traditional activation functions while maintaining only a softmax layer for final class probability distribution. This approach simplifies network architecture and provides unprecedented transparency into the network's decision-making process. Our comprehensive empirical evaluation across different datasets demonstrates that NMN consistently outperforms traditional MLPs. The results challenge the assumption that separate activation functions are necessary for effective deep-learning models. The implications of this work extend beyond immediate architectural benefits, by eliminating intermediate activation functions while preserving non-linear capabilities, yat-MLP establishes a new paradigm for neural network design that combines simplicity with effectiveness. Most importantly, our approach provides unprecedented insights into the traditionally opaque "black-box" nature of neural networks, offering a clearer understanding of how these models process and classify information.
- [145] arXiv:2411.08107 (cross-list from cond-mat.str-el) [pdf, html, other]
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Title: From $G_2$ to $SO(8)$: Emergence and reminiscence of supersymmetry and trialityComments: 13+9 pages, 4+1 figuresSubjects: Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We construct a (1+1)-dimension continuum model of 4-component fermions incorporating the exceptional Lie group symmetry $G_2$. Four gapped and five gapless phases are identified via the one-loop renormalization group analysis. The gapped phases are controlled by four different stable $SO(8)$ Gross-Neveu fixed points, among which three exhibit an emergent triality, while the rest one possesses the self-triality, i.e., invariant under the triality mapping. The gapless phases include three $SO(7)$ critical ones, a $G_2$ critical one, and a Luttinger liquid. Three $SO(7)$ critical phases correspond to different $SO(7)$ Gross-Neveu fixed points connected by the triality relation similar to the gapped SO(8) case. The $G_2$ critical phase is controlled by an unstable fixed point described by a direct product of the Ising and tricritical Ising conformal field theories with the central charges $c=\frac{1}{2}$ and $c=\frac{7}{10}$, respectively, while the latter one is known to possess spacetime supersymmetry. In the lattice realization with a Hubbard-type interaction, the triality is broken into the duality between two $SO(7)$ symmetries and the supersymmetric $G_2$ critical phase exhibits the degeneracy between bosonic and fermionic states, which are reminiscences of the continuum model.
- [146] arXiv:2411.08110 (cross-list from quant-ph) [pdf, html, other]
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Title: Characterising memory in quantum channel discrimination via constrained separability problemsComments: 32 pages, comments are welcome!Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Quantum memories are a crucial precondition in many protocols for processing quantum information. A fundamental problem that illustrates this statement is given by the task of channel discrimination, in which an unknown channel drawn from a known random ensemble should be determined by applying it for a single time. In this paper, we characterise the quality of channel discrimination protocols when the quantum memory, quantified by the auxiliary dimension, is limited. This is achieved by formulating the problem in terms of separable quantum states with additional affine constraints that all of their factors in each separable decomposition obey. We discuss the computation of upper and lower bounds to the solutions of such problems which allow for new insights into the role of memory in channel discrimination. In addition to the single-copy scenario, this methodological insight allows to systematically characterise quantum and classical memories in adaptive channel discrimination protocols. Especially, our methods enabled us to identify channel discrimination scenarios where classical or quantum memory is required, and to identify the hierarchical and non-hierarchical relationships within adaptive channel discrimination protocols.
- [147] arXiv:2411.08131 (cross-list from quant-ph) [pdf, html, other]
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Title: On some states minimizing uncertainty relationsComments: 9 pages, comments are welcomeSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Analyzing Heisenberg-Robertson (HR) and Schrödinger uncertainty relations we found, that there can exist large set of states of the quantum system under considerations, for which the lower bound of product of the standard deviations of a pair of non-commuting observables, $A$ and $B$, is zero. These states are not eigenstates of either the observable $A$ or $B$. We have also shown that the so-called "sum uncertainty relations" also do not provide any information about lower bounds on the standard deviations calculated for these states.
- [148] arXiv:2411.08179 (cross-list from cs.DM) [pdf, html, other]
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Title: On sampling two spin models using the local connective constantSubjects: Discrete Mathematics (cs.DM); Mathematical Physics (math-ph)
This work establishes novel, optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models using the powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. These bounds are expressed in terms of the local connective constant of the underlying graph $G$. This is a notion of effective degree of $G$ on a local scale.
Our results have some interesting consequences for bounded degree graphs:
(a) They include the max-degree bounds as a special case,
(b) They improve on the running time of the FPTAS considered in [Sinclair, Srivastava, Stefankonic and Yin: PTRF 2017],
(c) They allow us to obtain mixing bounds in terms of the spectral radius of the adjacency matrix and improve on results in [Hayes: FOCS 2006],
(d) They also allow us to refine the celebrated connection between the hardness of approximate counting and the phase transitions from statistical physics.
We obtain our mixing bounds by utilising the $k$-non-backtracking matrix $H_{G,k}$. This is a very interesting, alas technically intricate, object to work with. We upper bound the spectral radius of the pairwise influence matrix $I^{\Lambda,\tau}_{G}$ by means of the 2-norm of $H_{G,k}$. To our knowledge, obtaining mixing bound using $H_{G,k}$ has not been considered before in the literature. - [149] arXiv:2411.08232 (cross-list from cs.LG) [pdf, html, other]
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Title: Imitation Learning from Observations: An Autoregressive Mixture of Experts ApproachSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
This paper presents a novel approach to imitation learning from observations, where an autoregressive mixture of experts model is deployed to fit the underlying policy. The parameters of the model are learned via a two-stage framework. By leveraging the existing dynamics knowledge, the first stage of the framework estimates the control input sequences and hence reduces the problem complexity. At the second stage, the policy is learned by solving a regularized maximum-likelihood estimation problem using the estimated control input sequences. We further extend the learning procedure by incorporating a Lyapunov stability constraint to ensure asymptotic stability of the identified model, for accurate multi-step predictions. The effectiveness of the proposed framework is validated using two autonomous driving datasets collected from human demonstrations, demonstrating its practical applicability in modelling complex nonlinear dynamics.
- [150] arXiv:2411.08292 (cross-list from cs.CV) [pdf, html, other]
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Title: Noisy image decomposition: a new structure, texture and noise model based on local adaptivityComments: arXiv admin note: text overlap with arXiv:2411.05265Journal-ref: Journal of Mathematical Imaging and Vision (JMIV), Vol.28, No.3, 285-295, 2007Subjects: Computer Vision and Pattern Recognition (cs.CV); Image and Video Processing (eess.IV); Functional Analysis (math.FA)
These last few years, image decomposition algorithms have been proposed to split an image into two parts: the structures and the textures. These algorithms are not adapted to the case of noisy images because the textures are corrupted by noise. In this paper, we propose a new model which decomposes an image into three parts (structures, textures and noise) based on a local regularization scheme. We compare our results with the recent work of Aujol and Chambolle. We finish by giving another model which combines the advantages of the two previous ones.
- [151] arXiv:2411.08293 (cross-list from cs.CV) [pdf, html, other]
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Title: Choix d'un espace de repr\'esentation image adapt\'e \`a la d\'etection de r\'eseaux routiersComments: in French languageJournal-ref: Traitement et Analyse de l'Information: M\'ethode et Application (TAIMA), Hammamet, Tunisie, May 2007Subjects: Computer Vision and Pattern Recognition (cs.CV); Image and Video Processing (eess.IV); Functional Analysis (math.FA)
These last years, algorithms allowing to decompose an image into its structures and textures components have emerged. In this paper, we present an application of this type of decomposition to the problem road network detection in aerial or satelite imagery. The algorithmic procedure involves the image decomposition (using a unique property), an alignment detection step based on the Gestalt theory, and a refinement step using statistical active contours.
- [152] arXiv:2411.08297 (cross-list from cs.LG) [pdf, html, other]
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Title: TowerDebias: A Novel Debiasing Method based on the Tower PropertyComments: To be submitted to a journal soonSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Probability (math.PR); Applications (stat.AP); Machine Learning (stat.ML)
Decision-making processes have increasingly come to rely on sophisticated machine learning tools, raising concerns about the fairness of their predictions with respect to any sensitive groups. The widespread use of commercial black-box machine learning models necessitates careful consideration of their legal and ethical implications on consumers. In situations where users have access to these "black-box" models, a key question emerges: how can we mitigate or eliminate the influence of sensitive attributes, such as race or gender? We propose towerDebias (tDB), a novel approach designed to reduce the influence of sensitive variables in predictions made by black-box models. Using the Tower Property from probability theory, tDB aims to improve prediction fairness during the post-processing stage in a manner amenable to the Fairness-Utility Tradeoff. This method is highly flexible, requiring no prior knowledge of the original model's internal structure, and can be extended to a range of different applications. We provide a formal improvement theorem for tDB and demonstrate its effectiveness in both regression and classification tasks, underscoring its impact on the fairness-utility tradeoff.
- [153] arXiv:2411.08311 (cross-list from cond-mat.stat-mech) [pdf, html, other]
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Title: A generalization of the martingale property of entropy production in stochastic systemsSubjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Probability (math.PR)
By decoupling forward and backward stochastic trajectories, we develop a family of martingales and work theorems for the same stochastic process. We achieve this by introducing an alternative work theorem derivation that uses tools from stochastic calculus instead of path integrals. Our derivation applies to both overdamped and underdamped Langevin dynamics and generalizes work theorems so that they connect new quantities in stochastic processes, potentially revealing new applications in dissipative systems.
- [154] arXiv:2411.08332 (cross-list from cs.DS) [pdf, html, other]
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Title: Learning-Augmented Algorithms for Online Concave Packing and Convex Covering ProblemsComments: 38 pages. In submissionSubjects: Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Optimization and Control (math.OC)
Learning-augmented algorithms have been extensively studied across the computer science community in the recent years, driven by advances in machine learning predictors, which can provide additional information to augment classical algorithms. Such predictions are especially powerful in the context of online problems, where decisions have to be made without knowledge of the future, and which traditionally exhibits impossibility results bounding the performance of any online algorithm. The study of learning-augmented algorithms thus aims to use external advice prudently, to overcome classical impossibility results when the advice is accurate, and still perform comparably to the state-of-the-art online algorithms even when the advice is inaccurate.
In this paper, we present learning-augmented algorithmic frameworks for two fundamental optimizations settings, extending and generalizing prior works. For online packing with concave objectives, we present a simple but overarching strategy that switches between the advice and the state-of-the-art online algorithm. For online covering with convex objectives, we greatly extend primal-dual methods for online convex covering programs by Azar et al. (FOCS 2016) and previous learning-augmented framework for online covering linear programs from the literature, to many new applications. We show that our algorithms break impossibility results when the advice is accurate, while maintaining comparable performance with state-of-the-art classical online algorithms even when the advice is erroneous. - [155] arXiv:2411.08387 (cross-list from q-bio.PE) [pdf, html, other]
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Title: Steady-State and Dynamical Behavior of a PDE Model of Multilevel Selection with Pairwise Group-Level CompetitionComments: 60 pages. 12 figuresSubjects: Populations and Evolution (q-bio.PE); Analysis of PDEs (math.AP); Physics and Society (physics.soc-ph)
Evolutionary competition often occurs simultaneously at multiple levels of organization, in which traits or behaviors that are costly for an individual can provide collective benefits to groups to which the individual belongs. Building off of recent work that has used ideas from game theory to study evolutionary competition within and among groups, we study a PDE model for multilevel selection that considers group-level evolutionary dynamics through a pairwise conflict depending on the strategic composition of the competing groups. This model allows for incorporation of group-level frequency dependence, facilitating the exploration for how the form of probabilities for victory in a group-level conflict can impact the long-time support for cooperation via multilevel selection. We characterize well-posedness properties for measure-valued solutions of our PDE model and apply these properties to show that the population will converge to a delta-function at the all-defector equilibrium when between-group selection is sufficiently weak. We further provide necessary conditions for the existence of bounded steady state densities for the multilevel dynamics of Prisoners' Dilemma and Hawk-Dove scenarios, using a mix of analytical and numerical techniques to characterize the relative strength of between-group selection required to ensure the long-time survival of cooperation via multilevel selection. We also see that the average payoff at steady state appears to be limited by the average payoff of the all-cooperator group, even for games in which groups achieve maximal average payoff at intermediate levels of cooperation, generalizing behavior that has previously been observed in PDE models of multilevel selection with frequency-indepdent group-level competition.
- [156] arXiv:2411.08405 (cross-list from cs.CE) [pdf, other]
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Title: An Ising Machine Formulation for Design Updates in Topology Optimization of Flow ChannelsYudai Suzuki, Shiori Aoki, Fabian Key, Katsuhiro Endo, Yoshiki Matsuda, Shu Tanaka, Marek Behr, Mayu MuramatsuSubjects: Computational Engineering, Finance, and Science (cs.CE); Optimization and Control (math.OC)
Topology optimization is an essential tool in computational engineering, for example, to improve the design and efficiency of flow channels. At the same time, Ising machines, including digital or quantum annealers, have been used as efficient solvers for combinatorial optimization problems. Beyond combinatorial optimization, recent works have demonstrated applicability to other engineering tasks by tailoring corresponding problem formulations. In this study, we present a novel Ising machine formulation for computing design updates during topology optimization with the goal of minimizing dissipation energy in flow channels. We explore the potential of this approach to improve the efficiency and performance of the optimization process. To this end, we conduct experiments to study the impact of various factors within the novel formulation. Additionally, we compare it to a classical method using the number of optimization steps and the final values of the objective function as indicators of the time intensity of the optimization and the performance of the resulting designs, respectively. Our findings show that the proposed update strategy can accelerate the topology optimization process while producing comparable designs. However, it tends to be less exploratory, which may lead to lower performance of the designs. These results highlight the potential of incorporating Ising formulations for optimization tasks but also show their limitations when used to compute design updates in an iterative optimization process. In conclusion, this work provides an efficient alternative for design updates in topology optimization and enhances the understanding of integrating Ising machine formulations in engineering optimization.
- [157] arXiv:2411.08436 (cross-list from eess.SY) [pdf, other]
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Title: Robust performance for switched systems with constrained switching and its application to weakly hard real-time control systemsSubjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
Many cyber-physical systems can naturally be formulated as switched systems with constrained switching. This includes systems where one of the signals in the feedback loop may be lost. Possible sources for losses are shared or unreliable communication media in networked control systems, or signals which are discarded, e.g., when using a shared computation device such as a processor in real-time control applications. The use of switched systems with constrained switching is not limited to cyber-physical systems but, includes many other relevant applications such as power systems and modeling virus mutations. In this chapter, we introduce a framework for analyzing and designing controllers which guarantee robust quadratic performance for switched systems with constrained switching. The possible switching sequences are described by the language of a labeled graph where the labels are linked to the different subsystems. The subsystems are allowed to have different input and output dimensions, and their state-space representations can be affected by a broad class of uncertainties in a rational way. The proposed framework exploits ideas from dissipativity-based linear control theory to derive analysis and synthesis inequalities given by linear matrix inequalities. We demonstrate how the proposed framework can be applied to the design of controllers for uncertain weakly hard real-time control systems - a system class naturally appearing in networked and real-time control.
- [158] arXiv:2411.08480 (cross-list from cond-mat.stat-mech) [pdf, html, other]
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Title: Single impurity in the Totally Asymmetric Simple Exclusion ProcessComments: 20 pages, 12 figuresSubjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
We examine the behavior of a single impurity particle embedded within a Totally Asymmetric Simple Exclusion Process (TASEP). By analyzing the impurity's dynamics, characterized by two arbitrary hopping parameters $ \alpha $ and $\beta$, we investigate both its macroscopic impact on the system and its individual trajectory, providing new insights into the interaction between the impurity and the TASEP environment. We classify the induced hydrodynamic limit shapes based on the initial densities to the left and right of the impurity, along with the values of the parameters $\alpha$,$\beta$. We develop a new method that enables the analysis of the impurity's behavior within an arbitrary density field, thereby generalizing the traditional coupling technique used for second-class particles. With this tool, we extend to the impurity case under certain parameter conditions, Ferrari and Kipnis's results on the distribution of the asymptotic speed of a second-class particle within a rarefaction fan.
- [159] arXiv:2411.08491 (cross-list from stat.ME) [pdf, other]
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Title: Covariate Adjustment in Randomized Experiments Motivated by Higher-Order Influence FunctionsComments: 62 pages, 8 figuresSubjects: Methodology (stat.ME); Econometrics (econ.EM); Statistics Theory (math.ST)
Higher-Order Influence Functions (HOIF), developed in a series of papers over the past twenty years, is a fundamental theoretical device for constructing rate-optimal causal-effect estimators from observational studies. However, the value of HOIF for analyzing well-conducted randomized controlled trials (RCT) has not been explicitly explored. In the recent US Food \& Drug Administration (FDA) and European Medical Agency (EMA) guidelines on the practice of covariate adjustment in analyzing RCT, in addition to the simple, unadjusted difference-in-mean estimator, it was also recommended to report the estimator adjusting for baseline covariates via a simple parametric working model, such as a linear model. In this paper, we show that an HOIF-motivated estimator for the treatment-specific mean has significantly improved statistical properties compared to popular adjusted estimators in practice when the number of baseline covariates $p$ is relatively large compared to the sample size $n$. We also characterize the conditions under which the HOIF-motivated estimator improves upon the unadjusted estimator. Furthermore, we demonstrate that a novel debiased adjusted estimator proposed recently by Lu et al. is, in fact, another HOIF-motivated estimator under disguise. Finally, simulation studies are conducted to corroborate our theoretical findings.
- [160] arXiv:2411.08523 (cross-list from cs.FL) [pdf, other]
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Title: The Word Problem for $(\omega - 1)$-Terms over $\boldsymbol{\mathrm{DAb}}$Subjects: Formal Languages and Automata Theory (cs.FL); Group Theory (math.GR)
We give a ranker-based description using finite-index congruences for the variety $\boldsymbol{\mathrm{DAb}}$ of finite monoids whose regular $\mathcal{D}$-classes form Abelian groups. This combinatorial description yields a normal form for general pseudowords over $\boldsymbol{\mathrm{DAb}}$. For $(\omega - 1)$-terms, this normal form is computable, which yields an algorithm for the word problem for $(\omega - 1)$-terms of $\boldsymbol{\mathrm{DAb}}$.
- [161] arXiv:2411.08532 (cross-list from quant-ph) [pdf, html, other]
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Title: Conditional expectations in Quantum Mechanics and causal interpretations: the Bohm momentum as a best predictorSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Given a normalized state-vector $\psi $, we define the conditional expectation $\mathbb{E }_{\psi } (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to $\psi $, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique.
We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation. - [162] arXiv:2411.08540 (cross-list from hep-th) [pdf, html, other]
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Title: Flat limit of AdS/CFT from AdS geodesics: scattering amplitudes and antipodal matching of Li\'enard-Wiechert fieldsComments: 31 pages, 7 figuresSubjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
We revisit the flat limit of AdS/CFT from the point of view of geodesics in AdS. We show that the flat space scattering amplitudes can be constructed from operator insertions where the geodesics of the particles corresponding to the operators hit the conformal boundary of AdS. Further, we compute the Liénard-Wiechert solutions in AdS by boosting a static charge using AdS isometries and show that the solutions are antipodally matched between two regions, separated by a global time difference of $\Delta\tau=\pi$. Going to the boundary of AdS along null geodesics, in the flat limit, this antipodal matching leads to the flat space antipodal matching near spatial infinity.
- [163] arXiv:2411.08631 (cross-list from stat.ML) [pdf, html, other]
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Title: Deep Generative Demand Learning for Newsvendor and PricingComments: 30 pages, 6 figuresSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC)
We consider data-driven inventory and pricing decisions in the feature-based newsvendor problem, where demand is influenced by both price and contextual features and is modeled without any structural assumptions. The unknown demand distribution results in a challenging conditional stochastic optimization problem, further complicated by decision-dependent uncertainty and the integration of features. Inspired by recent advances in deep generative learning, we propose a novel approach leveraging conditional deep generative models (cDGMs) to address these challenges. cDGMs learn the demand distribution and generate probabilistic demand forecasts conditioned on price and features. This generative approach enables accurate profit estimation and supports the design of algorithms for two key objectives: (1) optimizing inventory for arbitrary prices, and (2) jointly determining optimal pricing and inventory levels. We provide theoretical guarantees for our approach, including the consistency of profit estimation and convergence of our decisions to the optimal solution. Extensive simulations-ranging from simple to complex scenarios, including one involving textual features-and a real-world case study demonstrate the effectiveness of our approach. Our method opens a new paradigm in management science and operations research, is adaptable to extensions of the newsvendor and pricing problems, and holds potential for solving other conditional stochastic optimization problems.
- [164] arXiv:2411.08668 (cross-list from econ.GN) [pdf, html, other]
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Title: A Machine Learning Algorithm for Finite-Horizon Stochastic Control Problems in EconomicsComments: arXiv admin note: substantial text overlap with arXiv:1611.01767Subjects: General Economics (econ.GN); Optimization and Control (math.OC); Machine Learning (stat.ML)
We propose a machine learning algorithm for solving finite-horizon stochastic control problems based on a deep neural network representation of the optimal policy functions. The algorithm has three features: (1) It can solve high-dimensional (e.g., over 100 dimensions) and finite-horizon time-inhomogeneous stochastic control problems. (2) It has a monotonicity of performance improvement in each iteration, leading to good convergence properties. (3) It does not rely on the Bellman equation. To demonstrate the efficiency of the algorithm, it is applied to solve various finite-horizon time-inhomogeneous problems including recursive utility optimization under a stochastic volatility model, a multi-sector stochastic growth, and optimal control under a dynamic stochastic integration of climate and economy model with eight-dimensional state vectors and 600 time periods.
- [165] arXiv:2411.08735 (cross-list from cs.NE) [pdf, html, other]
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Title: New advances in universal approximation with neural networks of minimal widthComments: 56 pages in the main text, 70 pages with AppendixSubjects: Neural and Evolutionary Computing (cs.NE); Functional Analysis (math.FA)
Deep neural networks have achieved remarkable success in diverse applications, prompting the need for a solid theoretical foundation. Recent research has identified the minimal width $\max\{2,d_x,d_y\}$ required for neural networks with input dimensions $d_x$ and output dimension $d_y$ that use leaky ReLU activations to universally approximate $L^p(\mathbb{R}^{d_x},\mathbb{R}^{d_y})$ on compacta. Here, we present an alternative proof for the minimal width of such neural networks, by directly constructing approximating networks using a coding scheme that leverages the properties of leaky ReLUs and standard $L^p$ results. The obtained construction has a minimal interior dimension of $1$, independent of input and output dimensions, which allows us to show that autoencoders with leaky ReLU activations are universal approximators of $L^p$ functions. Furthermore, we demonstrate that the normalizing flow LU-Net serves as a distributional universal approximator. We broaden our results to show that smooth invertible neural networks can approximate $L^p(\mathbb{R}^{d},\mathbb{R}^{d})$ on compacta when the dimension $d\geq 2$, which provides a constructive proof of a classical theorem of Brenier and Gangbo. In addition, we use a topological argument to establish that for FNNs with monotone Lipschitz continuous activations, $d_x+1$ is a lower bound on the minimal width required for the uniform universal approximation of continuous functions $C^0(\mathbb{R}^{d_x},\mathbb{R}^{d_y})$ on compacta when $d_x\geq d_y$.
- [166] arXiv:2411.08741 (cross-list from quant-ph) [pdf, html, other]
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Title: Unified analysis of non-Markovian open quantum systems in Gaussian environment using superoperator formalismComments: 46 pagesSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
We present perturbative error bounds for the non-Markovian dynamics of observables in open quantum systems interacting with Gaussian environments, governed by general Liouville dynamics. This extends the work of [Mascherpa et al., Phys. Rev. Lett. 118, 100401, 2017], which demonstrated qualitatively tighter bounds over the standard Grönwall-type analysis, where the joint system-environment evolution is unitary. Our results apply to systems with both bosonic and fermionic environments. Our approach utilizes a superoperator formalism, which avoids the need for formal coherent state path integral calculations, or the dilation of Lindblad dynamics into an equivalent unitary framework with infinitely many degrees of freedom. This enables a unified treatment of a wide range of open quantum systems. These findings provide a solid theoretical basis for various recently developed pseudomode methods in simulating open quantum system dynamics.
- [167] arXiv:2411.08773 (cross-list from cs.DS) [pdf, other]
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Title: Optimal Oblivious Subspace Embeddings with Near-optimal SparsitySubjects: Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Numerical Analysis (math.NA); Probability (math.PR); Machine Learning (stat.ML)
An oblivious subspace embedding is a random $m\times n$ matrix $\Pi$ such that, for any $d$-dimensional subspace, with high probability $\Pi$ preserves the norms of all vectors in that subspace within a $1\pm\epsilon$ factor. In this work, we give an oblivious subspace embedding with the optimal dimension $m=\Theta(d/\epsilon^2)$ that has a near-optimal sparsity of $\tilde O(1/\epsilon)$ non-zero entries per column of $\Pi$. This is the first result to nearly match the conjecture of Nelson and Nguyen [FOCS 2013] in terms of the best sparsity attainable by an optimal oblivious subspace embedding, improving on a prior bound of $\tilde O(1/\epsilon^6)$ non-zeros per column [Chenakkod et al., STOC 2024]. We further extend our approach to the non-oblivious setting, proposing a new family of Leverage Score Sparsified embeddings with Independent Columns, which yield faster runtimes for matrix approximation and regression tasks.
In our analysis, we develop a new method which uses a decoupling argument together with the cumulant method for bounding the edge universality error of isotropic random matrices. To achieve near-optimal sparsity, we combine this general-purpose approach with new traces inequalities that leverage the specific structure of our subspace embedding construction. - [168] arXiv:2411.08787 (cross-list from nlin.SI) [pdf, html, other]
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Title: Stability analysis of breathers for coupled nonlinear Schrodinger equationsComments: 59 pagesSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Dynamical Systems (math.DS); Pattern Formation and Solitons (nlin.PS)
We investigate the spectral stability of non-degenerate vector soliton solutions and the nonlinear stability of breather solutions for the coupled nonlinear Schrodinger (CNLS) equations. The non-degenerate vector solitons are spectrally stable despite the linearized operator admits either embedded or isolated eigenvalues of negative Krein signature. The nonlinear stability of breathers is obtained by the Lyapunov method with the help of the squared eigenfunctions due to integrability of the CNLS equations.
- [169] arXiv:2411.08798 (cross-list from cs.LG) [pdf, html, other]
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Title: Learning Gaussian Multi-Index Models with Gradient Flow: Time Complexity and Directional ConvergenceComments: 21 pages, 6 figures, under review by AISTATS 2025Subjects: Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE); Algebraic Geometry (math.AG)
This work focuses on the gradient flow dynamics of a neural network model that uses correlation loss to approximate a multi-index function on high-dimensional standard Gaussian data. Specifically, the multi-index function we consider is a sum of neurons $f^*(x) \!=\! \sum_{j=1}^k \! \sigma^*(v_j^T x)$ where $v_1, \dots, v_k$ are unit vectors, and $\sigma^*$ lacks the first and second Hermite polynomials in its Hermite expansion. It is known that, for the single-index case ($k\!=\!1$), overcoming the search phase requires polynomial time complexity. We first generalize this result to multi-index functions characterized by vectors in arbitrary directions. After the search phase, it is not clear whether the network neurons converge to the index vectors, or get stuck at a sub-optimal solution. When the index vectors are orthogonal, we give a complete characterization of the fixed points and prove that neurons converge to the nearest index vectors. Therefore, using $n \! \asymp \! k \log k$ neurons ensures finding the full set of index vectors with gradient flow with high probability over random initialization. When $ v_i^T v_j \!=\! \beta \! \geq \! 0$ for all $i \neq j$, we prove the existence of a sharp threshold $\beta_c \!=\! c/(c+k)$ at which the fixed point that computes the average of the index vectors transitions from a saddle point to a minimum. Numerical simulations show that using a correlation loss and a mild overparameterization suffices to learn all of the index vectors when they are nearly orthogonal, however, the correlation loss fails when the dot product between the index vectors exceeds a certain threshold.
- [170] arXiv:2411.08822 (cross-list from cs.CE) [pdf, html, other]
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Title: A probabilistic reduced-order modeling framework for patient-specific cardio-mechanical analysisSubjects: Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
Cardio-mechanical models can be used to support clinical decision-making. Unfortunately, the substantial computational effort involved in many cardiac models hinders their application in the clinic, despite the fact that they may provide valuable information. In this work, we present a probabilistic reduced-order modeling (ROM) framework to dramatically reduce the computational effort of such models while providing a credibility interval. In the online stage, a fast-to-evaluate generalized one-fiber model is considered. This generalized one-fiber model incorporates correction factors to emulate patient-specific attributes, such as local geometry variations. In the offline stage, Bayesian inference is used to calibrate these correction factors on training data generated using a full-order isogeometric cardiac model (FOM). A Gaussian process is used in the online stage to predict the correction factors for geometries that are not in the training data. The proposed framework is demonstrated using two examples. The first example considers idealized left-ventricle geometries, for which the behavior of the ROM framework can be studied in detail. In the second example, the ROM framework is applied to scan-based geometries, based on which the application of the ROM framework in the clinical setting is discussed. The results for the two examples convey that the ROM framework can provide accurate online predictions, provided that adequate FOM training data is available. The uncertainty bands provided by the ROM framework give insight into the trustworthiness of its results. Large uncertainty bands can be considered as an indicator for the further population of the training data set.
- [171] arXiv:2411.08853 (cross-list from nlin.SI) [pdf, html, other]
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Title: Rational solutions of Painlev\'e V from Hankel determinants and the asymptotics of their pole locationsComments: 44 pages, 11 figuresSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)
In this paper we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function with a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlevé equation. More specifically we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots fill a well-defined region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity. Moreover we provide an approximate location of these roots within the region in terms of suitable quantization conditions.
- [172] arXiv:2411.08861 (cross-list from stat.ME) [pdf, html, other]
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Title: Interaction Testing in Variation AnalysisSubjects: Methodology (stat.ME); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Statistics Theory (math.ST)
Relationships of cause and effect are of prime importance for explaining scientific phenomena. Often, rather than just understanding the effects of causes, researchers also wish to understand how a cause $X$ affects an outcome $Y$ mechanistically -- i.e., what are the causal pathways that are activated between $X$ and $Y$. For analyzing such questions, a range of methods has been developed over decades under the rubric of causal mediation analysis. Traditional mediation analysis focuses on decomposing the average treatment effect (ATE) into direct and indirect effects, and therefore focuses on the ATE as the central quantity. This corresponds to providing explanations for associations in the interventional regime, such as when the treatment $X$ is randomized. Commonly, however, it is of interest to explain associations in the observational regime, and not just in the interventional regime. In this paper, we introduce \text{variation analysis}, an extension of mediation analysis that focuses on the total variation (TV) measure between $X$ and $Y$, written as $\mathrm{E}[Y \mid X=x_1] - \mathrm{E}[Y \mid X=x_0]$. The TV measure encompasses both causal and confounded effects, as opposed to the ATE which only encompasses causal (direct and mediated) variations. In this way, the TV measure is suitable for providing explanations in the natural regime and answering questions such as ``why is $X$ associated with $Y$?''. Our focus is on decomposing the TV measure, in a way that explicitly includes direct, indirect, and confounded variations. Furthermore, we also decompose the TV measure to include interaction terms between these different pathways. Subsequently, interaction testing is introduced, involving hypothesis tests to determine if interaction terms are significantly different from zero. If interactions are not significant, more parsimonious decompositions of the TV measure can be used.
Cross submissions (showing 31 of 31 entries)
- [173] arXiv:1310.3636 (replaced) [pdf, other]
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Title: A note on injective envelopes and von Neumann algebrasComments: The proof of the main result contains some grave errors. The article has been overworked, replaced and extended by the recently submitted article "The Jordan lattice completion and a note on injective envelopes and von Neumann algebras" (arXiv:1804.05701) where the main result of the former paper is contained in a slightly modified version with an altogether different proofSubjects: Operator Algebras (math.OA)
The article exhibits certain relations between the injective envelope I(A) of a C*-algebra A and the von Neumann algebra generated by a representation lambda of A provided it is injective. More specifically we show that there exist positive retractions sigma : /lambda (A)'' ---> I(A) which are close to being *-homomorphisms in the sense that they are Jordan homomorphisms of the underlying Jordan algebras, and the kernel of /sigma is given by a twosided ideal.
- [174] arXiv:1804.06303 (replaced) [pdf, other]
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Title: Symmetry operators and generation of symmetry transformations of partial differential equationsComments: Enhanced and updated version; 23 pages; see also arXiv:0803.3688Journal-ref: Nausivios Chora Vol. 7 (2018) pp. C31-C48 ; https://nausivios.hna.gr/docs/2018C3.pdfSubjects: Analysis of PDEs (math.AP)
The study of symmetries of partial differential equations (PDEs) has been traditionally treated as a geometrical problem. Although geometrical methods have been proven effective with regard to finding infinitesimal symmetry transformations, they present certain conceptual difficulties in the case of matrix-valued PDEs; for example, the usual differential-operator representation of the symmetry-generating vector fields is not possible in this case. In this article an algebraic approach to the symmetry problem of PDEs - both scalar and matrix-valued - is described, based on abstract operators (characteristic derivatives) that admit a standard differential-operator representation in the case of scalar-valued PDEs. A number of examples are given.
- [175] arXiv:2104.04716 (replaced) [pdf, other]
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Title: Selecting Penalty Parameters of High-Dimensional M-Estimators using Bootstrapping after Cross-ValidationComments: 164 pages, 14 figuresSubjects: Statistics Theory (math.ST); Econometrics (econ.EM)
We develop a new method for selecting the penalty parameter for $\ell_{1}$-penalized M-estimators in high dimensions, which we refer to as bootstrapping after cross-validation. We derive rates of convergence for the corresponding $\ell_1$-penalized M-estimator and also for the post-$\ell_1$-penalized M-estimator, which refits the non-zero entries of the former estimator without penalty in the criterion function. We demonstrate via simulations that our methods are not dominated by cross-validation in terms of estimation errors and can outperform cross-validation in terms of inference. As an empirical illustration, we revisit Fryer Jr (2019), who investigated racial differences in police use of force, and confirm his findings.
- [176] arXiv:2108.08991 (replaced) [pdf, html, other]
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Title: Standard monomials and invariant theory for arc spaces III: special linear groupComments: Some corrections and improvements to the presentation. arXiv admin note: text overlap with arXiv:2108.06864Journal-ref: Journal of Algebra, Volume 664, (2025), Pages 289-327Subjects: Algebraic Geometry (math.AG)
This is the third in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the special linear group. This is more subtle than the results for the general linear and symplectic groups obtained in the first two papers because the arc space of the corresponding affine quotients can be nonreduced.
- [177] arXiv:2109.06693 (replaced) [pdf, html, other]
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Title: On the realization of subgroups of $PGL(2,F)$, and their automorphism groups, as Galois groups over function fieldsComments: This is uploaded again since we cannot delete version 1Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG); Group Theory (math.GR)
Let $F$ be any field. We give a short and elementary proof that any finite subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field $F(x)$. We also develop a theory of descent to subfields of $F$. This enables us to realize the automorphism groups of finite subgroups of $PGL(2,F)$ as Galois groups.
- [178] arXiv:2111.02297 (replaced) [pdf, html, other]
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Title: Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-valuesComments: 62 pages, to appear in American Journal of MathSubjects: Number Theory (math.NT)
Let $F$ be a totally real number field and $n\ge 3$. Let $\Pi$ and $\pi$ be cuspidal automorphic representations for $\mathrm{PGL}_{n+1}(F)$ and $\mathrm{PGL}_{n-1}(F)$, respectively, that are unramified and tempered at all finite places. We prove simultaneous non-vanishing of the Rankin--Selberg $L$-values $L(1/2,\Pi\otimes\widetilde{\sigma})$ and $L(1/2,\sigma\otimes\widetilde{\pi})$ for certain sequences of $\sigma$ varying over cuspidal automorphic representations for $\mathrm{PGL}_n(F)$ with conductor tending to infinity in the level aspect and bearing certain local conditions. Along the way, we also prove a reciprocity formula for the average of the product of Rankin--Selberg $L$-functions $L(1/2,\Pi\otimes\widetilde{\sigma})L(1/2,\sigma\otimes\widetilde{\pi})$ over a conductor aspect family of $\sigma$.
- [179] arXiv:2112.02871 (replaced) [pdf, other]
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Title: Variational inequality solutions and finite stopping time for a class of shear-thinning flowsJournal-ref: Annali di Matematica Pura ed Applicata, 2024, 203 (6), pp.2591-2612Subjects: Analysis of PDEs (math.AP)
The aim of this paper is to study the existence of a finite stopping time for solutions in the form of variational inequality to fluid flows following a power law (or Ostwald-DeWaele law) in dimension $N \in \{2,3\}$. We first establish the existence of solutions for generalized Newtonian flows, valid for viscous stress tensors associated with the usual laws such as Ostwald-DeWaele, Carreau-Yasuda, Herschel-Bulkley and Bingham, but also for cases where the viscosity coefficient satisfies a more atypical (logarithmic) form. To demonstrate the existence of such solutions, we proceed by applying a nonlinear Galerkin method with a double regularization on the viscosity coefficient. We then establish the existence of a finite stopping time for threshold fluids or shear-thinning power-law fluids, i.e. formally such that the viscous stress tensor is represented by a $p$-Laplacian for the symmetrized gradient for $p \in [1,2)$.
- [180] arXiv:2204.04946 (replaced) [pdf, html, other]
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Title: Birational boundedness of rationally connected log Calabi-Yau pairs with fixed indexComments: 12 pages, comments are very welcome; final version, to appear in Algebr. Geom. PhysSubjects: Algebraic Geometry (math.AG)
We show that the set of rationally connected projective varieties $X$ of a fixed dimension such that $(X,B)$ is klt, and $-l(K_X+B)$ is Cartier and nef for some fixed positive integer $l$, is bounded modulo flops.
- [181] arXiv:2208.01842 (replaced) [pdf, html, other]
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Title: Remarks on the determination of the Lorentzian metric by the lengths of geodesics or null-geodesicsComments: This article replaces articles arXiv:2208.01842 and arXiv:2205.05860, arXiv:2205.05860 is withdrawnSubjects: Analysis of PDEs (math.AP)
We consider a Lorentzian metric in $\mathbb{R}\times\mathbb{R}^n$. We show that if we know the lengths of the space-time geodesics starting at $(0,y,\eta)$ when $t=0$, then we can recover the metric at $y$. We prove the rigidity of Lorentzian metrics. We also prove a variant of the rigidity property for the case of null-geodesics: if two metrics are close and if corresponding null-geodesics have equal Euclidian lengths then the metrics are equal.
- [182] arXiv:2208.01921 (replaced) [pdf, html, other]
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Title: The invariants of the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$Comments: 36 pages; added some details; to appear in Commun. Contemp. MathSubjects: Number Theory (math.NT)
The transformation behaviour of the vector valued theta function of a positive-definite even lattice under the metaplectic group $\mathrm{Mp}_2(\mathbb{Z})$ is described by the Weil representation. We show that the invariants of this representation are induced from $5$ fundamental invariants. As an application we give simple generating sets for Jacobi forms of singular weight.
- [183] arXiv:2209.14929 (replaced) [pdf, html, other]
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Title: The Mittag-Leffler condition descents via pure monomorphismsComments: 6 pagesSubjects: Commutative Algebra (math.AC); Rings and Algebras (math.RA)
This notes aims to clarify the proof given by Raynaud and Gruson that the Mittag-Leffler property descents via pure rings monomorphism of commutative rings. A consequence of that is that projectivity dencents via such ring homomorphisms, a revision of the proof also allows to prove that the property of being pure-projective also descents via pure monomorphisms between commutative rings}.
- [184] arXiv:2212.02445 (replaced) [pdf, html, other]
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Title: Bounds on the covariance matrix of the Sherrington-Kirkpatrick modelComments: Improved bounds and added remarks based on referee feedbackSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We consider the Sherrington-Kirkpatrick model with no external field and inverse temperature $\beta<1$ and prove that the expected operator norm of the covariance matrix of the Gibbs measure is bounded by a constant depending only on $\beta$. This answers an open question raised by Talagrand, who proved a bound of $C(\beta) (\log n)^8$. Our result follows by establishing an approximate formula for the covariance matrix which we obtain by differentiating the TAP equations and then optimally controlling the associated error terms. We complement this result by showing diverging lower bounds on the operator norm, both at the critical and low temperatures.
- [185] arXiv:2212.02857 (replaced) [pdf, html, other]
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Title: Cutting planes for signomial programmingSubjects: Optimization and Control (math.OC)
Cutting planes are of crucial importance when solving nonconvex nonlinear programs to global optimality, for example using the spatial branch-and-bound algorithms. In this paper, we discuss the generation of cutting planes for signomial programming. Many global optimization algorithms lift signomial programs into an extended formulation such that these algorithms can construct relaxations of the signomial program by outer approximations of the lifted set encoding nonconvex signomial term sets, i.e., hypographs, or epigraphs of signomial terms. We show that any signomial term set can be transformed into the subset of the difference of two concave power functions, from which we derive two kinds of valid linear inequalities. Intersection cuts are constructed using signomial term-free sets which do not contain any point of the signomial term set in their interior. We show that these signomial term-free sets are maximal in the nonnegative orthant, and use them to derive intersection sets. We then convexify a concave power function in the reformulation of the signomial term set, resulting in a convex set containing the signomial term set. This convex outer approximation is constructed in an extended space, and we separate a class of valid linear inequalities by projection from this approximation. We implement the valid inequalities in a global optimization solver and test them on MINLPLib instances. Our results show that both types of valid inequalities provide comparable reductions in running time, number of search nodes, and duality gap.
- [186] arXiv:2212.11845 (replaced) [pdf, html, other]
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Title: p-Forms from SyzygiesComments: Minor changes. Comments welcome!Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)
These notes aim to develop a tool for constructing polynomial differential $p$-forms vanishing on prescribed loci through syzygies of homogeneous ideals. Examples are provided through implementing this method in Macaulay2, particularly examples of instanton bundles of charges $4$ and $5$ on $\mathbb{P}^3$ that arise in this construction.
- [187] arXiv:2212.14259 (replaced) [pdf, html, other]
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Title: Bipolar Theorems for Sets of Non-negative Random VariablesSubjects: Probability (math.PR); Functional Analysis (math.FA); Mathematical Finance (q-fin.MF)
This paper assumes a robust, in general not dominated, probabilistic framework and provides necessary and sufficient conditions for a bipolar representation of subsets of the set of all quasi-sure equivalence classes of non-negative random variables, without any further conditions on the underlying measure space. This generalizes and unifies existing bipolar theorems proved under stronger assumptions on the robust framework. Applications are in areas of robust financial modeling.
- [188] arXiv:2301.13655 (replaced) [pdf, html, other]
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Title: Zygmund dilations: bilinear analysis and commutator estimatesComments: To appear in Trans. Amer. Math. SocSubjects: Classical Analysis and ODEs (math.CA)
We develop both bilinear theory and commutator estimates in the context of entangled dilations, specifically Zygmund dilations $(x_1, x_2, x_3) \mapsto (\delta_1 x_1, \delta_2 x_2, \delta_1 \delta_2 x_3)$ in $\mathbb{R}^3$. We construct bilinear versions of recent dyadic multiresolution methods for Zygmund dilations and apply them to prove a paraproduct free $T1$ theorem for bilinear singular integrals invariant under Zygmund dilations. Independently, we prove linear commutator estimates even when the underlying singular integrals do not satisfy weighted estimates with Zygmund weights. This requires new paraproduct estimates.
- [189] arXiv:2302.08581 (replaced) [pdf, other]
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Title: Abstract Corrected IterationsSubjects: Logic (math.LO)
We consider $(<\lambda)$-support iterations of a version of $(<\lambda)$-strategically complete $\lambda^+$-c.c. definable forcing notions along partial orders. We show that such iterations can be corrected to yield an analog of a result by Judah and Shelah for finite support iterations of Suslin ccc forcing, namely that if $(\mathbb{P}_{\alpha}, \underset{\sim}{\mathbb Q_{\beta}} : \alpha \leq \delta, \beta <\delta)$ is a FS iteration of Suslin ccc forcing and $U\subseteq \delta$ is sufficiently closed, then letting $\mathbb{P}_U$ be the iteration along $U$, we have $\mathbb{P}_U \lessdot \mathbb{P}_{\delta}$.
- [190] arXiv:2303.00739 (replaced) [pdf, html, other]
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Title: Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic $p>0$Comments: 7 pages, comments welcome!Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)
The goal of this note is to prove that Hodge-de Rham degeneration holds for smooth and proper $\mathbf{F}_p$-schemes $X$ with $\dim(X)<p^n$ as soon as its category of quasicoherent sheaves admits a lift to the truncated Brown-Peterson spectrum $\mathrm{BP}\langle n-1\rangle$, and the Hochschild-Kostant-Rosenberg spectral sequence for $X$ degenerates at the $E_2$-page. This is obtained from a noncommutative version, whose proof is essentially the same as Mathew's argument in arXiv:1710.09045.
- [191] arXiv:2303.09432 (replaced) [pdf, other]
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Title: Chromatic aberrations of geometric Satake over the regular locusComments: Complete rewrite, new material added; now 134 pages. This is a preliminary version; comments and suggestions for improvements are greatly appreciated! I'll post major updates to the arXiv, but I'll upload minor edits to my website; so please see my website for the most up-to-date versionSubjects: Representation Theory (math.RT); Algebraic Topology (math.AT)
Let $G$ be a connected, simply-laced, almost simple algebraic group over $\mathbf{C}$, let $G_c$ be a maximal compact subgroup of $G(\mathbf{C})$, and let $T_c$ be a maximal torus therein. Let $\mathrm{Gr}_G$ denote the affine Grassmannian of $G$, and let $\check{G}$ denote the Langlands dual group to $G$ with Lie algebra $\check{\mathfrak{g}}$. The derived geometric Satake equivalence of Bezrukavnikov-Finkelberg gives an equivalence between the $\infty$-category $\mathrm{Loc}_{G_c}(\mathrm{Gr}_G; \mathbf{C})$ of $G_c$-equivariant local systems of $\mathbf{C}$-vector spaces on $\mathrm{Gr}_G$ and the $\infty$-category of quasicoherent sheaves on a large open substack of $\check{\mathfrak{g}}^\ast[2]/\check{G}$. In this article, we study the analogous story when $\mathrm{Loc}_{G_c}(\mathrm{Gr}_G; \mathbf{C})$ is replaced by the $\infty$-category of $T_c$-equivariant local systems of $k$-modules over $\mathrm{Gr}_G(\mathbf{C})$, where $k$ is ($2$-periodic) rational cohomology, (complex) K-theory, or elliptic cohomology. Crucial to our work is the genuine equivariant refinement of these cohomology theories. We show that, although there may not be an equivalence as in derived geometric Satake, the $\infty$-category $\mathrm{Loc}_{T_c}(\mathrm{Gr}_G; k)$ admits a 1-parameter degeneration to an $\infty$-category of quasicoherent sheaves built out of the geometry of various Langlands-dual stacks associated to $k$ and the $1$-dimensional group scheme computing $S^1$-equivariant $k$-cohomology. For example, when $k$ is an elliptic cohomology theory with elliptic curve $E$, the $\infty$-category $\mathrm{Loc}_{T_c}(\mathrm{Gr}_G; k)$ degenerates to the $\infty$-category of quasicoherent sheaves on a large open locus in the moduli stack of $\check{B}$-bundles of degree $0$ on $E$. We also study several applications of these equivalences.
- [192] arXiv:2303.10485 (replaced) [pdf, html, other]
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Title: The soliton resolution conjecture for the Boussinesq equationComments: 43 pages, 11 figuresSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We analyze the Boussinesq equation on the line with Schwartz initial data belonging to the physically relevant class of global solutions. In a recent paper, we determined ten main asymptotic sectors describing the large $(x,t)$-behavior of the solution, and for each of these sectors we provided the leading order asymptotics in the case when no solitons are present. In this paper, we give a formula valid in the asymptotic sector $x/t \in (1,M]$, where $M$ is a large positive constant, in the case when solitons are present. Combined with earlier results, this validates the soliton resolution conjecture for the Boussinesq equation everywhere in the $(x,t)$-plane except in a number of small transition zones.
- [193] arXiv:2304.12430 (replaced) [pdf, html, other]
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Title: Weak solutions for weak turbulence models in electrostatic plasmasSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Plasma Physics (physics.plasm-ph)
The weak turbulence model, also known as the quasilinear theory in plasma physics, has been a cornerstone in modeling resonant particle-wave interactions in plasmas. This reduced model stems from the Vlasov-Poisson/Maxwell system under the weak turbulence assumption, incorporating the random phase approximation and ergodicity. The interaction between particles and waves (plasmons) can be treated as a stochastic process, whose transition probability bridges the momentum space and the spectral space. Therefore, the operators on the right hand side resemble collision forms, such as those in Boltzmann and Landau interacting models. For them, there have been results on well-posedness and regularity of solutions. However, as far as we know, there is no such preceding work for the quasilinear theory addressed in this manuscript.
In this paper, we establish the existence of global weak solutions for the system modeling electrostatic plasmas in one dimension. Our key contribution consists of associating the original integral-differential system to a degenerate inhomogeneous porous medium equation(PME) with nonlinear source terms, and leveraging advanced techniques from the PME literature. This approach opens a novel pathway for analyzing weak turbulence models in plasma physics. Moreover, our work offers new tools for tackling related problems in the broader context of nonlinear nonlocal PDEs. - [194] arXiv:2305.00066 (replaced) [pdf, html, other]
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Title: The Kolmogorov N-width for linear transport: Exact representation and the influence of the dataSubjects: Numerical Analysis (math.NA)
The Kolmogorov $N$-width describes the best possible error one can achieve by elements of an $N$-dimensional linear space. Its decay has extensively been studied in Approximation Theory and for the solution of Partial Differential Equations (PDEs). Particular interest has occurred within Model Order Reduction (MOR) of parameterized PDEs e.g.\ by the Reduced Basis Method (RBM).
While it is known that the $N$-width decays exponentially fast (and thus admits efficient MOR) for certain problems, there are examples of the linear transport and the wave equation, where the decay rate deteriorates to $N^{-1/2}$. On the other hand, it is widely accepted that a smooth parameter dependence admits a fast decay of the $N$-width. However, a detailed analysis of the influence of properties of the data (such as regularity or slope) on the rate of the $N$-width seems to lack.
In this paper, we use techniques from Fourier Analysis to derive exact representations of the $N$-width in terms of initial and boundary conditions of the linear transport equation modeled by some function $g$ for half-wave symmetric data. For arbitrary functions $g$, we derive bounds and prove that these bounds are sharp. In particular, we prove that the $N$-width decays as ${c_r N^{-r}}$ for functions {with Sobolev regularity} $g\in H^{r{-\varepsilon}}$ for all $\varepsilon>0$ even if $g\not\in H^{r}$. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insight. - [195] arXiv:2305.14028 (replaced) [pdf, other]
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Title: Tiling, spectrality and aperiodicity of connected setsComments: 20 pages, 8 figures. A corrected and simplified folded bridge construction is given in the new version of the paper. One more open question added at the endSubjects: Classical Analysis and ODEs (math.CA); Combinatorics (math.CO)
Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's conjecture suggests that $\Omega$ admits an orthogonal basis of exponential functions if and only if it tiles $\mathbb{R}^d$ by translations. Both conjectures are known to be false in sufficiently high dimensions, with all the so-far-known counterexamples being highly disconnected. On the other hand, both conjectures are known to be true for convex sets. In this work we study these conjectures for connected sets. We show that the periodic tiling conjecture, as well as both directions of Fuglede's conjecture are false for connected sets in sufficiently high dimensions.
- [196] arXiv:2306.17599 (replaced) [pdf, html, other]
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Title: The cohomology of $BPU(p^l)$ and polynomial invariantsComments: 15 pagesSubjects: Algebraic Topology (math.AT); Representation Theory (math.RT)
For an odd prime $p$, we explore some connections between the cohomology algebra $H^*(BPU{p^l};\mathbb{F}_p)$ and the subrings of the polynomial ring $\mathbb{F}_p[\xi_1,\eta_1,\cdots,\xi_l,\eta_l]$ that are invariant under some $Sp_{2l}(\mathbb{F}_p)$-action and $GL_{2l}(\mathbb{F}_p)$-action.
- [197] arXiv:2308.00550 (replaced) [pdf, html, other]
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Title: Quantum difference equation for the affine type $A$ quiver varieties I: General ConstructionComments: 59 pages, typo corrected, Proposition 2.13 and Lemma 3.3 has been corrected, Update with an appendix of the rationality of the R-matrix. References addedSubjects: Representation Theory (math.RT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
In this article we use the philosophy in [OS22] to construct the quantum difference equation of affine type $A$ quiver varieties in terms of the quantum toroidal algebra $U_{q,t}(\hat{\hat{\mathfrak{sl}}}_{r})$. In the construction, and we define the set of wall for each quiver varieties by the action of the universal $R$-matrix, which is shown to be almost equivalent to that of the $K$-theoretic stable envelope on each interval in $H^2(X,\mathbb{Q})$. We also give the examples of the instanton moduli space $M(n,r)$ and the equivariant Hilbert scheme $\text{Hilb}_{n}([\mathbb{C}^2/\mathbb{Z}_{r}])$ to show the explicit form of the quantum difference operator.
- [198] arXiv:2308.10221 (replaced) [pdf, html, other]
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Title: Existence and uniqueness of the singular self-similar solutions of the fast diffusion equation and logarithmic diffusion equationComments: 34 pages, introduction rewritten and Remark 1.6 addedSubjects: Analysis of PDEs (math.AP)
Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\rho_1>0$, $\eta>0$, $\beta>\frac{m\rho_1}{n-2-nm}$, $\alpha=\alpha_m=\frac{2\beta+\rho_1}{1-m}$, $\beta_0>0$ and $\alpha_0=2\beta_0+1$. We use fixed point argument to give a new proof for the existence and uniqueness of radially symmetric singular solution $f=f^{(m)}$ of the elliptic equation $\Delta (f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{\alpha/\beta}f(x)=\eta$. We also prove the existence and uniqueness of radially symmetric singular solution $g$ of the equation $\Delta\log g+\alpha_0 g+\beta_0x\cdot\nabla g=0$, $g>0$, in $\mathbb{R}^n\setminus\{0\}$, satisfying $\displaystyle\lim_{|x|\to 0}|x|^{\alpha_0/\beta_0}g(x)=\eta$. Such equations arises from the study of backward singular self-similar solution of the fast diffusion equation $u_t=\Delta u^m$ and the logarithmic diffusion equation $u_t=\Delta\log u$ respectively. We will also prove the asymptotic decay rate of the function $f$ as $|x|\to\infty$.
- [199] arXiv:2308.12195 (replaced) [pdf, html, other]
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Title: A motivic Fundamental LemmaComments: 56 pagesSubjects: Algebraic Geometry (math.AG); Logic (math.LO); Number Theory (math.NT)
In this paper we prove motivic versions of the Langlands-Shelstad Fundamental Lemma and Ngô's Geometric Stabilization. To achieve this, we follow the strategy from the recent proof by Groechenig, Wyss and Ziegler which avoided the use of perverse sheaves using instead $p$-adic integration and Tate duality. We make a key use of a construction of Denef and Loeser which assigns a virtual motive to any definable set in the theory of pseudo-finite fields.
- [200] arXiv:2308.16362 (replaced) [pdf, html, other]
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Title: A Unified Analysis on the Subgradient Upper Bounds for the Subgradient Methods Minimizing Composite Nonconvex, Nonsmooth and Non-Lipschitz FunctionsSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
This paper presents a unified analysis for the proximal subgradient method (Prox-SubGrad) type approach to minimize an overall objective of $f(x)+r(x)$, subject to convex constraints, where both $f$ and $r$ are weakly convex, nonsmooth, and non-Lipschitz. Leveraging on the properties of the Moreau envelope of weakly convex functions, we are able to relate error-bound conditions, the growth conditions of the subgradients of the objective, and the behavior of the proximal subgradient iterates on some remarkably broad classes of objective functions. Various existing as well as new bounding conditions are studied, leading to novel iteration complexity results. The terrain of our exploration expands to stochastic proximal subgradient algorithms.
- [201] arXiv:2309.02013 (replaced) [pdf, html, other]
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Title: Empirical approximation of the gaussian distribution in $\mathbb{R}^d$Journal-ref: Advances in Mathematics, 2024+Subjects: Probability (math.PR); Functional Analysis (math.FA)
Let $G_1,\dots,G_m$ be independent copies of the standard gaussian random vector in $\mathbb{R}^d$. We show that there is an absolute constant $c$ such that for any $A \subset S^{d-1}$, with probability at least $1-2\exp(-c\Delta m)$, for every $t\in\mathbb{R}$, \[ \sup_{x \in A} \left| \frac{1}{m}\sum_{i=1}^m 1_{ \{\langle G_i,x\rangle \leq t \}} - \mathbb{P}(\langle G,x\rangle \leq t) \right| \leq \Delta + \sigma(t) \sqrt\Delta. \] Here $\sigma(t) $ is the variance of $1_{\{\langle G,x\rangle\leq t\}}$ and $\Delta\geq \Delta_0$, where $\Delta_0$ is determined by an unexpected complexity parameter of $A$ that captures the set's geometry (Talagrand's $\gamma_1$ functional). The bound, the probability estimate, and the value of $\Delta_0$ are all (almost) optimal.
We use this fact to show that if $\Gamma=\sum_{i=1}^m \langle G_i,x\rangle e_i$ is the random matrix that has $G_1,\dots,G_m$ as its rows, then the structure of $\Gamma(A)=\{\Gamma x: x\in A\}$ is far more rigid and well-prescribed than was previously expected. - [202] arXiv:2309.06059 (replaced) [pdf, html, other]
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Title: Dynamical Spin Limit Shape of Young Diagram and Spin Jucys-Murphy Elements for Symmetric GroupsComments: 55 pages, 12 figuresSubjects: Probability (math.PR); Representation Theory (math.RT)
The branching rule for spin irreducible representations of symmetric groups gives rise to a Markov chain on the spin dual $(\widetilde{\mathfrak{S}}_n)^\wedge_{\mathrm{spin}}$ of symmetric group $\mathfrak{S}_n$ through restriction and induction of spin irreducible representations. This further produces a continuous time random walk $(X_s^{(n)})_{s\geqq 0}$ on $(\widetilde{\mathfrak{S}}_n)^\wedge_{\mathrm{spin}}$ by introducing an appropriate pausing time. Taking diffusive scaling limit for these random walks under $s=tn$ and $1/\sqrt{n}$ reduction as $n\to\infty$, we consider a concentration phenomenon at each macroscopic time $t$. Since an element of $(\widetilde{\mathfrak{S}}_n)^\wedge_{\mathrm{spin}}$ is regarded as a strict partition of $n$ with $\pm 1$ indices, the limit shapes of profiles of strict partitions appear. In this paper, we give a framework in which initial concentration at $t=0$ is propagated to concentration at any $t>0$. We thus obtain the limit shape $\omega_t$ depending on macroscopic time $t$, and describe the time evolution by using devices in free probability theory. Included is the case where Kerov's transition measure has non-compact support but determinate moment problem. A spin version of Biane's formula for spin Jucys--Murphy elements is shown, which plays an important role in our analysis.
- [203] arXiv:2310.02809 (replaced) [pdf, html, other]
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Title: Persistence and neutrality in interacting replicator dynamicsSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the $N$-replicator system and the existence of invariant distributions for a class of associated McKean-Vlasov dynamics. In particular, our results show that, unlike typical models of neutral ecology, fitness equivalence does not need to be assumed but emerges as a condition for the persistence of the system. Further, neutrality is associated with a unique Dirichlet invariant probability measure. We illustrate our findings with some simple case studies, provide numerical results, and discuss our conclusions in the light of Neutral Theory in ecology.
- [204] arXiv:2310.20133 (replaced) [pdf, other]
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Title: Cohomological properties of multinorm-one toriComments: 28 pages, to appear in Results in MathSubjects: Number Theory (math.NT)
In this paper we investigate the Tate--Shafarevich group Sha^1(k, T) of a multinorm-one torus $T$ over a global field $k$. We establish a few functorial maps among cohomology groups and explore their relations. Using these properties and relations we obtain a few basic structural results for Sha^1(k, T) and extend a few results of Bayer-Fluckiger--Lee--Parimala [Adv. in Math., 2019] to some more general multinorm-one tori. We also give a uniform proof of a result of Demarche--Wei for a criterion of the vanishing of Sha^1(k, T), and of the main result of Pollio [Pure App. Math. Q., 2014] for the case where the étale $k$-algebra in question is a product of two abelian extensions. Moreover, we improve the explicit description of Sha^1(k, T) in Lee [J. Pure Appl. Alg., 2022] by removing an intersection condition.
- [205] arXiv:2311.05753 (replaced) [pdf, html, other]
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Title: A short computation of the Rouquier dimension for a cycle of projective linesComments: 10 pages, 5 figures. Minor changes in exposition following referee report. Version accepted to Proceedings of the AMSSubjects: Algebraic Geometry (math.AG)
Given a dg category $\mathcal C$, we introduce a new class of objects (weakly product bimodules) in $\mathcal C^{op}\otimes \mathcal C$ generalizing product bimodules. We show that the minimal generation time of the diagonal by weakly product bimodules provides an upper bound for the Rouquier dimension of $\mathcal C$. As an application, we give a purely algebro-geometric proof of a result of Burban and Drozd that the Rouquier dimension of the derived category of coherent sheaves on an $n$-cycle of projective lines is one. Our approach explicitly gives the generator realizing the minimal generation time.
- [206] arXiv:2311.06171 (replaced) [pdf, html, other]
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Title: Fast relaxation of the random field Ising dynamicsComments: Updated version based on referee feedback. 40 pages, 2 figuresSubjects: Probability (math.PR); Data Structures and Algorithms (cs.DS); Mathematical Physics (math-ph)
We study the convergence properties of Glauber dynamics for the random field Ising model (RFIM) with ferromagnetic interactions on finite domains of $\mathbb{Z}^d$, $d \ge 2$. Of particular interest is the Griffiths phase where correlations decay exponentially fast in expectation over the quenched disorder, but there exist arbitrarily large islands of weak fields where low-temperature behavior is observed. Our results are twofold:
1. Under weak spatial mixing (boundary-to-bulk exponential decay of correlations) in expectation, we show that the dynamics satisfy a weak Poincaré inequality implying algebraic relaxation to equilibrium over timescales polynomial in the volume $N$ of the domain, and polynomial time mixing from a warm start. From this we construct a polynomial-time approximate sampling algorithm based on running Glauber dynamics over an increasing sequence of approximations of the domain.
2. Under strong spatial mixing (exponential decay of correlations even near boundary pinnings) in expectation, we prove a full Poincaré inequality, implying exponential relaxation to equilibrium and $N^{o(1)}$-mixing time. Note by way of example, both weak and strong spatial mixing hold at any temperature, provided the external fields are strong enough.
Our proofs combine a stochastic localization technique which has the effect of increasing the variance of the field, with a field-dependent coarse graining which controls the resulting sub-critical percolation process of sites with weak fields. - [207] arXiv:2311.08285 (replaced) [pdf, html, other]
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Title: Energy-minimizing Mappings of Complex Projective SpacesComments: Expository changes. v2 matches the submitted versionSubjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Complex Variables (math.CV)
We show that in all homotopy classes of mappings from complex projective space to Riemannian manifolds, the infimum of the energy is proportional to the infimal area in the homotopy class of mappings of the 2-sphere which represents the induced homomorphism on the second homotopy group. We then establish a family of optimal lower bounds for a larger class of energy functionals for mappings from real and complex projective space to Riemannian manifolds and characterize the mappings which attain these lower bounds.
- [208] arXiv:2311.10618 (replaced) [pdf, html, other]
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Title: Metric viscosity solutions on metric spaces and on the Wasserstein spaceComments: 22 pagesSubjects: Analysis of PDEs (math.AP)
Viscosity solutions to the eikonal equation on a non-compact complete Riemannian manifold are crucial for understanding the underlying geometric and topological properties. The classical definition of viscosity solutions relies on the differential structure and does not directly extended to general metric spaces. In this work, we explore the (metric) viscosity solutions on metrics spaces, in particular, the Wasserstein space P_p(X) where X is a complete, separable, locally compact, non-compact geodesic space. We introduce a new and elegant definition of the metric viscosity solutions in this context and investigate their properties such as stability, uniqueness and comparison principle. Moreover, we study their relationship with two types of distance-like function and present two distinct methods to construct (strong) metric viscosity solutions on the Wasserstein space P_p(X).
- [209] arXiv:2311.13992 (replaced) [pdf, html, other]
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Title: Pro-$p$ Poincar\'{e}-Duality groups of infinite rankSubjects: Group Theory (math.GR)
We introduce the class of \textit{Generalized Poincaré-Duality groups}: i.e, pro-$p$ groups of infinite rank which satisfy a Poincaré-duality. We prove some basic properties of Generalized Poincaré-Duality groups, and show that under some conditions, a group that fits into an exact sequence of GPD groups must be a Demushkin group.
- [210] arXiv:2312.06370 (replaced) [pdf, html, other]
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Title: On the maximum degree of induced subgraphs of the Kneser graphComments: 31 pages. Clarifications in response to comments of two anonymous refereesSubjects: Combinatorics (math.CO)
For integers $n \geq k \geq 1$, the {\em Kneser graph} $K(n, k)$ is the graph with vertex-set consisting of all the $k$-element subsets of $\{1,2,\ldots,n\}$, where two $k$-element sets are adjacent in $K(n,k)$ if they are disjoint. We show that if $(n,k,s) \in \mathbb{N}^3$ with $n > 10000 k s^5$ and $\mathcal{F}$ is set of vertices of $K(n,k)$ of size larger than $\{A \subset \{1,2,\ldots,n\}:\ |A|=k,\ A \cap \{1,2,\ldots,s\} \neq \varnothing\}$, then the subgraph of $K(n,k)$ induced by $\mathcal{F}$ has maximum degree at least \[ \left(1 - O\left(\sqrt{s^3 k/n}\right)\right)\frac{s}{s+1} \cdot {n-k \choose k} \cdot \frac{|\mathcal{F}|}{\binom{n}{k}}.\] This is sharp up to the behaviour of the error term $O(\sqrt{s^3 k/n})$. In particular, if the triple of integers $(n, k, s)$ satisfies the condition above, then the minimum maximum degree does not increase `continuously' with $|\mathcal{F}|$. Instead, it has $s$ jumps, one at each time when $|\mathcal{F}|$ becomes just larger than the union of $i$ stars, for $i = 1, 2, \ldots, s$. An appealing special case of the above result is that if $\mathcal{F}$ is a family of $k$-element subsets of $\{1,2,\ldots,n\}$ with $|\mathcal{F}| = {n-1 \choose k-1}+1$, then there exists $A \in \mathcal{F}$ such that $\mathcal{F}$ is disjoint from at least $$\left(1/2-O\left(\sqrt{k/n}\right)\right){n-k-1 \choose k-1}$$ of the other sets in $\mathcal{F}$; this is asymptotically sharp if $k=o(n)$. Frankl and Kupavskii, using different methods, have recently proven closely related results under the hypothesis that $n$ is at least quadratic in $k$.
- [211] arXiv:2312.08742 (replaced) [pdf, other]
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Title: A note on the Casas-Alvero ConjectureDaniel Schaub (LAREMA), Mark Spivakovsky (IMT)Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
The Casas-Alvero conjecture predicts that every univariate polynomial $f$ over a field $K$ of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. Let $f=x^d+a_1x^{d-1}+\cdots+a_1x \in K[a_1,\ldots,a_{d-1}][x]$ and let $R_i = Res(f,H_i(f))\in K[a_1,\ldots,a_{d-1}]$ be the resultant of $f$ and $H_i(f)$, $i \in \{1,\ldots,d-1\}$. The Casas-Alvero Conjecture is equivalent to saying that $R_1,\ldots,R_{d-1}$ are ``independent'' in a certain sense, namely that the height $ht(R_1,\ldots,R_{d-1})=d-1$ in $K[a_1,\ldots,a_{d-1}]$. In this paper we prove a very partial result in this direction: if $i \in \{d-3,d-2,d-1\}$ then $R_i \notin \sqrt{(R_1,\ldots,\breve{R_i},\ldots,R_{d-1}}$.
- [212] arXiv:2312.13674 (replaced) [pdf, html, other]
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Title: Spanning trees for many different numbers of leavesComments: 8 pages, 1 figureSubjects: Combinatorics (math.CO)
Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.
- [213] arXiv:2312.13782 (replaced) [pdf, html, other]
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Title: One-nodal Fano threefolds with Picard number oneComments: 79 pages; v2: 1-cuspidal factorial Fano threefolds included into the classification; 83 pagesSubjects: Algebraic Geometry (math.AG)
We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds of higher Picard rank and with unprojections of complete intersection varieties.
- [214] arXiv:2312.14655 (replaced) [pdf, html, other]
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Title: Value Distributions of Derivatives of $K$-regular Polynomial FamiliesSubjects: Complex Variables (math.CV)
Let $\Omega \in \mathbb{C}$ be a domain such that $K:= \mathbb{C} \setminus \Omega$ is compact and non-polar. Let $g_\Omega$ be the Green's function with a logarithmic pole at infinity, and let $\omega = \omega_K$ be the equilibrium distribution on $K$. Let $(q_k)_{k>0}$ be a sequence of polynomials with $n_k$, the degree of $q_k$ satisfying $n_k \to \infty$, and let $(q_k^m)_k$ denote the sequence of $m$-th derivatives. We provide conditions, which ensure that the preimages $(q_k^m)^{-1}(\{a\})$ uniformly equidistribute on $\partial \Omega$, as $k \to \infty$, for every $a \in \mathbb{C}$ and every $m = 0, 1, \ldots$
- [215] arXiv:2401.00078 (replaced) [pdf, html, other]
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Title: Absolute concentration robustness: Algebra and geometryLuis David García Puente, Elizabeth Gross, Heather A Harrington, Matthew Johnston, Nicolette Meshkat, Mercedes Pérez Millán, Anne ShiuComments: Proof in Section 5 edited, in response to reviewer commentsSubjects: Algebraic Geometry (math.AG); Dynamical Systems (math.DS); Molecular Networks (q-bio.MN)
Motivated by the question of how biological systems maintain homeostasis in changing environments, Shinar and Feinberg introduced in 2010 the concept of absolute concentration robustness (ACR). A biochemical system exhibits ACR in some species if the steady-state value of that species does not depend on initial conditions. Thus, a system with ACR can maintain a constant level of one species even as the environment changes. Despite a great deal of interest in ACR in recent years, the following basic question remains open: How can we determine quickly whether a given biochemical system has ACR? Although various approaches to this problem have been proposed, we show that they are incomplete. Accordingly, we present new methods for deciding ACR, which harness computational algebra. We illustrate our results on several biochemical signaling networks.
- [216] arXiv:2401.02074 (replaced) [pdf, html, other]
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Title: On the boundary of the central quadratic hyperbolic componentComments: 9 pages,3 figuresSubjects: Dynamical Systems (math.DS)
We give a concrete description for the boundary of the central quadratic hyperbolic component. The connectedness of the Julia sets of the boundary maps are also considered.
- [217] arXiv:2401.03821 (replaced) [pdf, html, other]
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Title: On the degree of irrationality of low genus $K3$ surfacesComments: 32 pages, 1 figure. Comments welcome. Accepted version, to appear in Journal of the Institute of Mathematics of JussieauSubjects: Algebraic Geometry (math.AG)
Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$, we study projections $S\hookrightarrow \mathbb P^g\dashrightarrow \mathbb P^2$ of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$, and that for $g=7,8,9,11$ there are no rational maps $S\dashrightarrow \mathbb P^2$ of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques à la Lazarsfeld with derived category tools, and also make use of the rich theory of singular curves on $K3$ surfaces.
- [218] arXiv:2401.06486 (replaced) [pdf, html, other]
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Title: Cost-optimal adaptive FEM with linearization and algebraic solver for semilinear elliptic PDEsSubjects: Numerical Analysis (math.NA)
We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to a linear Poisson-type equation that is symmetric and positive definite. The resulting system is solved by a contractive algebraic solver such as a multigrid method with local smoothing. We formulate a fully adaptive algorithm that equibalances the various error components coming from mesh refinement, iterative linearization, and algebraic solver. We prove that the proposed adaptive iteratively linearized finite element method (AILFEM) guarantees convergence with optimal complexity, where the rates are understood with respect to the overall computational cost (i.e., the computational time). Numerical experiments investigate the involved adaptivity parameters.
- [219] arXiv:2401.08729 (replaced) [pdf, html, other]
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Title: Boundedness of operator-valued commutators involving martingale paraproductsSubjects: Operator Algebras (math.OA); Functional Analysis (math.FA)
Let $1<p<\infty$. We show the boundedness of operator-valued commutators $[\pi_a,M_b]$ on the noncommutative $L_p(L_\infty(\mathbb{R})\otimes \mathcal{M})$ for any von Neumann algebra $\mathcal{M}$, where $\pi_a$ is the $d$-adic martingale paraproduct with symbol $a\in BMO^d(\mathbb{R})$ and $M_b$ is the noncommutative left multiplication operator with $b\in BMO^d_\mathcal{M}(\mathbb{R})$. Besides, we consider the extrapolation property of semicommutative $d$-adic martingale paraproducts in terms of the $BMO^d_\mathcal{M}(\mathbb{R})$ space.
- [220] arXiv:2401.08731 (replaced) [pdf, html, other]
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Title: On the best constants of the noncommutative Littlewood-Paley-Stein inequalitiesSubjects: Operator Algebras (math.OA); Functional Analysis (math.FA)
Let $1<p<\infty$. Let $\{T_t\}_{t>0}$ be a noncommutative symmetric diffusion semigroup on a semifinite von Neumann algebra $\mathcal{M}$, and let $\{P_t\}_{t>0}$ be its associated subordinated Poisson semigroup. The celebrated noncommutative Littlewood-Paley-Stein inequality asserts that for any $x\in L_p(\mathcal{M})$,
\begin{equation*}
\alpha_p^{-1}\|x\|_{p}\le \|x\|_{p,P}\le \beta_p \|x\|_{p},
\end{equation*}
where $\|\cdot\|_{p,P}$ is the $L_p(\mathcal{M})$-norm of square functions associated with $\{P_t\}_{t>0}$, and $\alpha_p, \beta_p$ are the best constants only depending on $p$.
We show that as $p\to \infty$,
$$ \beta_p\lesssim p, $$
and $p$ is the optimal possible order of $\beta_p$ as well. We also obtain some lower and upper bounds of $\alpha_p$ and $\beta_p$ in the other cases. - [221] arXiv:2401.16344 (replaced) [pdf, other]
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Title: Trace estimates for harmonic functions along circular arcs with applications to domain decomposition on overlapping disksSubjects: Analysis of PDEs (math.AP)
In this paper we derive several (and in many cases sharp) estimates for the $\mathrm{L}^2$-trace norm of harmonic functions along circular arcs. More precisely, we obtain geometry-dependent estimates on the norm, spectral radius, and numerical range of the Dirchlet-to-Dirichlet (DtD) operator sending data on the boundary of the disk to the restriction of its harmonic extension along circular arcs inside the disk. The estimates we derive here have applications in the convergence analysis of the Schwarz domain decomposition method for overlapping disks in two dimensions. In particular, they allow us to establish a rigorous convergence proof for the discrete parallel Schwarz method applied to the Conductor-like Screening Model (COSMO) from theoretical chemistry in the two-disk case, and to derive error estimates with respect to the discretization parameter, the number of Schwarz iterations, and the geometry of the domain. Our analysis addresses challenges beyond classical domain decomposition theory, especially the weak enforcement of boundary conditions.
- [222] arXiv:2402.01611 (replaced) [pdf, other]
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Title: Hom $\omega$-categories of a computad are freeComments: 45 pages, updated to change the structure of the paper, add the suspension of $ω$-categories, change the title and abstract accordingly, add citations and correct a few typosSubjects: Category Theory (math.CT); Logic in Computer Science (cs.LO)
We provide a new description of the hom functor on weak $\omega$-categories, and we show that it admits a left adjoint that we call the suspension functor. We then show that the hom functor preserves the property of being free on a computad, in contrast to the hom functor for strict $\omega$-categories. Using the same technique, we define the opposite of an $\omega$-category with respect to a set of dimensions, and we show that this construction also preserves the property of being free on a computad. Finally, we show that the constructions of opposites and homs commute.
- [223] arXiv:2402.06081 (replaced) [pdf, html, other]
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Title: A Computer Search of New OBZCPs of Lengths up to 49Subjects: Information Theory (cs.IT); Number Theory (math.NT)
This paper aims to search for new optimal and sub-optimal Odd Binary Z-Complimentary Pairs (OBZCPs) for lengths up to 49. As an alternative to the celebrated binary Golay complementary pairs, optimal OBZCPs are the best almost-complementary sequence pairs having odd lengths. We introduce a computer search algorithm with time complexity $O(2^N)$, where $N$ denotes the sequence length and then show optimal results for all $27 \le N \le 33$ and $N=37,41,49$. For those sequence lengths (i.e., $N=35,39,43,45,47$) with no optimal pairs, we show OBZCPs with largest zero-correlation zone (ZCZ) widths (i.e., $Z$-optimal). Finally, based on the Pursley--Sarwate criterion (PSC), we present a table of OBZCPs with smallest combined auto-correlation and cross-correlation.
- [224] arXiv:2402.08058 (replaced) [pdf, html, other]
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Title: Colimits of Heyting Algebras through Esakia DualityComments: 27 pagesSubjects: Logic (math.LO); Rings and Algebras (math.RA)
In this note we generalize the construction, due to Ghilardi, of the free Heyting algebra generated by a finite distributive lattice, to the case of arbitrary distributive lattices. Categorically, this provides an explicit construction of a left adjoint to the inclusion of Heyting algebras in the category of distributive lattices This is shown to have several applications, both old and new, in the study of Heyting algebras: (1) it allows a more concrete description of colimits of Heyting algebras, as well as, via duality theory, limits of Esakia spaces, by knowing their description over distributive lattices and Priestley spaces; (2) It allows a direct proof of the amalgamation property for Heyting algebras, and of related facts; (3) it allows a proof of the fact that the category of Heyting algebras is co-distributive. We also study some generalizations and variations of this construction to different settings. First, we analyse some subvarieties of Heyting algebras -- such as Boolean algebras, $\mathsf{KC}$ and $\mathsf{LC}$ algebras, and show how the construction can be adapted to this setting. Second, we study the relationship between the category of image-finite posets with p-morphisms and the category of posets with monotone maps, showing that a variation of the above ideas provides us with an appropriate general idea.
- [225] arXiv:2402.13224 (replaced) [pdf, html, other]
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Title: Controlling Large Electric Vehicle Charging Stations via User Behavior Modeling and Stochastic ProgrammingSubjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Computational Engineering, Finance, and Science (cs.CE); Machine Learning (cs.LG); Systems and Control (eess.SY)
This paper introduces an Electric Vehicle Charging Station (EVCS) model that incorporates real-world constraints, such as slot power limitations, contract threshold overruns penalties, or early disconnections of electric vehicles (EVs). We propose a formulation of the problem of EVCS control under uncertainty, and implement two Multi-Stage Stochastic Programming approaches that leverage user-provided information, namely, Model Predictive Control and Two-Stage Stochastic Programming. The model addresses uncertainties in charging session start and end times, as well as in energy demand. A user's behavior model based on a sojourn-time-dependent stochastic process enhances cost reduction while maintaining customer satisfaction. The benefits of the two proposed methods are showcased against two baselines over a 22-day simulation using a real-world dataset. The two-stage approach demonstrates robustness against early disconnections by considering a wider range of uncertainty scenarios for optimization. The algorithm prioritizing user satisfaction over electricity cost achieves a 20% and 36% improvement in two user satisfaction metrics compared to an industry-standard baseline. Additionally, the algorithm striking the best balance between cost and user satisfaction exhibits a mere 3% relative cost increase compared to the theoretically optimal baseline - for which the nonanticipativity constraint is relaxed - while attaining 94% and 84% of the user satisfaction performance in the two used satisfaction metrics.
- [226] arXiv:2403.05737 (replaced) [pdf, html, other]
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Title: A survey of degree-boundednessSubjects: Combinatorics (math.CO)
Suppose a graph has no large balanced bicliques, but has large minimum degree. Then what can we say about its induced subgraphs? This question motivates the study of degree-boundedness, which is like $\chi$-boundedness but for minimum degree instead of chromatic number. We survey this area with an eye towards open problems.
- [227] arXiv:2403.07672 (replaced) [pdf, html, other]
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Title: Quantitative estimates in almost periodic homogenization of parabolic systemsSubjects: Analysis of PDEs (math.AP)
We consider a family of second-order parabolic operators $\partial_t+\mathcal{L}_\varepsilon$ in divergence form with rapidly oscillating, time-dependent and almost-periodic coefficients. We establish uniform interior and boundary Hölder and Lipschitz estimates as well as convergence rate. The estimates of fundamental solution and Green's function are also established. In contrast to periodic case, the main difficulty is that the corrector equation $(\partial_s+\mathcal{L}_1)(\chi^\beta_{j})=-\mathcal{L}_1(P^\beta_j) $ in $\mathbb{R}^{d+1}$ may not be solvable in the almost periodic setting for linear functions $P(y)$ and $\partial_t \chi_S$ may not in $B^2(\mathbb{R}^{d+1})$. Our results are new even in the case of time-independent coefficients.
- [228] arXiv:2403.09001 (replaced) [pdf, other]
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Title: Solving Partial Differential Equations Using Artificial Neural NetworksComments: Unofficial, revised, and English-only version of the original bilingual dissertation submitted to the University Repository prior to defense. The official version is available at this https URL. Erratum reports sent to this http URL@gmail.com are greatly appreciated. Last update: October 2024Subjects: Numerical Analysis (math.NA)
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays, among the most popular numerical methods for solving partial differential equations in engineering, we encounter the finite difference and finite element methods. An alternative numerical method that has recently gained popularity for numerically solving partial differential equations is the use of artificial neural networks.
Artificial neural networks, or neural networks for short, are mathematical structures with universal approximation properties. In addition, thanks to the extraordinary computational development of the last decade, neural networks have become accessible and powerful numerical methods for engineers and researchers. For example, imaging and language processing are applications of neural networks today that show sublime performance inconceivable years ago.
This dissertation contributes to the numerical solution of partial differential equations using neural networks with the following two-fold objective: investigate the behavior of neural networks as approximators of solutions of partial differential equations and propose neural-network-based methods for frameworks that are hardly addressable via traditional numerical methods.
As novel neural-network-based proposals, we first present a method inspired by the finite element method when applying mesh refinements to solve parametric problems. Secondly, we propose a general residual minimization scheme based on a generalized version of the Ritz method. Finally, we develop a memory-based strategy to overcome a usual numerical integration limitation when using neural networks to solve partial differential equations. - [229] arXiv:2403.10673 (replaced) [pdf, other]
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Title: Almost-Surely Convergent Randomly Activated Monotone Operator Splitting MethodsSubjects: Optimization and Control (math.OC)
We propose stochastic splitting algorithms for solving large-scale composite inclusion problems involving monotone and linear operators. They activate at each iteration blocks of randomly selected resolvents of monotone operators and, unlike existing methods, achieve almost sure convergence of the iterates to a solution without any regularity assumptions or knowledge of the norms of the linear operators. Applications to image recovery and machine learning are provided.
- [230] arXiv:2403.18179 (replaced) [pdf, html, other]
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Title: Tagged particles and size-biased dynamics in mean-field interacting particle systemsComments: 19 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.
- [231] arXiv:2404.02341 (replaced) [pdf, html, other]
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Title: Fully chaotic conservative models for some torus homeomorphismsComments: 95 pages, 16 figuresSubjects: Dynamical Systems (math.DS)
We study homotopic-to-the-identity torus homeomorphisms, whose rotation set has nonempty interior. We prove that any such map is monotonically semiconjugate to a homeomorphism that preserves the Lebesgue measure, and that has the same rotation set. Furthermore, the dynamics of the quotient map has several interesting chaotic traits: for instance, it is topologically mixing, has a dense set of periodic points and is continuum-wise expansive. In particular, this shows that a convex compact set of $\mathbb{R}^2$ with nonempty interior is the rotation set of the lift of a homeomorphism of $\mathbb{T}^2$ if and only if it is the rotation set of the lift of a conservative homeomorphism.
- [232] arXiv:2404.08383 (replaced) [pdf, html, other]
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Title: Optimal Transport and Wasserstein Barycenter for Radial Contoured DistributionsSubjects: Optimization and Control (math.OC)
The optimal transport and Wasserstein barycenter of Gaussian distributions have been solved. In literature, the closed form formulas of the Monge map, the Wasserstein distance and the Wasserstein barycenter have been given. Moreover, when Gaussian distributions extend more generally to elliptical contoured distributions, similar results also hold true. In this case, Gaussian distributions are regarded as elliptical contoured distribution with generator function $e^{-x/2}$. However, there are few results about optimal transport for elliptical contoured distributions with different generator functions. In this paper, we degenerate elliptical contoured distributions to radial contoured distributions and study their optimal transport and prove their Wasserstein barycenter is still radial contoured. For general elliptical contoured distributions, we give two numerical counterexamples to show that the Wasserstein barycenter of elliptical contoured distributions does not have to be elliptical contoured.
- [233] arXiv:2404.11314 (replaced) [pdf, html, other]
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Title: Destructive and constructive RIS beamforming in an ISAC-multi-user MIMO networkComments: submitted to IEEE ICC25Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Integrated sensing and communication (ISAC) has already established itself as a promising solution to the spectrum scarcity problem, even more so when paired with a reconfigurable intelligent surface (RIS), as RISs can shape the propagation environment by adjusting their phase-shift coefficients. Albeit the potential performance gain, a RIS is also a potential security threat to the system. In this paper, we explore both the positive and negative sides of having a RIS in a multi-user multiple-input multiple-output (MIMO) ISAC network. We first develop an alternating optimization algorithm, obtaining the active and passive beamforming vectors that maximize the sensing signal-to-noise ratio (SNR) under minimum signal-to-interference-plus-noise ratio (SINR) constraints for the communication users and finite power budget. We also investigate the destructive potential of the RIS by devising a RIS phase-shift optimization algorithm that minimizes the sensing SNR while preserving the same minimum communication SINR previously guaranteed by the system. We further investigate the impact of the RIS's individual element failures on the system performance. The simulation results show that the RIS performance-boosting potential is as good as its destructive one and that both of our optimization strategies are hindered by the investigated impairments.
- [234] arXiv:2404.13389 (replaced) [pdf, html, other]
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Title: Eigenvalues and graph minorsSubjects: Combinatorics (math.CO)
Let $spex(n,H_{minor})$ denote the maximum spectral radius of $n$-vertex $H$-minor free graphs. The problem on determining this extremal value can be dated back to the early 1990s. Up to now, it has been solved for $n$ sufficiently large and some special minors, such as $\{K_{2,3},K_4\}$, $\{K_{3,3},K_5\}$, $K_r$ and $K_{s,t}$. In this paper, we find some unified phenomena on general minors. Every graph $G$ on $n$ vertices with spectral radius $\rho\geq spex(n,H_{minor})$ contains either an $H$ minor or a spanning book $K_{\gamma_H}\nabla(n-\gamma_H)K_1$, where $\gamma_H=|H|-\alpha(H)-1$. Furthermore, assume that $G$ is $H$-minor free and $\Gamma^*_s(H)$ is the family of $s$-vertex irreducible induced subgraphs of $H$, then $G$ minus its $\gamma_H$ dominating vertices is $\Gamma^*_{\alpha(H)+1}(H)$-minor saturate, and it is further edge-maximal if $\Gamma^*_{\alpha(H)+1}(H)$ is a connected family. As applications, we obtain some known results on minors mentioned above. We also determine the extremal values for some other minors, such as flowers, wheels, generalized books and complete multi-partite graphs. Our results extend some conjectures on planar graphs, outer-planar graphs and $K_{s,t}$-minor free graphs. To obtain the results, we combine stability method, spectral techniques and structural analyses. Especially, we give an exploration of using absorbing method in spectral extremal problems.
- [235] arXiv:2404.13646 (replaced) [pdf, html, other]
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Title: Physics-informed Discretization-independent Deep Compositional Operator NetworkSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG)
Solving parametric Partial Differential Equations (PDEs) for a broad range of parameters is a critical challenge in scientific computing. To this end, neural operators, which \textcolor{black}{predicts the PDE solution with variable PDE parameter inputs}, have been successfully used. However, the training of neural operators typically demands large training datasets, the acquisition of which can be prohibitively expensive. To address this challenge, physics-informed training can offer a cost-effective strategy. However, current physics-informed neural operators face limitations, either in handling irregular domain shapes or in in generalizing to various discrete representations of PDE parameters. In this research, we introduce a novel physics-informed model architecture which can generalize to various discrete representations of PDE parameters and irregular domain shapes. Particularly, inspired by deep operator neural networks, our model involves a discretization-independent learning of parameter embedding repeatedly, and this parameter embedding is integrated with the response embeddings through multiple compositional layers, for more expressivity. Numerical results demonstrate the accuracy and efficiency of the proposed method. All the codes and data related to this work are available on GitHub: this https URL.
- [236] arXiv:2405.01399 (replaced) [pdf, html, other]
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Title: Algebraic types in Zilber's exponential fieldComments: Minor revisionsSubjects: Logic (math.LO); Number Theory (math.NT)
We characterise the model-theoretic algebraic closure in Zilber's exponential field. A key step involves showing that certain algebraic varieties have finite intersections with certain finite-rank subgroups of the graph of exponentiation. Mordell-Lang for algebraic tori (a theorem of Laurent) plays a central role in our proof.
- [237] arXiv:2405.02473 (replaced) [pdf, html, other]
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Title: From quantum difference equation to Dubrovin connection of affine type A quiver varietiesComments: 57 pages. Reference added. Some typo fixed. Comments are welcome!Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
This is the continuation of the article \cite{Z23}. In this article we will give a detailed analysis of the quantum difference equation of the equivariant $K$-theory of the affine type $A$ quiver varieties. We will give a good representation of the quantum difference operator $\mathbf{M}_{\mathcal{L}}(z)$ such that the monodromy operator $\mathbf{B}_{\mathbf{m}}(z)$in the formula can be written in the $U_{q}(\mathfrak{sl}_2)$-form or in the $U_{q}(\hat{\mathfrak{gl}}_1)$-form. We also give the detailed analysis of the connection matrix for the quantum difference equation in the nodal limit $p\rightarrow0$. Using these two results, we prove that the degeneration limit of the quantum difference equation is the Dubrovin connection for the quantum cohomology of the affine type $A$ quiver varieties, and the monodromy representation for the Dubrovin connection is generated by the monodromy operators $\mathbf{B}_{\mathbf{m}}$.
- [238] arXiv:2405.03016 (replaced) [pdf, html, other]
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Title: Pathwise uniform convergence of a full discretization for a three-dimensional stochastic Allen-Cahn equation with multiplicative noiseSubjects: Numerical Analysis (math.NA)
This paper analyzes a full discretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. The discretization uses the Euler scheme for temporal discretization and the finite element method for spatial discretization. A key contribution of this work is the introduction of a novel stability estimate for a discrete stochastic convolution, which plays a crucial role in establishing pathwise uniform convergence estimates for fully discrete approximations of nonlinear stochastic parabolic equations. By using this stability estimate in conjunction with the discrete stochastic maximal $L^p$-regularity estimate, the study derives a pathwise uniform convergence rate that encompasses general general spatial $L^q$-norms. Moreover, the theoretical convergence rate is verified by numerical experiments.
- [239] arXiv:2405.03335 (replaced) [pdf, html, other]
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Title: Spectral properties of the resolvent difference for singularly perturbed operatorsSubjects: Spectral Theory (math.SP); Analysis of PDEs (math.AP)
We obtain order sharp spectral estimates for the difference of resolvents of singularly perturbed elliptic operators $\mathbf{A}+\mathbf{V}_1$ and $\mathbf{A}+\mathbf{V}_2$ in a domain $\Omega\subseteq \mathbb{R}^\mathbf{N}$ with perturbations $\mathbf{V}_1, \mathbf{V}_2$ generated by $V_1\mu,V_2\mu,$ where $\mu$ is a measure singular with respect to the Lebesgue measure and satisfying two-sided or one-sided conditions of Ahlfors type, while $V_1,V_2$ are weight functions subject to some integral conditions. As an important special case, spectral estimates for the difference of resolvents of two Robin realizations of the operator $\mathbf{A}$ with different weight functions are obtained. For the case when the support of the measure is a compact Lipschitz hypersurface in $\Omega$ or, more generally, a rectifiable set of Haußdorff dimension $d=\mathbf{N}-1$, the Weyl type asymptotics for eigenvalues is justified.
- [240] arXiv:2405.09400 (replaced) [pdf, html, other]
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Title: Flow updates for domain decomposition of entropic optimal transportComments: Added acknowledgementsSubjects: Numerical Analysis (math.NA)
Domain decomposition has been shown to be a computationally efficient distributed method for solving large scale entropic optimal transport problems. However, a naive implementation of the algorithm can freeze in the limit of very fine partition cells (i.e. it asymptotically becomes stationary and does not find the global minimizer), since information can only travel slowly between cells. In practice this can be avoided by a coarse-to-fine multiscale scheme. In this article we introduce flow updates as an alternative approach. Flow updates can be interpreted as a variant of the celebrated algorithm by Angenent, Haker, and Tannenbaum, and can be combined canonically with domain decomposition. We prove convergence to the global minimizer and provide a formal discussion of its continuity limit. We give a numerical comparison with naive and multiscale domain decomposition, and show that the hybrid method does not suffer from freezing in the regime of very many cells. While the multiscale scheme is observed to be faster than the hybrid approach in general, the latter could be a viable alternative in cases where a good initial coupling is available. Our numerical experiments are based on a novel GPU implementation of domain decomposition that we describe in the appendix.
- [241] arXiv:2405.11818 (replaced) [pdf, html, other]
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Title: A Rate-Distortion Analysis for Composite Sources Under Subsource-Dependent Fidelity CriteriaComments: 17 pages, 8 figures, a revised version for IEEE Journal on Selected Areas in CommunicationsSubjects: Information Theory (cs.IT)
A composite source, consisting of multiple subsources and a memoryless switch, outputs one symbol at a time from the subsource selected by the switch. If some data should be encoded more accurately than other data from an information source, the composite source model is suitable because in this model different distortion constraints can be put on the subsources. In this context, we propose subsource-dependent fidelity criteria for composite sources and use them to formulate a rate-distortion problem. We solve the problem and obtain a single-letter expression for the rate-distortion function. Further rate-distortion analysis characterizes the performance of classify-then-compress (CTC) coding, which is frequently used in practice when subsource-dependent fidelity criteria are considered. Our analysis shows that CTC coding generally has performance loss relative to optimal coding, even if the classification is perfect. We also identify the cause of the performance loss, that is, class labels have to be reproduced in CTC coding. Last but not least, we show that the performance loss is negligible for asymptotically small distortion if CTC coding is appropriately designed and some mild conditions are satisfied.
- [242] arXiv:2406.01097 (replaced) [pdf, other]
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Title: A multiplicative inequality of Riesz transform type on general Riemannian manifoldsEl Maati Ouhabaz (IMB)Comments: Expanded version, three new section are addedSubjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
Given any complete Riemannian manifold $M$, we prove that for every $p \in (1, 2]$ and every $\epsilon > 0$, $$ \| \nabla f \|_p^2 \le C_\epsilon \| \Delta^{\frac{1}{2} + \epsilon} f \|_{p}\| \Delta^{\frac{1}{2} - \epsilon} f \|_{p}.$$The estimate is dimension free. This inequality is even proved in the abstract setting of generators of sub-Markov semigroups.
- [243] arXiv:2406.04191 (replaced) [pdf, html, other]
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Title: Strong Approximations for Empirical Processes Indexed by Lipschitz FunctionsSubjects: Statistics Theory (math.ST); Econometrics (econ.EM); Probability (math.PR); Methodology (stat.ME)
This paper presents new uniform Gaussian strong approximations for empirical processes indexed by classes of functions based on $d$-variate random vectors ($d\geq1$). First, a uniform Gaussian strong approximation is established for general empirical processes indexed by possibly Lipschitz functions, improving on previous results in the literature. In the setting considered by Rio (1994), and if the function class is Lipschitzian, our result improves the approximation rate $n^{-1/(2d)}$ to $n^{-1/\max\{d,2\}}$, up to a $\operatorname{polylog}(n)$ term, where $n$ denotes the sample size. Remarkably, we establish a valid uniform Gaussian strong approximation at the rate $n^{-1/2}\log n$ for $d=2$, which was previously known to be valid only for univariate ($d=1$) empirical processes via the celebrated Hungarian construction (Komlós et al., 1975). Second, a uniform Gaussian strong approximation is established for multiplicative separable empirical processes indexed by possibly Lipschitz functions, which addresses some outstanding problems in the literature (Chernozhukov et al., 2014, Section 3). Finally, two other uniform Gaussian strong approximation results are presented when the function class is a sequence of Haar basis based on quasi-uniform partitions. Applications to nonparametric density and regression estimation are discussed.
- [244] arXiv:2406.09218 (replaced) [pdf, html, other]
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Title: Cohomological integrality for weakly symmetric representations of reductive groupsComments: v5: 27pages. Many improvements following feedback. The main result now holds for weakly symmetric representations. Title adapted; v4: Abstract and introduction rewritten and exposition improved; v3: 24 pages, corrected Lemma 4.5, added Proposition 1.4; v2: fixed example section and proof of finite dimensionality; v1:22 pages, comments very welcomeSubjects: Representation Theory (math.RT); Algebraic Geometry (math.AG)
In this paper, we prove the integrality conjecture for quotient stacks arising from weakly symmetric representations of reductive groups. Our main result is a decomposition of the cohomology of the stack into finite-dimensional components indexed by some equivalence classes of cocharacters of a maximal torus. This decomposition enables the definition of new enumerative invariants associated with the stack, which we begin to explore.
- [245] arXiv:2406.19463 (replaced) [pdf, html, other]
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Title: Many-body Fu-Kane-Mele indexComments: v1 --> v2: Equivalence with non-interacting case refined and extendedSubjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)
We define a $\mathbb{Z}_2$-valued index for stably short-range entangled states of two-dimensional fermionic lattice systems with charge conservation and time reversal symmetry. The index takes its non-trivial value precisely if the `fluxon', the state obtained by inserting a $\pi$-flux through the system, transforms under time reversal as part of a Kramers pair. This index extends the Fu-Kane-Mele index of free fermionic topological insulators to interacting systems.
- [246] arXiv:2406.19697 (replaced) [pdf, html, other]
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Title: A Note on Ordinally Concave FunctionsSubjects: Combinatorics (math.CO)
The notion of ordinal concavity of utility functions has recently been considered by Hafalir, Kojima, Yenmez, and Yokote in economics while there exist earlier related works in discrete optimization and operations research. In the present note we consider functions satisfying ordinal concavity and introduce a weaker notion of ordinal weak-concavity as well. We also investigate useful behaviors of ordinally (weak-)concave functions and related choice correspondences, show a characterization of ordinally weak-concave functions, and give an efficient algorithm for maximizing ordinally concave functions. We further examine a duality in ordinally (weak-)concave functions and introduce the lexicographic composition of ordinally weak-concave functions.
- [247] arXiv:2406.19715 (replaced) [pdf, other]
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Title: A conjectural basis for the $(1,2)$-bosonic-fermionic coinvariant ringComments: 31 pages, 8 figures, 4 tables. Corrected typos and updated referencesSubjects: Combinatorics (math.CO)
We give the first conjectural construction of a monomial basis for the coinvariant ring $R_n^{(1,2)}$, for the symmetric group $S_n$ acting on one set of bosonic (commuting) and two sets of fermionic (anticommuting) variables. Our construction interpolates between the modified Motzkin path basis for $R_n^{(0,2)}$ of Kim-Rhoades (2022) and the super-Artin basis for $R_n^{(1,1)}$ conjectured by Sagan-Swanson (2024) and proven by Angarone et al. (2024). We prove that our proposed basis has cardinality $2^{n-1}n!$, aligning with a conjecture of Zabrocki (2020) on the dimension of $R_n^{(1,2)}$, and show how it gives a combinatorial expression for the Hilbert series. We also conjecture a Frobenius series for $R_n^{(1,2)}$. We show that these proposed Hilbert and Frobenius series are equivalent to conjectures of Iraci, Nadeau, and Vanden Wyngaerd (2024) on $R_n^{(1,2)}$ in terms of segmented Smirnov words, by exhibiting a weight-preserving bijection between our proposed basis and their segmented permutations. We extend some of their results on the sign character to hook characters, and give a formula for the $m_\mu$ coefficients of the conjectural Frobenius series. Finally, we conjecture a monomial basis for the analogous ring in type $B_n$, and show that it has cardinality $4^nn!$.
- [248] arXiv:2407.01170 (replaced) [pdf, html, other]
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Title: Geometric singularities and Hodge theoryComments: Added examples of rough metricsSubjects: Differential Geometry (math.DG); Functional Analysis (math.FA)
We consider smooth vector bundles over smooth manifolds equipped with non-smooth geometric data. For nilpotent differential operators acting on these bundles, we show that the kernels of induced Hodge-Dirac-type operators remain isomorphic under uniform perturbations of the geometric data. We consider applications of this to the Hodge-Dirac operator on differential forms induced by so-called rough Riemannian metrics, which can be of only measurable coefficient in regularity, on both compact and non-compact settings. As a consequence, we show that the kernel of the associated non-smooth Hodge-Dirac operator with respect to a rough Riemannian metric remains isomorphic to smooth and singular cohomology when the underlying manifold is compact.
- [249] arXiv:2407.06034 (replaced) [pdf, html, other]
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Title: About Wess-Zumino-Witten equation and Harder-Narasimhan potentialsComments: v2. Added Theorem 1.6. Streamlined the presentation. Changed the title, and added referencesSubjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Complex Variables (math.CV)
For a polarized family of complex projective manifolds, we identify the algebraic obstructions that govern the existence of approximate solutions to the Wess-Zumino-Witten equation. When this is specialized to the fibration associated with a projectivization of a vector bundle, we recover a version of Kobayashi-Hitchin correspondence.
More broadly, we demonstrate that a certain auxiliary Monge-Ampère type equation, generalizing the Wess-Zumino-Witten equation by taking into account the weighted Bergman kernel associated with the Harder-Narasimhan filtrations of direct image sheaves, admits approximate solutions over any polarized family. These approximate solutions are shown to be the closest counterparts to true solutions of the Wess-Zumino-Witten equation whenever the latter do not exist, as they minimize the associated Yang-Mills functional.
As an application, in a fibered setting, we prove an asymptotic converse to the Andreotti-Grauert theorem conjectured by Demailly. - [250] arXiv:2407.20954 (replaced) [pdf, html, other]
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Title: On the dimension of observable sets for the heat equationSubjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Complex Variables (math.CV)
We consider the heat equation on a bounded $C^1$ domain in $\mathbb{R}^n$ with Dirichlet boundary conditions. The primary aim of this paper is to prove that the heat equation is observable from any measurable set with a Hausdorff dimension strictly greater than $n - 1$. The proof relies on a novel spectral estimate for linear combinations of Laplace eigenfunctions, achieved through the propagation of smallness for solutions to Cauchy-Riemann systems as established by Malinnikova, and uses the Lebeau-Robbiano method. While this observability result is sharp regarding the Hausdorff dimension scale, our secondary goal is to construct families of sets with dimensions less than $n - 1$ from which the heat equation is still observable.
- [251] arXiv:2408.00615 (replaced) [pdf, html, other]
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Title: Quantum difference equations and Maulik-Okounkov quantum affine algebras of affine type $A$Comments: 58 pages. Some typo fixed. The proof of Proposition 4.2 refined. Reference added. Comments are welcome!Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
In this paper we prove the isomorphism of the positive half of the quantum toroidal algebra and the positive half of the Maulik-Okounkov quantum affine algebra of affine type $A$ via the monodromy representation for the Dubrovin connection. The main tool is on the proof of the fact that the degeneration limit of the algebraic quantum difference equation is the same as that of the Okounkov-Smirnov geometric quantum difference equation.
- [252] arXiv:2408.01213 (replaced) [pdf, html, other]
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Title: Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spacesComments: 54 pagesSubjects: Representation Theory (math.RT); Differential Geometry (math.DG)
In this paper we classify and construct differential symmetry breaking operators $\mathbb{D}$ from a line bundle over the real projective space $\mathbb{R}\mathbb{P}^n$ to a vector bundle over $\mathbb{R}\mathbb{P}^{n-1}$. We further determine the factorization identities of $\mathbb{D}$ and the branching laws of the corresponding generalized Verma modules of $\mathfrak{sl}(n+1,\mathbb{C})$. By utilizing the factorization identities, the $SL(n,\mathbb{R})$-representations realized on the image $\text{Im}(\mathbb{D})$ are also investigated.
- [253] arXiv:2408.06830 (replaced) [pdf, html, other]
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Title: A Functional Central Limit Theorem for the General Brownian Motion on the Half-LineComments: 31 pages, 12 figuresSubjects: Probability (math.PR)
In this work, we establish a Trotter-Kato type theorem. More precisely, we characterize the convergence in distribution of Feller processes by examining the convergence of their generators. The main novelty lies in providing quantitative estimates in the vague topology at any fixed time. As important applications, we deduce functional central limit theorems for random walks on the positive integers with boundary conditions, which converge to Brownian motions on the positive half-line with boundary conditions at zero.
- [254] arXiv:2408.07133 (replaced) [pdf, html, other]
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Title: Normalizer Quotients of Symmetric Groups and Inner HolomorphsSubjects: Group Theory (math.GR)
We show that every finite group $T$ is isomorphic to a normalizer quotient $N_{S_n}(H)/H$ for some $n$ and a subgroup $H\leq S_n$. We show that this holds for all large enough $n\ge n_0(T)$ and also with $S_n$ replaced by $A_n$. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group $G$ with $\mathrm{Out}(G)\cong T$ (for any given finite group $T$) and the determination of the normalizer in $\mathrm{Sym(G)}$ of the inner holomorph $\mathrm{InHol}(G)\leq\mathrm{Sym}(G)$ for any centerless indecomposable finite group $G$, which may be of independent interest.
- [255] arXiv:2408.09584 (replaced) [pdf, other]
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Title: From classes in the Weyl group to strataComments: 17 pages. A section on dihedral groups has been addedSubjects: Representation Theory (math.RT)
In a 2015 paper we have defined a map from the set of conjugacy classes in a Weyl group W to the set of irreducible representations of W (its image parametrizes the strata of a reductive group with Weyl group W). In this paper we provide evidence that this map makes sense even when W is replaced by a noncrystallographic Coxeter group.
- [256] arXiv:2408.12876 (replaced) [pdf, other]
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Title: The local limit theorem for complex valued sequences: the parabolic caseJean-François Coulombel (IMT), Grégory Faye (IMT)Comments: Comptes Rendus. Mathématique, In pressSubjects: Numerical Analysis (math.NA)
We give a complete expansion, at any accuracy order, for the iterated convolution of a complex valued integrable sequence in one space dimension. The remainders are estimated sharply with generalized Gaussian bounds. The result applies in probability theory for random walks as well as in numerical analysis for studying the large time behavior of numerical schemes.
- [257] arXiv:2409.00605 (replaced) [pdf, html, other]
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Title: Average-case optimization analysis for distributed consensus algorithms on regular graphsSubjects: Optimization and Control (math.OC); Distributed, Parallel, and Cluster Computing (cs.DC)
The consensus problem in distributed computing involves a network of agents aiming to compute the average of their initial vectors through local communication, represented by an undirected graph. This paper focuses on the studying of this problem using an average-case analysis approach, particularly over regular graphs. Traditional algorithms for solving the consensus problem often rely on worst-case performance evaluation scenarios, which may not reflect typical performance in real-world applications. Instead, we apply average-case analysis, focusing on the expected spectral distribution of eigenvalues to obtain a more realistic view of performance. Key contributions include deriving the optimal method for consensus on regular graphs, showing its relation to the Heavy Ball method, analyzing its asymptotic convergence rate, and comparing it to various first-order methods through numerical experiments.
- [258] arXiv:2409.03235 (replaced) [pdf, other]
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Title: $SLE_6$ and 2-d critical bond percolation on the square latticeComments: Minor revision. 113 pages, 10 figuresSubjects: Probability (math.PR); Mathematical Physics (math-ph); Complex Variables (math.CV)
Through the rotational invariance of the 2-d critical bond percolation exploration path on the square lattice we express Smirnov's edge parafermionic observable as a sum of two new edge observables. With the help of these two new edge observables we can apply the discrete harmonic analysis and conformal mapping theory to prove the convergence of the 2-d critical bond percolation exploration path on the square lattice to the trace of $SLE_6$ as the mesh size of the lattice tends to zero.
- [259] arXiv:2409.08624 (replaced) [pdf, html, other]
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Title: Borel graphable equivalence relationsComments: 60 pages, updated to add referencesSubjects: Logic (math.LO); Dynamical Systems (math.DS)
This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible ordinals and show that it is Borel graphable if and only if there is a non-constructible real. As a corollary of the proof, we construct an analytic equivalence relation which is (provably in ZFC) not Borel graphable and an effectively analytic equivalence relation which is Borel graphable but not effectively Borel graphable. Next, we study analytic equivalence relations given by the isomorphism relation for some class of countable structures. We show that all such equivalence relations are Borel graphable, which implies that for every Borel action of $S_\infty$, the associated orbit equivalence relation is Borel graphable. This leads us to study the class of Polish groups whose Borel actions always give rise to Borel graphable orbit equivalence relations; we refer to such groups as graphic groups. We show that besides $S_\infty$, the class of graphic groups includes all connected Polish groups and is closed under countable products. We finish by studying structural properties of the class of Borel graphable analytic equivalence relations and by considering two variations on Borel graphability: a generalization with hypegraphs instead of graphs and an analogue of Borel graphability in the setting of computably enumerable equivalence relations.
- [260] arXiv:2409.08961 (replaced) [pdf, html, other]
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Title: A curious dynamical system in the planeComments: 19 pages, 9 figuresSubjects: Dynamical Systems (math.DS)
For any irrational $\alpha > 0$ and any initial value $z_{-1} \in \mathbb{C}$, we define a sequence of complex numbers $(z_n)_{n=0}^{\infty}$ as follows: $z_n$ is $z_{n-1} + e^{2 \pi i \alpha n}$ or $z_{n-1} - e^{2 \pi i \alpha n}$, whichever has the smaller absolute value. If both numbers have the same absolute value, the sequence terminates at $z_{n-1}$ but this happens rarely. This dynamical system has astonishingly intricate behavior: the choice of signs in $z_{n-1} \pm e^{2 \pi i \alpha n}$ appears to eventually become periodic (though the period can be large). We prove that if one observes periodic signs for a sufficiently long time (depending on $z_{-1}, \alpha$), the signs remain periodic for all time. The surprising complexity of the system is illustrated through examples.
- [261] arXiv:2409.16219 (replaced) [pdf, html, other]
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Title: Equiangular lines via improved eigenvalue multiplicityComments: We added an additional result giving a correct answer up to lower order terms already from the superpolynomial regimeSubjects: Combinatorics (math.CO)
A family of lines passing through the origin in an inner product space is said to be equiangular if every pair of lines defines the same angle. In 1973, Lemmens and Seidel raised what has since become a central question in the study of equiangular lines in Euclidean spaces. They asked for the maximum number of equiangular lines in $\mathbb{R}^r$ with a common angle of $\arccos{\frac{1}{2k-1}}$ for any integer $k \geq 2$. We show that the answer equals $r-1+\left\lfloor\frac{r-1}{k-1}\right\rfloor,$ provided that $r$ is at least exponential in a polynomial in $k$. This improves upon a recent breakthrough of Jiang, Tidor, Yao, Zhang, and Zhao [Ann. of Math. (2) 194 (2021), no. 3, 729-743], who showed that this holds for $r$ at least doubly exponential in a polynomial in $k$. We also show that for any common angle $\arccos{\alpha}$, the answer equals $r+o(r)$ already when $r$ is superpolynomial in $1/\alpha \to \infty$.
The key new ingredient underlying our results is an improved upper bound on the multiplicity of the second-largest eigenvalue of a graph. In one of the regimes, this improves and significantly extends a result of McKenzie, Rasmussen, and Srivastava [STOC 2021, pp. 396-407]. - [262] arXiv:2409.18659 (replaced) [pdf, html, other]
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Title: Explainable Enrichment-Driven GrAph Reasoner (EDGAR) for Large Knowledge Graphs with Applications in Drug RepurposingOlawumi Olasunkanmi, Evan Morris, Yaphet Kebede, Harlin Lee, Stanley Ahalt, Alexander Tropsha, Chris BizonComments: 10 pages, 5 figures, 4 tablesSubjects: Information Theory (cs.IT); Information Retrieval (cs.IR)
Knowledge graphs (KGs) represent connections and relationships between real-world entities. We propose a link prediction framework for KGs named Enrichment-Driven GrAph Reasoner (EDGAR), which infers new edges by mining entity-local rules. This approach leverages enrichment analysis, a well-established statistical method used to identify mechanisms common to sets of differentially expressed genes. EDGAR's inference results are inherently explainable and rankable, with p-values indicating the statistical significance of each enrichment-based rule.
We demonstrate the framework's effectiveness on a large-scale biomedical KG, ROBOKOP, focusing on drug repurposing for Alzheimer disease (AD) as a case study. Initially, we extracted 14 known drugs from the KG and identified 20 contextual biomarkers through enrichment analysis, revealing functional pathways relevant to shared drug efficacy for AD. Subsequently, using the top 1000 enrichment results, our system identified 1246 additional drug candidates for AD treatment. The top 10 candidates were validated using evidence from medical literature.
EDGAR is deployed within ROBOKOP, complete with a web user interface. This is the first study to apply enrichment analysis to large graph completion and drug repurposing. - [263] arXiv:2409.18748 (replaced) [pdf, html, other]
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Title: On NP-Hardness of $L_1/L_2$ Minimization and Bound Theory of Nonzero Entries in SolutionsSubjects: Optimization and Control (math.OC)
The \(L_1/L_2\) norm ratio has gained significant attention as a measure of sparsity due to three merits: sharper approximation to the \(L_0\) norm compared to the \(L_1\) norm, being parameter-free and scale-invariant, and exceptional performance with highly coherent matrices. These properties have led to its successful application across a wide range of fields. While several efficient algorithms have been proposed to compute stationary points for \(L_1/L_2\) minimization problems, their computational complexity has remained open. In this paper, we prove that finding the global minimum of both constrained and unconstrained \(L_1/L_2\) models is strongly NP-hard.
In addition, we establish uniform upper bounds on the \(L_2\) norm for any local minimizer of both constrained and unconstrained \(L_1/L_2\) minimization models. We also derive upper and lower bounds on the magnitudes of the nonzero entries in any local minimizer of the unconstrained model, aiding in classifying nonzero entries. Finally, we extend our analysis to demonstrate that the constrained and unconstrained \(L_p/L_q\) (\(0 < p \leq 1, 1 < q < +\infty\)) models are also strongly NP-hard. - [264] arXiv:2409.19981 (replaced) [pdf, other]
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Title: On the effectiveness of twisted pluricanonical bundlesComments: The proof of the main theorem is invalidSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV)
In this paper, we prove a conjecture proposed by Schnell, which implies the equivalence between the non-vanishing conjecture and its generalized form, as proposed by Campana and Peternell.
- [265] arXiv:2410.00219 (replaced) [pdf, html, other]
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Title: Improved performance guarantees for Tukey's medianComments: Strengthened results related to the size of the depth regions and added lower bounds for their diameterSubjects: Statistics Theory (math.ST); Probability (math.PR)
Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey's depth has found applications in robust statistics, graph theory, and the study of elections and social choice. We present improved performance guarantees for empirical Tukey's median, a deepest point associated with the given sample, when the data-generating distribution is elliptically symmetric and possibly anisotropic. Some of our results remain valid in the wider class of affine equivariant estimators. As a corollary of our bounds, we show that the diameter of the set of all empirical Tukey's medians scales like $o(n^{-1/2})$ where $n$ is the sample size. Moreover, when the data are 2-dimensional, we prove that the diameter is of order $O(n^{-3/4}\log^{3/2}(n))$.
- [266] arXiv:2410.08859 (replaced) [pdf, html, other]
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Title: Domain decomposition for entropic unbalanced optimal transportComments: Added acknowledgementsSubjects: Optimization and Control (math.OC)
Solving large scale entropic optimal transport problems with the Sinkhorn algorithm remains challenging, and domain decomposition has been shown to be an efficient strategy for problems on large grids. Unbalanced optimal transport is a versatile variant of the balanced transport problem and its entropic regularization can be solved with an adapted Sinkhorn algorithm. However, it is a priori unclear how to apply domain decomposition to unbalanced problems since the independence of the cell problems is lost. In this article we show how this difficulty can be overcome at a theoretical and practical level and demonstrate with experiments that domain decomposition is also viable and efficient on large unbalanced entropic transport problems.
- [267] arXiv:2410.11659 (replaced) [pdf, html, other]
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Title: A quasilinear elliptic equation with absorption term and Hardy potentialComments: 47 pages,4 figuresSubjects: Analysis of PDEs (math.AP)
Here we study the positive solutions of the equation \begin{equation*} -\Delta _{p}u+\mu \frac{u^{p-1}}{\left\vert x\right\vert ^{p}}+\left\vert x\right\vert ^{\theta }u^{q}=0,\qquad x\in \mathbb{R}^{N}\backslash \left\{ 0\right\} \end{equation*}% where $\Delta _{p}u={div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u) $ and $1<p<N,q>p-1,\mu ,\theta \in \mathbb{R}.$ We give a complete description of the existence and the asymptotic behaviour of the solutions near the singularity $0,$ or in an exterior domain. We show that the global solutions $\mathbb{R}^{N}\backslash \left\{ 0\right\} $ are radial and give their expression according to the position of the Hardy coefficient $\mu $ with respect to the critical exponent $\mu _{0}=-(\frac{N-p}{p})^{p}.$ Our method consists into proving that any nonradial solution can be compared to a radial one, then making exhaustive radial study by phase-plane techniques. Our results are optimal, extending the known results when $\mu =0$ or $p=2$, with new simpler this http URL make in evidence interesting phenomena of nonuniqueness when $\theta +p=0$, and of existence of locally constant solutions when moreover $p>2$ .
- [268] arXiv:2410.13606 (replaced) [pdf, html, other]
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Title: Arthur packets for metaplectic groupsComments: 90 pages with an index. Minor improvementsSubjects: Representation Theory (math.RT); Number Theory (math.NT)
For metaplectic groups over a local field of characteristic zero, we define the Arthur packet attached to any Arthur parameter $\psi$ as a multi-set of unitary genuine irreducible representations, characterized by endoscopic character relations. Over number fields, we obtain a multiplicity formula for the genuine discrete $L^2$-automorphic spectrum in terms of global Arthur parameters and $\epsilon$-factors, by leveraging the trace formula for metaplectic groups. This confirms a conjecture of Gan, and extends earlier results of Gan-Ichino on the Shimura-Waldspurger correspondences, whereas their works play a critical role in our proof. Furthermore, all these are shown to be compatible with existing results in rank one (Waldspurger) and two (Gan-Ichino).
- [269] arXiv:2410.14443 (replaced) [pdf, other]
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Title: Sensitivity analysis for linear changes of the constraint matrix of a linear programSubjects: Optimization and Control (math.OC)
Understanding the variation of the optimal value with respect to change in the data is an old problem of mathematical optimisation. This paper focuses on the linear problem $f(\lambda) = \min c^t x$ such that $(A+\lambda D)x \leq b$, where $\lambda$ is an unknown parameter that varies within an interval and $D$ is a matrix modifying the coefficients of the constraint matrix $A$. This problem is used to analyse the impact of multiple affine changes in the constraint matrix on the objective function. The function $f(\lambda)$ can have an erratic behaviour and computing it for a large number of points is computationally heavy while not providing any guarantees in between the computed points. As a new approach to the problem, we derive several bounding methods that provide guarantees on the objective function's behaviour. Those guarantees can be exploited to avoid recomputing the problem for numerous $\lambda$. The bounding methods are based on approaches from robust optimisation or Lagrangian relaxations. For each approach, we derive three methods of increasing complexity and precision, one that provides constant bounds, one that provides $\lambda$-dependant bounds and envelope bounds. We assess each bounding method in terms of precision, availability and timing. We show that for a large number of problems, the bound approach outperforms the naive sampling approach considered with 100 points while still providing a good precision and stronger guarantees on a large dataset of problems. We also introduce an iterative algorithm that uses these bounds to compute an approximation of the original function within a given error threshold and discuss its performances.
- [270] arXiv:2410.17895 (replaced) [pdf, html, other]
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Title: Smullyan's truth and provabilityComments: 22 pagesSubjects: Logic (math.LO)
We revisit Smullyan's paper ``Truth and Provability'' for three purposes. First, we introduce the notion of Smullyan models to give a precise definition for Smullyan's framework discussed in that paper. Second, we clarify the relationship between Theorems F, T, and G proved by Smullyan and other newly introduced properties for Smullyan models in terms of both implications and non-implications. Third, we construct two Smullyan models based on arithmetic and show the correspondence between the properties of these Smullyan models and those concerning truth and provability in arithmetic.
- [271] arXiv:2410.19287 (replaced) [pdf, html, other]
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Title: A mathematical theory of topological invariants of quantum spin systemsComments: 45 pages. v2: typos corrected, other minor changesSubjects: Mathematical Physics (math-ph); Strongly Correlated Electrons (cond-mat.str-el); Quantum Algebra (math.QA)
We show that Hall conductance and its non-abelian and higher-dimensional analogs are obstructions to promoting a symmetry of a state to a gauge symmetry. To do this, we define a local Lie algebra over a Grothendieck site as a pre-cosheaf of Lie algebras with additional properties and propose that a gauge symmetry should be described by such an object. We show that infinitesimal symmetries of a gapped state of a quantum spin system form a local Lie algebra over a site of semilinear sets and use it to construct topological invariants of the state. Our construction applies to lattice systems on arbitrary asymptotically conical subsets of a Euclidean space including those which cannot be studied using field theory.
- [272] arXiv:2410.23406 (replaced) [pdf, html, other]
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Title: On a Rigidity Result in Positive Scalar Curvature GeometryComments: comments welcomeSubjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that geodesic balls of radii strictly less than $\frac{\pi}{2}$ in $n+1~(n\geq 2)$ dimensional unit sphere are rigid under smooth perturbations that increase scalar curvature preserving the intrinsic geometry and the mean curvature of the boundary, and such rigidity result fails for the hemisphere. The proof of this assertion requires the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. The result serves as the sharpest refinement of the now-disproven Min-Oo conjecture.
- [273] arXiv:2410.23816 (replaced) [pdf, html, other]
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Title: A theoretical analysis of mass scaling techniquesComments: 26 pages, 15 figuresSubjects: Numerical Analysis (math.NA)
Mass scaling is widely used in finite element models of structural dynamics for increasing the critical time step of explicit time integration methods. While the field has been flourishing over the years, it still lacks a strong theoretical basis and mostly relies on numerical experiments as the only means of assessment. This contribution thoroughly reviews existing methods and connects them to established linear algebra results to derive rigorous eigenvalue bounds and condition number estimates. Our results cover some of the most successful mass scaling techniques, unraveling for the first time well-known numerical observations.
- [274] arXiv:2411.01247 (replaced) [pdf, html, other]
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Title: Probabilistic Effectivity in the Subspace TheoremSubjects: Number Theory (math.NT)
The Subspace Theorem due to Schmidt (1972) is a broad generalisation of Roth's Theorem in Diophantine Approximation (1955) which, in the same way as the latter, suffers a notorious lack of effectivity. This problem is tackled from a probabilistic standpoint by determining the proportion of algebraic linear forms of bounded heights and degrees for which there exists a solution to the Subspace Inequality lying in a subspace of large height. The estimates are established for a class of height functions emerging from an analytic parametrisation of the projective space. They are pertinent in the regime where the heights of the algebraic quantities are larger than those of the rational solutions to the inequality under consideration, and are valid for approximation functions more general than the power functions intervening in the original Subspace Theorem. These estimates are further refined in the case of Roth's Theorem so as to yield a Khintchin-type density version of the so-called Waldschmidt conjecture (which is known to fail pointwise). This answers a question raised by Beresnevich, Bernik and Dodson (2009).
- [275] arXiv:2411.02050 (replaced) [pdf, html, other]
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Title: The 2-burning number of a graphSubjects: Combinatorics (math.CO)
We study a discrete-time model for the spread of information in a graph, motivated by the idea that people believe a story when they learn of it from two different origins. Similar to the burning number, in this problem, information spreads in rounds and a new source can appear in each round. For a graph $G$, we are interested in $b_2(G)$, the minimum number of rounds until the information has spread to all vertices of graph $G$. We are also interested in finding $t_2(G)$, the minimum number of sources necessary so that the information spreads to all vertices of $G$ in $b_2(G)$ rounds. In addition to general results, we find $b_2(G)$ and $t_2(G)$ for the classes of spiders and wheels and show that their behavior differs with respect to these two parameters. We also provide examples and prove upper bounds for these parameters for Cartesian products of graphs.
- [276] arXiv:2411.03167 (replaced) [pdf, html, other]
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Title: Tight closure of products and F-rational singularitiesSubjects: Commutative Algebra (math.AC)
We prove a characterization of F-rationality in terms of tight closure of products of parameter ideals. Our results are inspired by the theory of complete ideals for surfaces and, in particular, the fundamental results of Lipman-Teissier and Cutkosky characterizing rational surface singularities in terms of products of complete ideals, but are valid also in higher dimensions.
- [277] arXiv:2411.03801 (replaced) [pdf, html, other]
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Title: 1-loop equals torsion for two-bridge knotsComments: 24 pages, 20 figuresSubjects: Geometric Topology (math.GT)
Motivated by the conjectured asymptotics of the Kashaev invariant, Dimofte and the first author introduced a power series associated to a suitable ideal triangulation of a cusped hyperbolic 3-manifold, proved that its constant (1-loop) term is a topological invariant and conjectured that it equals to the adjoint Reidemeister torsion. We prove this conjecture for hyperbolic 2-bridge knots by combining the work of Ohtsuki--Takata with an explicit computation.
- [278] arXiv:2411.04033 (replaced) [pdf, html, other]
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Title: Energy transport in a free Euler-Bernoulli beam in terms of Schr\"odinger's wave functionComments: 5 pagesSubjects: Mathematical Physics (math-ph); Classical Physics (physics.class-ph); Quantum Physics (quant-ph)
The Schrödinger equation is not frequently used in the framework of the classical mechanics, though historically this equation was derived as a simplified equation, which is equivalent to the classical Germain-Lagrange dynamic plate equation. The question concerning the exact meaning of this equivalence is still discussed in modern literature. In this note, we consider the one-dimensional case, where the Germain-Lagrange equation reduces to the Euler-Bernoulli equation, which is used in the classical theory of a beam. We establish a one-to-one correspondence between the set of all solutions (i.e., wave functions $\psi$) of the 1D time-dependent Schrödinger equation for a free particle with arbitrary complex valued initial data and the set of ordered pairs of quantities (the beam strain and the particle velocity), which characterize solutions $u$ of the beam equation with arbitrary real valued initial data. Thus, the dynamics of a free infinite Euler-Bernoulli beam can be described by the Schrödinger equation for a free particle and vice versa. Finally, we show that for two corresponding solutions $u$ and $\psi$ the mechanical energy density calculated for $u$ propagates in the beam exactly in the same way as the probability density calculated for $\psi$.
- [279] arXiv:2411.05488 (replaced) [pdf, html, other]
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Title: Pathwise Optimal Control and Rough Fractional Hamilton-Jacobi-Bellman Equations for Rough-Fractional DynamicsSubjects: Optimization and Control (math.OC)
We use a rough path-based approach to investigate the degeneracy problem in the context of pathwise control. We extend the framework developed in arXiv:1902.05434 to treat admissible controls from a suitable class of Hölder continuous paths and simultaneously to handle a broader class of noise terms. Our approach uses fractional calculus to augment the original control equation, resulting in a system with added fractional dynamics. We adapt the existing analysis of fractional systems from the work of Gomoyunov arXiv:1908.01747, arXiv:2111.14400v1 , arXiv:2109.02451 to this new setting, providing a notion of a rough fractional viscosity solution for fractional systems that involve a noise term of arbitrarily low regularity. In this framework, following the method outlined in arXiv:1902.05434, we derive sufficient conditions to ensure that the control problem remains non-degenerate.
- [280] arXiv:2411.05810 (replaced) [pdf, html, other]
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Title: Complex median method and Schatten class membership of commutatorsComments: Partial results have been announced in the paper arXiv:2407.14988. New results on the weak-type Schatten class are included in this paperSubjects: Functional Analysis (math.FA)
This article is devoted to the study of the Schatten class membership of commutators involving singular integral operators. We utilize martingale paraproducts and Hytönen's dyadic martingale technique to obtain sufficient conditions on the weak-type and strong-type Schatten class membership of commutators in terms of Sobolev spaces and Besov spaces respectively.
We also establish the complex median method, which is applicable to complex-valued functions. We apply it to get the optimal necessary conditions on the weak-type and strong-type Schatten class membership of commutators associated with non-degenerate kernels. This resolves the problem on the characterization of the weak-type and strong-type Schatten class membership of commutators.
Our new approach is built on Hytönen's dyadic martingale technique and the complex median method. Compared with all the previous ones, this new one is more powerful in several aspects: $(a)$ it permits us to deal with more general singular integral operators with little smoothness; $(b)$ it allows us to deal with commutators with complex-valued kernels; $(c)$ it turns out to be powerful enough to deal with the weak-type and strong-type Schatten class of commutators in a universal way. - [281] arXiv:2411.06978 (replaced) [pdf, html, other]
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Title: Cancellation in sums over special sequences on $\mathbf{{\rm{GL}}_{m}}$ and their applicationsComments: 33 pages. Comments welcomeSubjects: Number Theory (math.NT)
Let $a(n)$ be the $n$-th Dirichlet coefficient of the automorphic $L$-function or the Rankin--Selberg $L$-function. We investigate the cancellation of $a(n)$ over specific sequences linked to the Waring--Goldbach problem, by establishing a nontrivial bound for the additive twisted sums over primes on ${\mathrm{GL}}_m$ and ${\mathrm{GL}}_m\times{\mathrm{GL}}_m .$ The bound does not depend on the generalized Ramanujan conjecture or the nonexistence of Landau--Siegel zeros. Furthermore, we present an application associated with the Sato--Tate conjecture and propose a conjecture about Goldbach's conjecture on average bound.
- [282] arXiv:2411.07361 (replaced) [pdf, html, other]
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Title: Reduced Sample Complexity in Scenario-Based Control System Design via Constraint ScalingSubjects: Optimization and Control (math.OC); Probability (math.PR)
The scenario approach is widely used in robust control system design and chance-constrained optimization, maintaining convexity without requiring assumptions about the probability distribution of uncertain parameters. However, the approach can demand large sample sizes, making it intractable for safety-critical applications that require very low levels of constraint violation. To address this challenge, we propose a novel yet simple constraint scaling method, inspired by large deviations theory. Under mild nonparametric conditions on the underlying probability distribution, we show that our method yields an exponential reduction in sample size requirements for bilinear constraints with low violation levels compared to the classical approach, thereby significantly improving computational tractability. Numerical experiments on robust pole assignment problems support our theoretical findings.
- [283] arXiv:2411.07363 (replaced) [pdf, html, other]
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Title: Intuitionistic logic, dual intuitionistic logic, and modalityComments: Fixed a critical error in example 4.0.1Subjects: Logic (math.LO)
We explore various semantic understandings of dual intuitionistic logic by exploring the relationship between co-Heyting algebras and topological spaces. First, we discuss the relevant ideas in the setting of Heyting algebras and intuitionistic logic, showing organically the progression from the primordial example of lattices of open sets of topological spaces to more general ways of thinking about Heyting algebras. Then, we adapt the ideas to the dual intuitionistic setting, and use them to prove a number of interesting properties, including a deep relationship both intuitionistic and dual intuitionistic logic share to the modal logic $\mathsf{S4}$.
- [284] arXiv:2411.07422 (replaced) [pdf, html, other]
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Title: Impact of Numerical Fluxes on High Order Semidiscrete WENO-DeC Finite Volume SchemesSubjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
The numerical flux determines the performance of numerical methods for solving hyperbolic partial differential equations (PDEs). In this work, we compare a selection of 8 numerical fluxes in the framework of nonlinear semidiscrete finite volume (FV) schemes, based on Weighted Essentially Non-Oscillatory (WENO) spatial reconstruction and Deferred Correction (DeC) time discretization. The methodology is implemented and systematically assessed for order of accuracy in space and time up to seven. The numerical fluxes selected in the present study represent the two existing classes of fluxes, namely centred and upwind. Centred fluxes do not explicitly use wave propagation information, while, upwind fluxes do so from the solution of the Riemann problem via a wave model containing $A$ waves. Upwind fluxes include two subclasses: complete and incomplete fluxes. For complete upwind fluxes, $A=E$, where $E$ is the number of characteristic fields in the exact problem. For incomplete upwind ones, $A<E$. Our study is conducted for the one- and two-dimensional Euler equations, for which we consider the following numerical fluxes: Lax-Friedrichs (LxF), First-Order Centred (FORCE), Rusanov (Rus), Harten-Lax-van Leer (HLL), Central-Upwind (CU), Low-Dissipation Central-Upwind (LDCU), HLLC, and the flux computed through the exact Riemann solver (this http URL). We find that the numerical flux has an effect on the performance of the methods. The magnitude of the effect depends on the type of numerical flux and on the order of accuracy of the scheme. It also depends on the type of problem; that is, whether the solution is smooth or discontinuous, whether discontinuities are linear or nonlinear, whether linear discontinuities are fast- or slowly-moving, and whether the solution is evolved for short or long time.
- [285] arXiv:2209.00832 (replaced) [pdf, other]
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Title: Efficiency of estimators for locally asymptotically normal quantum statistical modelsJournal-ref: Ann. Statist. 51 (3) 1159 - 1182, June 2023Subjects: Quantum Physics (quant-ph); Statistics Theory (math.ST)
We herein establish an asymptotic representation theorem for locally asymptotically normal quantum statistical models. This theorem enables us to study the asymptotic efficiency of quantum estimators such as quantum regular estimators and quantum minimax estimators, leading to a universal tight lower bound beyond the i.i.d. assumption. This formulation complements the theory of quantum contiguity developed in the previous paper [Fujiwara and Yamagata, Bernoulli 26 (2020) 2105-2141], providing a solid foundation of the theory of weak quantum local asymptotic normality.
- [286] arXiv:2211.09417 (replaced) [pdf, html, other]
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Title: Some Results on Digital Segments and Balanced WordsComments: 17 pages, 5 figuresJournal-ref: Theoretical Computer Science 1021 (2024) 114935Subjects: Formal Languages and Automata Theory (cs.FL); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
We exhibit combinatorial results on Christoffel words and binary balanced words that are motivated by their geometric interpretation as approximations of digital segments. We give a closed formula for counting the exact number of balanced words with $a$ zeroes and $b$ ones. We also study minimal non-balanced words.
- [287] arXiv:2309.16824 (replaced) [pdf, html, other]
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Title: The fork and its role in unification of closure algebrasComments: Updated after reviewers' reportsSubjects: Logic in Computer Science (cs.LO); Rings and Algebras (math.RA)
We consider the two-pronged fork frame $F$ and the variety $\mathbf{Eq}(B_F)$ generated by its dual closure algebra $B_F$. We describe the finite projective algebras in $\mathbf{Eq}(B_F)$ and give a purely semantic proof that unification in $\mathbf{Eq}(B_F)$ is finitary and not unitary.
- [288] arXiv:2310.09673 (replaced) [pdf, html, other]
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Title: Robust Quickest Change Detection in Non-Stationary ProcessesComments: A corrected version of the previous submissionSubjects: Methodology (stat.ME); Signal Processing (eess.SP); Statistics Theory (math.ST)
Optimal algorithms are developed for robust detection of changes in non-stationary processes. These are processes in which the distribution of the data after change varies with time. The decision-maker does not have access to precise information on the post-change distribution. It is shown that if the post-change non-stationary family has a distribution that is least favorable in a well-defined sense, then the algorithms designed using the least favorable distributions are robust and optimal. Non-stationary processes are encountered in public health monitoring and space and military applications. The robust algorithms are applied to real and simulated data to show their effectiveness.
- [289] arXiv:2310.10545 (replaced) [pdf, html, other]
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Title: Optimal vintage factor analysis with deflation varimaxSubjects: Machine Learning (stat.ML); Information Theory (cs.IT); Machine Learning (cs.LG); Signal Processing (eess.SP)
Vintage factor analysis is one important type of factor analysis that aims to first find a low-dimensional representation of the original data, and then to seek a rotation such that the rotated low-dimensional representation is scientifically meaningful. The most widely used vintage factor analysis is the Principal Component Analysis (PCA) followed by the varimax rotation. Despite its popularity, little theoretical guarantee can be provided to date mainly because varimax rotation requires to solve a non-convex optimization over the set of orthogonal matrices.
In this paper, we propose a deflation varimax procedure that solves each row of an orthogonal matrix sequentially. In addition to its net computational gain and flexibility, we are able to fully establish theoretical guarantees for the proposed procedure in a broader context. Adopting this new deflation varimax as the second step after PCA, we further analyze this two step procedure under a general class of factor models. Our results show that it estimates the factor loading matrix in the minimax optimal rate when the signal-to-noise-ratio (SNR) is moderate or large. In the low SNR regime, we offer possible improvement over using PCA and the deflation varimax when the additive noise under the factor model is structured. The modified procedure is shown to be minimax optimal in all SNR regimes. Our theory is valid for finite sample and allows the number of the latent factors to grow with the sample size as well as the ambient dimension to grow with, or even exceed, the sample size. Extensive simulation and real data analysis further corroborate our theoretical findings. - [290] arXiv:2402.08621 (replaced) [pdf, html, other]
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Title: A Unified Framework for Analyzing Meta-algorithms in Online Convex OptimizationSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
In this paper, we analyze the problem of online convex optimization in different settings, including different feedback types (full-information/semi-bandit/bandit/etc) in either stochastic or non-stochastic setting and different notions of regret (static adversarial regret/dynamic regret/adaptive regret). This is done through a framework which allows us to systematically propose and analyze meta-algorithms for the various settings described above. We show that any algorithm for online linear optimization with fully adaptive adversaries is an algorithm for online convex optimization. We also show that any such algorithm that requires full-information feedback may be transformed to an algorithm with semi-bandit feedback with comparable regret bound. We further show that algorithms that are designed for fully adaptive adversaries using deterministic semi-bandit feedback can obtain similar bounds using only stochastic semi-bandit feedback when facing oblivious adversaries. We use this to describe general meta-algorithms to convert first order algorithms to zeroth order algorithms with comparable regret bounds. Our framework allows us to analyze online optimization in various settings, recovers several results in the literature with a simplified proof technique, and provides new results.
- [291] arXiv:2402.15322 (replaced) [pdf, html, other]
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Title: Optimal Transport on the Lie Group of Roto-translationsSubjects: Computer Vision and Pattern Recognition (cs.CV); Differential Geometry (math.DG); Optimization and Control (math.OC)
The roto-translation group SE2 has been of active interest in image analysis due to methods that lift the image data to multi-orientation representations defined on this Lie group. This has led to impactful applications of crossing-preserving flows for image de-noising, geodesic tracking, and roto-translation equivariant deep learning. In this paper, we develop a computational framework for optimal transportation over Lie groups, with a special focus on SE2. We make several theoretical contributions (generalizable to matrix Lie groups) such as the non-optimality of group actions as transport maps, invariance and equivariance of optimal transport, and the quality of the entropic-regularized optimal transport plan using geodesic distance approximations. We develop a Sinkhorn like algorithm that can be efficiently implemented using fast and accurate distance approximations of the Lie group and GPU-friendly group convolutions. We report valuable advancements in the experiments on 1) image barycentric interpolation, 2) interpolation of planar orientation fields, and 3) Wasserstein gradient flows on SE2. We observe that our framework of lifting images to SE2 and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image. This yields sharper and more meaningful interpolations compared to their counterparts on R^2
- [292] arXiv:2403.16336 (replaced) [pdf, html, other]
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Title: Predictive Inference in Multi-environment ScenariosSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST); Methodology (stat.ME)
We address the challenge of constructing valid confidence intervals and sets in problems of prediction across multiple environments. We investigate two types of coverage suitable for these problems, extending the jackknife and split-conformal methods to show how to obtain distribution-free coverage in such non-traditional, potentially hierarchical data-generating scenarios. We demonstrate a novel resizing method to adapt to problem difficulty, which applies both to existing approaches for predictive inference and the methods we develop; this reduces prediction set sizes using limited information from the test environment, a key to the methods' practical performance, which we evaluate through neurochemical sensing and species classification datasets. Our contributions also include extensions for settings with non-real-valued responses, a theory of consistency for predictive inference in these general problems, and insights on the limits of conditional coverage.
- [293] arXiv:2404.07361 (replaced) [pdf, html, other]
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Title: Gradient NetworksComments: To be published in IEEE Transactions on Signal ProcessingSubjects: Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE); Signal Processing (eess.SP); Optimization and Control (math.OC)
Directly parameterizing and learning gradients of functions has widespread significance, with specific applications in inverse problems, generative modeling, and optimal transport. This paper introduces gradient networks (GradNets): novel neural network architectures that parameterize gradients of various function classes. GradNets exhibit specialized architectural constraints that ensure correspondence to gradient functions. We provide a comprehensive GradNet design framework that includes methods for transforming GradNets into monotone gradient networks (mGradNets), which are guaranteed to represent gradients of convex functions. Our results establish that our proposed GradNet (and mGradNet) universally approximate the gradients of (convex) functions. Furthermore, these networks can be customized to correspond to specific spaces of potential functions, including transformed sums of (convex) ridge functions. Our analysis leads to two distinct GradNet architectures, GradNet-C and GradNet-M, and we describe the corresponding monotone versions, mGradNet-C and mGradNet-M. Our empirical results demonstrate that these architectures provide efficient parameterizations and outperform existing methods by up to 15 dB in gradient field tasks and by up to 11 dB in Hamiltonian dynamics learning tasks.
- [294] arXiv:2406.12763 (replaced) [pdf, html, other]
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Title: Implicit Bias of Mirror Flow on Separable DataComments: Neurips camera ready. Minor changes from the previous versions. Mainly added full iterate trajectories (Figure 4)Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC)
We examine the continuous-time counterpart of mirror descent, namely mirror flow, on classification problems which are linearly separable. Such problems are minimised `at infinity' and have many possible solutions; we study which solution is preferred by the algorithm depending on the mirror potential. For exponential tailed losses and under mild assumptions on the potential, we show that the iterates converge in direction towards a $\phi_\infty$-maximum margin classifier. The function $\phi_\infty$ is the \textit{horizon function} of the mirror potential and characterises its shape `at infinity'. When the potential is separable, a simple formula allows to compute this function. We analyse several examples of potentials and provide numerical experiments highlighting our results.
- [295] arXiv:2407.09154 (replaced) [pdf, html, other]
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Title: Simple Linear Loops: Algebraic Invariants and ApplicationsComments: 29 pages, full version of a POPL 2025 paperSubjects: Computational Complexity (cs.CC); Algebraic Geometry (math.AG)
The automatic generation of loop invariants is a fundamental challenge in software verification. While this task is undecidable in general, it is decidable for certain restricted classes of programs. This work focuses on invariant generation for (branching-free) loops with a single linear update.
Our primary contribution is a polynomial-space algorithm that computes the strongest algebraic invariant for simple linear loops, generating all polynomial equations that hold among program variables across all reachable states. The key to achieving our complexity bounds lies in mitigating the blowup associated with variable elimination and Gröbner basis computation, as seen in prior works. Our procedure runs in polynomial time when the number of program variables is fixed.
We examine various applications of our results on invariant generation, focusing on invariant verification and loop synthesis. The invariant verification problem investigates whether a polynomial ideal defining an algebraic set serves as an invariant for a given linear loop. We show that this problem is coNP-complete and lies in PSPACE when the input ideal is given in dense or sparse representations, respectively. In the context of loop synthesis, we aim to construct a loop with an infinite set of reachable states that upholds a specified algebraic property as an invariant. The strong synthesis variant of this problem requires the construction of loops for which the given property is the strongest invariant. In terms of hardness, synthesising loops over integers (or rationals) is as hard as Hilbert's Tenth problem (or its analogue over the rationals). When the constants of the output are constrained to bit-bounded rational numbers, we demonstrate that loop synthesis and its strong variant are both decidable in PSPACE, and in NP when the number of program variables is fixed. - [296] arXiv:2407.21164 (replaced) [pdf, other]
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Title: Extending choice assessments to choice functions: An algorithm for computing the natural extensionComments: 40 pages, 8 figures, pre-print for International Journal of Approximate ReasoningSubjects: Artificial Intelligence (cs.AI); Probability (math.PR)
We study how to infer new choices from prior choices using the framework of choice functions, a unifying mathematical framework for decision-making based on sets of preference orders. In particular, we define the natural (most conservative) extension of a given choice assessment to a coherent choice function -- whenever possible -- and use this natural extension to make new choices. We provide a practical algorithm for computing this natural extension and various ways to improve scalability. Finally, we test these algorithms for different types of choice assessments.
- [297] arXiv:2408.01600 (replaced) [pdf, html, other]
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Title: Physics-Informed Geometry-Aware Neural OperatorComments: arXiv admin note: text overlap with arXiv:2404.13646Subjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
Engineering design problems often involve solving parametric Partial Differential Equations (PDEs) under variable PDE parameters and domain geometry. Recently, neural operators have shown promise in learning PDE operators and quickly predicting the PDE solutions. However, training these neural operators typically requires large datasets, the acquisition of which can be prohibitively expensive. To overcome this, physics-informed training offers an alternative way of building neural operators, eliminating the high computational costs associated with Finite Element generation of training data. Nevertheless, current physics-informed neural operators struggle with limitations, either in handling varying domain geometries or varying PDE parameters. In this research, we introduce a novel method, the Physics-Informed Geometry-Aware Neural Operator (PI-GANO), designed to simultaneously generalize across both PDE parameters and domain geometries. We adopt a geometry encoder to capture the domain geometry features, and design a novel pipeline to integrate this component within the existing DCON architecture. Numerical results demonstrate the accuracy and efficiency of the proposed method. All the codes and data related to this work are available on GitHub: this https URL.
- [298] arXiv:2409.08266 (replaced) [pdf, other]
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Title: On the space of $2d$ integrable modelsComments: v1, 44 pages; v2, small discussions added; v3, corrections to some results, paper restructured, 50 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We study infinite dimensional Lie algebras, whose infinite dimensional mutually commuting subalgebras correspond with the symmetry algebra of $2d$ integrable models. These Lie algebras are defined by the set of infinitesimal, nonlinear, and higher derivative symmetry transformations present in theories with a left(right)-moving or (anti)-holomorphic current. We study a large class of such Lagrangian theories. We study the commuting subalgebras of the $2d$ free massless scalar, and find the symmetries of the known integrable models such as sine-Gordon, Liouville, Bullough-Dodd, and Korteweg-de Vries. Along the way, we find several new sequences of commuting charges, which we conjecture are charges of integrable models which are new deformations of a single scalar. After quantizing, the Lie algebra is deformed, and so are their commuting subalgebras.
- [299] arXiv:2410.16561 (replaced) [pdf, html, other]
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Title: Gradient Normalization Provably Benefits Nonconvex SGD under Heavy-Tailed NoiseSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
This paper investigates the roles of gradient normalization and clipping in ensuring the convergence of Stochastic Gradient Descent (SGD) under heavy-tailed noise. While existing approaches consider gradient clipping indispensable for SGD convergence, we theoretically demonstrate that gradient normalization alone without clipping is sufficient to ensure convergence. Furthermore, we establish that combining gradient normalization with clipping offers significantly improved convergence rates compared to using either technique in isolation, particularly as gradient noise diminishes. With these results, our work provides the first theoretical evidence demonstrating the benefits of gradient normalization in SGD under heavy-tailed noise. Finally, we introduce an accelerated SGD variant that incorporates both gradient normalization and clipping, further enhancing convergence rates under heavy-tailed noise.
- [300] arXiv:2411.05502 (replaced) [pdf, other]
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Title: Infection Pressure on Fish in CagesWilliam Waites, Philip Gillibrand, Thomas Adams, Rek Bell, Duncan Guthrie, Tróndur Kragesteen, Crawford Revie, Meadhbh MoriartySubjects: Populations and Evolution (q-bio.PE); Dynamical Systems (math.DS)
We address the question of how to connect predictions by hydrodynamic models of how sea lice move in water to observable measures that count the number of lice on each fish in a cage in the water. This question is important for management and regulation of aquacultural practice that tries to maximise food production and minimise risk to the environment. We do this through a simple rule-based model of interaction between sea lice and caged fish. The model is simple: sea lice can attach and detach from a fish. The model has a novel feature, encoding what is known as a master equation producing a time-series of distributions of lice on fish that one might expect to find if a cage full of fish were placed at any given location. To demonstrate how this works, and to arrive at a rough estimate of the interaction rates, we fit a simplified version of the model with three free parameters to publicly available data about an experiment with sentinel cages in Loch Linnhe in Scotland. Our construction, coupled to the hydrodynamic models driven by surveillance data from industrial farms, quantifies the environmental impact as: what would the infection burden look like in a notional cage at any location and how does it change with time?
- [301] arXiv:2411.06190 (replaced) [pdf, html, other]
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Title: Gravitational reheating formulas and bounds in oscillating backgrounds II: Constraints on the spectral index and gravitational dark matter productionComments: 13 pages (including references), 2 figures, 6 tables; comments are welcomeSubjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
The reheating temperature plays a crucial role in the early universe's evolution, marking the transition from inflation to the radiation-dominated era. It directly impacts the number of $e$-folds and, consequently, the observable parameters of inflation, such as the spectral index of scalar perturbations. By establishing a relationship between the gravitational reheating temperature and the spectral index, we can derive constraints on inflationary models. Specifically, the range of viable reheating temperatures imposes bounds on the spectral index, which can then be compared with observational data, such as those from the Planck satellite, to test the consistency of various models with cosmological observations. Additionally, in the context of dark matter production, we demonstrate that gravitational reheating provides a viable mechanism when there is a relationship between the mass of the dark matter particles and the mass of the particles responsible for reheating. This connection offers a pathway to link dark matter genesis with inflationary and reheating parameters, allowing for a unified perspective on early universe dynamics.
- [302] arXiv:2411.07558 (replaced) [pdf, html, other]
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Title: Discrete-Valued Signal Estimation via Low-Complexity Message Passing Algorithm for Highly Correlated MeasurementsComments: Submission notification added, Fig size adjusted, typos correctedSubjects: Signal Processing (eess.SP); Information Theory (cs.IT); Statistics Theory (math.ST)
This paper considers a discrete-valued signal estimation scheme based on a low-complexity Bayesian optimal message passing algorithm (MPA) for solving massive linear inverse problems under highly correlated measurements. Gaussian belief propagation (GaBP) can be derived by applying the central limit theorem (CLT)-based Gaussian approximation to the sum-product algorithm (SPA) operating on a dense factor graph (FG), while matched filter (MF)-expectation propagation (EP) can be obtained based on the EP framework tailored for the same FG. Generalized approximate message passing (GAMP) can be found by applying a rigorous approximation technique for both of them in the large-system limit, and these three MPAs perform signal detection using MF by assuming large-scale uncorrelated observations. However, each of them has a different inherent self-noise suppression mechanism, which makes a significant difference in the robustness against the correlation of the observations when we apply an annealed discrete denoiser (ADD) that adaptively controls its nonlinearity with the inverse temperature parameter corresponding to the number of iterations. In this paper, we unravel the mechanism of this interesting phenomenon, and further demonstrate the practical applicability of the low-complexity Bayesian optimal MPA with ADD under highly correlated measurements.
- [303] arXiv:2411.07900 (replaced) [pdf, html, other]
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Title: Hybrid finite element implementation of two-potential constitutive modeling of dielectric elastomersSubjects: Computational Physics (physics.comp-ph); Numerical Analysis (math.NA)
Dielectric elastomers are increasingly studied for their potential in soft robotics, actuators, and haptic devices. Under time-dependent loading, they dissipate energy via viscous deformation and friction in electric polarization. However, most constitutive models and finite element (FE) implementations consider only mechanical dissipation because mechanical relaxation times are much larger than electric ones. Accounting for electric dissipation is crucial when dealing with alternating electric fields. Ghosh et al. (2021) proposed a fully coupled three-dimensional constitutive model for isotropic, incompressible dielectric elastomers. We critically investigate their numerical scheme for solving the initial boundary value problem (IBVP) describing the time-dependent behavior. We find that their fifth-order explicit Runge-Kutta time discretization may require excessively small or unphysical time steps for realistic simulations due to the stark contrast in mechanical and electric relaxation times. To address this, we present a stable implicit time-integration algorithm that overcomes these constraints. We implement this algorithm with a conforming FE discretization to solve the IBVP and present the mixed-FE formulation implemented in FEniCSx. We demonstrate that the scheme is robust, accurate, and capable of handling finite deformations, incompressibility, and general time-dependent loading. Finally, we validate our code against experimental data for VHB 4910 under complex time-dependent electromechanical loading, as studied by Hossain et al. (2015).