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- [1] arXiv:2411.13561 [pdf, html, other]
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Title: Model discovery on the fly using continuous data assimilationSubjects: Numerical Analysis (math.NA); Dynamical Systems (math.DS); Data Analysis, Statistics and Probability (physics.data-an)
We review an algorithm developed for parameter estimation within the Continuous Data Assimilation (CDA) approach. We present an alternative derivation for the algorithm presented in a paper by Carlson, Hudson, and Larios (CHL, 2021). This derivation relies on the same assumptions as the previous derivation but frames the problem as a finite dimensional root-finding problem. Within the approach we develop, the algorithm developed in (CHL, 2021) is simply a realization of Newton's method. We then consider implementing other derivative based optimization algorithms; we show that the Levenberg Maqrquardt algorithm has similar performance to the CHL algorithm in the single parameter estimation case and generalizes much better to fitting multiple parameters. We then implement these methods in three example systems: the Lorenz '63 model, the two-layer Lorenz '96 model, and the Kuramoto-Sivashinsky equation.
- [2] arXiv:2411.13567 [pdf, html, other]
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Title: Why the p-norms $p{=}1$, $p{=}2$ and $p{=}\infty$ are so special? An answer based on spatial uniformitySubjects: Statistics Theory (math.ST); Cryptography and Security (cs.CR); Numerical Analysis (math.NA)
Among all metrics based on p-norms, the Manhattan (p=1), euclidean (p=2) and Chebyshev distances (p=infinity) are the most widely used for their interpretability, simplicity and technical convenience. But these are not the only arguments for the ubiquity of these three p-norms. This article proves that there is a volume-surface correspondence property that is unique to them. More precisely, it is shown that sampling uniformly from the volume of an n-dimensional p-ball and projecting to its surface is equivalent to directly sampling uniformly from its surface if and only if p is 1, 2 or infinity. Sampling algorithms and their implementations in Python are also provided.
- [3] arXiv:2411.13569 [pdf, html, other]
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Title: Unconditionally stable symplectic integrators for the Navier-Stokes equations and other dissipative systemsSutthikiat Sungkeetanon, Joseph S. Gaglione, Robert L. Chapman, Tyler M. Kelly, Howard A. Cushman, Blakeley H. Odom, Bryan MacGavin, Gafar A. Elamin, Nathan J. Washuta, Jonathan E. Crosmer, Adam C. DeVoria, John W. SandersComments: 18 pages, 7 figuresSubjects: Numerical Analysis (math.NA); Dynamical Systems (math.DS); Fluid Dynamics (physics.flu-dyn)
Symplectic integrators offer vastly superior performance over traditional numerical techniques for conservative dynamical systems, but their application to \emph{dissipative} systems is inherently difficult due to dissipative systems' lack of symplectic structure. Leveraging the intrinsic variational structure of higher-order dynamics, this paper presents a general technique for applying existing symplectic integration schemes to dissipative systems, with particular emphasis on viscous fluids modeled by the Navier-Stokes equations. Two very simple such schemes are developed here. Not only are these schemes unconditionally stable for dissipative systems, they also outperform traditional methods with a similar degree of complexity in terms of accuracy for a given time step. For example, in the case of viscous flow between two infinite, flat plates, one of the schemes developed here is found to outperform both the implicit Euler method and the explicit fourth-order Runge-Kutta method in predicting the velocity profile. To the authors' knowledge, this is the very first time that a symplectic integration scheme has been applied successfully to the Navier-Stokes equations. We interpret the present success as direct empirical validation of the canonical Hamiltonian formulation of the Navier-Stokes problem recently published by Sanders~\emph{et al.} More sophisticated symplectic integration schemes are expected to exhibit even greater performance. It is hoped that these results will lead to improved numerical methods in computational fluid dynamics.
- [4] arXiv:2411.13571 [pdf, html, other]
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Title: A low-rank balanced truncation approach for large-scale RLCk model order reduction based on extended Krylov subspace and a frequency-aware convergence criterionComments: arXiv admin note: substantial text overlap with arXiv:2311.08478Subjects: Numerical Analysis (math.NA); Hardware Architecture (cs.AR); Computational Engineering, Finance, and Science (cs.CE)
Model order reduction (MOR) is essential in integrated circuit design, particularly when dealing with large-scale electromagnetic models extracted from complex designs. The numerous passive elements introduced in these models pose significant challenges in the simulation process. MOR methods based on balanced truncation (BT) help address these challenges by producing compact reduced-order models (ROMs) that preserve the original model's input-output port behavior. In this work, we present an extended Krylov subspace-based BT approach with a frequency-aware convergence criterion and efficient implementation techniques for reducing large-scale models. Experimental results indicate that our method generates accurate and compact ROMs while achieving up to x22 smaller ROMs with similar accuracy compared to ANSYS RaptorX ROMs for large-scale benchmarks.
- [5] arXiv:2411.13573 [pdf, other]
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Title: Higher-Order Spectral Element Methods for Electromagnetic Modeling of Complex Anisotropic WaveguidesComments: Ph.D. Thesis in Electrical Engineering at the Pontifical Catholic University of Rio de JaneiroJournal-ref: Maxwell, PUC-Rio, 2024Subjects: Numerical Analysis (math.NA); Computational Engineering, Finance, and Science (cs.CE)
This research thesis presents a novel higher-order spectral element method (SEM) formulated in cylindrical coordinates for analyzing electromagnetic fields in waveguides filled with complex anisotropic media. In this study, we consider a large class of cylindrical waveguides: radially-bounded and radially-unbounded domains; homogeneous and inhomogeneous waveguides; concentric and non-concentric geometries; Hermitian and non-Hermitian anisotropic media tensors. This work explores different wave equation formulations for one-layer eccentric and multilayer cylindrical waveguides. For the first case, we can define a new normalized scalar Helmholtz equation for decoupling TM and TE modes, and for the second, a vectorial Helmholtz equation for hybrid modes in multilayered anisotropic structures. Additionally, we formulate a transformation optics (TO) framework to include non-symmetric and non-Hermitian media tensors for non-concentric multilayer waveguides. Lastly, we model excitation sources for logging sensors applied in geophysical problems using the fields obtained by SEM. We validate the proposed approach against analytical solutions, perturbation-based and mode-matching-based methods, finite-elements, and finite-integration numerical methods. Our technique obtains accurate results with fewer elements and degrees of freedom (DoF) than Cartesian-based SEM and ordinary finite-element approaches. To this end, we use higher-order two-dimensional basis functions associated with the zeros of the completed Lobatto polynomial to model the fields in each reference element. The convergence analysis demonstrates the absence of the Runge effect as the expansion order increases. Numerical results show that our formulation is efficient and accurate for modeling cylindrical waveguided geometries filled with complex media.
- [6] arXiv:2411.13575 [pdf, html, other]
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Title: On reconstruction from imaginary part for radiation solutions in two dimensionsComments: arXiv admin note: substantial text overlap with arXiv:2405.10333Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We consider a radiation solution $\psi$ for the Helmholtz equation in an exterior region in $\mathbb R^2$. We show that $\psi$ in the exterior region is uniquely determined by its imaginary part $Im(\psi)$ on an interval of a line $L$ lying in the exterior region. This result has holographic prototype in the recent work [Nair, Novikov, arXiv:2408.08326]. Some other curves for measurements instead of the lines $L$ are also considered. Applications to the Gelfand-Krein-Levitan inverse problem and passive imaging are also indicated.
- [7] arXiv:2411.13589 [pdf, html, other]
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Title: Bicomplex Mittag-Leffler DistributionSubjects: Probability (math.PR)
Probability distribution theory helps in studying the impact of various dimensions in life while the Mittag-Leffler function and bicomplex are used in electromagnetism, quantum mechanics, and signal theory. Considering the importance of both, the purpose of this paper is to introduce bicomplex Mittag-Leffler distribution theory with the help of the bicomplex Mittag-Leffler function. Moreover, it also tells us about the moment-generating function, the first four moments, the mean, and the variance of this endeavor.
- [8] arXiv:2411.13617 [pdf, html, other]
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Title: Maximum-norm a posteriori error bounds for parabolic equations discretised by the extrapolated Euler method in time and FEM in spaceSubjects: Numerical Analysis (math.NA)
A class of linear parabolic equations is considered. We derive a framework for the a posteriori error analysis of time discretisations by Richardson extrapolation of arbitrary order combined with finite element discretisations in space. We use the idea of elliptic reconstructions and certain bounds for the Green's function of the parabolic operator. The crucial point in the analysis is the design of suitable polynomial reconstructions in time from approximations that are given only in the mesh points.
- [9] arXiv:2411.13624 [pdf, html, other]
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Title: A Priori Bounds for H\'enon-like RenormalizationComments: 47 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:2411.08317Subjects: Dynamical Systems (math.DS)
We formulate and prove $\textit{a priori}$ bounds for the renormalization of Hénon-like maps (under certain regularity assumptions). This provides a certain uniform control on the small-scale geometry of the dynamics, and ensures pre-compactness of the renormalization sequence. In a sequel to this paper, a priori bounds are used in the proof of the main results, including renormalization convergence, finite-time checkability of the required regularity conditions and regular unicriticality of the dynamics.
- [10] arXiv:2411.13656 [pdf, html, other]
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Title: On vertex sets inducing tanglesSubjects: Combinatorics (math.CO)
Diestel, Hundertmark and Lemanczyk asked whether every $k$-tangle in a graph is induced by a set of vertices by majority vote. We reduce their question to graphs whose size is bounded by a function in $k$. Additionally, we show that if for any fixed $k$ this problem has a positive answer, then every $k$-tangle is induced by a vertex set whose size is bounded in $k$. More generally, we prove for all $k$ that every $k$-tangle in a graph $G$ is induced by a weight function $V(G) \to \mathbb{N}$ whose total weight is bounded in $k$. As the key step of our proofs, we show that any given $k$-tangle in a graph $G$ is the lift of a $k$-tangle in some topological minor of $G$ whose size is bounded in $k$.
- [11] arXiv:2411.13662 [pdf, other]
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Title: Logarithmic geometry beyond fsComments: 69 pages, 2 figures, comments welcome!Subjects: Algebraic Geometry (math.AG)
We develop the foundations of logarithmic structures beyond the standard finiteness conditions. The motivation is the study of semistable models over general valuation rings. The key new notion is that of a morphism of finite presentation up to saturation (sfp), which is one that is qcqs and which is locally isomorphic to the saturated base change of a finitely presented morphism between fs log schemes. As in the case of schemes, sfp maps can (locally on the base) be approximated by maps between fs log schemes of finite type over $\mathbb{Z}$. Based on sfp maps, we define smooth, étale, and Kummer étale maps. Importantly, the maps of schemes underlying such maps are no longer of finite type in general, though surprisingly they are if the base is the spectrum of a valuation ring with algebraically closed field of fractions. These foundations allow us to extend beyond the fs case the theory of the Kummer étale site and of the Kummer étale fundamental group.
- [12] arXiv:2411.13671 [pdf, html, other]
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Title: Extremal functions on moduli spaces and applicationsComments: 6 pagesSubjects: Number Theory (math.NT)
We construct and study extremal functions, which are defined by distance functions of convex bodies. Such functions take values in the moduli spaces of geometric objects associated with these convex bodies. Examples of such functions are the homogeneous arithmetic minimum of a function in a lattice, the Hermite constant, and the critical determinant of a body. We define and investigate extremal functions that yield optimal packings of bodies, the best values of covering constants, and optimal solutions to Diophantine approximation problems.
- [13] arXiv:2411.13678 [pdf, html, other]
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Title: Approximation spaces, greedy classes and Lorentz spacesSubjects: Functional Analysis (math.FA)
We characterize the approximation spaces of a broad class of bases - which includes almost greedy bases - in terms of weighted Lorentz spaces. For those bases, we also find necessary and sufficient conditions under which the approximation spaces and greedy classes are the same.
- [14] arXiv:2411.13679 [pdf, html, other]
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Title: Characterising exchange of stability in scalar reaction-diffusion equations via geometric blow-upSubjects: Dynamical Systems (math.DS); Analysis of PDEs (math.AP)
We study the exchange of stability in scalar reaction-diffusion equations which feature a slow passage through transcritical and pitchfork type singularities in the reaction term, using a novel adaptation of the geometric blow-up method. Our results are consistent with known results on bounded spatial domains which were obtained by Butuzov, Nefedov & Schneider using comparison principles like upper and lower solutions in [7], however, from a methodological point of view, the approach is motivated by the analysis of closely related ODE problems using geometric blow-up presented by Krupa & Szmolyan in [34]. After applying the blow-up transformation, we obtain a system of PDEs which can be studied in local coordinate charts. Importantly, the blow-up procedure resolves a spectral degeneracy in which continuous spectrum along the entire negative real axis is 'pushed back' so as to create a spectral gap in the linearisation about particular steady states which arise within the so-called entry and exit charts. This makes it possible to extend slow-type invariant manifolds into and out of a neighbourhood of the singular point using center manifold theory, in a manner which is conceptually analogous to the established approach in the ODE setting. We expect that the approach can be adapted and applied to the study of dynamic bifurcations in PDEs in a wide variety of different contexts.
- [15] arXiv:2411.13702 [pdf, html, other]
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Title: Veronese polytopes: Extending the framework of cyclic polytopesComments: 33 pages, 8 figuresSubjects: Combinatorics (math.CO); Differential Geometry (math.DG)
This article introduces the theory of Veronese polytopes, a broad generalisation of cyclic polytopes. These arise as convex hulls of points on curves with one or more connected components, obtained as the image of the rational normal curve in affine charts. We describe their facial structure by extending Gale's evenness condition, and provide a further combinatorial characterisation of facets via $\sigma$-parity alternating sequences. Notably, we establish a bijective correspondence between combinatorial types of Veronese polytopes and partitions of finite sets equipped with a cyclic order, called circular compositions. We show that, although the only Veronese $3$-polytopes are the cyclic $3$-polytopes and the octahedron, in general dimension they form a rich and diverse class including all combinatorial types of simplicial $d$-polytopes with at most $d+3$ vertices, the cross-polytope and particular stacked polytopes. In addition, we characterise which curves defining Veronese polytopes are $d$-order curves, and provide a closed formula for the number of facets of any Veronese polytope.
- [16] arXiv:2411.13706 [pdf, html, other]
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Title: Closed subcategories of quotient categoriesComments: 25 pagesSubjects: Rings and Algebras (math.RA); Category Theory (math.CT); Quantum Algebra (math.QA)
We study the spectrum of closed subcategories in a quasi-scheme, i.e. a Grothendieck category $X$. The closed subcategories are the direct analogs of closed subschemes in the commutative case, in the sense that when $X$ is the category of quasi-coherent sheaves on a quasi-projective scheme $S$, then the closed subschemes of $S$ correspond bijectively to the closed subcategories of $X$. Many interesting quasi-schemes, such as the noncommutative projective scheme Qgr-$B$ = Gr-$B$/Tors-$B$ associated to a graded algebra $B$, arise as quotient categories of simpler abelian categories. In this paper we will show how to describe the closed subcategories of any quotient category $X/Y$ in terms of closed subcategories of $X$ with special properties, when $X$ is a category with a set of compact projective generators.
- [17] arXiv:2411.13707 [pdf, html, other]
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Title: High-degree cubature on Wiener space through unshuffle expansionsSubjects: Probability (math.PR)
Utilising classical results on the structure of Hopf algebras, we develop a novel approach for the construction of cubature formulae on Wiener space based on unshuffle expansions. We demonstrate the effectiveness of this approach by constructing the first explicit degree-7 cubature formula on $d$-dimensional Wiener space with drift, in the sense of Lyons and Victoir. The support of our degree-7 formula is significantly smaller than that of currently implemented or proposed constructions.
- [18] arXiv:2411.13723 [pdf, html, other]
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Title: The elementary theory of free Steiner triple systemsSubjects: Logic (math.LO)
Free Steiner triple systems (STS) are infinite structures that are naturally characterised by a universal property. We consider the class of free STSs from a model theoretic viewpoint. We show that free STSs on any number of generators are elementarily equivalent. We axiomatise their theory and show that it is stable.
- [19] arXiv:2411.13726 [pdf, html, other]
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Title: A priori estimates for the linearized relativistic Euler equations with a physical vacuum boundary and an ideal gas equation of stateSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
In this paper, we will provide a result on the relativistic Euler equations for an ideal gas equation of state and a physical vacuum boundary. More specifically, we will prove a priori estimates for the linearized system in weighted Sobolev spaces. Our focus will be on choosing the correct thermodynamic variables, developing a weighted book-keeping scheme, and then proving energy estimates for the linearized system.
- [20] arXiv:2411.13735 [pdf, html, other]
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Title: $L^p$-spectral triples and $p$-quantum compact metric spacesComments: AMSLaTeX; 19 pagesSubjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
In this paper we generalize the concept of classical spectral triples by extending the framework from Hilbert spaces to $L^p$-spaces, and from C*-algebras to $L^p$-operator algebras, $p \in [1, \infty)$. Specifically, we construct $L^p$-spectral triples for reduced group $L^p$-operator algebras and for $L^p$ UHF-algebras of infinite tensor product type. Furthermore, inspired by Christensen and Ivan's construction of a Dirac operator on AF C*-algebras, we provide an $L^p$ version of this construction for $L^p$ UHF-algebras. This leads to the construction of a $p$-quantum compact metric space structure on the state space of the $L^p$ UHF-algebra.
- [21] arXiv:2411.13736 [pdf, html, other]
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Title: $2\times 2$ Laguerre-type differential operator with triangular eigenvalueSubjects: Classical Analysis and ODEs (math.CA)
In this paper, we present a comprehensive account of all Laguerre-type differential operators $D$ that are symmetric with respect to a $2\times 2$ irreducible weight $W$ on the interval $(0, \infty)$. These operators are associated with monic orthogonal polynomials ${P_n}$, which satisfy the equation $DP_n = P_n\Delta_n$ for a certain lower triangular eigenvalue $\Delta_n$. We introduce three distinct families of operators and weights, each characterized by explicit expressions depending on two or three parameters, along with a new expression based on a single parameter.
- [22] arXiv:2411.13758 [pdf, html, other]
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Title: On parametric formulations for the Asymmetric Traveling Salesman ProblemSubjects: Optimization and Control (math.OC)
The traveling salesman problem is a widely studied classical combinatorial problem for which there are several integer linear formulations. In this work, we consider the Miller-Tucker-Zemlin (MTZ), Desrochers-Laporte (DL) and Single Commodity Flow (SCF) formulations. We argue that the choice of some parameters of these formulations is arbitrary and, therefore, there are families of formulations of which each of MTZ, DL, and SCF is a particular case. We analyze these families for different choices of the parameters, noting that in general the formulations involved are not comparable to each other and there is no one that dominates the rest. Then we define and study the closure of each family, that is, the set obtained by considering all the associated formulations simultaneously. In particular, we give an explicit integer linear formulation for the closure of each of the families we have defined and then show how they compare to each other.
- [23] arXiv:2411.13763 [pdf, html, other]
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Title: Active Subsampling for Measurement-Constrained M-Estimation of Individualized Thresholds with High-Dimensional DataSubjects: Statistics Theory (math.ST); Methodology (stat.ME); Machine Learning (stat.ML)
In the measurement-constrained problems, despite the availability of large datasets, we may be only affordable to observe the labels on a small portion of the large dataset. This poses a critical question that which data points are most beneficial to label given a budget constraint. In this paper, we focus on the estimation of the optimal individualized threshold in a measurement-constrained M-estimation framework. Our goal is to estimate a high-dimensional parameter $\theta$ in a linear threshold $\theta^T Z$ for a continuous variable $X$ such that the discrepancy between whether $X$ exceeds the threshold $\theta^T Z$ and a binary outcome $Y$ is minimized. We propose a novel $K$-step active subsampling algorithm to estimate $\theta$, which iteratively samples the most informative observations and solves a regularized M-estimator. The theoretical properties of our estimator demonstrate a phase transition phenomenon with respect to $\beta\geq 1$, the smoothness of the conditional density of $X$ given $Y$ and $Z$. For $\beta>(1+\sqrt{3})/2$, we show that the two-step algorithm yields an estimator with the parametric convergence rate $O_p((s \log d /N)^{1/2})$ in $l_2$ norm. The rate of our estimator is strictly faster than the minimax optimal rate with $N$ i.i.d. samples drawn from the population. For the other two scenarios $1<\beta\leq (1+\sqrt{3})/2$ and $\beta=1$, the estimator from the two-step algorithm is sub-optimal. The former requires to run $K>2$ steps to attain the same parametric rate, whereas in the latter case only a near parametric rate can be obtained. Furthermore, we formulate a minimax framework for the measurement-constrained M-estimation problem and prove that our estimator is minimax rate optimal up to a logarithmic factor. Finally, we demonstrate the performance of our method in simulation studies and apply the method to analyze a large diabetes dataset.
- [24] arXiv:2411.13765 [pdf, html, other]
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Title: Schr\"odinger Bridge Problem for Jump DiffusionsSubjects: Probability (math.PR); Information Theory (cs.IT); Optimization and Control (math.OC)
The Schrödinger bridge problem (SBP) seeks to find the measure $\hat{\mathbf{P}}$ on a certain path space which interpolates between state-space distributions $\rho_0$ at time $0$ and $\rho_T$ at time $T$ while minimizing the KL divergence (relative entropy) to a reference path measure $\mathbf{R}$. In this work, we tackle the SBP in the case when $\mathbf{R}$ is the path measure of a jump diffusion. Under mild assumptions, with both the operator theory approach and the stochastic calculus techniques, we establish an $h$-transform theory for jump diffusions and devise an approximation method to achieve the jump-diffusion SBP solution $\hat{\mathbf{P}}$ as the strong-convergence limit of a sequence of harmonic $h$-transforms. To the best of our knowledge, these results are novel in the study of SBP. Moreover, the $h$-transform framework and the approximation method developed in this work are robust and applicable to a relatively general class of jump diffusions. In addition, we examine the SBP of particular types of jump diffusions under additional regularity conditions and extend the existing results on the SBP from the diffusion case to the jump-diffusion setting.
- [25] arXiv:2411.13767 [pdf, html, other]
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Title: Improved Upper Bounds on Key Invariants of Erd\H{o}s-R\'enyi Numerical SemigroupsComments: 17 pages, 4 figuresSubjects: Commutative Algebra (math.AC); Combinatorics (math.CO); Number Theory (math.NT)
De Loera, O'Neill and Wilburne introduced a general model for random numerical semigroups in which each positive integer is chosen independently with some probability p to be a generator, and proved upper and lower bounds on the expected Frobenius number and expected embedding dimensions. We use a range of probabilistic methods to improve the upper bounds to within a polylogarithmic factor of the lower bounds in each case. As one of the tools to do this, we prove that for any prime q, if A is a random subset of the cyclic group Z_q whose size is of order log(q) and k is also of order log(q), then with high probability the k-fold sumset kA is all of Z_q.
- [26] arXiv:2411.13772 [pdf, html, other]
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Title: A Characteristic Mapping Method with Source Terms: Applications to Ideal MagnetohydrodynamicsComments: The preprint has not been revised yet!Subjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph); Fluid Dynamics (physics.flu-dyn); Plasma Physics (physics.plasm-ph)
This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We derive a recursive formula for the time decomposition of the map and the source term integral, enhancing computational efficiency. Benchmark computations are presented for a test case with an exact solution and for two-dimensional ideal incompressible magnetohydrodynamics (MHD). Results demonstrate third-order accuracy in both space and time. The submap decomposition method achieves exceptionally high resolution, as illustrated by zooming into fine-scale current sheets. An error estimate is performed and suggests third order convergence in space and time.
- [27] arXiv:2411.13780 [pdf, html, other]
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Title: Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equationsComments: 45 pagesSubjects: Analysis of PDEs (math.AP)
We study the asymptotic behavior of solutions of an equation of the form \begin{equation}\label{abs}\tag{*} G\big(x, D_x u,\lambda u(x)\big) = c_0\qquad\hbox{in $M$} \end{equation} on a closed Riemannian manifold $M$, where $G\in C(T^*M\times\mathbb{R})$ is convex and superlinear in the gradient variable, is globally Lipschitz but not monotone in the last argument, and $c_0$ is the critical constant associated with the Hamiltonian $H:=G(\cdot,\cdot,0)$. By assuming that $\partial_u G(\cdot,\cdot,0)$ satisfies a positivity condition of integral type on the Mather set of $H$, we prove that any equi-bounded family of solutions of \eqref{abs} uniformly converges to a distinguished critical solution $u_0$ as $\lambda \to 0^+$. We furthermore show that any other possible family of solutions uniformly diverges to $+\infty$ or $-\infty$. We then look into the linear case $G(x,p,u):=a(x)u + H(x,p)$ and prove that the family $(u_\lambda)_{\lambda \in (0,\lambda_0)}$ of maximal solutions to \eqref{abs} is well defined and equi-bounded for $\lambda_0>0$ small enough. When $a$ changes sign and enjoys a stronger localized positivity assumption, we show that equation \eqref{abs} does admit other solutions too, and that they all uniformly diverge to $-\infty$ as $\lambda \to 0^+$. This is the first time that converging and diverging families of solutions are shown to coexist in such a generality.
- [28] arXiv:2411.13781 [pdf, html, other]
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Title: Asymptotic speeds of spreading for the Lotka-Volterra system with strong competition in $\mathbb{R}^N$Subjects: Analysis of PDEs (math.AP)
This paper is concerned with the asymptotic spreading behavior of solutions of the Lotka-Volterra system with strong competition in $\mathbb{R}^{N}$. Two types of initial conditions are proposed: (C1) two species initially occupy bounded domains; (C2) two species initially occupy the whole space separately. The spreading dynamics for (C1) (C2) is strongly depending on the speeds of traveling fronts of the scalar equations with no competition and the system. We give the asymptotic speeds of spreading for both (C1) (C2).
- [29] arXiv:2411.13785 [pdf, html, other]
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Title: Throughput Maximization for Movable Antenna Systems with Movement Delay ConsiderationSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
In this paper, we model the minimum achievable throughput within a transmission block of restricted duration and aim to maximize it in movable antenna (MA)-enabled multiuser downlink communications. Particularly, we account for the antenna moving delay caused by mechanical movement, which has not been fully considered in previous studies, and reveal the trade-off between the delay and signal-to-interference-plus-noise ratio at users. To this end, we first consider a single-user setup to analyze the necessity of antenna movement. By quantizing the virtual angles of arrival, we derive the requisite region size for antenna moving, design the initial MA position, and elucidate the relationship between quantization resolution and moving region size. Furthermore, an efficient algorithm is developed to optimize MA position via successive convex approximation, which is subsequently extended to the general multiuser setup. Numerical results demonstrate that the proposed algorithms outperform fixed-position antenna schemes and existing ones without consideration of movement delay. Additionally, our algorithms exhibit excellent adaptability and stability across various transmission block durations and moving region sizes, and are robust to different antenna moving speeds. This allows the hardware cost of MA-aided systems to be reduced by employing low rotational speed motors.
- [30] arXiv:2411.13788 [pdf, html, other]
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Title: Gradient bounds and Liouville property for a class of hypoelliptic diffusion via couplingComments: Modify minor errors after acceptance by Forum MathSubjects: Probability (math.PR)
In this paper, we obtain the reverse Bakry-Émery type estimates for a class of hypoelliptic diffusion operator by coupling method. The (right and reverse) Poincaré inequalities and the (right and reverse) logarithmic Sobolev inequalities are presented as consequences of such estimates. Wang-Harnack inequality, Hamilton's gradient estimate and Liouville property are also presented by reverse logarithmic Sobolev inequality.
- [31] arXiv:2411.13791 [pdf, html, other]
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Title: Zero-density estimates and the optimality of the error term in the prime number theoremComments: 8 pagesSubjects: Number Theory (math.NT)
We demonstrate the impact of a generic zero-free region and zero-density estimate on the error term in the prime number theorem. Consequently, we are able to improve upon previous work of Pintz and provide an essentially optimal error term for some choices of the zero-free region. As an example, we show that if there are no zeros $\rho=\beta+it$ of $\zeta(s)$ for $$\beta>1-\frac{1}{c(\log t)^{2/3}(\log\log t)^{1/3}},$$ then $$\frac{|\psi(x)-x|}{x}\ll\exp(-\omega(x))\frac{(\log x)^9}{(\log\log x)^3},$$ where $\psi(x)$ is the Chebyshev prime-counting function, and $$\omega(x)=\left(\frac{5^6}{2^2\cdot 3^4\cdot c^3}\right)^{1/5}\frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}.$$ This improves upon the best known error term for the prime number theorem, previously given by $$\frac{|\psi(x)-x|}{x}\ll_{\varepsilon}\exp(-(1-\varepsilon)\omega(x))$$ for any $\varepsilon>0$.
- [32] arXiv:2411.13798 [pdf, html, other]
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Title: Nonlinear stability of the one dimensional screened Vlasov Poisson equationComments: 18 pagesSubjects: Analysis of PDEs (math.AP)
We study the asymptotic behavior of small data solutions to the screened Vlasov Poisson(i.e. Vlasov-Yukawa) equation on ${\mathbb R}\times{\mathbb R}$ near vacuum. We show that for initial data small in Gevrey-2 regularity, the derivative of the density of order $n$ decays like $(t+1)^{-n-1}$.
- [33] arXiv:2411.13804 [pdf, html, other]
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Title: The Brown Measure of Non-Hermitian Sums of ProjectionsComments: 31 pages, 2 figuresSubjects: Operator Algebras (math.OA); Probability (math.PR)
We compute the Brown measure of the non-normal operators $X = p + i q$, where $p$ and $q$ are Hermitian, freely independent, and have spectra consisting of $2$ atoms. The computation relies on the model of the non-trivial part of the von Neumann algebra generated by 2 projections as $2 \times 2$ random matrices. We observe that these measures are supported on hyperbolas and note some other properties related to their atoms and symmetries.
- [34] arXiv:2411.13805 [pdf, html, other]
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Title: On Representing Convex Quadratically Constrained Quadratic Programs via Graph Neural NetworksSubjects: Optimization and Control (math.OC)
Convex quadratically constrained quadratic programs (QCQPs) involve finding a solution within a convex feasible region defined by quadratic constraints while minimizing a convex quadratic objective function. These problems arise in various industrial applications, including power systems and signal processing. Traditional methods for solving convex QCQPs primarily rely on matrix factorization, which quickly becomes computationally prohibitive as the problem size increases. Recently, graph neural networks (GNNs) have gained attention for their potential in representing and solving various optimization problems such as linear programs and linearly constrained quadratic programs. In this work, we are the first to investigate the representation power of GNNs in the context of QCQP tasks. Specifically, we propose a new tripartite graph representation for general convex QCQPs and properly associate it with message-passing GNNs. We demonstrate that there exist GNNs capable of reliably representing key properties of convex QCQPs, including feasibility, optimal value, and optimal solution. Our result deepens the understanding of the connection between QCQPs and GNNs, paving the way for future machine learning approaches to efficiently solve QCQPs.
- [35] arXiv:2411.13812 [pdf, html, other]
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Title: When are off-diagonal hypergraph Ramsey numbers polynomial?David Conlon, Jacob Fox, Benjamin Gunby, Xiaoyu He, Dhruv Mubayi, Andrew Suk, Jacques Verstraëte, Hung-Hsun Hans YuComments: 12 pagesSubjects: Combinatorics (math.CO)
A natural open problem in Ramsey theory is to determine those $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H, K_n^{(3)})$ grows polynomially with $n$. We make substantial progress on this question by showing that if $H$ is tightly connected or has at most two tight components, then $r(H, K_n^{(3)})$ grows polynomially if and only if $H$ is not contained in an iterated blowup of an edge.
- [36] arXiv:2411.13818 [pdf, html, other]
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Title: Proof of Merca's stronger conjecture on truncated Jacobi triple product seriesSubjects: Number Theory (math.NT); Combinatorics (math.CO)
The truncated series start off with Andrews and Merca's truncated version of Euler's pentagonal number theorem in 2012. Moreover, Andrews--Merca and Guo--Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been proved analytically by Mao and combinatorially by Yee. In 2021, Merca gave a stronger version for the truncated Jacobi triple product series(JTPS). Some very spcial cases of the conjecture have been proved by Ballantine--Feigon, Ding-Sun and Zhou. On the one hand, we consider the partial finite denomintor of the stronger truncated JTPS series with Residue theorem and prove that the coefficients of $ q^{n} $ in the series are positive when $ n $ greater than a finite constant. On the other hand, for the infinte denomintor, we deduce an asymptotic of nonmodular infinite products and therefore prove Merca's conjecture when $ n $ greater than a constant. We also show than when $ k $ is large enough, the infinte denomintor can be ignored and the Conjecture holds as a corollary.
- [37] arXiv:2411.13828 [pdf, html, other]
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Title: Cosmetic surgery on satellite knotsComments: 5 pagesSubjects: Geometric Topology (math.GT)
We show that if there exists a knot in $S^3$ that admits purely cosmetic surgeries, then there exists a hyperbolic one with this property.
- [38] arXiv:2411.13830 [pdf, html, other]
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Title: Rational contact instantons and Legendrian Fukaya categoryComments: 45 pages, comments welcome!Subjects: Symplectic Geometry (math.SG)
This is the first of a series of papers in preparation on the Fukaya-type $A_\infty$ category generated by tame Legendrian submanifolds, called the Legendrian contact instanton Fukaya category (abbreviated as the Legendrian CI Fukaya category) and its applications to contact dynamics and topology. In the present paper, we give the construction of an $A_\infty$ category whose objects are Legendrian links and whose structure maps are defined by the moduli spaces of finite energy contact instantons on tame contact manifolds in the sense of [Oh21b]. In a sequel [KO], jointed by Jongmyeong Kim, we will explain the relationships with various previous results in the literature concerning Rabinowitz Fukaya categories [CF09, CFO10], [GGV], [BJK] on the Liouville manifolds with ideal boundary of contact manifolds, and the Floer theory of Lagrangian cobordism [CDRGG20], [EES05].
- [39] arXiv:2411.13838 [pdf, html, other]
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Title: Mathematical Analysis of Regularity, Bifurcations, and Turbulence in Fluid Dynamics via Sobolev, Besov, and Triebel-Lizorkin SpacesComments: 27 pagesSubjects: Analysis of PDEs (math.AP); Fluid Dynamics (physics.flu-dyn)
This article presents a comprehensive mathematical framework for the study of regularity, bifurcations, and turbulence in fluid dynamics, leveraging the power of Sobolev and Besov function spaces. We delve into the detailed definitions, properties, and notations of these spaces, illustrating their relevance in the context of partial differential equations governing fluid flow. The work emphasizes the intricate connections between Sobolev, Besov, and Triebel-Lizorkin spaces, highlighting their interplay in the analysis of fluid systems. We propose new regularity criteria for solutions to the Navier-Stokes equations, based on the interaction of low and high-frequency modes in turbulent regimes. These criteria offer a novel perspective on the conditions under which singularities may form, providing critical insights into the structure of turbulent flows. The article further explores the applications of these function spaces to the analysis of bifurcations in fluid systems, offering a deeper understanding of the mechanisms that lead to complex flow phenomena such as turbulence. Through the development of rigorous theorems and proofs, the paper aims to bridge the gap between abstract mathematical theory and practical fluid dynamics. In particular, the results contribute to ongoing efforts in solving the Navier-Stokes existence and smoothness problem, a key challenge in the field, and have potential implications for the Millennium Prize Problem. The conclusion underscores the significance of these findings, offering a pathway for future research in the analysis of fluid behavior at both small and large scales.
- [40] arXiv:2411.13843 [pdf, other]
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Title: Non-parametric structural shape optimization of piecewise developable surfaces using discrete differential geometryComments: Presented at Asian Congress of Structural and Multidisciplinary Optimization (ACSMO 2024)Subjects: Optimization and Control (math.OC)
We propose a two-level structural optimization method for obtaining an approximate optimal shape of piecewise developable surface without specifying internal boundaries between surface patches. The condition for developability of a polyhedral surface onto a plane is formulated using the area of discrete Gauss map formed by unit normal vectors at the faces adjacent to each vertex. The objective function of the lower-level optimization problem is the sum of square errors for developability at all interior vertices. The contribution of large error to the objective function is underestimated by filtering with hyperbolic tangent function so that the internal boundary between the surface patches can naturally emerge as a result of optimization. Vertices are located non-periodically to generate the internal boundaries in various unspecified directions. Simulated annealing is used for the upper-level optimization problem for maximizing stiffness evaluated by the compliance under the specified vertical loads. The design variables are the heights of the specified points. It is shown in the numerical examples that the compliance values of the surfaces with a square and a rectangular plan are successfully reduced by the proposed method while keeping the developability of each surface patch. Thus, a new class of structural shape optimization problem of shell surfaces is proposed by limiting the feasible surface to piecewise developable surfaces which have desirable geometrical characteristics in view of fabrication and construction.
- [41] arXiv:2411.13857 [pdf, html, other]
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Title: Effective actions, cutoff regularization, quasi-locality, and gluing of partition functionsComments: LaTeX, 30 pages, 1 figure. Firstly appeared in Russian, November 15, 2024, see this https URLSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)
The paper studies a regularization of the quantum (effective) action for a scalar field theory in a general position on a compact smooth Riemannian manifold. As the main method, we propose the use of a special averaging operator, which leads to a quasi-locality and is a natural generalization of a cutoff regularization in the coordinate representation in the case of a curved metric. It is proved that the regularization method is consistent with a process of gluing of manifolds and partition functions, that is, with the transition from submanifolds to the main manifold using an additional functional integration. It is shown that the method extends to other models, and is also consistent with the process of multiplicative renormalization. Additionally, we discuss issues related to the correct introduction of regularization and the locality.
- [42] arXiv:2411.13858 [pdf, html, other]
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Title: Zimmer's conjecture for non-split semisimple Lie groupsSubjects: Dynamical Systems (math.DS)
We prove many new cases of Zimmer's conjecture for actions by lattices in non-$\mathbb{R}$-split semisimple Lie groups $G$. By prior arguments, Zimmer's conjecture reduces to studying certain probability measures invariant under a minimal parabolic subgroup for the induced $G$-action.
Two techniques are introduced to give lower bounds on the dimension of a manifold $M$ admitting a non-isometric action. First, when the Levi component of the stabilizer of the measure has higher-rank simple factors, cocycle superrigidity provides a lower bound on the dimension of $M$. Second, when certain fiberwise coarse Lyapunov distributions are one-dimensional, a measure rigidity argument provides additional invariance of the measure if the associated root spaces are higher-dimensional. - [43] arXiv:2411.13864 [pdf, html, other]
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Title: Einstein metrics on homogeneous superspacesComments: 46 pagesSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Differential Geometry (math.DG)
This paper initiates the study of the Einstein equation on homogeneous supermanifolds. First, we produce explicit curvature formulas for graded Riemannian metrics on these spaces. Next, we present a construction of homogeneous supermanifolds by means of Dynkin diagrams, resembling the construction of generalised flag manifolds in classical (non-super) theory. We describe the Einstein metrics on several classes of spaces obtained through this approach. Our results provide examples of compact homogeneous supermanifolds on which the Einstein equation has no solutions, discrete families of solutions, and continuous families of Ricci-flat solutions among invariant metrics. These examples demonstrate that the finiteness conjecture from classical homogeneous geometry fails on supermanifolds, and challenge the intuition furnished by Bochner's vanishing theorem.
- [44] arXiv:2411.13869 [pdf, other]
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Title: Topology optimization of periodic lattice structures for specified mechanical properties using machine learning considering member connectivityComments: Presented at Asian Congress of Structural and Multidisciplinary Optimization (ACSMO 2024)Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
This study proposes a methodology to utilize machine learning (ML) for topology optimization of periodic lattice structures. In particular, we investigate data representation of lattice structures used as input data for ML models to improve the performance of the models, focusing on the filtering process and feature selection. We use the filtering technique to explicitly consider the connectivity of lattice members and perform feature selection to reduce the input data size. In addition, we propose a convolution approach to apply pre-trained models for small structures to structures of larger sizes. The computational cost for obtaining optimal topologies by a heuristic method is reduced by incorporating the prediction of the trained ML model into the optimization process. In the numerical examples, a response prediction model is constructed for a lattice structure of 4x4 units, and topology optimization of 4x4-unit and 8x8-unit structures is performed by simulated annealing assisted by the trained ML model. The example demonstrates that ML models perform higher accuracy by using the filtered data as input than by solely using the data representing the existence of each member. It is also demonstrated that a small-scale prediction model can be constructed with sufficient accuracy by feature selection. Additionally, the proposed method can find the optimal structure in less computation time than the pure simulated annealing.
- [45] arXiv:2411.13875 [pdf, html, other]
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Title: Large deviations at the origin of random walk in random environmentComments: 22 pages, 2 figuresSubjects: Probability (math.PR)
We consider a random walk in an i.i.d. random environment on Zd and study properties of its large deviation rate function at the origin. It was proved by Comets, Gantert and Zeitouni in dimension d = 1 in 1999 and later by Varadhan in dimensions d >= 2 in 2003 that, for uniformly elliptic i.i.d. random environments, the quenched and the averaged large deviation rate functions coincide at the origin. Here we provide a description of an atypical event realizing the correct quenched large deviation rate in the nestling and marginally nestling setting: the random walk seeks regions of space where the environment emulates the element in the convex hull of the support of the law of the environment at a site which minimizes the rate function. Periodic environments play a natural role in this description.
- [46] arXiv:2411.13876 [pdf, other]
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Title: Iterative decoding of short BCH codes and its post-processingComments: 6 pages, 6 figures, 2 tablesSubjects: Information Theory (cs.IT)
Effective iterative decoding of short BCH codes faces two primary challenges: identifying an appropriate parity-check matrix and accelerating decoder convergence. To address these issues, we propose a systematic scheme to derive an optimized parity-check matrix through a heuristic approach. This involves a series of binary sum and row shift operations, resulting in a low-density, quasi-regular column weight distribution with a reduced number of shortest cycles in the underlying redundant Tanner graph. For the revised normalized min-sum decoder, we concurrently integrate three types of random permutations into the alternated messages across iterations, leading to significantly faster convergence compared to existing methods. Furthermore, by utilizing the iterative trajectories of failed normalized min-sum decoding, we enhance the reliability measurement of codeword bits with the assistance of a neural network model from prior work, which accommodates more failures for the post-processing of ordered statistics decoding. Additionally, we report the types of undetected errors for the design of iterative decoders for short BCH codes, which potentially challenge efforts to approach the maximum likelihood limit. Extensive simulations demonstrate that the proposed hybrid framework achieves an attractive balance between performance, latency, and complexity.
- [47] arXiv:2411.13877 [pdf, html, other]
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Title: Inequalities on six points in a $\mathrm{CAT}(0)$ spaceComments: 10 pagesSubjects: Metric Geometry (math.MG)
We establish a family of inequalities that hold true on any $6$ points in any $\mathrm{CAT}(0)$ space. We prove that the validity of these inequalities does not follow from any properties of $5$-point subsets of $\mathrm{CAT}(0)$ spaces. In particular, the validity of these inequalities does not follow from the $\mathrm{CAT}(0)$ $4$-point condition.
- [48] arXiv:2411.13878 [pdf, html, other]
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Title: Sparse Zero Correlation Zone Arrays for Training Design in Spatial Modulation SystemsSubjects: Information Theory (cs.IT)
This paper presents a novel training matrix design for spatial modulation (SM) systems, by introducing a new class of two-dimensional (2D) arrays called sparse zero correlation zone (SZCZ) arrays. An SZCZ array is characterized by a majority of zero entries and exhibits the zero periodic auto- and cross-correlation zone properties across any two rows. With these unique properties, we show that SZCZ arrays can be effectively used as training matrices for SM systems. Additionally, direct constructions of SZCZ arrays with large ZCZ widths and controllable sparsity levels based on 2D restricted generalized Boolean functions (RGBFs) are proposed. Compared with existing training schemes, the proposed SZCZ-based training matrices have larger ZCZ widths, thereby offering greater tolerance for delay spread in multipath channels. Simulation results demonstrate that the proposed SZCZ-based training design exhibits superior channel estimation performance over frequency-selective fading channels compared to existing alternatives.
- [49] arXiv:2411.13880 [pdf, html, other]
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Title: Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy SpacesSubjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP)
In harmonic analysis, studies of inequalities of Riesz potential in various function spaces have a very important place. Variable exponent Morrey type spaces and the examines of the boundedness of such operators on these spaces have an important place in harmonic analysis and have become an interesting field. In this work, we obtain the boundedness of Riesz potential on homogeneous variable exponent Herz-Morrey-Hardy spaces under some conditions.
- [50] arXiv:2411.13884 [pdf, html, other]
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Title: Reinforcement Learning for Jointly Optimal Coding and Control over a Communication ChannelComments: 8 pages, 3 figures, submitted to American Control Conference 2025Subjects: Optimization and Control (math.OC)
We develop rigorous approximation and near optimality results for the optimal control of a system which is connected to a controller over a finite rate noiseless channel. While structural results on the optimal encoding and control have been obtained in the literature, their implementation has been prohibitive in general, except for linear models. We develop regularity and structural properties, followed by approximations and reinforcement learning results. Notably, we establish near optimality of finite model approximations as well as sliding finite window coding policies and their reinforcement learning convergence to near optimality.
- [51] arXiv:2411.13887 [pdf, html, other]
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Title: A cohomology-based Gromov-Hausdorff metric approach for quantifying molecular similarityComments: 14 pages, 3 figuresSubjects: Algebraic Topology (math.AT); Materials Science (cond-mat.mtrl-sci); Computational Geometry (cs.CG); Metric Geometry (math.MG); Machine Learning (stat.ML)
We introduce, for the first time, a cohomology-based Gromov-Hausdorff ultrametric method to analyze 1-dimensional and higher-dimensional (co)homology groups, focusing on loops, voids, and higher-dimensional cavity structures in simplicial complexes, to address typical clustering questions arising in molecular data analysis. The Gromov-Hausdorff distance quantifies the dissimilarity between two metric spaces. In this framework, molecules are represented as simplicial complexes, and their cohomology vector spaces are computed to capture intrinsic topological invariants encoding loop and cavity structures. These vector spaces are equipped with a suitable distance measure, enabling the computation of the Gromov-Hausdorff ultrametric to evaluate structural dissimilarities. We demonstrate the methodology using organic-inorganic halide perovskite (OIHP) structures. The results highlight the effectiveness of this approach in clustering various molecular structures. By incorporating geometric information, our method provides deeper insights compared to traditional persistent homology techniques.
- [52] arXiv:2411.13894 [pdf, html, other]
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Title: On the double critical Maxwell equationsSubjects: Analysis of PDEs (math.AP)
In this paper, we focus on (no)existence and asymptotic behavior of solutions for the double critical Maxwell equation involving with the Hardy, Hardy-Sobolev, Sobolev critical exponents. The existence and noexistence of solutions completely depend on the power exponents and coefficients of equation. On one hand, based on the concentration-compactness ideas, applying the Nehari manifold and the mountain pass theorem, we prove the existence of the ground state solutions for the critical Maxwell equation for three different scenarios. On the other hand, for the case $\lambda<0$ and $0\leq s_2<s_1<2$, which is a type open problem raised by Li and Lin. Draw support from a changed version of Caffarelli-Kohn-Nirenberg inequality, we find that there exists a constant $\lambda^*$ which is a negative number having explicit expression, such that the problem has no nontrivial solution as the coefficient $\lambda<\lambda^*$. Moreover, there exists a constant $\lambda^*<\lambda^{**}<0$ such that, as $\lambda^{**}<\lambda<0$, the equation has a nontrivial solution using truncation methods. Furthermore, we establish the asymptotic behavior of solutions of equation as coefficient converges to zero for the all cases above.
- [53] arXiv:2411.13896 [pdf, html, other]
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Title: A blow up solution of the Navier-Stokes equations with a critical forceSubjects: Analysis of PDEs (math.AP)
A forced solution $v$ of the Navier-Stokes equation in any open domain with no slip boundary condition is constructed. The scaling factor of the forcing term is the critical order $-2$. The velocity, which is smooth until its final blow up moment, is in the energy space through out. Since most physical forces from a point source in nature are regarded as order $-2$, such as Coulomb force, Yukawa force, this result indicates possible singularity formation under these kind of forces. The result also holds for some log subcritical forces or some forces in the standard critical space $L^\infty_t L^{3/2}_x$.
- [54] arXiv:2411.13906 [pdf, other]
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Title: Structure-preserving model reduction of Hamiltonian systems by learning a symplectic autoencoderSubjects: Numerical Analysis (math.NA)
Evolutionary partial differential equations play a crucial role in many areas of science and engineering. Spatial discretization of these equations leads to a system of ordinary differential equations which can then be solved by numerical time integration. Such a system is often of very high dimension, making the simulation very time consuming. One way to reduce the computational cost is to approximate the large system by a low-dimensional model using a model reduction approach. This master thesis deals with structure-preserving model reduction of Hamiltonian systems by using machine learning techniques. We discuss a nonlinear approach based on the construction of an encoder-decoder pair that minimizes the approximation error and satisfies symplectic constraints to guarantee the preservation of the structure inherent in Hamiltonian systems. More specifically, we study an autoencoder network that learns a symplectic encoder-decoder pair. Symplecticity poses some additional difficulties, as we need to ensure this structure in each network layer. Since these symplectic constraints are described by the (symplectic) Stiefel manifold, we use manifold optimization techniques to ensure the symplecticity of the encoder and decoder. A particular challenge is to adapt the ADAM optimizer to the manifold structure. We present a modified ADAM optimizer that works directly on the Stiefel manifold and compare it to the existing approach based on homogeneous spaces. In addition, we propose several modifications to the network and training setup that significantly improve the performance and accuracy of the autoencoder. Finally, we numerically validate the modified optimizer and different learning configurations on two Hamiltonian systems, the 1D wave equation and the sine-Gordon equation, and demonstrate the improved accuracy and computational efficiency of the presented learning algorithms.
- [55] arXiv:2411.13911 [pdf, html, other]
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Title: Analytic torsion for irreducible holomorphic symplectic fourfolds with involution, II: the singularity of the invariant (with an Appendix by Ken-Ichi Yoshikawa)Comments: 57 pages. arXiv admin note: text overlap with arXiv:1007.2835 by other authorsSubjects: Algebraic Geometry (math.AG)
We study the boundary behavior of the invariant of $K3^{[2]}$-type manifolds with antisymplectic involution, which we obtained using equivariant analytic torsion. We show the algebraicity of the singularity of the invariant by using the asymptotic of equivariant Quillen metrics and equivariant $L^2$-metrics. We prove that, in some cases, the invariant coincides with Yoshikawa's invariant for 2-elementary K3 surfaces. Hence, in these cases, our invariant is expressed as the Petersson norm of a Borcherds product and a Siegel modular form.
- [56] arXiv:2411.13912 [pdf, html, other]
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Title: Einstein manifolds of negative lower bounds on curvature operator of the second KindComments: All comments are welcomeSubjects: Differential Geometry (math.DG)
We demonstrate that $n$-dimension closed Einstein manifolds, whose smallest eigenvalue of the curvature operator of the second kind of $\mathring{R}$ satisfies $\lambda_1 \ge -\theta(n) \bar\lambda$, are either flat or round spheres, where $\bar \lambda$ is the average of the eigenvalues of $\mathring{R}$, and $\theta(n)$ is defined as in equation (1.2). Our result improves a celebrated result (Theorem 1.1) concerning Einstein manifolds with nonnegative curvature operator of the second kind.
- [57] arXiv:2411.13913 [pdf, html, other]
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Title: Generalizing subdiffusive Black-Scholes model by variable exponent: Model transformation and numerical approximationSubjects: Numerical Analysis (math.NA)
This work generalizes the subdiffusive Black-Scholes model by introducing the variable exponent in order to provide adequate descriptions for the option pricing, where the variable exponent may account for the variation of the memory property. In addition to standard nonlinear-to-linear transformation, we apply a further spatial-temporal transformation to convert the model to a more tractable form in order to circumvent the difficulties caused by the ``non-positive, non-monotonic'' variable-exponent memory kernel. An interesting phenomenon is that the spatial transformation not only eliminates the advection term but naturally turns the original noncoercive spatial operator into a coercive one due to the specific structure of the Black-Scholes model, which thus avoids imposing constraints on coefficients. Then we perform numerical analysis for both the semi-discrete and fully discrete schemes to support numerical simulation. Numerical experiments are carried out to substantiate the theoretical results.
- [58] arXiv:2411.13923 [pdf, html, other]
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Title: Fourier dimension of Gaussian multiplicative chaosComments: This is the first version of our work on Fourier dimension of GMC. New version with more comprehensive and simpler proof, together with illustrative pictures and applications, generalizations of the main result will be updated soonSubjects: Probability (math.PR); Mathematical Physics (math-ph); Dynamical Systems (math.DS); Functional Analysis (math.FA)
We obtain the precise Fourier dimension of the Gaussian multiplicative chaos on the unit interval. Our main result confirms a conjecture of Garban-Vargas.
- [59] arXiv:2411.13926 [pdf, html, other]
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Title: Random walks on random walks: non-perturbative results in high dimensionsComments: 36 pagesSubjects: Probability (math.PR)
Consider the dynamic environment governed by a Poissonian field of independent particles evolving as simple random walks on $\mathbb{Z}^d$. The random walk on random walks model refers to a particular stochastic process on $\mathbb{Z}^d$ whose evolution at time $t$ depends on the number of such particles at its location. We derive classical limit theorems for this instrumental model of a random walk in a dynamic random environment, applicable in sufficiently high dimensions. More precisely, for $d \geq 5$, we prove a strong law of large numbers and large deviation estimates. Further, for $d\geq 9$, we obtain a functional central limit theorem under the annealed law. These results are non-perturbative in the sense that they hold for any positive density of the Poissonian field. Under the aforementioned assumptions on the dimension they therefore improve on previous work on the model. Moreover, they stand in contrast to the anomalous behaviour predicted in low dimensions.
- [60] arXiv:2411.13939 [pdf, html, other]
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Title: Filtering and Statistical Properties of Unimodal Maps Perturbed by Heteroscedastic NoisesSubjects: Statistics Theory (math.ST); Dynamical Systems (math.DS); Probability (math.PR)
We propose a theory of unimodal maps perturbed by an heteroscedastic Markov chain noise and experiencing another heteroscedastic noise due to uncertain observation. We address and treat the filtering problem showing that by collecting more and more observations, one would predict the same distribution for the state of the underlying Markov chain no matter one's initial guess. Moreover we give other limit theorems, emphasizing in particular concentration inequalities and extreme value and Poisson distributions. Our results apply to a family of maps arising from a model of systemic risk in finance.
- [61] arXiv:2411.13957 [pdf, html, other]
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Title: Variational Multiscale Evolve and Filter Strategies for Convection-Dominated FlowsSubjects: Numerical Analysis (math.NA); Fluid Dynamics (physics.flu-dyn)
The evolve-filter (EF) model is a filter-based numerical stabilization for under-resolved convection-dominated flows. EF is a simple, modular, and effective strategy for both full-order models (FOMs) and reduced-order models (ROMs). It is well-known, however, that when the filter radius is too large, EF can be overdiffusive and yield inaccurate results. To alleviate this, EF is usually supplemented with a relaxation step. The relaxation parameter, however, is very sensitive with respect to the model parameters. In this paper, we propose a novel strategy to alleviate the EF overdiffusivity for a large filter radius. Specifically, we leverage the variational multiscale (VMS) framework to separate the large resolved scales from the small resolved scales in the evolved velocity, and we use the filtered small scales to correct the large scales. Furthermore, in the new VMS-EF strategy, we use two different ways to decompose the evolved velocity: the VMS Evolve-Filter-Filter-Correct (VMS-EFFC) and the VMS Evolve-Postprocess-Filter-Correct (VMS-EPFC) algorithms. The new VMS-based algorithms yield significantly more accurate results than the standard EF in both the FOM and the ROM simulations of a flow past a cylinder at Reynolds number Re = 1000.
- [62] arXiv:2411.13959 [pdf, other]
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Title: On the multivariate multifractal formalism: examples and counter-examplesStéphane Seuret (UPEC UP12)Subjects: Metric Geometry (math.MG)
In this article, we investigate the bivariate multifractal analysis of pairs of Borel probability measures. We prove that, contrarily to what happens in the univariate case, the natural extension of the Legendre spectrum does not yield an upper bound for the bivariate multifractal spectrum. For this we build a pair of measures for which the two spectra have disjoint supports. Then we study the bivariate multifractal behavior of an archetypical pair of randomly correlated measures, which give new, surprising, behaviors, enriching the narrow class of measures for which such an analysis is achieved.
- [63] arXiv:2411.13964 [pdf, other]
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Title: Long-time analysis of a pair of on-lattice and continuous run-and-tumble particles with jamming interactionsSubjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech)
Run-and-Tumble Particles (RTPs) are a key model of active matter. They are characterized by alternating phases of linear travel and random direction reshuffling. By this dynamic behavior, they break time reversibility and energy conservation at the microscopic level. It leads to complex out-of-equilibrium phenomena such as collective motion, pattern formation, and motility-induced phase separation (MIPS). In this work, we study two fundamental dynamical models of a pair of RTPs with jamming interactions and provide a rigorous link between their discrete- and continuous-space descriptions. We demonstrate that as the lattice spacing vanishes, the discrete models converge to a continuous RTP model on the torus, described by a Piecewise Deterministic Markov Process (PDMP). This establishes that the invariant measures of the discrete models converge to that of the continuous model, which reveals finite mass at jamming configurations and exponential decay away from them. This indicates effective attraction, which is consistent with MIPS. Furthermore, we quantitatively explore the convergence towards the invariant measure. Such convergence study is critical for understanding and characterizing how MIPS emerges over time. Because RTP systems are non-reversible, usual methods may fail or are limited to qualitative results. Instead, we adopt a coupling approach to obtain more accurate, non-asymptotic bounds on mixing times. The findings thus provide deeper theoretical insights into the mixing times of these RTP systems, revealing the presence of both persistent and diffusive regimes.
- [64] arXiv:2411.13966 [pdf, html, other]
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Title: On comass and stable systolic inequalitiesComments: 11 pages. To appear in Differential Geometry and Its ApplicationsSubjects: Differential Geometry (math.DG)
We study the maximum ratio of the Euclidean norm to the comass norm of p-covectors in Euclidean n-space and improve the known upper bound found in the standard references by Whitney and Federer. We go on to prove stable systolic inequalities when the fundamental cohomology class of the manifold is a cup product of forms of lower degree.
- [65] arXiv:2411.13967 [pdf, other]
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Title: A description of and an upper bound on the set of bad primes in the study of the Casas-Alvero ConjectureDaniel Schaub (LAREMA), Mark Spivakovsky (IMT, LaSol)Comments: arXiv admin note: text overlap with arXiv:2307.05997Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
The Casas--Alvero conjecture predicts that every univariate polynomial over a field of characteristic zero having a common factor with each of its derivatives $H_i(f)$ is a power of a linear polynomial. One approach to proving the conjecture is to first prove it for polynomials of some small degree $n$, compile a list of bad primes for that degree (namely, those primes $p$ for which the conjecture fails in degree $n$ and characteristic $p$) and then deduce the conjecture for all degrees of the form $np^\ell$, $\ell\in \mathbb{N}$, where $p$ is a good prime for $n$. In this paper we give an explicit description of the set of bad primes in any given degree $n$. In particular, we show that if the conjecture holds in degree $n$ then the bad primes for $n$ are bounded above by $\binom{\frac{n^2-n}2}{n-2}!\prod\limits_{i=1}^{n-1}\binom{i+n-2}{n-2}^{\binom{d-i+n-2}{n-2}}$.
- [66] arXiv:2411.13969 [pdf, html, other]
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Title: Continuum of coupled Wasserstein gradient flowsComments: 36 pages, 8 figuresSubjects: Analysis of PDEs (math.AP)
We study a system of drift-diffusion PDEs for a potentially infinite number of incompressible phases, subject to a joint pointwise volume constraint. Our analysis is based on the interpretation as a collection of coupled Wasserstein gradient flows or, equivalently, as a gradient flow in the space of couplings under a `fibered' Wasserstein distance. We prove existence of weak solutions, long-time asymptotics, and stability with respect to the mass distribution of the phases, including the discrete to continuous limit. A key step is to establish convergence of the product of pressure gradient and density, jointly over the infinite number of phases. The underlying energy functional is the objective of entropy regularized optimal transport, which allows us to interpret the model as the relaxation of the classical Angenent-Haker-Tannenbaum (AHT) scheme to the entropic setting. However, in contrast to the AHT scheme's lack of convergence guarantees, the relaxed scheme is unconditionally convergent. We conclude with numerical illustrations of the main results.
- [67] arXiv:2411.13972 [pdf, other]
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Title: The stochastic Bessel operator at high temperaturesHugo Magaldi (CEREMADE)Subjects: Probability (math.PR)
We know from Ram{í}rez and Rider that the hard edge of the spectrum of the Beta-Laguerre ensemble converges, in the high-dimensional limit, to the bottom of the spectrum of the stochastic Bessel operator. Using stochastic analysis techniques, we show that, in the high temperatures limit, the rescaled eigenvalues point process of the stochastic Bessel operator converges to a limiting point process characterized with coupled stochastic dierential equations.
- [68] arXiv:2411.13974 [pdf, other]
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Title: Distributional regression: CRPS-error bounds for model fitting, model selection and convex aggregationClément Dombry (LMB), Ahmed Zaoui (LMB)Journal-ref: 38th Conference on Neural Information Processing Systems (NeurIPS 2024), Dec 2024, Vancoucer, CanadaSubjects: Statistics Theory (math.ST)
Distributional regression aims at estimating the conditional distribution of a targetvariable given explanatory co-variates. It is a crucial tool for forecasting whena precise uncertainty quantification is required. A popular methodology consistsin fitting a parametric model via empirical risk minimization where the risk ismeasured by the Continuous Rank Probability Score (CRPS). For independentand identically distributed observations, we provide a concentration result for theestimation error and an upper bound for its expectation. Furthermore, we considermodel selection performed by minimization of the validation error and provide aconcentration bound for the regret. A similar result is proved for convex aggregationof models. Finally, we show that our results may be applied to various models suchas Ensemble Model Output Statistics (EMOS), distributional regression networks,distributional nearest neighbors or distributional random forests and we illustrateour findings on two data sets (QSAR aquatic toxicity and Airfoil self-noise).
- [69] arXiv:2411.13976 [pdf, html, other]
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Title: Blow-up result for a piezoelectric beams system with magnetic effectsComments: 13 pages, no figuresSubjects: Analysis of PDEs (math.AP)
The system under studying is for a piezoelectric beams system with magnetic effects, frictional dampings and source terms. We use the concavity method to study the competition of the dampings and the sources that leads to a blow-up result for solutions with negative initial energy.
- [70] arXiv:2411.13977 [pdf, html, other]
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Title: Long-range effects in asymptotic fields and angular momentum of classical field electrodynamicsComments: 44 pages; this is an old article (1995) which may be of interest in connection with more recent works (in particular, arXiv:2403.09234)Journal-ref: J. Math. Phys. 36 (1995) 4044-4086Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)
Asymptotic properties of classical field electrodynamics are considered. Special attention is paid to the long-range structure of the electromagnetic field. It is shown that conserved Poincare quantities may be expressed in terms of the asymptotic fields. Long-range variables are shown to be responsible for an angular momentum contribution which mixes Coulomb and infrared free field characteristics; otherwise angular momentum and energy-momentum separate into electromagnetic and matter fields contributions.
- [71] arXiv:2411.13980 [pdf, other]
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Title: Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spacesIgor Honoré (UCBL, ICJ, PSPM)Subjects: Analysis of PDEs (math.AP)
We obtain new controls for the Leray solutions $u$ of the incompressible Navier-Stokes equation in $\mathbb{R}^3$. Specifically, we estimate $u$, $\nabla u$, and $\nabla^2 u$ in suitable Lebesgue spaces $L^{\tilde r}_TL^r$, $r <+ \infty$ with some constraints on $\tilde r>0$. Our method is based on a Duhamel formula around a perturbed heat equation, allowing to thoroughly exploit the well-known energy estimates which balances the potential singularities. We also perform a new Bihari-LaSalle argument in this context. Eventually, we adapt our strategy to prove that $\sup_{t \in [0,T]} \int_{0}^t (t-s)^{-\theta} \|\nabla^k u(s,\cdot)\|_{L^r} ds<+ \infty$, for all $\theta< \frac{3-kr}{2r}$, $k \in [0,2]$, and $1<r<\frac{3}{k}$.
- [72] arXiv:2411.13999 [pdf, html, other]
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Title: Accelerated zero-order SGD under high-order smoothness and overparameterized regimeComments: 10 pages, 1 figureSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
We present a novel gradient-free algorithm to solve a convex stochastic optimization problem, such as those encountered in medicine, physics, and machine learning (e.g., adversarial multi-armed bandit problem), where the objective function can only be computed through numerical simulation, either as the result of a real experiment or as feedback given by the function evaluations from an adversary. Thus we suppose that only a black-box access to the function values of the objective is available, possibly corrupted by adversarial noise: deterministic or stochastic. The noisy setup can arise naturally from modeling randomness within a simulation or by computer discretization, or when exact values of function are forbidden due to privacy issues, or when solving non-convex problems as convex ones with an inexact function oracle. By exploiting higher-order smoothness, fulfilled, e.g., in logistic regression, we improve the performance of zero-order methods developed under the assumption of classical smoothness (or having a Lipschitz gradient). The proposed algorithm enjoys optimal oracle complexity and is designed under an overparameterization setup, i.e., when the number of model parameters is much larger than the size of the training dataset. Overparametrized models fit to the training data perfectly while also having good generalization and outperforming underparameterized models on unseen data. We provide convergence guarantees for the proposed algorithm under both types of noise. Moreover, we estimate the maximum permissible adversarial noise level that maintains the desired accuracy in the Euclidean setup, and then we extend our results to a non-Euclidean setup. Our theoretical results are verified on the logistic regression problem.
- [73] arXiv:2411.14015 [pdf, html, other]
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Title: On the geometry of isomonodromic deformations on the torus and the elliptic Calogero-Moser systemComments: 25 pagesSubjects: Mathematical Physics (math-ph); Symplectic Geometry (math.SG); Exactly Solvable and Integrable Systems (nlin.SI)
We consider isomonodromic deformations of connections with a simple pole on the torus, motivated by the elliptic version of the sixth Painlevé equation. We establish an extended symmetry, complementing known results. The Calogero-Moser system in its elliptic version is shown to fit nicely in the geometric framework, the extended symplectic two-form is introduced and shown to be closed.
- [74] arXiv:2411.14020 [pdf, html, other]
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Title: Pointwise convergence of solutions of the Schr\"odinger equation along general curves with radial initial data on Damek-Ricci spacesComments: arXiv admin note: substantial text overlap with arXiv:2411.04084Subjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP)
In this article, we consider the Schrödinger equation corresponding to the Laplace-Beltrami operator with radial initial data on Damek-Ricci spaces and study the Carleson's problem of pointwise convergence of the solution to its initial data along general curves that satisfy certain Hölder conditions and bilipschitz conditions in the distance from the identity. We obtain a sufficient condition on the regularity of the initial data for the above pointwise convergence to hold, which is sharp upto the endpoint. Certain Euclidean analogues are also obtained.
- [75] arXiv:2411.14021 [pdf, html, other]
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Title: Orientation Determination of Cryo-EM Images Using Block Stochastic Riemannian Subgradient MethodsSubjects: Optimization and Control (math.OC)
The determination of molecular orientations is crucial for the three-dimensional reconstruction of Cryo-EM images. Traditionally addressed using the common-line method, this challenge is reformulated as a self-consistency error minimization problem constrained to rotation groups. In this paper, we consider the least-squared deviation (LUD) formulation and employ a Riemannian subgradient method to effectively solve the orientation determination problem. To enhance computational efficiency, a block stochastic version of the method is proposed, and its convergence properties are rigorously established. Extensive numerical evaluations reveal that our method not only achieves accuracy comparable to that of state-of-the-art methods but also delivers an average 20-fold speedup. Additionally, we implement a modified formulation and algorithm specifically designed to address scenarios characterized by very low SNR.
- [76] arXiv:2411.14024 [pdf, html, other]
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Title: Classification of traveling wave solutions of the modified Zakharov--Kuznetsov equationSubjects: Analysis of PDEs (math.AP)
The $\mathcal{C}^{\infty}$-structure-based method of integration of distributions of vector fields is used to classify all the traveling wave solutions of the modified Zakharov--Kuznetsov equation. This work unifies and generalizes the particular results obtained in the recent literature by using specific ansatz-based methods.
- [77] arXiv:2411.14027 [pdf, html, other]
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Title: An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicityComments: 38 pagesSubjects: Operator Algebras (math.OA)
We generalize the Li-Yang notion of self-similar $k$-graph $(G,\Lambda)$ and its $C^*$-algebra $\mathcal{O}_{G,\Lambda}$ to any finitely aligned $k$-graph $\Lambda$. We then introduce an inverse semigroup model for $\mathcal{O}_{G,\Lambda}$ and analyze its tight groupoid and $C^*$-algebra via inverse semigroup methods.
- [78] arXiv:2411.14028 [pdf, html, other]
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Title: Global well-posedness in a Hartree-Fock model for grapheneComments: 21 pagesSubjects: Mathematical Physics (math-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Analysis of PDEs (math.AP)
Graphene is a monolayer graphitic film where electrons behave like two-dimensional Dirac fermions without mass. Its study has attracted a wide interest in the domain of condensed matter physics. In particular, it represents an ideal system to test the comprehension of 2D massless relativistic particles in a laboratory, the Fermi velocity being $300$ times smaller than the speed of light. In this work, we present a global well-posedness result for graphene in the Hartree-Fock approximation. The model allows to describe the time evolution of graphene in the presence of external electric fields, such as those induced by local defects in the monolayer of carbon atoms. Our approach is based on a well established non-perturbative framework originating from the study of three-dimensional quantum electrodynamics.
- [79] arXiv:2411.14030 [pdf, html, other]
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Title: Performance Analysis of STAR-RIS-Assisted Cell-Free Massive MIMO Systems with Electromagnetic Interference and Phase ErrorsComments: 13 pages, 6 figures. This work has been submitted to the IEEE for possible publicationSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Simultaneous Transmitting and Reflecting Reconfigurable Intelligent Surfaces (STAR-RISs) are being explored for the next generation of sixth-generation (6G) networks. A promising configuration for their deployment is within cell-free massive multiple-input multiple-output (MIMO) systems. However, despite the advantages that STAR-RISs could bring, challenges such as electromagnetic interference (EMI) and phase errors may lead to significant performance degradation. In this paper, we investigate the impact of EMI and phase errors on STAR-RIS-assisted cell-free massive MIMO systems and propose techniques to mitigate these effects. We introduce a novel projected gradient descent (GD) algorithm for STAR-RIS coefficient matrix design by minimizing the local channel estimation normalised mean square error. We also derive the closed-form expressions of the uplink and downlink spectral efficiency (SE) to analyze system performance with EMI and phase errors, in which fractional power control methods are applied for performance improvement. The results reveal that the projected GD algorithm can effectively tackle EMI and phase errors to improve estimation accuracy and compensate for performance degradation with nearly $10\%\sim20\%$ SE improvement. Moreover, increasing access points (APs), antennas per AP, and STAR-RIS elements can also improve SE performance. Applying STAR-RIS in the proposed system achieves a larger $25\%$-likely SE than conventional RISs. However, the advantages of employing more STAR-RIS elements are reduced when EMI is severe.
- [80] arXiv:2411.14031 [pdf, html, other]
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Title: Numerical null controllability of parabolic PDEs using Lagrangian methodsSubjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
In this paper, we study several theoretical and numerical questions concerning the null controllability problems for linear parabolic equations and systems for several dimensions. The control is distributed and acts on a small subset of the domain. The main goal is to compute numerically a control that drives a numerical approximation of the state from prescribed initial data exactly to zero. We introduce a methodology for solving numerical controllability problems that is new in some sense. The main idea is to apply classical Lagrangian and Augmented Lagrangian techniques to suitable constrained extremal formulations that involve unbounded weights in time that make global Carleman inequalities possible. The theoretical results are validated by satisfactory numerical experiments for spatially 2D and 3D problems.
- [81] arXiv:2411.14036 [pdf, html, other]
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Title: Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjectureComments: 23 pages, 4 figuresSubjects: Combinatorics (math.CO); Algebraic Topology (math.AT)
We prove that a Murai sphere is flag if and only if it is a nerve complex of a flag nestohedron and classify all the polytopes arising in this way. Our classification implies that flag Murai spheres satisfy the Nevo-Petersen conjecture on $\gamma$-vectors of flag homology spheres. We continue by showing that a Bier sphere is minimally non-Golod if and only if it is a nerve complex of a truncation polytope different from a simplex and classify all the polytopes arising in this way. Finally, the notion of a cubical Bier sphere is introduced based on the polyhedral product construction, and we study combinatorial and geometrical properties of these cubical complexes.
- [82] arXiv:2411.14057 [pdf, html, other]
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Title: Characterizing and Transforming DAGs within the I-LCA FrameworkComments: 9 pages, 3 figures. arXiv admin note: text overlap with arXiv:2411.00708Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
We explore the connections between clusters and least common ancestors (LCAs) in directed acyclic graphs (DAGs), focusing on DAGs with unique LCAs for specific subsets of their leaves. These DAGs are important in modeling phylogenetic networks that account for reticulate processes or horizontal gene transfer. Phylogenetic DAGs inferred from genomic data are often complex, obscuring evolutionary insights, especially when vertices lack support as LCAs for any subset of taxa. To address this, we focus on $I$-lca-relevant DAGs, where each vertex serves as the unique LCA for a subset $A$ of leaves of specific size $|A|\in I$. We characterize DAGs with the so-called $I$-lca-property and establish their close relationship to pre-$I$-ary and $I$-ary set systems. Moreover, we build upon recently established results that use a simple operator $\ominus$, enabling the transformation of arbitrary DAGs into $I$-lca-relevant DAGs. This process reduces unnecessary complexity while preserving the key structural properties of the original DAG. The set $C_G$ consists of all clusters in a DAG $G$, where clusters correspond to the descendant leaves of vertices. While in some cases $C_H = C_G$ when transforming $G$ into an $I$-lca-relevant DAG $H$, it often happens that certain clusters in $C_G$ do not appear as clusters in $H$. To understand this phenomenon in detail, we characterize the subset of clusters in $C_G$ that remain in $H$ for DAGs $G$ with the $I$-lca-property. Furthermore, we show that the set $W$ of vertices required to transform $G$ into $H = G \ominus W$ is uniquely determined for such DAGs. This, in turn, allows us to show that the transformed DAG $H$ is always a tree or a galled-tree whenever $C_G$ represents the clustering system of a tree or galled-tree and $G$ has the $I$-lca-property. In the latter case $C_H = C_G$ always holds.
- [83] arXiv:2411.14059 [pdf, html, other]
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Title: Bounded cohomology of diffeomorphism groups of higher dimensional spheresComments: Comments are welcomed!Subjects: Geometric Topology (math.GT)
In this paper we prove the vanishing of the bounded cohomology of $\text{Diff}^r_+(S^n)$ with real coefficients when $n\geq 4$ and $1\leq r\leq \infty$. This answers the question raised in \cite{FNS24} for $\geq 4$ dimensional spheres.
- [84] arXiv:2411.14066 [pdf, html, other]
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Title: Monochromatic configurations induced from the set of sums of two squaresComments: 12 pagesSubjects: Combinatorics (math.CO)
The set of sums of two squares plays an important role in the elementary number theory. Di Nasso investigated several infinite monochromatic patterns in integers considering operations induced from affine maps and asked whether one can find different sets of infinite monochromatic configurations in natural numbers with the structure induced from the set of sums of two squares. This article investigates Nasso's question and along the way proves several classical Ramsey-type theorems in this new setting.
- [85] arXiv:2411.14080 [pdf, html, other]
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Title: Tensors in algebraic statisticsComments: Overview paperSubjects: Statistics Theory (math.ST); Algebraic Geometry (math.AG); History and Overview (math.HO)
Tensors are ubiquitous in statistics and data analysis. The central object that links data science to tensor theory and algebra is that of a model with latent variables. We provide an overview of tensor theory, with a particular emphasis on its applications in algebraic statistics. This high-level treatment is supported by numerous examples to illustrate key concepts. Additionally, an extensive literature review is included to guide readers toward more detailed studies on the subject.
- [86] arXiv:2411.14081 [pdf, other]
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Title: Prandtl Equations and Related Boundary Layer EquationsSubjects: Analysis of PDEs (math.AP)
This book aims to present some recent results on Prandtl equations and MHD boundary layer equations.
This book is essentially divided into two parts. Chapter 1 as the first part systematically surveys the results till 2020 on Prandtl equations and MHD boundary layer equations. Chapter 2 to 6 are the main part of the book, which presents the local and the global well-posedness of solutions to the Prandtl equations and MHD boundary layer equations. In detail, Chapter 2 is concerned with global well-posedness of solutions to the 2D Prandtl-Hartmann equations in an analytic framework. Chapter 3 investigates the local existence of solutions to the 2D Prandtl equations in a weighted Sobolev space. Chapter 4 studies the local well-posedness of solutions to the 2D mixed Prandtl equations in a Sobolev space without monotonicity and lower bound. Chapter 5 is concerned with global existence of solutions to the 2D magnetic Prandtl equations in the Prandtl-Hartmann regime. Chapter 6 proves the local existence of solutions to the 3D Prandtl equations with a special structure.
Mathematicians and physicists who are interested in fluid dynamics will find this book helpful. - [87] arXiv:2411.14083 [pdf, html, other]
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Title: Existence and Non-existence for Exchange-Driven Growth ModelSubjects: Analysis of PDEs (math.AP)
The exchange-driven growth (EDG) model describes the evolution of clusters through the exchange of single monomers between pairs of interacting clusters. The dynamics of this process are primarily influenced by the interaction kernel $K_{j,k}$. In this paper, the global existence of classical solutions to the EDG equations is established for non-negative, symmetric interaction kernels satisfying $K_{j,k} \leq C(j^{\mu}k^{\nu} + j^{\nu}k^{\mu}) $, where $\mu, \nu \leq 2$, $\mu + \nu \leq 3$, and $C>0$, with a broader class of initial data. This result extends the previous existence results obtained by Esenturk [10], Schlichting [23], and Eichenberg \& Schlichting [7].
Furthermore, the local existence of classical solutions to the EDG equations is demonstrated for symmetric interaction kernels that satisfy $K_{j,k} \leq C j^{2} k^{2}$ with $C > 0$, considering a broader class of initial data. In the intermediate regime $3 < \mu + \nu \leq 4$, the occurrence of finite-time gelation is established for symmetric interaction kernels satisfying $C_{1}\left(j^{2}k^{\alpha}+j^{\alpha}k^{2}\right)\leq K_{j,k}\leq Cj^{2}k^{2}$, where $1 < \alpha \leq 2$, $C>0$, and $C_{1} > 0$, as conjectured in [10]. In this case, the non-existence of the global solutions is ensured by the occurrence of finite-time gelation. Finally, the occurrence of instantaneous gelation of the solutions to EDG equations for symmetric interaction kernels satisfying $K_{j,k}\geq C\left(j^{\beta}+k^{\beta}\right)$ ($\beta>2, C>0)$ is shown, which also implies the non-existence of solutions in this case. - [88] arXiv:2411.14084 [pdf, html, other]
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Title: Neural numerical homogenization based on Deep Ritz correctionsSubjects: Numerical Analysis (math.NA)
Numerical homogenization methods aim at providing appropriate coarse-scale approximations of solutions to (elliptic) partial differential equations that involve highly oscillatory coefficients. The localized orthogonal decomposition (LOD) method is an effective way of dealing with such coefficients, especially if they are non-periodic and non-smooth. It modifies classical finite element basis functions by suitable fine-scale corrections. In this paper, we make use of the structure of the LOD method, but we propose to calculate the corrections based on a Deep Ritz approach involving a parametrization of the coefficients to tackle temporal variations or uncertainties. Numerical examples for a parabolic model problem are presented to assess the performance of the approach.
- [89] arXiv:2411.14087 [pdf, html, other]
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Title: Determining the covering radius of all generalized Zetterberg codes in odd characteristicSubjects: Information Theory (cs.IT)
For an integer $s\ge 1$, let $\mathcal{C}_s(q_0)$ be the generalized Zetterberg code of length $q_0^s+1$ over the finite field $\F_{q_0}$ of odd characteristic. Recently, Shi, Helleseth, and Özbudak (IEEE Trans. Inf. Theory 69(11): 7025-7048, 2023) determined the covering radius of $\mathcal{C}_s(q_0)$ for $q_0^s \not \equiv 7 \pmod{8}$, and left the remaining case as an open problem. In this paper, we develop a general technique involving arithmetic of finite fields and algebraic curves over finite fields to determine the covering radius of all generalized Zetterberg codes for $q_0^s \equiv 7 \pmod{8}$, which therefore solves this open problem. We also introduce the concept of twisted half generalized Zetterberg codes of length $\frac{q_0^s+1}{2}$, and show the same results hold for them. As a result, we obtain some quasi-perfect codes.
- [90] arXiv:2411.14088 [pdf, html, other]
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Title: Channel Customization for Low-Complexity CSI Acquisition in Multi-RIS-Assisted MIMO SystemsComments: Accepted by IEEE JSAC special issue on Next Generation Advanced Transceiver TechnologiesSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
The deployment of multiple reconfigurable intelligent surfaces (RISs) enhances the propagation environment by improving channel quality, but it also complicates channel estimation. Following the conventional wireless communication system design, which involves full channel state information (CSI) acquisition followed by RIS configuration, can reduce transmission efficiency due to substantial pilot overhead and computational complexity. This study introduces an innovative approach that integrates CSI acquisition and RIS configuration, leveraging the channel-altering capabilities of the RIS to reduce both the overhead and complexity of CSI acquisition. The focus is on multi-RIS-assisted systems, featuring both direct and reflected propagation paths. By applying a fast-varying reflection sequence during RIS configuration for channel training, the complex problem of channel estimation is decomposed into simpler, independent tasks. These fast-varying reflections effectively isolate transmit signals from different paths, streamlining the CSI acquisition process for both uplink and downlink communications with reduced complexity. In uplink scenarios, a positioning-based algorithm derives partial CSI, informing the adjustment of RIS parameters to create a sparse reflection channel, enabling precise reconstruction of the uplink channel. Downlink communication benefits from this strategically tailored reflection channel, allowing effective CSI acquisition with fewer pilot signals. Simulation results highlight the proposed methodology's ability to accurately reconstruct the reflection channel with minimal impact on the normalized mean square error while simultaneously enhancing spectral efficiency.
- [91] arXiv:2411.14090 [pdf, html, other]
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Title: Exponential Ergodicity in $\W_1$ for SDEs with Distribution Dependent Noise and Partially Dissipative DriftsComments: 18 pagesSubjects: Probability (math.PR)
We establish a general result on exponential ergodicity via $L^1$-Wasserstein distance for McKean--Vlasov SDEs. The result is successfully applied in non-degenerate and multiplicative Brownian motion cases and degenerate second order systems, where the diffusion coefficients are allowed to be distribution dependent and the drifts are only assumed to be partially dissipative. Our approach overcomes the essential difficulty caused by the distribution dependent diffusion coefficient and our results considerably improve existing ones in which the diffusion coefficient is distribution-free.
- [92] arXiv:2411.14093 [pdf, html, other]
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Title: Desingularization of bounded-rank tensor setsComments: 41 pages, 10 figures, 1 tableSubjects: Optimization and Control (math.OC); Algebraic Geometry (math.AG); Numerical Analysis (math.NA)
Low-rank tensors appear to be prosperous in many applications. However, the sets of bounded-rank tensors are non-smooth and non-convex algebraic varieties, rendering the low-rank optimization problems to be challenging. To this end, we delve into the geometry of bounded-rank tensor sets, including Tucker and tensor train formats. We propose a desingularization approach for bounded-rank tensor sets by introducing slack variables, resulting in a low-dimensional smooth manifold embedded in a higher-dimensional space while preserving the structure of low-rank tensor formats. Subsequently, optimization on tensor varieties can be reformulated to optimization on smooth manifolds, where the methods and convergence are well explored. We reveal the relationship between the landscape of optimization on varieties and that of optimization on manifolds. Numerical experiments on tensor completion illustrate that the proposed methods are in favor of others under different rank parameters.
- [93] arXiv:2411.14097 [pdf, html, other]
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Title: On Larsen's conjecture on the ranks of Elliptic CurvesSubjects: Number Theory (math.NT)
Let $E$ be an elliptic curve over $\mathbb{Q}$ and $G=\langle\sigma_1, \dots, \sigma_n\rangle$ be a finitely generated subgroup of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$. Larsen's conjecture claims that the rank of the Mordell-Weil group $E(\overline{\mathbb{Q}}^G)$ is infinite where ${\overline{\mathbb Q}}^G$ is the $G$-fixed sub-field of $\overline{\mathbb Q}$. In this paper we prove the conjecture for the case in which $\sigma_i$ for each $i=1, \dots, n$ is an element of some infinite families of elements of $\operatorname{Gal}(\overline{\mathbb{Q}}/ \mathbb{Q})$.
- [94] arXiv:2411.14102 [pdf, html, other]
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Title: Vertices of the monotone path polytopes of hypersimpliciesComments: 25 pages, 14 figuresSubjects: Combinatorics (math.CO)
The monotone path polytope of a polytope $P$ encapsulates the combinatorial behavior of the shadow vertex rule (a pivot rule used in linear programming) on $P$. On the other hand, computing monotone path polytopes is the entry door to the larger subject of fiber polytopes, for which explicitly computing examples remains a challenge.
Monotone path polytopes of cubes and simplices have been known since the seminal article of Billera and Strumfels 1992. We (partially) extend these results to hypersimplices by linking this problem to the combinatorics of lattice paths. Indeed, we give a combinatorial model to describe the vertices of the monotone path polytope of the hypersimplex $\Delta(n, 2)$ (for any generic direction). With this model, we give a precise count of these vertices, and furthermore count the number of coherent monotone paths on $\Delta(n, 2)$ according to their lengths.
We prove that some of the results obtained also hold for hypersimplices $\Delta(n, k)$ for $k\geq 2$. - [95] arXiv:2411.14105 [pdf, html, other]
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Title: Simultaneous replica-symmetry breaking for vector spin glassesComments: 35 pagesSubjects: Probability (math.PR); Disordered Systems and Neural Networks (cond-mat.dis-nn)
We consider mean-field vector spin glasses with possibly non-convex interactions. Up to a small perturbation of the parameters defining the model, the asymptotic behavior of the Gibbs measure is described in terms of a critical point of an explicit functional. In this paper, we study some properties of these critical points. Under modest assumptions ensuring that different types of spins interact, we show that the replica-symmetry-breaking structures of the different types of spins are in one-to-one correspondence with one another. For instance, if some type of spins displays one level of replica-symmetry breaking, then so do all the other types of spins. This extends the recent results of [Electronic Journal of Probability, 27:1-75, 2022] and [Comm. Math. Phys., 394(3):1101-1152, 2022] that were obtained in the case of multi-species spherical spin glasses with convex interactions.
- [96] arXiv:2411.14111 [pdf, other]
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Title: Stochastic processes on preferential attachment modelsComments: Doctoral thesisSubjects: Probability (math.PR)
In real life, networks are dynamic in nature; they grow over time and often exhibit power-law degree sequences. To model the evolving structure of the internet, Barabási and Albert introduced a simple dynamic model with a power-law degree distribution. This model has since been generalised, leading to a broad class of affine preferential attachment models, where each new vertex connects to existing vertices with a probability proportional to the current degree of the vertex. While numerous studies have explored the global and local properties of these random graphs, their dynamic nature and the dependencies in edge-connection probabilities have posed significant analytical challenges. The first part of this thesis identifies the local limit of preferential attachment models in considerable generality. The second part focuses on stochastic processes on preferential attachment models, introducing an additional layer of randomness to the random graphs. Examples of such processes include bond and site percolation, random walks, the Ising and Potts models, and Gaussian processes on random graphs. In this thesis, we specifically examine percolation and the Ising model, exploring these processes using the local limit identified earlier.
- [97] arXiv:2411.14112 [pdf, html, other]
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Title: Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space FormsComments: 14 pages, any comments are welcomeSubjects: Differential Geometry (math.DG)
For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $\alpha(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a topological sphere for the maximal bound $\alpha(n,[\frac{n}{2}],H,c)$, or has up to $k$-th homology groups vanishing. This gives an almost complete (except for the differentiable sphere theorem) characterization of compact submanifolds with pinched Ricci curvature, generalizing celebrated rigidity results obtained by Ejiri, Xu-Tian, Xu-Gu, Xu-Leng-Gu, Vlachos, Dajczer-Vlachos.
- [98] arXiv:2411.14123 [pdf, html, other]
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Title: Multi-terminal Strong Coordination subject to Secrecy ConstraintsComments: Extended version of ISIT 2024 paperSubjects: Information Theory (cs.IT)
A fundamental problem in decentralized networked systems is to coordinate actions of different agents so that they reach a state of agreement. In such applications, it is additionally desirable that the actions at various nodes may not be anticipated by malicious eavesdroppers. Motivated by this, we investigate the problem of secure multi-terminal strong coordination aided by a multiple-access wiretap channel. In this setup, independent and identically distributed copies of correlated sources are observed by two transmitters who encode the channel inputs to the MAC-WT. The legitimate receiver observing the channel output and side information correlated with the sources must produce approximately i.i.d. copies of an output variable jointly distributed with the sources. Furthermore, we demand that an external eavesdropper learns essentially nothin g about the sources and the simulated output sequence by observing its own MAC-WT output. This setting is aided by the presence of independent pairwise shared randomness between each encoder and the legitimate decoder, that is unavailable to the eavesdropper. We derive an achievable rate region based on a combination of coordination coding and wiretap coding, along with an outer bound. The inner bound is shown to be tight and a complete characterization is derived for the special case when the sources are conditionally independent given the decoder side information and the legitimate channel is composed of deterministic links. Further, we also analyze a more general scenario with possible encoder cooperation, where one of the encoders can non-causally crib from the other encoders input, for which an achievable rate region is proposed. We then explicitly compute the rate regions for an example both with and without cribbing between the encoders, and demonstrate that cribbing strictly improves upon the achievable rate region.
- [99] arXiv:2411.14124 [pdf, html, other]
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Title: Quadrature domains packingSubjects: Spectral Theory (math.SP); Complex Variables (math.CV)
Given a finite family of compact subsets of the complex plane we propose a certificate of mutual non-overlapping with respect to area measure. The criterion is stated as a couple of positivity conditions imposed on a four argument analytic/anti-analytic kernel defined in a neighborhood of infinity. In case the compact sets are closures of quadrature domains the respective kernel is rational, enabling an effective matrix analysis algorithm for the non-overlapping decision. The simplest situation of two disks is presented in detail from a matrix model perspective as well as from a Riemann surface potential theoretic interpretation.
- [100] arXiv:2411.14129 [pdf, html, other]
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Title: On averaged self-distances in finite dimensional Banach spacesComments: 3 pagesSubjects: Functional Analysis (math.FA); Metric Geometry (math.MG)
Assume that $\mathfrak A$ is a real Banach space of finite dimension $n\geq2$. Consider any Borel probability measure $\nu$ supported on the unit ball $K$ of $\mathfrak A$. We show that \[\Delta(\nu)=\int_{x \in K}\int_{ y\in K}|x-y|_{\mathfrak A} \,\,\,\nu(x)\,\nu(y)\leq 2(1-2^{-n}f(n)),\] where $f:\mathbb N\setminus \{0,1\}\rightarrow (0,1]$ is a concrete universal function such that $f(n)\sim \frac{2}{\mathrm e n^2\log n}$. It is hoped that in the estimate`$f(n)$' can be replaced by `$1$'.
- [101] arXiv:2411.14132 [pdf, html, other]
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Title: Transients versus network interactions give rise to multistability through trapping mechanismComments: Submitted to ChaosSubjects: Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)
In networked systems, the interplay between the dynamics of individual subsystems and their network interactions has been found to generate multistability in various contexts. Despite its ubiquity, the specific mechanisms and ingredients that give rise to multistability from such interplay remain poorly understood. In a network of coupled excitable units, we show that this interplay generating multistability occurs through a competition between the units' transient dynamics and their coupling. Specifically, the diffusive coupling between the units manages to reinject them in the excitability region of their individual state space and effectively trap them there. We show that this trapping mechanism leads to the coexistence of multiple types of oscillations: periodic, quasiperiodic, and even chaotic, although the units separately do not oscillate. Interestingly, we show that the attractors emerge through different types of bifurcations - in particular, the periodic attractors emerge through either saddle-node of limit cycles bifurcations or homoclinic bifurcations - but in all cases the reinjection mechanism is present.
- [102] arXiv:2411.14138 [pdf, html, other]
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Title: Sharp Thresholds for Factors in Random GraphsComments: 17 pagesSubjects: Combinatorics (math.CO); Probability (math.PR)
Let $F$ be a graph on $r$ vertices and let $G$ be a graph on $n$ vertices. Then an $F$-factor in $G$ is a subgraph of $G$ composed of $n/r$ vertex-disjoint copies of $F$, if $r$ divides $n$. In other words, an $F$-factor yields a partition of the $n$ vertices of $G$. The study of such $F$-factors in the Erdős-Rényi random graph dates back to Erdős himself. Decades later, in 2008, Johansson, Kahn and Vu established the thresholds for the existence of an $F$-factor for strictly 1-balanced $F$ -- up to the leading constant. The sharp thresholds, meaning the leading constants, were obtained only recently by Riordan and Heckel, but only for complete graphs $F=K_r$ and for so-called nice graphs. Their results rely on sophisticated couplings that utilize the recent, celebrated solution of Shamir's problem by Kahn.
We extend the couplings by Riordan and Heckel to any strictly 1-balanced $F$ and thereby obtain the sharp threshold for the existence of an $F$-factor. In particular, we confirm the thirty year old conjecture by Rucínski that this sharp threshold indeed coincides with the sharp threshold for the disappearance of the last vertices which are not contained in a copy of $F$. - [103] arXiv:2411.14139 [pdf, html, other]
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Title: On the Classification of the L\'evy-Leblond SpinorsComments: 8 pages; based on the L. M.'s talk at ISQS28, Prague, July 1-5, 2024; to appear in the ProceedingsSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph)
The first-order Lévy-Leblond differential equations (LLEs) are the non-relativistic analogous of the Dirac equation: they are the "square roots" of the Schrödinger equation in ($1+d$) dimensions and admit spinor solutions. In this paper we show how to extend to the Lévy-Leblond spinors the real/complex/quaternionic classification of the relativistic spinors (which leads to the notions of Dirac, Weyl, Majorana, Majorana-Weyl, Quaternionic spinors). Besides the free equations, we also consider the presence of potential terms. Applied to a conformal potential, the simplest $(1+1)$-dimensional LLE induces a new differential realization of the $osp(1|2)$ superalgebra in terms of differential operators depending on the time and space coordinates.
- [104] arXiv:2411.14141 [pdf, html, other]
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Title: Differentiable SVD based on Moore-Penrose Pseudoinverse for Inverse Imaging ProblemsComments: 11 pagesSubjects: Numerical Analysis (math.NA); Artificial Intelligence (cs.AI); Computer Vision and Pattern Recognition (cs.CV)
Low-rank regularization-based deep unrolling networks have achieved remarkable success in various inverse imaging problems (IIPs). However, the singular value decomposition (SVD) is non-differentiable when duplicated singular values occur, leading to severe numerical instability during training. In this paper, we propose a differentiable SVD based on the Moore-Penrose pseudoinverse to address this issue. To the best of our knowledge, this is the first work to provide a comprehensive analysis of the differentiability of the trivial SVD. Specifically, we show that the non-differentiability of SVD is essentially due to an underdetermined system of linear equations arising in the derivation process. We utilize the Moore-Penrose pseudoinverse to solve the system, thereby proposing a differentiable SVD. A numerical stability analysis in the context of IIPs is provided. Experimental results in color image compressed sensing and dynamic MRI reconstruction show that our proposed differentiable SVD can effectively address the numerical instability issue while ensuring computational precision. Code is available at this https URL.
- [105] arXiv:2411.14142 [pdf, html, other]
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Title: Some new module-theoretic characterizations of $S$-Noetherian rings and $S$-coherent ringsComments: arXiv admin note: substantial text overlap with arXiv:2108.06851Subjects: Commutative Algebra (math.AC)
Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we first introduce and study the notions of $S$-pure exact sequences and $S$-absolutely pure modules which extend the classical notions of pure exact sequences and absolutely pure modules. And then, we give some new characterizations of $S$-Noetherian rings and $S$-coherent rings in terms of $S$-absolutely pure modules.
- [106] arXiv:2411.14143 [pdf, html, other]
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Title: Volume preservation of Butcher series methods from the operad viewpointComments: 25 pages, comments are welcomeSubjects: Category Theory (math.CT); K-Theory and Homology (math.KT); Numerical Analysis (math.NA); Quantum Algebra (math.QA)
We study a coloured operad involving rooted trees and directed cycles of rooted trees that generalizes the operad of rooted trees of Chapoton and Livernet. We describe all the relations between the generators of a certain suboperad of that operad, and compute the Chevalley-Eilenberg homology of two naturally arising differential graded Lie algebras. This allows us to give short and conceptual new proofs of two important results on Butcher series methods of numerical solution of ODEs: absence of volume-preserving integration schemes and the acyclicity of the aromatic bicomplex, the key step in a complete classification of volume-preserving integration schemes using the so called aromatic Butcher series.
- [107] arXiv:2411.14145 [pdf, html, other]
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Title: The structure of sets with cube-avoiding sumsetsComments: 12 pagesSubjects: Combinatorics (math.CO); Number Theory (math.NT)
We prove that if $d \ge 2$ is an integer, $G$ is a finite abelian group, $Z_0$ is a subset of $G$ not contained in any strict coset in $G$, and $E_1,\dots,E_d$ are dense subsets of $G^n$ such that the sumset $E_1+\dots+E_d$ avoids $Z_0^n$ then $E_1, \dots, E_d$ essentially have bounded dimension. More precisely, they are almost entirely contained in sets $E_1' \times G^{I^c}, \dots, E_d' \times G^{I^c}$, where the size of $I \subset [n]$ is non-zero and independent of $n$, and $E_1',\dots,E_d'$ are subsets of $G^{I}$ such that the sumset $E_1'+\dots+E_d'$ avoids $Z_0^I$.
- [108] arXiv:2411.14151 [pdf, html, other]
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Title: Error Analysis of the Deep Mixed Residual Method for High-order Elliptic EquationsComments: 40 pages, none figuresSubjects: Numerical Analysis (math.NA)
This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM with two types of loss functions, referred to as first-order and second-order least squares systems. By providing boundedness and coercivity analysis, we leverage Céa's Lemma to decompose the total error into the approximation, generalization, and optimization errors. Utilizing the Barron space theory and Rademacher complexity, an a priori error is derived regarding the training samples and network size that are exempt from the curse of dimensionality. Our results reveal that MIM significantly reduces the regularity requirements for activation functions compared to the deep Ritz method, implying the effectiveness of MIM in solving high-order equations.
- [109] arXiv:2411.14156 [pdf, html, other]
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Title: Statistical Biharmonicity of Identity MapsComments: All comments are welcome!; 19 pagesSubjects: Differential Geometry (math.DG)
The tension field of the identity map from a statistical manifold to a Riemannian statistical manifold, which shares the same Riemannian metric, is the Tchevychev vector field multiplied by negative one. We derive a new class of statistical manifolds that satisfy the semi-equiaffine condition based on the statistical biharmonicity of the identity map. Furthermore, we determine the statistical structures of this class, when the pair of the manifold and the Riemannian metric is a simply connected complete Riemannian manifold of constant curvature.
- [110] arXiv:2411.14161 [pdf, other]
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Title: Complex line fields on almost-complex manifoldsComments: 42 pages, 6 figuresSubjects: Algebraic Topology (math.AT); Geometric Topology (math.GT)
We study linearly independent complex line fields on almost-complex manifolds, which is a topic of long-standing interest in differential topology and complex geometry. A necessary condition for the existence of such fields is the vanishing of appropriate virtual Chern classes. We prove that this condition is also sufficient for the existence of one, two, or three linearly independent complex line fields over certain manifolds. More generally, our results hold for a wider class of complex bundles over CW complexes.
- [111] arXiv:2411.14166 [pdf, other]
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Title: SPARKLE: A Unified Single-Loop Primal-Dual Framework for Decentralized Bilevel OptimizationComments: 73 pages, the Thirty-Eighth Annual Conference on Neural Information Processing Systems (2024)Subjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Machine Learning (stat.ML)
This paper studies decentralized bilevel optimization, in which multiple agents collaborate to solve problems involving nested optimization structures with neighborhood communications. Most existing literature primarily utilizes gradient tracking to mitigate the influence of data heterogeneity, without exploring other well-known heterogeneity-correction techniques such as EXTRA or Exact Diffusion. Additionally, these studies often employ identical decentralized strategies for both upper- and lower-level problems, neglecting to leverage distinct mechanisms across different levels. To address these limitations, this paper proposes SPARKLE, a unified Single-loop Primal-dual AlgoRithm frameworK for decentraLized bilEvel optimization. SPARKLE offers the flexibility to incorporate various heterogeneitycorrection strategies into the algorithm. Moreover, SPARKLE allows for different strategies to solve upper- and lower-level problems. We present a unified convergence analysis for SPARKLE, applicable to all its variants, with state-of-the-art convergence rates compared to existing decentralized bilevel algorithms. Our results further reveal that EXTRA and Exact Diffusion are more suitable for decentralized bilevel optimization, and using mixed strategies in bilevel algorithms brings more benefits than relying solely on gradient tracking.
- [112] arXiv:2411.14170 [pdf, html, other]
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Title: The Quantum Bruhat Graph for $\widehat{SL}_2$ and Double Affine Demazure ProductsComments: 25 pages. Comments welcomeSubjects: Representation Theory (math.RT); Quantum Algebra (math.QA)
We investigate the Demazure product in a double affine setting. Work by Muthiah and Puskás gives a conjectural way to define this in terms of the $q=0$ specialisation of these Hecke algebras. We instead take a different approach generalising work by Felix Schremmer, who gave an equivalent formula for the (single) affine Demazure product in terms of the quantum Bruhat graph. We focus on type $\widehat{SL}_2$, where we prove that the quantum Bruhat graph of this type satisfies some nice properties, which allows us to construct a well-defined associative Demazure product for the double affine Weyl semigroup $W_{\mathcal{T}}$ (for level greater than one). We give results regarding the Demazure product and Muthiah and Orr's length function for $W_{\mathcal{T}}$, and we verify that our proposal matches specific examples computed by Muthiah and Puskás using the Kac-Moody affine Hecke algebra
- [113] arXiv:2411.14171 [pdf, other]
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Title: A rigorous Peierls-Onsager effective dynamics for semimetals in long-range magnetic fieldsComments: 59 pages, 2 figuresSubjects: Mathematical Physics (math-ph)
We consider periodic (pseudo)differential {elliptic operators of Schrödinger type} perturbed by weak magnetic fields not vanishing at infinity, and extend our previous analysis in \cite{CIP,CHP-2,CHP-4} to the case {of a semimetal having a finite family of Bloch eigenvalues whose range may overlap with the other Bloch bands but remains isolated at each fixed quasi-momentum.} We do not make any assumption of triviality for the associated Bloch bundle. In this setting, we formulate a general form of the Peierls-Onsager substitution {via strongly localized tight-frames and magnetic matrices. We also} prove the existence of an approximate time evolution for initial states supported inside the range of the isolated Bloch family, with a precise error control.
- [114] arXiv:2411.14173 [pdf, html, other]
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Title: Nodal sets and continuity of eigenfunctions of Kre\u{\i}-Feller operatorsSubjects: Analysis of PDEs (math.AP)
Let $\mu$ be a compactly supported positive finite Borel measure on $\R^{d}$. Let $0<\lambda_{1}\leq\lambda_{2}\leq\ldots$ be eigenvalues of the Kre$\breve{ı}$n-Feller operator $\Delta_{\mu}$. We prove that, on a bounded domain, the nodal set of a continuous $\lambda_{n}$-eigenfunction of a Kre$\breve{ı}$n-Feller operator divides the domain into at least 2 and at most $n+r-1$ subdomains, where $r$ is the multiplicity of $\lambda_{n}$. This work generalizes the nodal set theorem of the classical Laplace operator to Kre$\breve{ı}$n-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Kre$\breve{ı}$n-Feller operator are continuous.
- [115] arXiv:2411.14175 [pdf, html, other]
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Title: Chebyshev polynomials in the complex plane and on the real lineSubjects: Complex Variables (math.CV); Classical Analysis and ODEs (math.CA)
We present a survey of key developments in the study of Chebyshev polynomials, first introduced by P. L. Chebyshev and later significantly expanded upon by G. Faber to the complex setting. Our primary focus is on their defining property: being the polynomial with a specified leading coefficient that minimizes the supremum norm on a given set.
- [116] arXiv:2411.14177 [pdf, html, other]
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Title: Invariant Sublinear ExpectationsComments: 12Subjects: Probability (math.PR)
We first give a decomposition for a $T$-invariant sublinear expectation $\mathbb{E}=\sup_{P\in\Theta}\mathrm{E}_P$, and show that each component $\mathbb{E}^{(d)}=\sup_{P\in\Theta^{(d)}}\mathrm{E}_P$ of the decomposition has a finite period $p_d\in\mathbb{N}$, i.e., \[\mathbb{E}^{(d)}\left[f-f\circ T^{p_d}\right]=0, \quad f\in\mathcal{H}.\] Then we prove that a continuous invariant sublinear expectation that is strongly ergodic has a finite period $p_{\mathbb{E}}$, and each component $\Theta^{(d)}$ of its periodic decomposition is the convex hull of a finite set of $T^{p_d}$-ergodic probabilities.
As an application of the characterization, we prove an ergodicity result which shows that the limit of the $p_{\mathbb{E}}$-step time means achieves the upper expectation. - [117] arXiv:2411.14178 [pdf, html, other]
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Title: Mathematical aspects of space-time horizontal ray methodSubjects: Mathematical Physics (math-ph)
The following development of the well-known "vertical modes and horizontal rays" approach for acoustic waves propagation in shallow water, introduced in different works, is studied. In this approach we study so-called space-time horizontal rays, constructed on the base of decomposition of the sound field, depending on time, over adiabatic vertical modes (solutions of the Sturm-Liouville problem). Using this technique we obtain different properties of signals, propagating in underwater waveguide, such as space-time caustics, and provide rather simple method for the prediction of the form of the signal and all its parameters (amplitude and frequency modulation, different front angles, etc.) at some point of observation.
- [118] arXiv:2411.14181 [pdf, html, other]
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Title: Average sizes of mixed character sumsComments: 14 pagesSubjects: Number Theory (math.NT); Probability (math.PR)
We prove that the average size of a mixed character sum $$\sum_{1\le n \le x} \chi(n) e(n\theta) w(n/x)$$ (for a suitable smooth function $w$) is on the order of $\sqrt{x}$ for all irrational real $\theta$ satisfying a weak Diophantine condition, where $\chi$ is drawn from the family of Dirichlet characters modulo a large prime $r$ and where $x\le r$. In contrast, it was proved by Harper that the average size is $o(\sqrt{x})$ for rational $\theta$. Certain quadratic Diophantine equations play a key role in the present paper.
- [119] arXiv:2411.14186 [pdf, html, other]
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Title: Harmonic maps to the circle with higher dimensional singular setComments: 38 pagesSubjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
In a closed, oriented ambient manifold $(M^n,g)$ we consider the problem of finding $\mathbb{S}^1$-valued harmonic maps with prescribed singular set. We show that the boundary of any oriented $(n-1)$-submanifold can be realised as the singular set of an $\mathbb{S}^1$-valued map, which is classically harmonic away from the singularity and distributionally harmonic across. If the singular set $\Gamma$ is also embedded and $C^{1,1}$, we consider three variational relaxations of the same problem and show that the energy of minimisers converges, after renormalisation, to the volume $\mathcal{H}^{n-2}(\Gamma)$ plus a lower-order "renormalised energy" -- common to all relaxations -- describing an energetic interaction between different components of the singular set.
- [120] arXiv:2411.14188 [pdf, html, other]
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Title: An exact way to verify whether n is a congruent number using Heegner pointsComments: 10 pages,0 figuresSubjects: Number Theory (math.NT)
We introduce the relationship between congruent numbers and elliptic curves, and compute the conductor of the elliptic curve $y^2 = x^3 - n^2 x$ associated with it. Furthermore, we prove that its $L$-series coefficient $a_m = 0$ when $m \equiv 3 \mod 4$.By using the invariants of the elliptic curve introduced above, we calculate Heegner points to quickly verify whether $n$ is a congruent number.
- [121] arXiv:2411.14195 [pdf, html, other]
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Title: On $k$-convex hullsComments: 7 pagesSubjects: Metric Geometry (math.MG)
For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam} Q_{k-1}(K)\geq R\cdot\text{diam} Q_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull of $K$. The purpose of this short note is to show that this result due to E. Kopecká is impossible to obtain if one additionally requires that all isometric images of $K$ satisfy the same inequality. To this end, we introduce the dual construction to the $k$-convex hull of $K$, which we call $k$-cross approximation.
- [122] arXiv:2411.14203 [pdf, html, other]
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Title: Piecewise quasiconformal dynamical systems of the unit circleComments: 38 pages, 9 figuresSubjects: Dynamical Systems (math.DS); Complex Variables (math.CV)
We study piecewise quasiconformal covering maps of the unit circle. We provide sufficient conditions so that a conjugacy between two such dynamical systems has a quasiconformal or David extension to the unit disk. Our main result generalizes the main result of arXiv:2010.11256, which deals with piecewise analytic maps. As applications, we provide a classification of piecewise quasiconformal maps of the circle up to quasisymmetric conjugacy, we prove a general conformal mating theorem for Blaschke products, and we study the quasiconformal geometry of parabolic basins.
- [123] arXiv:2411.14217 [pdf, html, other]
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Title: Regularity results for a class of mixed local and nonlocal singular problems involving distance functionSubjects: Analysis of PDEs (math.AP)
This article deals with the study of the following mixed local nonlocal singular quasilinear equation \begin{eqnarray*} \begin{split} -\Delta_pu+(-\Delta)_q^s u&=\frac{f(x)}{u^{\delta}}\text { in } \Omega, \\u&>0 \text{ in } \Omega,\\u&=0 \text { in }\mathbb{R}^n \backslash \Omega; \end{split} \end{eqnarray*} where \begin{equation*} (-\Delta )_q^s u(x)= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+sq}} d y, \end{equation*} $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with $C^2$ boundary, $1<q\leq p<\infty$, $s\in(0,1)$, $\delta>0$ and $f\in L^\infty_{\mathrm{loc}}(\Omega)$ is a non-negative function which behaves like $dist(x,\partial \Omega)^{-\beta}$, $\beta\geq 0$ near $\partial \Omega$. We prove the existence of a weak solution in $W_{\mathrm{loc}}^{1,p}(\Omega)$ and its behavior near $\partial \Omega$ which consequently gives the optimal Sobolev regularity. Moreover, we establish several Hölder and gradient Hölder regularity results for a more general class of quasilinear operators involving singular as well as regular nonlinearities. As an application, we obtain Hölder regularity of weak solutions to the singular problem up to the boundary, of course with different exponents depending on $\beta+\delta$. Further, we include uniqueness and non-existence results for weak solutions to the problem. We also include an a priori boundedness and Hölder regularity result for the singular equation with critical exponent perturbation.
- [124] arXiv:2411.14221 [pdf, html, other]
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Title: An additive application of the resonance methodComments: 12 pagesSubjects: Number Theory (math.NT)
We improve upon an Omega result due to Soundararajan with respect to general trigonometric polynomials having positive Fourier coefficients. Instead of Dirichlet's approximation theorem we employ the resonance method and this leads to better extreme results in lattice point problems such as Dirichlet's divisor problem and Gauss' circle problem. Moreover, the present approach shows that the resonance method can also be viewed as an additive device, which has been used in multiplicative problems so far. Its extension to trigonometric polynomials with complex coefficients is also discussed and its connection to Bohr and Jessen's proof of Kronecker's theorem is highlighted.
- [125] arXiv:2411.14226 [pdf, html, other]
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Title: Regularization and passivity-preserving model reduction of quasilinear magneto-quasistatic coupled problemsComments: 33 pages, 5 figuresSubjects: Numerical Analysis (math.NA)
We consider the quasilinear magneto-quasistatic field equations that arise in the simulation of low-frequency electromagnetic devices coupled to electrical circuits. Spatial discretization of these equations on 3D domains using the finite element method results in a singular system of differential-algebraic equations (DAEs). First, we analyze the structural properties of this system and present a novel regularization approach based on projecting out the singular state components. Next, we explore the passivity of the variational magneto-quasistatic problem and its discretization by defining suitable storage functions. For model reduction of the magneto-quasistatic system, we employ the proper orthogonal decomposition (POD) technique combined with the discrete empirical interpolation method (DEIM), to facilitate efficient evaluation of the system's nonlinearities. Our model reduction approach involves the transformation of the regularized DAE into a system of ordinary differential equations, leveraging a special block structure inherent in the problem, followed by applying standard model reduction techniques to the transformed system. We prove that the POD-reduced model preserves passivity, and for the POD-DEIM-reduced model, we propose to enforce passivity by perturbing the output in a way that accounts for DEIM errors. Numerical experiments illustrate the effectiveness of the presented model reduction methods and the passivity enforcement technique.
- [126] arXiv:2411.14227 [pdf, html, other]
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Title: On the strong persistence property and normally torsion-freeness of square-free monomial idealsSubjects: Commutative Algebra (math.AC)
In this paper, we first show that any square-free monomial ideal in $K[x_1, x_2, x_3, x_4, x_5]$ has the strong persistence property. Next we will provide a criterion for a minimal counterexample to the Conforti-Cornuejols conjecture. Finally we give a necessary and sufficient condition to determine the normally torsion-freeness of a linear combination of two normally torsion-free square-free monomial ideals.
- [127] arXiv:2411.14232 [pdf, html, other]
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Title: Counting 3-uple Veronese surfacesSubjects: Algebraic Geometry (math.AG)
This paper culminates in the count of the number of 3-Veronese surfaces passing through 13 general points. This follows the case of 2-Veronese surfaces discovered by Coble in the 1920's. One important element of the calculation is a direct construction of a space of "complete triangles." Our construction is different from the classical ordered constructions of Schubert, Collino and Fulton, as it occurs directly on the Hilbert scheme of length 3 subschemes of the plane. We transport the enumerative problem into a 26-dimensional Grassmannian bundle over our space of complete triangles, where we perform Atiyah-Bott localization. Several important questions arise, which we collect at the end of the paper.
- [128] arXiv:2411.14236 [pdf, other]
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Title: Size of chaos for Gibbs measures of mean field interacting diffusionsComments: 35 pagesSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We investigate Gibbs measures for diffusive particles interacting through a two-body mean field energy. By uncovering a gradient structure for the conditional law, we derive sharp bounds on the size of chaos, providing a quantitative characterization of particle independence. To handle unbounded interaction forces, we study the concentration of measure phenomenon for Gibbs measures via a defective Talagrand inequality, which may hold independent interest. Our approach provides a unified framework for both the flat semi-convex and displacement convex cases. Additionally, we establish sharp chaos bounds for the quartic Curie-Weiss model in the sub-critical regime, demonstrating the generality of this method.
- [129] arXiv:2411.14237 [pdf, html, other]
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Title: Closed geodesics on compact Lorentzian solvmanifoldsSubjects: Differential Geometry (math.DG)
The aim of this work is the study of geodesics on Lorentzian homogeneous spaces of the form $M=G/\Lambda$, where $G$ is a solvable Lie group endowed with a bi-invariant Lorentzian metric and $\Lambda < G$ is a cocompact lattice. Conditions to assert closedness of light, time or spacelike geodesics on the compact quotient spaces are given. This study implicitly requires additional information about the lattices in each case. We found conditions for which every lightlight geodesic on the quotient space is closed. And more important, this situation depends on the lattice. Moreover, even in dimension four, there are examples of compact solvmanifolds for which not every lightlike geodesic is closed. For time and spacelike geodesics, the conclusion are different. Finally, we study isometry groups of those compact spaces and show some computations in dimension six.
- [130] arXiv:2411.14238 [pdf, html, other]
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Title: Computing the permanental polynomial of $4k$-intercyclic bipartite graphsComments: 9 pagesJournal-ref: American Journal of Combinatorics, 3:35-43, (2024)Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
Let $G$ be a bipartite graph with adjacency matrix $A(G)$. The characteristic polynomial $\phi(G,x)=\det(xI-A(G))$ and the permanental polynomial $\pi(G,x) = \text{per}(xI-A(G))$ are both graph invariants used to distinguish graphs. For bipartite graphs, we define the modified characteristic polynomial, which is obtained by changing the signs of some of the coefficients of $\phi(G,x)$. For $4k$-intercyclic bipartite graphs, i.e., those for which the removal of any $4k$-cycle results in a $C_{4k}$-free graph, we provide an expression for $\pi(G,x)$ in terms of the modified characteristic polynomial of the graph and its subgraphs. Our approach is purely combinatorial in contrast to the Pfaffian orientation method found in the literature to compute the permanental polynomial.
- [131] arXiv:2411.14239 [pdf, other]
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Title: Duality for Evolutionary Equations with Applications to Control TheoryComments: 19 pagesSubjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA); Optimization and Control (math.OC)
We study evolutionary equations in exponentially weighted $\mathrm{L}^{2}$-spaces as introduced by Picard in 2009. First, for a given evolutionary equation, we explicitly describe the $\nu$-adjoint system, which turns out to describe a system backwards in time. We prove well-posedness for the $\nu$-adjoint system. We then apply the thus obtained duality to introduce and study notions of null-controllability for evolutionary equations.
- [132] arXiv:2411.14240 [pdf, html, other]
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Title: Modeling and dynamics near irregular elongated asteroidsSubjects: Dynamical Systems (math.DS)
We investigate the qualitative characteristics of a test particle attracted to an irregular elongated body, modeled as a non-homogeneous straight segment with a variable linear density. By deriving the potential function in closed form, we formulate the Hamiltonian equations of motion for this system. Our analysis reveals a family of periodic circular orbits parameterized by angular momentum. Additionally, we utilize the axial symmetry resulting from rotations around the segment's axis to consider the corresponding reduced system. This approach identifies several reduced-periodic orbits by analyzing appropriate Poincaré sections. These periodic orbits are then reconstructed into quasi-periodic orbits within the full dynamical system.
- [133] arXiv:2411.14248 [pdf, html, other]
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Title: The dib-chromatic number of digraphsComments: 10 pages, 1 figureSubjects: Combinatorics (math.CO)
We study an extension to directed graphs of the parameter called the $b$-chromatic number of a graph in terms of acyclic vertex colorings: the dib-chromatic number. We give general bounds for this parameter. We also show some results about tournaments and regular digraphs.
- [134] arXiv:2411.14260 [pdf, html, other]
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Title: Existence and global behaviour of solutions of a parabolic problem involving the fractional $p$-Laplacian in porous mediumSubjects: Analysis of PDEs (math.AP)
In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian:
\begin{equation*} \begin{cases} \partial_t u+(-\Delta)^s_p(|u|^{m-1}u)=h(t,x,|u|^{m-1}u) & \text{in} \; (0,T)\times \Omega,\\ u=0 & \text{in} \; (0,T) \times \mathbb{R}^d\backslash \Omega, \\ u(0,\cdot)=u_0 & \text{in} \; \Omega . \end{cases}\ \end{equation*} We also study further the the homogeneous case $h(u)=|u|^{q-1}u$ with $q>0$. In particular we investigate global time existence, uniqueness, global behaviour of weak solutions and stabilization. - [135] arXiv:2411.14262 [pdf, html, other]
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Title: Accelerating Construction of Non-Intrusive Nonlinear Structural Dynamics Reduced Order Models through HyperreductionSubjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
We present a novel technique to significantly reduce the offline cost associated to non-intrusive nonlinear tensors identification in reduced order models (ROMs) of geometrically nonlinear, finite elements (FE)-discretized structural dynamics problems. The ROM is obtained by Galerkin-projection of the governing equations on a reduction basis (RB) of Vibration Modes (VMs) and Static Modal Derivatives (SMDs), resulting in reduced internal forces that are cubic polynomial in the reduced coordinates. The unknown coefficients of the nonlinear tensors associated with this polynomial representation are identified using a modified version of Enhanced Enforced Displacement (EED) method which leverages Energy Conserving Sampling and Weighting (ECSW) as hyperreduction technique for efficiency improvement. Specifically, ECSW is employed to accelerate the evaluations of the nonlinear reduced tangent stiffness matrix that are required within EED. Simulation-free training sets of forces for ECSW are obtained from displacements corresponding to quasi-random samples of a nonlinear second order static displacement manifold. The proposed approach is beneficial for the investigation of the dynamic response of structures subjected to acoustic loading, where multiple VMs must be added in the RB, resulting in expensive nonlinear tensor identification. Superiority of the novel method over standard EED is demonstrated on FE models of a shallow curved clamped panel and of a nine-bay aeronautical reinforced panel modelled, using the commercial finite element program Abaqus.
- [136] arXiv:2411.14266 [pdf, html, other]
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Title: Quantitative Propagation of Chaos for 2D Viscous Vortex Model with General Circulations on the Whole SpaceSubjects: Analysis of PDEs (math.AP); Probability (math.PR)
We derive the quantitative propagation of chaos in the sense of relative entropy for the 2D viscous vortex model with general circulations, approximating the vorticity formulation of the 2D Navier-Stokes equation on the whole Euclidean space, which is an extension of the previous works \cite{fournier2014propagation,jabin2018quantitative}. We improve our previous results in \cite{feng2023quantitative} to the more general case that the vorticity may change sign, allowing the circulations to be in different orientations. We also extend the results of \cite{wang2024sharp} to the whole space to obtain the sharp local propagation of chaos in the high viscosity regime. Both results of convergence rates are global-in-time on any finite time horizon and optimal among existing literature, thanks to the careful estimates using the Grigor'yan parabolic maximum principle and a new ODE hierarchy and iterated integral estimates.
- [137] arXiv:2411.14287 [pdf, html, other]
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Title: Constructing strictly sign regular matrices of all sizes and sign patternsComments: 18 pages, no figureSubjects: Rings and Algebras (math.RA)
The class of strictly sign regular (SSR) matrices has been extensively studied by many authors over the past century, notably by Schoenberg, Motzkin, Gantmacher, and Krein. A classical result of Gantmacher-Krein assures the existence of SSR matrices for any dimension and sign pattern. In this article, we provide an algorithm to explicitly construct an SSR matrix of any given size and sign pattern. (We also provide in an Appendix, the Python code implementing our algorithm.) To develop this algorithm, we show that one can extend an SSR matrix by adding an extra row (column) to its border, resulting in a higher order SSR matrix. Furthermore, we show how inserting a suitable new row/column between any two successive rows/columns of an SSR matrix results in a matrix that remains SSR. We also establish analogous results for strictly sign regular $m \times n$ matrices of order $p$ for any $p \in [1, \min\{m,n\}]$.
- [138] arXiv:2411.14297 [pdf, html, other]
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Title: Limitations of the Generalized Pareto Distribution-based estimators for the local dimensionSubjects: Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)
Two dynamical indicators, the local dimension and the extremal index, used to quantify persistence in phase space have been developed and applied to different data across various disciplines. These are computed using the asymptotic limit of exceedances over a threshold, which turns to be a Generalized Pareto Distribution in many cases. However the derivation of the asymptotic distribution requires mathematical properties which are not present even in highly idealized dynamical systems, and unlikely to be present in real data. Here we examine in detail issues that arise when estimating these quantities for some known dynamical systems with a particular focus on how the geometry of an invariant set can affect the regularly varying properties of the invariant measure. We demonstrate that singular measures supported on sets of non-integer dimension are typically not regularly varying and that the absence of regular variation makes the estimates resolution dependent. We show as well that the most common extremal index estimation method is ambiguous for continuous time processes sampled at fixed time steps, which is an underlying assumption in its application to data.
- [139] arXiv:2411.14308 [pdf, html, other]
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Title: New results similar to Lagrange's four-square theoremComments: 11 pagesSubjects: Number Theory (math.NT)
In this paper we establish some new results similar to Lagrange's four-square theorem. For example, we prove that any integer $n>1$ can be written $w(5w+1)/2+x(5x+1)/2+y(5y+1)/2+z(5z+1)/2$ with $w,x,y,z\in\mathbb Z$. Here we state two general results:
(1) If $a$ and $b$ are odd integers with $a>0$, $b>-a$ and $\gcd(a,b)=1$, then all sufficient large integers can be written as $$\frac{w(aw+b)}2+\frac{x(ax+b)}2+\frac{y(ay+b)}2+\frac{z(az+b)}2$$ with $w,x,y,z$ nonnegative integers.
(2) If $a$ and $b$ are integers with $a>0$, $b>-a$, $2\mid a$ and $\gcd(a,b)=1$, then all sufficient large integers can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers. - [140] arXiv:2411.14312 [pdf, html, other]
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Title: Density of Stable Interval Translation MapsSubjects: Dynamical Systems (math.DS)
Assume that the interval $I=[0,1)$ is partitioned into finitely many intervals $I_1,\dots,I_r$ and consider a map $T\colon I\to I$ so that $T_{\vert I_s}$ is a translation for each $1 \le s \le r$. We do not assume that the images of these intervals are disjoint. Such maps are called Interval Translation Maps. Let $ITM(r)$ be the space of all such transformations, where we fix $r$ but not the intervals $I_1,\dots,I_r$, nor the translations. The set $X(T):=\bigcap_{n\ge 0} T^n[0,1)$ can be a finite union of intervals (in which case the map is called of finite type), or is a disjoint union of finitely many intervals and a Cantor set (in which case the map is called of infinite type). In this paper we show that there exists an open and dense subset $\mathcal{S}(r)$ of $ITM(r)$ consisting of stable maps, i.e. each $T\in \mathcal{S}(r)$ is of finite type, the first return map to any component of $X(T)$ corresponds to a circle rotation and $\mathcal{S}(r) \ni T \mapsto X(T)$ is continuous in the Hausdorff topology.
- [141] arXiv:2411.14315 [pdf, html, other]
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Title: Introducing a Harmonic Balance Navier-Stokes Finite Element Solver to Accelerate Cardiovascular SimulationsSubjects: Numerical Analysis (math.NA); Fluid Dynamics (physics.flu-dyn)
The adoption of cardiovascular simulations for diagnosis and surgical planning on a patient-specific basis requires the development of faster methods than the existing state-of-the-art techniques. To address this need, we leverage the periodic nature of these flows to accurately capture their time-dependence using spectral discretization. Owing to the reduced size of the discrete problem, the resulting approach, known as the harmonic balance method, significantly lowers the solution cost when compared against the conventional time marching methods. This study describes a stabilized finite element implementation of the harmonic balanced method that targets the simulation of physically-stable time-periodic flows. That stabilized method is based on the Galerkin/least-squares formulation that permits stable solution in convection-dominant flows and convenient use of the same interpolation functions for velocity and pressure. We test this solver against its equivalent time marching method using three common physiological cases where blood flow is modeled in a Glenn operation, a cerebral artery, and a left main coronary artery. Using the conventional time marching solver, simulating these cases takes more than ten hours. That cost is reduced by up to two orders of magnitude when the proposed harmonic balance solver is utilized, where a solution is produced in approximately 30 minutes. We show that that solution is in excellent agreement with the conventional solvers when the number of modes is sufficiently large to accurately represent the imposed boundary conditions.
- [142] arXiv:2411.14316 [pdf, html, other]
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Title: Quasistatic nonassociative plasticity at finite strainsSubjects: Analysis of PDEs (math.AP)
We investigate finite-strain elastoplastic evolution in the nonassociative setting. The constitutive material model is formulated in variational terms and coupled with the quasistatic equilibrium system. We introduce measure-valued energetic solutions and prove their existence via a time discretization approach. The existence theory hinges on a suitable regularization of the dissipation term via a space-time mollification. Eventually, we discuss the possibility of solving the problem in the setting of functions, instead of measures.
- [143] arXiv:2411.14320 [pdf, html, other]
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Title: Robust Energy System Design via Semi-infinite ProgrammingComments: manuscript (32 pages, 6 figures), supplementary materials (24 pages, 2 figures, 2 tables)Subjects: Optimization and Control (math.OC)
Time-series information needs to be incorporated into energy system optimization to account for the uncertainty of renewable energy sources. Typically, time-series aggregation methods are used to reduce historical data to a few representative scenarios but they may neglect extreme scenarios, which disproportionally drive the costs in energy system design. We propose the robust energy system design (RESD) approach based on semi-infinite programming and use an adaptive discretization-based algorithm to identify worst-case scenarios during optimization. The RESD approach can guarantee robust designs for problems with nonconvex operational behavior, which current methods cannot achieve. The RESD approach is demonstrated by designing an energy supply system for the island of La Palma. To improve computational performance, principal component analysis is used to reduce the dimensionality of the uncertainty space. The robustness and costs of the approximated problem with significantly reduced dimensionality approximate the full-dimensional solution closely. Even with strong dimensionality reduction, the RESD approach is computationally intense and thus limited to small problems.
- [144] arXiv:2411.14325 [pdf, other]
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Title: Bounded minimizers of double phase problems at nearly linear growthComments: 46 pages, comments are welcomeSubjects: Analysis of PDEs (math.AP)
Bounded minimizers of double phase problems at nearly linear growth have locally Hölder continuous gradient within the sharp maximal nonuniformity range $q<1+\alpha$.
- [145] arXiv:2411.14326 [pdf, html, other]
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Title: Nuclear C*-algebras and similarity problemSubjects: Operator Algebras (math.OA)
We prove that the minimal tensor product of a $C^*$-algebra $\cl A$ satisfying Kadison's similarity property and a nuclear $C^*$-algebra $\cl B,$ satisfies Kadison's similarity property and its length $\ell\left(\cl A \tens \limits_{min} \cl B\right)$ is less than $3\,\ell(\cl A).$
- [146] arXiv:2411.14332 [pdf, html, other]
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Title: Continuous nonlinear adaptive experimental design with gradient flowSubjects: Numerical Analysis (math.NA)
Identifying valuable measurements is one of the main challenges in computational inverse problems, often framed as the optimal experimental design (OED) problem. In this paper, we investigate nonlinear OED within a continuously-indexed design space. This is in contrast to the traditional approaches on selecting experiments from a finite measurement set. This formulation better reflects practical scenarios where measurements are taken continuously across spatial or temporal domains. However, optimizing over a continuously-indexed space introduces computational challenges. To address these, we employ gradient flow and optimal transport techniques, complemented by adaptive strategy for interactive optimization. Numerical results on the Lorenz 63 system and Schrödinger equation demonstrate that our solver identifies valuable measurements and achieves improved reconstruction of unknown parameters in inverse problems.
- [147] arXiv:2411.14333 [pdf, html, other]
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Title: Generalized Finite Difference Method for Solving Stochastic Diffusion EquationsComments: 22 pagesSubjects: Numerical Analysis (math.NA)
Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency, stability and convergence in mean-square, showing that the proposed method preserves stability and demonstrates favorable convergence characteristics under suitable assumptions. In order to validate the methodology, we present numerical results in one-, two-, and three-dimensional space domains.
- [148] arXiv:2411.14334 [pdf, html, other]
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Title: Well-Posedness for Dean-Kawasaki Models of Vlasov-Fokker-Planck TypeSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Probability (math.PR)
We consider systems of interacting particles which are described by a second order Langevin equation, i.e., particles experiencing inertia. We introduce an associated equation of fluctuating hydrodynamics, which can be interpreted as stochastic version of a Vlasov-Fokker-Planck equation. We show that this stochastic partial differential equation exhibits the same dichotomy as the corresponding first order (inertial-free) equation, the so-called Dean-Kawasaki equation: Solutions exist only for suitable atomic initial data, but not for smooth initial data. The class of systems covered includes several models of active matter.
- [149] arXiv:2411.14336 [pdf, html, other]
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Title: Finding the root in random nearest neighbor treesComments: 22 pages, 7 figuresSubjects: Probability (math.PR); Data Structures and Algorithms (cs.DS); Social and Information Networks (cs.SI)
We study the inference of network archaeology in growing random geometric graphs. We consider the root finding problem for a random nearest neighbor tree in dimension $d \in \mathbb{N}$, generated by sequentially embedding vertices uniformly at random in the $d$-dimensional torus and connecting each new vertex to the nearest existing vertex. More precisely, given an error parameter $\varepsilon > 0$ and the unlabeled tree, we want to efficiently find a small set of candidate vertices, such that the root is included in this set with probability at least $1 - \varepsilon$. We call such a candidate set a $\textit{confidence set}$. We define several variations of the root finding problem in geometric settings -- embedded, metric, and graph root finding -- which differ based on the nature of the type of metric information provided in addition to the graph structure (torus embedding, edge lengths, or no additional information, respectively).
We show that there exist efficient root finding algorithms for embedded and metric root finding. For embedded root finding, we derive upper and lower bounds (uniformly bounded in $n$) on the size of the confidence set: the upper bound is subpolynomial in $1/\varepsilon$ and stems from an explicit efficient algorithm, and the information-theoretic lower bound is polylogarithmic in $1/\varepsilon$. In particular, in $d=1$, we obtain matching upper and lower bounds for a confidence set of size $\Theta\left(\frac{\log(1/\varepsilon)}{\log \log(1/\varepsilon)} \right)$. - [150] arXiv:2411.14339 [pdf, html, other]
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Title: On Dual of LMIs for Absolute Stability Analysis of Nonlinear Feedback Systems with Static O'Shea-Zames-Falb MultipliersComments: 8 pages, 5 figures, submitted to European Control Conference 2025Subjects: Optimization and Control (math.OC)
This study investigates the absolute stability criteria based on the framework of integral quadratic constraint (IQC) for feedback systems with slope-restricted nonlinearities. In existing works, well-known absolute stability certificates expressed in the IQC-based linear matrix inequalities (LMIs) were derived, in which the input-to-output characteristics of the slope-restricted nonlinearities were captured through static O'Shea-Zames-Falb multipliers. However, since these certificates are only sufficient conditions, they provide no clue about the absolute stability in the case where the LMIs are infeasible. In this paper, by taking advantage of the duality theory of LMIs, we derive a condition for systems to be not absolutely stable when the above-mentioned LMIs are infeasible. In particular, we can identify a destabilizing nonlinearity within the assumed class of slope-restricted nonlinearities as well as a non-zero equilibrium point of the resulting closed-loop system, by which the system is proved to be not absolutely stable. We demonstrate the soundness of our results by numerical examples.
- [151] arXiv:2411.14340 [pdf, html, other]
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Title: Canonical foliation of bubblesheetsSubjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
We introduce a new curvature condition for high-codimension submanifolds of a Riemannian ambient space, called quasi-parallel mean curvature (QPMC). The class of submanifolds with QPMC includes all CMC hypersurfaces and submanifolds with parallel mean curvature. We use our notion of QPMC to prove that certain kinds of high-curvature regions which appear in geometric flows, called bubblesheets, can be placed in a suitable normal form. This follows from a more general result asserting that the manifold $\mathbb{R}^k \times \mathbb{S}^{n-k}$, equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded $(n-k)$-spheres with QPMC.
- [152] arXiv:2411.14342 [pdf, html, other]
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Title: A Note on Complexity for Two Classes of Structured Non-Smooth Non-Convex Compositional OptimizationSubjects: Optimization and Control (math.OC)
This note studies numerical methods for solving compositional optimization problems, where the inner function is smooth, and the outer function is Lipschitz continuous, non-smooth, and non-convex but exhibits one of two special structures that enable the design of efficient first-order methods. In the first structure, the outer function allows for an easily solvable proximal mapping. We demonstrate that, in this case, a smoothing compositional gradient method can find a $(\delta,\epsilon)$-stationary point--specifically defined for compositional optimization--in $O(1/(\delta \epsilon^2))$ iterations. In the second structure, the outer function is expressed as a difference-of-convex function, where each convex component is simple enough to allow an efficiently solvable proximal linear subproblem. In this case, we show that a prox-linear method can find a nearly ${\epsilon}$-critical point in $O(1/\epsilon^2)$ iterations.
- [153] arXiv:2411.14350 [pdf, html, other]
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Title: Randomized Geodesic Flow on Hyperbolic GroupsSubjects: Probability (math.PR); Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)
Motivated by Gromov's geodesic flow problem on hyperbolic groups $G$, we develop in this paper an analog using random walks. This leads to a notion of a harmonic analog $\Theta$ of the Bowen-Margulis-Sullivan measure on $\partial^2 G$. We provide three different but related constructions of $\Theta$: 1) by moving the base-point along a quasigeodesic ray 2) by moving the base-point along random walk trajectories 3) directly as a push-forward under the boundary map to $\partial^2 G$ of a measure inherited from studying all bi-infinite random walk trajectories (with no restriction on base-point) on $G^{\mathbb{Z}}$.
Of these, the third construction is the most involved and needs new techniques. It relies on developing a framework where we can treat bi-infinite random walk trajectories as analogs of bi-infinite geodesics on complete simply connected negatively curved manifolds. Geodesic flow on a hyperbolic group is typically not well-defined due to non-uniqueness of geodesics. We circumvent this problem in the random walk setup by considering \emph{all} trajectories. We thus get a well-defined \emph{discrete flow} given by the $\mathbb{Z}-$shift on bi-infinite random walk trajectories. The $\mathbb{Z}-$shift is the random analog of the time one map of the geodesic flow. As an analog of ergodicity of the geodesic flow on a closed negatively curved manifold, we establish ergodicity of the $G$-action on $(\partial^2G, \Theta)$. - [154] arXiv:2411.14352 [pdf, html, other]
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Title: Particle systems, Dipoles and Besov spacesComments: 16 pagesSubjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA)
In a measure space with a very mild structure, a good grid, we introduced a scale of Besov Banach spaces of distributions with negative smoothness. We establish an atomic decomposition in terms of Dirac masses and Dipoles, that is, pairs of Diracs masses with equal charge but opposite signal.
- [155] arXiv:2411.14362 [pdf, html, other]
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Title: On the geometry of K\"ahler--Frobenius manifolds and their classificationSubjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG)
The purpose of this article is to show that flat compact Kähler manifolds exhibit the structure of a Frobenius manifold, a structure originating in 2D Topological Quantum Field Theory and closely related to Joyce structure. As a result, we classify all such manifolds. It can be deduced that Kähler--Frobenius manifolds include certain Calabi--Yau manifolds, complex tori $T=\mathbb{C}^n/\mathbb{Z}^n$, Hantzsche--Wendt manifolds, hyperelliptic manifolds and manifolds of type $T/G$, where $G$ is a finite group acting on $T$ freely and containing no translations. An explicit study is provided for the two-dimensional case. Additionally, we can prove that Chern's conjecture for Kähler pre-Frobenius manifolds holds. Lastly, we establish that certain classes of Kähler-Frobenius manifolds share a direct relationship with theta functions which are important objects in number theory as well as complex analysis.
- [156] arXiv:2411.14364 [pdf, html, other]
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Title: A new lower bound for the multicolor Ramsey number $r_k(K_{2, t + 1})$Comments: 8 pagesSubjects: Combinatorics (math.CO)
In this short note, we provide a new infinite family of $K_{2, t+1}$-free graphs for each prime power $t$. Using these graphs, we show that it is possible to partition the edges of $K_n$ into parts, such that each part is isomorphic to our $K_{2, t+1}$-free graph. This yields an improved lower bound to the multicolor Ramsey number $r_k(K_{2, t+1})$ when $k$ and $t$ are powers of the same prime. For these values of $k$ and $t$, our coloring implies that $$ tk^2 + 1 \leq r_k(K_{2, t+1}) \leq tk^2 + k + 2. $$ where the upper bound is due to Chung and Graham.
- [157] arXiv:2411.14370 [pdf, html, other]
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Title: Convergence and Stability Analysis of the Extended Infinite Horizon Model Predictive ControlComments: 23 pagesSubjects: Optimization and Control (math.OC)
Model Predictive Control (MPC) is a popular technology to operate industrial systems. It refers to a class of control algorithms that use an explicit model of the system to obtain the control action by minimizing a cost function. At each time step, MPC solves an optimization problem that minimizes the future deviation of the outputs which are calculated from the model. The solution of the optimization problem is a sequence of control inputs, the first input is applied to the system, and the optimization process is repeated at subsequent time steps. In the context of MPC, convergence and stability are fundamental issues. A common approach to obtain MPC stability is by setting the prediction horizon as infinite. For stable open-loop systems, the infinite horizon can be reduced to a finite horizon MPC with a terminal weight computed through the solution of a Lyapunov equation. This paper presents a rigorous analysis of convergence and stability of the extended nominally stable MPC developed by Odloak [Odloak, D. Extended robust model predictive control, AIChE J. 50 (8) (2004) 1824-1836] and the stable MPC with zone control [González, A.H., Odloak, D. A stable MPC with zone control, J. Proc. Cont. 19 (2009) 110-122]. The mathematical proofs consider that the system is represented by a general gain matrix $D_0$, i.e., not necessarily regular, and they are developed for any input horizon $m$. The proofs are based on elementary geometric and algebraic tools and we believe that they can be adapted to the derived MPC approaches, as well as future studies.
- [158] arXiv:2411.14376 [pdf, html, other]
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Title: Solutions to the minimal surface system with large singular setsComments: 18 pagesSubjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
Lawson and Osserman proved that the Dirichlet problem for the minimal surface system is not always solvable in the class of Lipschitz maps. However, it is known that minimizing sequences (for area) of Lipschitz graphs converge to objects called Cartesian currents. Essentially nothing is known about these limits. We show that such limits can have surprisingly large interior vertical and non-minimal portions. This demonstrates a striking discrepancy between the parametric and non-parametric area minimization problems in higher codimension. Moreover, our construction has the smallest possible dimension ($n = 3$) and codimension $(m = 2)$.
- [159] arXiv:2411.14379 [pdf, other]
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Title: Rationality of singular cubic threefolds over $\mathbb R$Comments: 27 pagesSubjects: Algebraic Geometry (math.AG)
We study rationality properties of real singular cubic threefolds.
- [160] arXiv:2411.14382 [pdf, html, other]
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Title: Sampling Observability for Heat Equations with MemorySubjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP)
This paper studies the sampling observability for the heat equations with memory in the lower-order term, where the observation is conducted at a finite number of time instants and on a small open subset at each time instant. We present a two-sided sampling observability inequality and give a sharp sufficient condition to ensure the aforementioned inequality. We also provide a method to select the time instants and then to design the observation regions, based on a given memory kernel, such that the above-mentioned inequality holds for these time instants and observation regions. Additionally, we demonstrate that the positions of these time instants depend significantly on the memory kernel.
- [161] arXiv:2411.14391 [pdf, html, other]
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Title: Phase Space Representation of the Density Operator: Bopp Pseudodifferential Calculus and Moyal ProductSubjects: Mathematical Physics (math-ph); Operator Algebras (math.OA); Quantum Physics (quant-ph)
Bopp shifts were introduced in 1956 in the study of statistical interpretations of quantum mechanics. They lead to a phase space view of quantum mechanics closely related to the Moyal star product and its interpretation as a deformation quantization. In the present paper we pursue our study of Bopp quantization by initiated in previous work and apply it to give a new phase space description of the density operator, that is of the mixed states of quantum mechanics.
- [162] arXiv:2411.14399 [pdf, html, other]
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Title: DiscoTEX 1.0: Discontinuous collocation and implicit-turned-explicit (IMTEX) integration symplectic, symmetric numerical algorithms with high order jumps for differential equations II: extension to higher-orders of numerical convergenceComments: 15 pages, 5 figures, 2 tables. Second paper of a series of papers. See this http URL:2401.08758 for application of these algorithms to numerical black hole perturbation theory. Comments welcomed. arXiv admin note: text overlap with arXiv:2401.08758Subjects: Numerical Analysis (math.NA); Instrumentation and Methods for Astrophysics (astro-ph.IM); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
\texttt{DiscoTEX} is a highly accurate numerical algorithm for computing numerical weak-form solutions to distributionally sourced partial differential equations (PDE)s. The aim of this second paper, succeeding \cite{da2024discotex}, is to present its extension up to twelve orders. This will be demonstrated by computing numerical weak-form solutions to the distributionally sourced wave equation and comparing it to its exact solutions. The full details of the numerical scheme at higher orders will be presented.
- [163] arXiv:2411.14409 [pdf, html, other]
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Title: Inexact Generalized Golub-Kahan Methods for Large-Scale Bayesian Inverse ProblemsSubjects: Numerical Analysis (math.NA)
Solving large-scale Bayesian inverse problems presents significant challenges, particularly when the exact (discretized) forward operator is unavailable. These challenges often arise in image processing tasks due to unknown defects in the forward process that may result in varying degrees of inexactness in the forward model. Moreover, for many large-scale problems, computing the square root or inverse of the prior covariance matrix is infeasible such as when the covariance kernel is defined on irregular grids or is accessible only through matrix-vector products. This paper introduces an efficient approach by developing an inexact generalized Golub-Kahan decomposition that can incorporate varying degrees of inexactness in the forward model to solve large-scale generalized Tikhonov regularized problems. Further, a hybrid iterative projection scheme is developed to automatically select Tikhonov regularization parameters. Numerical experiments on simulated tomography reconstructions demonstrate the stability and effectiveness of this novel hybrid approach.
- [164] arXiv:2411.14417 [pdf, other]
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Title: Construction of Lie algebra weight system kernel via Vogel algebraComments: 18 pagesSubjects: Quantum Algebra (math.QA); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT); Representation Theory (math.RT)
We develop a method of constructing a kernel of Lie algebra weight system. A main tool we use in the analysis is Vogel's $\Lambda$ algebra and the surrounding framework. As an example of a developed technique we explicitly provide all Jacobi diagrams lying in the kernel of $\mathfrak{sl}_N$ weight system at low orders. We also discuss consequences of the presence of the kernel in Lie algebra weight systems for detection of correlators in the 3D Chern-Simons topological field theory and for distinguishing of knots by the corresponding quantum knot invariants.
New submissions (showing 164 of 164 entries)
- [165] arXiv:2411.12190 (cross-list from cond-mat.mes-hall) [pdf, html, other]
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Title: Higher-dimensional magnetic SkyrmionsComments: LaTeX: 40 pages, 14 figuresSubjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We propose a generalization of the theory of magnetic Skyrmions in chiral magnets in two dimensions to a higher-dimensional theory with magnetic Skyrmions in three dimensions and an $S^3$ target space, requiring a 4-dimensional magnetization vector. A physical realization of our theory necessitates the use of a synthetic dimension, recently promoted and realized in condensed matter physics. In the simplest incarnation of the theory, we find a Skyrmion and a sphaleron - the latter being an unstable soliton. Including also the Skyrme term in theory enriches the spectrum to a small metastable Skyrmion, an unstable sphaleron and a large stable Skyrmion.
- [166] arXiv:2411.13558 (cross-list from q-fin.CP) [pdf, html, other]
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Title: Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized marketSubjects: Computational Finance (q-fin.CP); Probability (math.PR); Mathematical Finance (q-fin.MF)
The relative arbitrage problem in Stochastic Portfolio Theory seeks to generate an investment strategy that almost surely outperforms a benchmark portfolio at the end of a certain time horizon. The highest relative return in relative arbitrage opportunities is related to the smallest nonnegative continuous solution of a Cauchy PDE. However, solving the PDE poses analytical and numerical challenges, due to the high dimensionality and its non-unique solutions. In this paper, we discuss numerical methods to address the relative arbitrage problem and the associated PDE in a volatility-stabilized market using time-changed Bessel bridges. We present a practical algorithm and demonstrate numerical results through an example in volatility-stabilized markets.
- [167] arXiv:2411.13579 (cross-list from q-fin.MF) [pdf, other]
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Title: Optimal portfolio under ratio-type periodic evaluation in stochastic factor models under convex trading constraintsComments: Keywords: Periodic evaluation, relative portfolio performance, incomplete market, stochastic factor model, convex trading constraints, convex duality approach. This manuscript combines two previous preprints arXiv:2311.12517 and arXiv:2401.14672 into one paper with more general and improved resultsSubjects: Mathematical Finance (q-fin.MF); Optimization and Control (math.OC); Portfolio Management (q-fin.PM)
This paper studies a type of periodic utility maximization problems for portfolio management in incomplete stochastic factor models with convex trading constraints. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon, featuring the dynamic adjustments in portfolio decision according to past achievements. Under power utility, we transform the original infinite horizon optimal control problem into an auxiliary terminal wealth optimization problem under a modified utility function. To cope with the convex trading constraints, we further introduce an auxiliary unconstrained optimization problem in a modified market model and develop the martingale duality approach to establish the existence of the dual minimizer such that the optimal unconstrained wealth process can be obtained using the dual representation. With the help of the duality results in the auxiliary problems, the relationship between the constrained and unconstrained models as well as some fixed point arguments, we finally derive and verify the optimal constrained portfolio process in a periodic manner for the original problem over an infinite horizon.
- [168] arXiv:2411.13605 (cross-list from hep-th) [pdf, html, other]
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Title: Quantum Field Measurements in the Fewster-Verch FrameworkComments: 21 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
The Fewster-Verch (FV) framework was introduced as a prescription to define local operations within a quantum field theory (QFT) that are free from Sorkin-like causal paradoxes. In this framework the measurement device is modeled via a probe QFT that, after interacting with the target QFT, is subject to an arbitrary local measurement. While the FV framework is rich enough to carry out quantum state tomography, it has two drawbacks. First, it is unclear if the FV framework allows conducting arbitrary local measurements. Second, if the probe field is interpreted as physical and the FV framework as fundamental, then one must demand the probe measurement to be itself implementable within the framework. That would involve a new probe, which should also be subject to an FV measurement, and so on. It is unknown if there exist non-trivial FV measurements for which such an "FV-Heisenberg cut" can be moved arbitrarily far away. In this work, we advance the first problem by proving that measurements of locally smeared fields fit within the FV framework. We solve the second problem by showing that any such field measurement admits a movable FV-Heisenberg cut.
- [169] arXiv:2411.13606 (cross-list from cond-mat.stat-mech) [pdf, html, other]
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Title: The stabilizing role of multiplicative noise in non-confining potentialsComments: 14 pages, 8 figuresSubjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Statistics Theory (math.ST); Adaptation and Self-Organizing Systems (nlin.AO)
We provide a simple framework for the study of parametric (multiplicative) noise, making use of scale parameters. We show that for a large class of stochastic differential equations increasing the multiplicative noise intensity surprisingly causes the mass of the stationary probability distribution to become increasingly concentrated around the minima of the multiplicative noise term, whilst under quite general conditions exhibiting a kind of intermittent burst like jumps between these minima. If the multiplicative noise term has one zero this causes on-off intermittency. Our framework relies on first term expansions, which become more accurate for larger noise intensities. In this work we show that the full width half maximum in addition to the maximum is appropriate for quantifying the stationary probability distribution (instead of the mean and variance, which are often undefined). We define a corresponding new kind of weak sense stationarity. We consider a double well potential as an example of application, demonstrating relevance to tipping points in noisy systems.
- [170] arXiv:2411.13646 (cross-list from hep-th) [pdf, html, other]
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Title: CFTs with Large Gap from Barnes-Wall Lattice OrbifoldsComments: 27 pages. Ancillary Mathematica and Magma filesSubjects: High Energy Physics - Theory (hep-th); Quantum Algebra (math.QA)
We investigate orbifolds of lattice conformal field theories with the goal of constructing theories with large gap. We consider Barnes-Wall lattices, which are a family of lattices with no short vectors, and orbifold by an extraspecial 2-group of lattice automorphisms. To construct the orbifold CFT, we investigate the orbifold vertex operator algebra and its twisted modules. To obtain a holomorphic CFT, a certain anomaly 3-cocycle $\omega$ needs to vanish; based on evidence we provide, we conjecture that it indeed does. Granting this conjecture, we construct a holomorphic CFT of central charge 128 with gap 4.
- [171] arXiv:2411.13680 (cross-list from q-bio.QM) [pdf, html, other]
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Title: Long-term predictive models for mosquito borne diseases: a narrative reviewSubjects: Quantitative Methods (q-bio.QM); Dynamical Systems (math.DS); Biological Physics (physics.bio-ph)
In face of climate change and increasing urbanization, the predictive mosquito-borne diseases (MBD) transmission models require constant updates. Thus, is urgent to comprehend the driving forces of this non stationary behavior, observed through spatial and incidence expansion. We observed that temperature is a critical driver in predictive models for MBD transmission, also being consistently used in multiple reviewed papers with considerable incidence predictive capacity. Rainfall, however, have more subtle importance as moderate precipitation creates breeding sites for mosquitoes, but excessive rainfall can reduce larvae populations. We highlight the frequent use of mechanistic models, particularly those that integrate temperature-dependent biological parameters of disease transmission in incidence proxies as the Vectorial Capacity (VC) and temperature-based basic reproduction number $R_0(t)$, for example. These models show the importance of climate variables, but the socio-demographic factors are often not considered. This gap is a significant opportunity for future research to incorporate socio-demographic data into long-term predictive models for more comprehensive and reliable forecasts. With this survey, we outline the most promising paths to be followed by long-term MBD transmission research and highlighting the potential facing challenges. Thus, we offer a valuable foundation for enhancing disease forecasting models and supporting more effective public health interventions, specially in the long term.
- [172] arXiv:2411.13708 (cross-list from cs.DS) [pdf, html, other]
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Title: Comments on "$\mathcal{O}(m\cdot n)$ algorithms for the recognition and isomorphism problems on circular-arc graphs"Comments: Comment on doi:https://doi.org/10.1137/S0097539793260726Subjects: Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
In the work [$\mathcal{O}(m\cdot n)$ algorithms for the recognition and isomorphism problems on circular-arc graphs, SIAM J. Comput. 24(3), 411--439, (1995)], Wen-Lian Hsu claims three results concerning the class of circular-arc graphs: - the design of so-called \emph{decomposition trees} that represent the structure of all normalized intersection models of circular-arc graphs, - an $\mathcal{O}(m\cdot n)$ recognition algorithm for circular-arc graphs, - an $\mathcal{O}(m\cdot n)$ isomorphism algorithm for circular-arc graphs. In [Discrete Math. Theor. Comput. Sci., 15(1), 157--182, 2013] Curtis, Lin, McConnell, Nussbaum, Soulignac, Spinrad, and Szwarcfiter showed that Hsu's isomorphism algorithm is incorrect. In this note, we show that the other two results -- namely, the construction of decomposition trees and the recognition algorithm -- are also flawed.
- [173] arXiv:2411.13711 (cross-list from cs.LG) [pdf, html, other]
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Title: Almost Sure Convergence Rates and Concentration of Stochastic Approximation and Reinforcement Learning with Markovian NoiseSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
This paper establishes the first almost sure convergence rate and the first maximal concentration bound with exponential tails for general contractive stochastic approximation algorithms with Markovian noise. As a corollary, we also obtain convergence rates in $L^p$. Key to our successes is a novel discretization of the mean ODE of stochastic approximation algorithms using intervals with diminishing (instead of constant) length. As applications, we provide the first almost sure convergence rate for $Q$-learning with Markovian samples without count-based learning rates. We also provide the first concentration bound for off-policy temporal difference learning with Markovian samples.
- [174] arXiv:2411.13713 (cross-list from nlin.SI) [pdf, html, other]
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Title: Closed-form solutions of the nonlinear Schr\"odinger equation with arbitrary dispersion and potentialComments: 28Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
For the first time, the general nonlinear Schrödinger equation is investigated, in which the chromatic dispersion and potential are specified by two arbitrary functions. The equation in question is a natural generalization of a wide class of related nonlinear partial differential equations that are often used in various areas of theoretical physics, including nonlinear optics, superconductivity and plasma physics. To construct exact solutions, a combination of the method of functional constraints and methods of generalized separation of variables is used. Exact closed-form solutions of the general nonlinear Schrödinger equation, which are expressed in quadratures or elementary functions, are found. One-dimensional non-symmetry reductions are described, which lead the considered nonlinear partial differential equation to a simpler ordinary differential equation or a system of such equations. The exact solutions obtained in this work can be used as test problems intended to assess the accuracy of numerical and approximate analytical methods for integrating nonlinear equations of mathematical physics.
- [175] arXiv:2411.13764 (cross-list from stat.ME) [pdf, html, other]
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Title: Selective inference is easier with p-valuesSubjects: Methodology (stat.ME); Statistics Theory (math.ST)
Selective inference is a subfield of statistics that enables valid inference after selection of a data-dependent question. In this paper, we introduce selectively dominant p-values, a class of p-values that allow practitioners to easily perform inference after arbitrary selection procedures. Unlike a traditional p-value, whose distribution must stochastically dominate the uniform distribution under the null, a selectively dominant p-value must have a post-selection distribution that stochastically dominates that of a uniform having undergone the same selection process; moreover, this property must hold simultaneously for all possible selection processes. Despite the strength of this condition, we show that all commonly used p-values (e.g., p-values from two-sided testing in parametric families, one-sided testing in monotone likelihood ratio and exponential families, $F$-tests for linear regression, and permutation tests) are selectively dominant. By recasting two canonical selective inference problems-inference on winners and rank verification-in our selective dominance framework, we provide simpler derivations, a deeper conceptual understanding, and new generalizations and variations of these methods. Additionally, we use our insights to introduce selective variants of methods that combine p-values, such as Fisher's combination test.
- [176] arXiv:2411.13783 (cross-list from econ.GN) [pdf, html, other]
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Title: Process and Policy Insights from Intercomparing Electricity System Capacity Expansion ModelsGreg Schivley, Michael Blackhurst, Patricia Hidalgo-Gonzalez, Jesse Jenkins, Oleg Lugovoy, Qian Luo, Michael J. Roberts, Rangrang Zheng, Cameron Wade, Matthias FrippSubjects: General Economics (econ.GN); Optimization and Control (math.OC)
This study undertakes a detailed intercomparison of four open-source electricity system capacity expansion models--Temoa, Switch, GenX, and USENSYS--to examine their suitability for guiding U.S. power sector decarbonization policies. We isolate the effects of model-specific differences on policy outcomes and investment decisions by harmonizing empirical inputs via PowerGenome and systematically defining "scenarios" (policy conditions) and "configurations" (model setup choices). Our framework allows each model to be tested on identical assumptions for policy, technology costs, and operational constraints, thus distinguishing results that arise from data inputs or configuration versus inherent model structure. Key findings highlight that, when harmonized, models produce very similar capacity portfolios under each current policies and net-zero configuration, with less than 1 percent difference in system costs for most configurations. This agreement across models allows us to examine the impact of configuration choices. For example, configurations that assume unit commitment constraints or economic retirement of generators reveal the difference in investment decisions and system costs that arise from these modeling choices, underscoring the need for clear scenario and configuration definitions in policy guidance. Through this study, we identify critical structural assumptions that influence model outcomes and demonstrate the advantages of a standardized approach when using capacity expansion models. This work offers a valuable benchmark and identifies a few key modeling choices for policymakers, which ultimately will enhance transparency and reliability in modeling efforts to inform the clean energy transition for clean energy planning.
- [177] arXiv:2411.13848 (cross-list from cs.LG) [pdf, html, other]
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Title: Exact and approximate error bounds for physics-informed neural networksComments: 10 pages, 1 figure, accepted to NeurIPS 2024 Workshop on Machine Learning and the Physical SciencesSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
The use of neural networks to solve differential equations, as an alternative to traditional numerical solvers, has increased recently. However, error bounds for the obtained solutions have only been developed for certain equations. In this work, we report important progress in calculating error bounds of physics-informed neural networks (PINNs) solutions of nonlinear first-order ODEs. We give a general expression that describes the error of the solution that the PINN-based method provides for a nonlinear first-order ODE. In addition, we propose a technique to calculate an approximate bound for the general case and an exact bound for a particular case. The error bounds are computed using only the residual information and the equation structure. We apply the proposed methods to particular cases and show that they can successfully provide error bounds without relying on the numerical solution.
- [178] arXiv:2411.13868 (cross-list from cs.LG) [pdf, html, other]
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Title: Robust Detection of Watermarks for Large Language Models Under Human EditsSubjects: Machine Learning (cs.LG); Computation and Language (cs.CL); Statistics Theory (math.ST); Methodology (stat.ME); Machine Learning (stat.ML)
Watermarking has offered an effective approach to distinguishing text generated by large language models (LLMs) from human-written text. However, the pervasive presence of human edits on LLM-generated text dilutes watermark signals, thereby significantly degrading detection performance of existing methods. In this paper, by modeling human edits through mixture model detection, we introduce a new method in the form of a truncated goodness-of-fit test for detecting watermarked text under human edits, which we refer to as Tr-GoF. We prove that the Tr-GoF test achieves optimality in robust detection of the Gumbel-max watermark in a certain asymptotic regime of substantial text modifications and vanishing watermark signals. Importantly, Tr-GoF achieves this optimality \textit{adaptively} as it does not require precise knowledge of human edit levels or probabilistic specifications of the LLMs, in contrast to the optimal but impractical (Neyman--Pearson) likelihood ratio test. Moreover, we establish that the Tr-GoF test attains the highest detection efficiency rate in a certain regime of moderate text modifications. In stark contrast, we show that sum-based detection rules, as employed by existing methods, fail to achieve optimal robustness in both regimes because the additive nature of their statistics is less resilient to edit-induced noise. Finally, we demonstrate the competitive and sometimes superior empirical performance of the Tr-GoF test on both synthetic data and open-source LLMs in the OPT and LLaMA families.
- [179] arXiv:2411.13922 (cross-list from stat.ML) [pdf, html, other]
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Title: Exponentially Consistent Nonparametric Clustering of Data StreamsSubjects: Machine Learning (stat.ML); Information Theory (cs.IT); Machine Learning (cs.LG); Signal Processing (eess.SP)
In this paper, we consider nonparametric clustering of $M$ independent and identically distributed (i.i.d.) data streams generated from unknown distributions. The distributions of the $M$ data streams belong to $K$ underlying distribution clusters. Existing results on exponentially consistent nonparametric clustering algorithms, like single linkage-based (SLINK) clustering and $k$-medoids distribution clustering, assume that the maximum intra-cluster distance ($d_L$) is smaller than the minimum inter-cluster distance ($d_H$). First, in the fixed sample size (FSS) setting, we show that exponential consistency can be achieved for SLINK clustering under a less strict assumption, $d_I < d_H$, where $d_I$ is the maximum distance between any two sub-clusters of a cluster that partition the cluster. Note that $d_I < d_L$ in general. Our results show that SLINK is exponentially consistent for a larger class of problems than $k$-medoids distribution clustering. We also identify examples where $k$-medoids clustering is unable to find the true clusters, but SLINK is exponentially consistent. Then, we propose a sequential clustering algorithm, named SLINK-SEQ, based on SLINK and prove that it is also exponentially consistent. Simulation results show that the SLINK-SEQ algorithm requires fewer expected number of samples than the FSS SLINK algorithm for the same probability of error.
- [180] arXiv:2411.14043 (cross-list from quant-ph) [pdf, html, other]
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Title: From classical probability densities to quantum states: quantization of Gaussian for arbitrary orderingsComments: 16 Pages, one figureSubjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
The primary focus of this work is to investigate how the most emblematic classical probability density, namely a Gaussian, can be mapped to a valid quantum states. To explore this issue, we consider a Gaussian whose squared variance depends on a parameter $\lambda$. Specifically, depending on the value of $\lambda$, we study what happens in the classical-quantum correspondence as we change the indeterminacy of the classical particle. Furthermore, finding a correspondence between a classical state and a quantum state is not a trivial task. Quantum observables, described by Hermitian operators, do not generally commute, so a precise ordering must be introduced to resolve this ambiguity. In this work, we study two different arbitrary orderings: the first is an arbitrary ordering of the position and momentum observables; the second, which is the main focus of the present work, is an arbitrary ordering of the annihilation and creation operators. In this latter case, we find the interesting result that even a $\delta$-function, which in general has no quantum correspondence, can be mapped into a valid quantum state for a particular ordering, specifically the antinormal one (all creation operators are to the right of all annihilation operators in the product). This means that the Gaussian probability density corresponds to a valid quantum state, regardless of how localized classical particles are in phase space.
- [181] arXiv:2411.14089 (cross-list from gr-qc) [pdf, html, other]
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Title: Is there any Trinity of Gravity, to start with?Comments: 11 pages; prepared for Proceedings of 11th Mathematical Physics Meeting, September 2024 in Belgrade, SerbiaSubjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); History and Philosophy of Physics (physics.hist-ph)
In recent years, it has been rather fashionable to talk about geometric trinity of gravity. The main idea is that one can formally present the gravity equations in different terms, those of either torsion or nonmetricity instead of curvature. It starts from a very erroneous claim that the Levi-Civita connection, and therefore the (pseudo-)Riemannian geometry itself, are nothing but an arbitrary choice. The point is that, as long as we admit the need of having a metric for describing gravity, the standard approach does not involve any additional independent geometric structures on top of that. At the same time, any other metric-affine model does go for genuinely new stuff. In particular, the celebrated teleparallel framework introduces a notion of yet another parallel transport which is flat. It gives us curious new ways of modifying gravity, even though very often quite problematic. However, in GR-equivalent models, we only get a new language for describing the same physics, in terms of absolutely unobservable and unpredictable geometrical inventions. For sure, one can always safely create novel constructions which do not influence the physical equations of motion, but in itself it does not make much sense and blatantly goes against the Occam's razor.
- [182] arXiv:2411.14185 (cross-list from stat.ME) [pdf, html, other]
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Title: A note on numerical evaluation of conditional Akaike information for nonlinear mixed-effects modelsSubjects: Methodology (stat.ME); Statistics Theory (math.ST)
We propose two methods to evaluate the conditional Akaike information (cAI) for nonlinear mixed-effects models with no restriction on cluster size. Method 1 is designed for continuous data and includes formulae for the derivatives of fixed and random effects estimators with respect to observations. Method 2, compatible with any type of observation, requires modeling the marginal (or prior) distribution of random effects as a multivariate normal distribution. Simulations show that Method 1 performs well with Gaussian data but struggles with skewed continuous distributions, whereas Method 2 consistently performs well across various distributions, including normal, gamma, negative binomial, and Tweedie, with flexible link functions. Based on our findings, we recommend Method 2 as a distributionally robust cAI criterion for model selection in nonlinear mixed-effects models.
- [183] arXiv:2411.14204 (cross-list from quant-ph) [pdf, html, other]
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Title: Exact solution for a class of quantum models of interacting bosonsComments: 11 pages, no figuresSubjects: Quantum Physics (quant-ph); Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
Quantum models of interacting bosons have wide range of applications, among them the propagation of optical modes in nonlinear media, such as the $k$-photon down conversion. Many of such models are related to nonlinear deformations of finite group algebras, thus, in this sense, they are exactly solvable. Whereas the advanced group-theoretic methods have been developed to study the eigenvalue spectrum of exactly solvable Hamiltonians, in quantum optics the prime interest is not the spectrum of the Hamiltonian, but the evolution of an initial state, such as the generation of optical signal modes by a strong pump mode propagating in a nonlinear medium. I propose a simple and general method of derivation of the solution to such a state evolution problem, applicable to a wide class of quantum models of interacting bosons. For the $k$-photon down conversion model and its generalizations, the solution to the state evolution problem is given in the form of an infinite series expansion in the powers of propagation time with the coefficients defined by a recursion relation with a single polynomial function, unique for each nonlinear model. As an application, I compare the exact solution to the parametric down conversion process with the semiclassical parametric approximation.
- [184] arXiv:2411.14224 (cross-list from cs.ET) [pdf, html, other]
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Title: Thermodynamic Algorithms for Quadratic ProgrammingPatryk-Lipka Bartosik, Kaelan Donatella, Maxwell Aifer, Denis Melanson, Marti Perarnau-Llobet, Nicolas Brunner, Patrick J. ColesComments: 13 pages, 4 figuresSubjects: Emerging Technologies (cs.ET); Statistical Mechanics (cond-mat.stat-mech); Optimization and Control (math.OC)
Thermodynamic computing has emerged as a promising paradigm for accelerating computation by harnessing the thermalization properties of physical systems. This work introduces a novel approach to solving quadratic programming problems using thermodynamic hardware. By incorporating a thermodynamic subroutine for solving linear systems into the interior-point method, we present a hybrid digital-analog algorithm that outperforms traditional digital algorithms in terms of speed. Notably, we achieve a polynomial asymptotic speedup compared to conventional digital approaches. Additionally, we simulate the algorithm for a support vector machine and predict substantial practical speedups with only minimal degradation in solution quality. Finally, we detail how our method can be applied to portfolio optimization and the simulation of nonlinear resistive networks.
- [185] arXiv:2411.14261 (cross-list from hep-th) [pdf, html, other]
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Title: On braid statistics versus parastatisticsComments: 11 pages. Based on a plenary talk at ISQS28, Prague, July 1-5, 2024; to appear in the ProceedingsSubjects: High Energy Physics - Theory (hep-th); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
I report the recent advances in applying (graded) Hopf algebras with braided tensor product in two scenarios: i) paraparticles beyond bosons and fermions living in any space dimensions and transforming under the permutation group; ii) physical models of anyons living in two space-dimensions and transforming under the braid group. In the first scenario simple toy models based on the so-called $2$-bit parastatistics show that, in the multiparticle sector, certain observables can discriminate paraparticles from ordinary bosons/fermions (thus, providing a counterexample to the widespread belief of the "conventionality of parastatistics" argument). In the second scenario the notion of (braided) Majorana qubit is introduced as the simplest building block to implement the Kitaev's proposal of a topological quantum computer which protects from decoherence.
- [186] arXiv:2411.14276 (cross-list from cs.CC) [pdf, html, other]
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Title: A $k^{\frac{q}{q-2}}$ Lower Bound for Odd Query Locally Decodable Codes from Bipartite Kikuchi GraphsSubjects: Computational Complexity (cs.CC); Information Theory (cs.IT)
A code $C \colon \{0,1\}^k \to \{0,1\}^n$ is a $q$-query locally decodable code ($q$-LDC) if one can recover any chosen bit $b_i$ of the message $b \in \{0,1\}^k$ with good confidence by querying a corrupted string $\tilde{x}$ of the codeword $x = C(b)$ in at most $q$ coordinates. For $2$ queries, the Hadamard code is a $2$-LDC of length $n = 2^k$, and this code is in fact essentially optimal. For $q \geq 3$, there is a large gap in our understanding: the best constructions achieve $n = \exp(k^{o(1)})$, while prior to the recent work of [AGKM23], the best lower bounds were $n \geq \tilde{\Omega}(k^{\frac{q}{q-2}})$ for $q$ even and $n \geq \tilde{\Omega}(k^{\frac{q+1}{q-1}})$ for $q$ odd.
The recent work of [AGKM23] used spectral methods to prove a lower bound of $n \geq \tilde{\Omega}(k^3)$ for $q = 3$, thus achieving the "$k^{\frac{q}{q-2}}$ bound" for an odd value of $q$. However, their proof does not extend to any odd $q \geq 5$. In this paper, we prove a $q$-LDC lower bound of $n \geq \tilde{\Omega}(k^{\frac{q}{q-2}})$ for any odd $q$. Our key technical idea is the use of an imbalanced bipartite Kikuchi graph, which gives a simpler method to analyze spectral refutations of odd arity XOR without using the standard "Cauchy-Schwarz trick", a trick that typically produces random matrices with correlated entries and makes the analysis for odd arity XOR significantly more complicated than even arity XOR. - [187] arXiv:2411.14285 (cross-list from stat.ME) [pdf, html, other]
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Title: Stochastic interventions, sensitivity analysis, and optimal transportComments: 37 pages, 1 figureSubjects: Methodology (stat.ME); Statistics Theory (math.ST)
Recent methodological research in causal inference has focused on effects of stochastic interventions, which assign treatment randomly, often according to subject-specific covariates. In this work, we demonstrate that the usual notion of stochastic interventions have a surprising property: when there is unmeasured confounding, bounds on their effects do not collapse when the policy approaches the observational regime. As an alternative, we propose to study generalized policies, treatment rules that can depend on covariates, the natural value of treatment, and auxiliary randomness. We show that certain generalized policy formulations can resolve the "non-collapsing" bound issue: bounds narrow to a point when the target treatment distribution approaches that in the observed data. Moreover, drawing connections to the theory of optimal transport, we characterize generalized policies that minimize worst-case bound width in various sensitivity analysis models, as well as corresponding sharp bounds on their causal effects. These optimal policies are new, and can have a more parsimonious interpretation compared to their usual stochastic policy analogues. Finally, we develop flexible, efficient, and robust estimators for the sharp nonparametric bounds that emerge from the framework.
- [188] arXiv:2411.14288 (cross-list from cs.LG) [pdf, html, other]
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Title: On the Sample Complexity of One Hidden Layer Networks with Equivariance, Locality and Weight SharingSubjects: Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
Weight sharing, equivariance, and local filters, as in convolutional neural networks, are believed to contribute to the sample efficiency of neural networks. However, it is not clear how each one of these design choices contribute to the generalization error. Through the lens of statistical learning theory, we aim to provide an insight into this question by characterizing the relative impact of each choice on the sample complexity. We obtain lower and upper sample complexity bounds for a class of single hidden layer networks. It is shown that the gain of equivariance is directly manifested in the bound, while getting a similar increase for weight sharing depends on the sharing mechanism. Among our results, we obtain a completely dimension-free bound for equivariant networks for a class of pooling operations. We show that the bound depends merely on the norm of filters, which is tighter than using the spectral norm of the respective matrix. We also characterize the trade-off in sample complexity between the parametrization of filters in spatial and frequency domains, particularly when spatial filters are localized as in vanilla convolutional neural networks.
- [189] arXiv:2411.14292 (cross-list from quant-ph) [pdf, html, other]
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Title: Hypothesis testing of symmetry in quantum dynamicsComments: 14 pagesSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT)
Symmetry plays a crucial role in quantum physics, dictating the behavior and dynamics of physical systems. In this paper, We develop a hypothesis-testing framework for quantum dynamics symmetry using a limited number of queries to the unknown unitary operation and establish the quantum max-relative entropy lower bound for the type-II error. We construct optimal ancilla-free protocols that achieve optimal type-II error probability for testing time-reversal symmetry (T-symmetry) and diagonal symmetry (Z-symmetry) with limited queries. Contrasting with the advantages of indefinite causal order strategies in various quantum information processing tasks, we show that parallel, adaptive, and indefinite causal order strategies have equal power for our tasks. We establish optimal protocols for T-symmetry testing and Z-symmetry testing for 6 and 5 queries, respectively, from which we infer that the type-II error exhibits a decay rate of $\mathcal{O}(m^{-2})$ with respect to the number of queries $m$. This represents a significant improvement over the basic repetition protocols without using global entanglement, where the error decays at a slower rate of $\mathcal{O}(m^{-1})$.
- [190] arXiv:2411.14302 (cross-list from cond-mat.quant-gas) [pdf, html, other]
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Title: Electrodynamics of Vortices in Quasi-2D Scalar Bose-Einstein CondensatesComments: 15 pages, 1 figureSubjects: Quantum Gases (cond-mat.quant-gas); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Plasma Physics (physics.plasm-ph); Quantum Physics (quant-ph)
In two spatial dimensions, vortex-vortex interactions approximately vary with the logarithm of the inter-vortex distance, making it possible to describe an ensemble of vortices as a Coulomb gas. We introduce a duality between vortices in a quasi-two-dimensional (quasi-2D) scalar Bose-Einstein condensates (BEC) and effective Maxwell's electrodynamics. Specifically, we address the general scenario of inhomogeneous, time-dependent BEC number density with dissipation or rotation. Starting from the Gross-Pitaevskii equation (GPE), which describes the mean-field dynamics of a quasi-2D scalar BEC without dissipation, we show how to map vortices in a quasi-2D scalar BEC to 2D electrodynamics beyond the point-vortex approximation, even when dissipation is present or in a rotating system. The physical meaning of this duality is discussed.
- [191] arXiv:2411.14317 (cross-list from cond-mat.stat-mech) [pdf, html, other]
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Title: Model-free learning of probability flows: Elucidating the nonequilibrium dynamics of flockingSubjects: Statistical Mechanics (cond-mat.stat-mech); Machine Learning (cs.LG); Probability (math.PR)
Active systems comprise a class of nonequilibrium dynamics in which individual components autonomously dissipate energy. Efforts towards understanding the role played by activity have centered on computation of the entropy production rate (EPR), which quantifies the breakdown of time reversal symmetry. A fundamental difficulty in this program is that high dimensionality of the phase space renders traditional computational techniques infeasible for estimating the EPR. Here, we overcome this challenge with a novel deep learning approach that estimates probability currents directly from stochastic system trajectories. We derive a new physical connection between the probability current and two local definitions of the EPR for inertial systems, which we apply to characterize the departure from equilibrium in a canonical model of flocking. Our results highlight that entropy is produced and consumed on the spatial interface of a flock as the interplay between alignment and fluctuation dynamically creates and annihilates order. By enabling the direct visualization of when and where a given system is out of equilibrium, we anticipate that our methodology will advance the understanding of a broad class of complex nonequilibrium dynamics.
- [192] arXiv:2411.14329 (cross-list from hep-th) [pdf, html, other]
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Title: Peierls substitution and Hall motion in exotic Carroll dynamicsComments: 29 pages, 2 figuresSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
The particle with first-order dynamics proposed by Dunne, Jackiw and Trugenberger (DJT) to justify the "Peierls substitution" is obtained by reduction from both of two-parameter centrally extended Galilean and Carroll systems. In the latter case the extension parameters $\kappa_{exo}$ and $\kappa_{mag}$ generate non-commutativity of the coordinates resp. behave as an internal magnetic field. The position and momentum follow uncoupled anomalous Hall motions. Consistently with partial immobility, one of the Carroll boost generators is broken but the other remains a symmetry. Switching off $\kappa_{exo}$, the immobility of unextended Carroll particles is recovered. The Carroll system is dual to an uncharged anyon on the horizon of a black hole which exhibits the spin-Hall effect.
- [193] arXiv:2411.14361 (cross-list from cs.CC) [pdf, html, other]
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Title: Improved Lower Bounds for all Odd-Query Locally Decodable CodesSubjects: Computational Complexity (cs.CC); Combinatorics (math.CO)
We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in a sequence of prior works. For $q=3$, the above bound was achieved in a recent work of Alrabiah, Guruswami, Kothari and Manohar using an argument that crucially exploits known exponential lower bounds for $2$-LDCs. Their strategy hits an inherent bottleneck for $q \geq 5$.
Our key insight is identifying a general sufficient condition on the hypergraph of local decoding sets called $t$-approximate strong regularity. This condition demands that 1) the number of hyperedges containing any given subset of vertices of size $t$ (i.e., its co-degree) be equal to the same but arbitrary value $d_t$ up to a multiplicative constant slack, and 2) all other co-degrees be upper-bounded relative to $d_t$. This condition significantly generalizes related proposals in prior works that demand absolute upper bounds on all co-degrees.
We give an argument based on spectral bounds on Kikuchi Matrices that lower bounds the blocklength of any LDC whose local decoding sets satisfy $t$-approximate strong regularity for any $t \leq q$. Crucially, unlike prior works, our argument works despite having no non-trivial absolute upper bound on the co-degrees of any set of vertices. To apply our argument to arbitrary $q$-LDCs, we give a new, greedy, approximate strong regularity decomposition that shows that arbitrary, dense enough hypergraphs can be partitioned (up to a small error) into approximately strongly regular pieces satisfying the required relative bounds on the co-degrees. - [194] arXiv:2411.14390 (cross-list from cond-mat.dis-nn) [pdf, html, other]
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Title: Persistent Homology for Structural Characterization in Disordered SystemsComments: 19 pages, 17 figuresSubjects: Disordered Systems and Neural Networks (cond-mat.dis-nn); Materials Science (cond-mat.mtrl-sci); Machine Learning (cs.LG); Mathematical Physics (math-ph); Algebraic Topology (math.AT)
We propose a unified framework based on persistent homology (PH) to characterize both local and global structures in disordered systems. It can simultaneously generate local and global descriptors using the same algorithm and data structure, and has shown to be highly effective and interpretable in predicting particle rearrangements and classifying global phases. Based on this framework, we define a non-parametric metric, the Separation Index (SI), which not only outperforms traditional bond-orientational order parameters in phase classification tasks but also establishes a connection between particle environments and the global phase structure. Our methods provide an effective framework for understanding and analyzing the properties of disordered materials, with broad potential applications in materials science and even wider studies of complex systems.
- [195] arXiv:2411.14396 (cross-list from hep-th) [pdf, other]
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Title: Topological Twisting of 4d $\mathcal{N}=2$ Supersymmetric Field TheoriesComments: 46 pages + appendices = 97 pages, 4 figuresSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Algebraic Topology (math.AT); Differential Geometry (math.DG); Geometric Topology (math.GT)
We discuss what topological data must be provided to define topologically twisted partition functions of four-dimensional $\mathcal{N}=2$ supersymmetric field theories. The original example of Donaldson-Witten theory depends only on the diffeomorphism type of the spacetime and 't Hooft fluxes (characteristic classes of background gerbe connections, a.k.a. "one-form symmetry connections.") The example of $\mathcal{N}=2^*$ theories shows that, in general, the twisted partition functions depend on further topological data. We describe topological twisting for general four-dimensional $\mathcal{N}=2$ theories and argue that the topological partition functions depend on (a): the diffeomorphism type of the spacetime, (b): the characteristic classes of background gerbe connections and (c): a "generalized spin-c structure," a concept we introduce and define. The main ideas are illustrated with both Lagrangian theories and class $\mathcal{S}$ theories. In the case of class $\mathcal{S}$ theories of $A_1$ type, we note that the different $S$-duality orbits of a theory associated with a fixed UV curve $C_{g,n}$ can have different topological data.
- [196] arXiv:2411.14424 (cross-list from cs.LG) [pdf, html, other]
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Title: Learning Fair Robustness via Domain MixupSubjects: Machine Learning (cs.LG); Cryptography and Security (cs.CR); Information Theory (cs.IT)
Adversarial training is one of the predominant techniques for training classifiers that are robust to adversarial attacks. Recent work, however has found that adversarial training, which makes the overall classifier robust, it does not necessarily provide equal amount of robustness for all classes. In this paper, we propose the use of mixup for the problem of learning fair robust classifiers, which can provide similar robustness across all classes. Specifically, the idea is to mix inputs from the same classes and perform adversarial training on mixed up inputs. We present a theoretical analysis of this idea for the case of linear classifiers and show that mixup combined with adversarial training can provably reduce the class-wise robustness disparity. This method not only contributes to reducing the disparity in class-wise adversarial risk, but also the class-wise natural risk. Complementing our theoretical analysis, we also provide experimental results on both synthetic data and the real world dataset (CIFAR-10), which shows improvement in class wise disparities for both natural and adversarial risks.
Cross submissions (showing 32 of 32 entries)
- [197] arXiv:1309.3940 (replaced) [pdf, html, other]
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Title: The convergence Newton polygon of a $p$-adic differential equation V : local index theoremsComments: 88 pagesSubjects: Number Theory (math.NT)
In this paper and its sequel we consider locally-free $\mathscr{O}_X$-modules together with a connection over a quasi-smooth Berkovich curve $X$. We obtain necessary and sufficient conditions for the finite dimensionality of their de Rham cohomology over local domains such as disks and annuli. We deal with both analytic and meromorphic connections and we derive index formulas relating the index to the behavior of the radii of convergence of their solutions at the boundary of the curve $X$. We introduce the notion of absolute local index. We prove that it is an intrinsic notion extending the previous notions of Robba's generalized index and that of $p$-adic exponents. This condition arises at the boundary of the curve $X$ and it is an exact condition for the finite dimensionality of the de Rham cohomology. We derive comparison results between formal, meromorphic and analytic de Rham cohomologies.
- [198] arXiv:1809.00603 (replaced) [pdf, html, other]
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Title: Uniform boundedness on extremal subsets in Alexandrov spacesComments: rewritten introduction, added references, and other minor changes; to appear in J. Geom. AnalSubjects: Differential Geometry (math.DG); Metric Geometry (math.MG)
In this paper, we study extremal subsets in Alexandrov spaces with dimension $n$, curvature $\ge\kappa$, and diameter $\le D$. We show that the following three quantities are uniformly bounded above in terms of $n$, $\kappa$, and $D$: (1) the number of extremal subsets in an Alexandrov space; (2) the Betti numbers of an extremal subset; (3) the volume of an extremal subset. The proof is an application of essential coverings introduced by Yamaguchi.
- [199] arXiv:2012.10915 (replaced) [pdf, html, other]
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Title: On the formality of nearly K\"ahler manifolds and of Joyce's examples in $G_2$-holonomyComments: 19 pagesJournal-ref: Moscow Math. J. 24:4 (2024), 495-512Subjects: Differential Geometry (math.DG); Algebraic Topology (math.AT)
It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kähler manifolds. A connection between these two results can be found in the fact that both "special holonomy" and "nearly Kähler" naturally generalize compact Kähler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.
- [200] arXiv:2103.16557 (replaced) [pdf, other]
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Title: Diamantine Picard functors of rigid spacesComments: v4: This is the accepted version. v3: Sections 4,5 of v1 now part of arXiv:2207.13657Subjects: Algebraic Geometry (math.AG)
For a connected smooth proper rigid space $X$ over a perfectoid field extension of $\mathbb Q_p$, we show that the étale Picard functor of $X$ defined on perfectoid test objects is the diamondification of the rigid analytic Picard functor. In particular, it is represented by a rigid analytic group variety if and only if the rigid analytic Picard functor is.
Second, we study the $v$-Picard functor that parametrises line bundles in the finer $v$-topology on the diamond associated to $X$ and relate this to the rigid analytic Picard functor by a geometrisation of the multiplicative Hodge--Tate sequence.
The motivation is an application to the $p$-adic Simpson correspondence, namely our results pave the way towards the first instance of a new moduli theoretic perspective. - [201] arXiv:2104.14678 (replaced) [pdf, html, other]
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Title: Locally moving groups and laminar actions on the lineComments: 205 pages, 11 figures; v2 is a major revision after report: title changed (previously 'Locally moving groups acting on the line and $\mathbb{R}$-focal actions'), structure reworked (chapters organized into 3 parts, each devoted to a single main theorem), many results strengthend to nearly optimal statements (requiring different approaches), digressions removed. To appear as an Astérisque volumeSubjects: Group Theory (math.GR); Dynamical Systems (math.DS)
We prove various results that, given a sufficiently rich subgroup $G$ of the group of homeomorphisms on the real line, describe the structure of the other possible actions of $G$ on the line, and address under which conditions such actions must be semi-conjugate to the natural defining action of $G$. The main assumption is that $G$ should be locally moving, meaning that for every open interval the subgroup of elements fixing pointwise its complement, acts on it without fixed points. We show that when $G$ is a locally moving group, every $C^1$ action of $G$ on the real line is semi-conjugate to its standard action or to a non-faithful action. The situation is much wilder when considering actions by homeomorphisms: for a large class of groups, we describe uncountably many conjugacy classes of faithful minimal actions. Next, we prove structure theorems for $C^0$ actions, based on the study of laminar actions, which are actions on the line preserving a lamination. When $G$ is a group of homeomorphisms of the line acting minimally, and with a non-trivial compactly supported element, then any faithful minimal action of $G$ on the line is either laminar or conjugate to its standard action. Moreover, when $G$ is a locally moving group with a suitable finite generation condition, for any faithful minimal laminar action there is a map from the lamination to the line, called a horograding, which is equivariant with respect to the action on the lamination and the standard one, and with some extra suitable conditions. This establishes a tight relation between all minimal actions on the line of such groups, and their standard actions. Finally, based on an analysis of the space of harmonic actions, we show that for a large class of locally moving groups, the standard action is locally rigid, in the sense that sufficiently small perturbations in the compact-open topology give semi-conjugate actions.
- [202] arXiv:2105.05230 (replaced) [pdf, html, other]
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Title: Line bundles on perfectoid covers: case of good reductionComments: Added Thm 5.16; minor edits reflecting new/updated literatureSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We study Picard groups and Picard functors of perfectoid spaces which are limits of rigid spaces. For sufficiently large covers that are limits of rigid spaces of good reduction, we show that the Picard functor can be represented by the special fibre. We use our results to answer several open questions about Picard groups of perfectoid spaces from the literature, for example we show that these are not always $p$-divisible. Along the way, we construct a "Hodge--Tate spectral sequence for $\mathbb G_m$" of independent interest.
- [203] arXiv:2106.13138 (replaced) [pdf, html, other]
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Title: Generalized indefinite strings with purely discrete spectrumComments: This version includes an additional appendix that provides explicit two-sided estimates on Schatten-von Neumann norms that are needed in arXiv:2310.06658Journal-ref: "From Complex Analysis to Operator Theory: A Panorama. In Memory of Sergey Naboko", M. Brown (ed.) et al., Oper. Theory Adv. Appl. 291, 435-474 (2023)Subjects: Spectral Theory (math.SP)
We establish criteria for the spectrum of a generalized indefinite string to be purely discrete and to satisfy Schatten-von Neumann properties. The results can be applied to the isospectral problem associated with the conservative Camassa-Holm flow and to Schrödinger operators with $\delta'$-interactions.
- [204] arXiv:2205.13823 (replaced) [pdf, html, other]
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Title: Fourier-Stieltjes algebras, decomposable Fourier multipliers and amenabilityComments: 73 pages, revision, Section 5.2, 5.3, 5.4, 5.6 are new, Theorem 1.9 was improved using a strong Iwasawa spitting theoremSubjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
We prove that the Fourier-Stieltjes algebra $\mathrm{B}(G)$ of a discrete group $G$ is isometrically isomorphic to the algebra $\mathfrak{M}^{\infty,\mathrm{dec}}(G)$ of decomposable Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$. In contrast, we show that $\mathfrak{M}^{\infty,\mathrm{dec}}(G) \neq \mathrm{B}(G)$ for some classes of non-discrete locally compact groups, while we prove that $\mathfrak{M}^{\infty,\mathrm{dec}}(G) = \mathrm{B}(G)$ holds for any (unimodular) inner amenable locally compact group. To prove these results, we leverage groupoid theory and investigate the problem of whether a contractive projection exists, preserving complete positivity, from the space of normal completely bounded operators on $\mathrm{VN}(G)$ onto the space $\mathfrak{M}^{\infty,\mathrm{cb}}(G)$ of completely bounded Fourier multipliers. We provide an affirmative solution in the inner amenable case and demonstrate that such projections do not exist for non-amenable connected locally compact groups. Further, we investigate whether the space $\mathfrak{M}^{p,\mathrm{cb}}(G)$ of completely bounded Fourier multipliers on the noncommutative $\mathrm{L}^p$-space $\mathrm{L}^p(\mathrm{VN}(G))$ is complemented in the space of completely bounded operators, where $1 \leq p \leq \infty$. Using doubling metrics on Lie groups and structural results from the solution to Hilbert's fifth problem, we establish that any (unimodular) amenable locally compact group admits compatible bounded projections at the levels $p=1$ and $p=\infty$, which has applications to decomposable Fourier multipliers. Moreover, we present a new characterization of amenability.
- [205] arXiv:2207.02030 (replaced) [pdf, html, other]
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Title: Uniform convergence of the Fleming-Viot process in a hard killing metastable caseSubjects: Probability (math.PR)
We study the long-time convergence of a Fleming-Viot process, in the case where the underlying process is a metastable diffusion killed when it reaches some level set. Through a coupling argument, we establish the long-time convergence of the Fleming-Viot process toward some stationary measure at an exponential rate independent of $N$, the size of the system, as well as uniform in time propagation of chaos estimates.
- [206] arXiv:2207.10554 (replaced) [pdf, html, other]
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Title: Solvability of the Poisson-Dirichlet problem with interior data in $L^{p'}$-Carleson spaces and its applications to the $L^{p}$-regularity problemComments: 69 pages. To appear in the Journal of the European Mathematical SocietySubjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)
We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\operatorname{div} F$, which is defined in terms of an $L^{p'}$ estimate on the non-tangential maximal function, assuming that $\operatorname{dist}(\cdot, \partial \Omega) H$ and $F$ lie in certain $L^{p'}$-Carleson-type spaces, and that the domain $\Omega\subset\mathbb R^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular boundary. In turn, we use this result to show that, in a bounded domain with uniformly $n$-rectifiable boundary that satisfies the corkscrew condition, $L^{p'}$-solvability of the homogeneous Dirichlet problem for an operator $L=-\operatorname{div} A\nabla$ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the $L^p$-regularity problem for the adjoint operator $L^*=-\operatorname{div} A^T \nabla$, where $1/p+1/p'=1$ and $A^T$ is the transpose matrix of $A$. This result for Dahlberg-Kenig-Pipher operators is new even if $\Omega$ is the unit ball, despite the fact that the $L^{p'}$-solvability of the Dirichlet problem for these operators in Lipschitz domains has been known since 2001.
Further novel applications include i) new local estimates for the Green's function and its gradient in rough domains, ii) a local $T1$-type theorem for the $L^{p}$-solvability of the ``Poisson-Regularity problem'', itself equivalent to the $L^{p'}$-solvability of the homogeneous Dirichlet problem, in terms of certain gradient estimates for local landscape functions, and iii) new $L^p$ estimates for the eigenfunctions (and their gradients) of symmetric operators $L$ on bounded rough domains. - [207] arXiv:2209.06079 (replaced) [pdf, html, other]
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Title: Effective upper bounds on the number of resonance in potential scatteringComments: Revised versionJournal-ref: Mathematika, 2024Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Functional Analysis (math.FA)
We prove upper bounds on the number of resonances and eigenvalues of Schrödinger operators $-\Delta+V$ with complex-valued potentials, where $d\geq 3$ is odd. The novel feature of our upper bounds is that they are \emph{effective}, in the sense that they only depend on an exponentially weighted norm of V. Our main focus is on potentials in the Lorentz space $L^{(d+1)/2,1/2}$, but we also obtain new results for compactly supported or pointwise decaying potentials. The main technical innovation, possibly of independent interest, are singular value estimates for Fourier-extension type operators. The obtained upper bounds not only recover several known results in a unified way, they also provide new bounds for potentials which are not amenable to previous methods.
- [208] arXiv:2211.11234 (replaced) [pdf, html, other]
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Title: The measure transfer for subshifts induced by a morphism of free monoidsSubjects: Dynamical Systems (math.DS)
Every non-erasing monoid morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ induces a {\em measure transfer map} $\sigma_X^{\mathcal{M}}: \mathcal{M}(X) \to \mathcal{M}(\sigma(X))$ between the measure cones $\mathcal{M}(X)$ and $\mathcal{M}(\sigma(X))$, associated to any subshift $X \subset \mathcal{A}^{\mathbb{Z}}$ and its image subshift $\sigma(X) \subset \mathcal{B}^{\mathbb{Z}}$ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective \cite{BHL2.8-II}. Furthermore, an efficient technique to compute the value of the transferred measure $\sigma_X^{\mathcal{M}(\mu)}$ on any cylinder $[w]$ (for $w \in \mathcal{B}^*$) is presented.
\smallskip \noindent {\bf Theorem:} If a non-erasing morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ is injective on the shift-orbits of some subshift $X \subset \mathcal{A}^\mathbb{Z}$, then $\sigma^{\mathcal{M}_X}$ is injective.
\smallskip
The assumption on $\sigma$ that it is ``injective on the shift-orbits of $X$'' is strictly weaker than ``recognizable in $X$'', and strictly stronger than ``recognizable for aperiodic points in $X$''. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map $\sigma_X^{\mathcal{M}}$. - [209] arXiv:2211.16664 (replaced) [pdf, other]
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Title: Exotic Dehn twists on sums of two contact 3-manifoldsComments: Final version; accepted in Geometry & TopologySubjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)
We exhibit the first examples of exotic contactomorphisms with infinite order as elements of the contact mapping class group. These are given by certain Dehn twists on the separating sphere in a connected sum of two closed contact 3-manifolds. We detect these by a combination of hard and soft techniques. On the one hand, we make essential use of an invariant for families of contact structures which generalises the Kronheimer--Mrowka contact invariant in monopole Floer homology. We then exploit an h-principle for families of convex spheres in tight contact 3-manifolds, from which we establish a parametric version of Colin's decomposition theorem. As a further application, we also exhibit new exotic 1-parametric phenomena in overtwisted contact 3-manifolds.
- [210] arXiv:2212.02444 (replaced) [pdf, html, other]
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Title: Homotopy type theory as a language for diagrams of $\infty$-logosesSubjects: Category Theory (math.CT); Logic in Computer Science (cs.LO); Logic (math.LO)
We show that certain diagrams of $\infty$-logoses are reconstructed in homotopy type theory extended with some lex, accessible modalities, which enables us to use plain homotopy type theory to reason about not only a single $\infty$-logos but also a diagram of $\infty$-logoses. This also provides a higher dimensional version of Sterling's synthetic Tait computability -- a type theory for higher dimensional logical relations.
- [211] arXiv:2212.03780 (replaced) [pdf, other]
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Title: Multiple Landau level filling for a mean field limit of 2D fermionsJournal-ref: Journal of Mathematical Physics volume: 65 number: 2 pages: 021902 year: 2024Subjects: Mathematical Physics (math-ph); Quantum Gases (cond-mat.quant-gas); Functional Analysis (math.FA); Quantum Physics (quant-ph)
Motivated by the quantum hall effect, we study N two dimensional interacting fermions in a large magnetic field limit. We work in a bounded domain, ensuring finite degeneracy of the Landau levels. In our regime, several levels are fully filled and inert: the density in these levels is constant. We derive a limiting mean-field and semi classical description of the physics in the last, partially filled Landau level.
- [212] arXiv:2302.01202 (replaced) [pdf, html, other]
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Title: Linear independence of coherent systems associated to discrete subgroupsSubjects: Functional Analysis (math.FA)
This note considers the finite linear independence of coherent systems associated to discrete subgroups. We show by simple arguments that such coherent systems of amenable groups are linearly independent whenever the associated twisted group ring does not contain any nontrivial zero divisors. We verify the latter for discrete subgroups in nilpotent Lie groups. For the particular case of time-frequency translates of Euclidean space, our approach provides a simple and self-contained proof of the Heil--Ramanathan--Topiwala (HRT) conjecture for subsets of arbitrary discrete subgroups.
- [213] arXiv:2303.03902 (replaced) [pdf, other]
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Title: An Optimal Minimization Problem In The Lowest Landau Level And Related QuestionsValentin Schwinte (IECL)Subjects: Analysis of PDEs (math.AP)
We solve a minimization problem related to the cubic Lowest Landau level equation, which is used in the study of Bose-Einstein condensation. We provide an optimal condition for the Gaussian to be the unique global minimizer. This extends previous results from P. G{é}rard, P. Germain and L. Thomann. We then provide another condition so that the second special Hermite function is a global minimizer.
- [214] arXiv:2303.10038 (replaced) [pdf, html, other]
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Title: Semilinear Feynman-Kac Formulae for $B$-Continuous Viscosity SolutionsComments: Accepted for publication in Stoch. Anal. ApplSubjects: Probability (math.PR); Analysis of PDEs (math.AP)
We prove the existence of a $B$-continuous viscosity solution for a class of infinite dimensional semilinear partial differential equations (PDEs) using probabilistic methods. Our approach also yields a stochastic representation formula for the solution in terms of a scalar-valued backward stochastic differential equation. The uniqueness is proved under additional assumptions using a comparison theorem for viscosity solutions. Our results constitute the first nonlinear Feynman-Kac formula using the notion of $B$-continuous viscosity solutions and thus introduces a framework allowing for generalizations to the case of fully nonlinear PDEs.
- [215] arXiv:2303.14907 (replaced) [pdf, html, other]
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Title: Weakly invertible cells in a weak $\omega$-categoryComments: 21 pages. Published version. Comments welcome!Journal-ref: Higher Structures 8(2):386-415, 2024Subjects: Category Theory (math.CT)
We study weakly invertible cells in weak $\omega$-categories in the sense of Batanin-Leinster, adopting the coinductive definition of weak invertibility. We show that weakly invertible cells in a weak $\omega$-category are closed under globular pasting. Using this, we generalise elementary properties of weakly invertible cells known to hold in strict $\omega$-categories to weak $\omega$-categories, and show that every weak $\omega$-category has a largest weak $\omega$-subgroupoid.
- [216] arXiv:2304.00699 (replaced) [pdf, html, other]
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Title: $\hat{Z}_b$ for plumbed manifolds and splice diagramsComments: 39 pages, comments welcomeSubjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Quantum Algebra (math.QA)
We study $q$-series invariants of 3-manifolds $\hat{Z}_b$ defined by Gukov--Pei--Putrov--Vafa using techniques from the theory of normal surface singularities such as splice diagrams. This provides a link between algebraic geometry with quantum topology. We show that the (suitably normalized) sum of all $\hat{Z}_b$ depends only on the splice diagram, and in particular, it agrees for manifolds with the same universal Abelian cover. Using these ideas we find simple formulas for $\hat{Z}_b$ invariants of Seifert manifolds that resemble equivariant Poincaré series of corresponding quasihomogeneous singularity. Applications include a better understanding of the vanishing of the $q$-series $\hat{Z}_b$.
- [217] arXiv:2305.06204 (replaced) [pdf, html, other]
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Title: Immersions of directed graphs in tournamentsComments: 11 pages, 2 figures, author accepted manuscript, to appear in Random Structures & AlgorithmsSubjects: Combinatorics (math.CO)
Recently, Draganić, Munhá Correia, Sudakov and Yuster showed that every tournament on $(2+o(1))k^2$ vertices contains a $1$-subdivision of a transitive tournament on $k$ vertices, which is tight up to a constant factor. We prove a counterpart of their result for immersions. Let $f(k)$ be the smallest integer such that any tournament on at least $f(k)$ vertices must contain a $1$-immersion of a transitive tournament on $k$ vertices. We show that $f(k)=O(k)$, which is clearly tight up to a multiplicative factor. If one insists in finding an immersion of a complete directed graph on $k$ vertices then an extra condition on the tournament is necessary. Indeed, we show that every tournament with minimum out-degree at least $Ck$ must contain a $2$-immersion of a complete digraph on $k$ vertices. This is again tight up to the value of $C$ and tight on the order of the paths in the immersion.
- [218] arXiv:2305.09614 (replaced) [pdf, html, other]
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Title: Algebraic periodic points of transcendental entire functionsSubjects: Number Theory (math.NT)
We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, and in addition having a prescribed number of $k$-periodic algebraic orbits, for all $k\geq 1$. Under a suitable topology, such functions are shown to be dense in the set of all entire transcendental functions.
- [219] arXiv:2305.19964 (replaced) [pdf, html, other]
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Title: The list-Ramsey threshold for families of graphsJournal-ref: Combinatorics, Probability and Computing , Volume 33 , Issue 6 , November 2024 , pp. 829 - 851Subjects: Combinatorics (math.CO)
Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey property in the binomial random graph, where $\mathcal{F}$ consists of one graph, was located in the celebrated work of Rödl and Ruciński.
In this paper, we offer a twofold generalisation to the Rödl--Ruciński theorem. First, we show that the list-colouring version of the property has the same threshold. Second, we extend this result to finite families $\mathcal{F}$, where the threshold statements might also diverge. This also confirms further special cases of the Kohayakawa--Kreuter conjecture. Along the way, we supply a short(-ish), self-contained proof of the $0$-statement of the Rödl--Ruciński theorem. - [220] arXiv:2306.11824 (replaced) [pdf, html, other]
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Title: A criterion for absolute continuity relative to the law of fractional Brownian motionComments: 10 pagesSubjects: Probability (math.PR)
Let $X$ be the sum of a fractional Brownian motion with Hurst parameter $H$ and an absolutely continuous and adapted drift process. We establish a simple criterion that guarantees that the law of $X$ is absolutely continuous with respect to the law of the original fractional Brownian motion. For $H<1/2$, the trajectories of the derivative of the drift need to be bounded by an almost surely finite random variable; for $H>1/2$, they need to satisfy a Hölder condition with some exponent larger than $2H-1$. These are almost-sure conditions, and no expectation requirements are imposed. For the case in which $X$ arises as the solution of a nonlinear stochastic integral equation driven by fractional Brownian motion, we provide a simple criterion on the drift coefficient under which the law of $X$ is automatically equivalent to the one of fractional Brownian motion.
- [221] arXiv:2306.15973 (replaced) [pdf, html, other]
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Title: Artin's Primitive Root Conjecture in Number Fields and For MatricesSubjects: Number Theory (math.NT)
In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are a primitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967, Hooley showed that this conjecture follows from the Generalized Riemann Hypothesis (GRH). In this paper we consider variants of the primitive root conjecture for number fields and for matrices. All results are conditional on GRH.
For an algebraic number field $K$ and some element $\alpha \in K$, we examine the order of $\alpha$ modulo various rational primes $p$. We extend previous results of Roskam which only worked for quadratic extensions $K/\mathbb{Q}$ to more general field extensions of higher degree. Specifically, under some constraints on the Galois group of $K/\mathbb{Q}$ and on the element $\alpha\in K$, we show that $\alpha$ is of almost maximal order mod $p$ for almost all rational primes $p$ which factor into primes of degree 2 in $K$.
We also consider Artin's primitive root conjecture for matrices. Given a matrix $A\in\text{GL}_n(\mathbb{Q})$, we examine the order of $A\bmod p$ in $\text{GL}_n(\mathbb{F}_p)$ for various primes $p$, which turns out to be equivalent to the number field setting. - [222] arXiv:2307.11229 (replaced) [pdf, other]
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Title: A Convergent Finite Element Scheme for the Q-Tensor Model of Liquid Crystals Subjected to an Electric FieldSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which we show that it is possible to simulate the Fréedericksz transition with this scheme.
- [223] arXiv:2308.04458 (replaced) [pdf, html, other]
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Title: Numerical calculations of primes in short intervalsSubjects: Number Theory (math.NT)
The author sharpens a result of Baker, Harman and Pintz (2001), showing that the interval $[x - x^{0.524},x]$ contains prime numbers for all large $x$. This gives an affirmative answer to Pintz's argument.
- [224] arXiv:2308.07266 (replaced) [pdf, html, other]
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Title: Full Duplex Joint Communications and Sensing for 6G: Opportunities and ChallengesChandan Kumar Sheemar, Sourabh Solanki, George C. Alexandropoulos, Eva Lagunas, Jorge Querol, Symeon Chatzinotas, Björn OtterstenSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
The paradigm of joint communications and sensing (JCAS) envisions a revolutionary integration of communication and radar functionalities within a unified hardware platform. This novel concept not only opens up unprecedented interoperability opportunities, but also exhibits unique design challenges. To this end, the success of JCAS is highly dependent on efficient full-duplex (FD) operation, which has the potential to enable simultaneous transmission and reception within the same frequency band. While JCAS research is lately expanding, there still exist relevant directions of investigation that hold tremendous potential to profoundly transform the sixth generation (6G), and beyond, cellular networks. This article presents new opportunities and challenges brought up by FD-enabled JCAS, taking into account the key technical peculiarities of FD systems. Unlike simplified JCAS scenarios, we delve into the most comprehensive configuration, encompassing uplink and downlink users, as well as monostatic and bistatic radars, all harmoniously coexisting to jointly push the boundaries of both communications and sensing. The performance improvements resulting from this advancement bring forth numerous new challenges, each meticulously examined and expounded upon.
- [225] arXiv:2310.06821 (replaced) [pdf, html, other]
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Title: Spherical sets avoiding orthonormal basesComments: 8 pages, added a new resultSubjects: Metric Geometry (math.MG); Combinatorics (math.CO)
We show that there exists an absolute constant $c_0<1$ such that for all $n \ge 2$, any set $A \subset S^{n-1}$ of density at least $c_0$ contains $n$ pairwise orthogonal vectors. The result is sharp up to the value of the constant $c_0$.
Moreover, we show that for all $2\le k \le n$ a set $A$ avoiding $k$ pairwise orthogonal vectors has measure at most $\exp(-c \min\{\sqrt{n}, n/k\})$ for some $c>0$. Proofs rely on the harmonic analysis on the sphere and the hypercontractive inequality. - [226] arXiv:2310.10609 (replaced) [pdf, html, other]
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Title: Bounds on the M\"obius-signed partition numbersComments: Updated to reflect grammatical/typographical edits made during publication; statement and proof of Proposition 5.4 have been greatly simplified, allowing for the deletion of the appendix; 35 pages, 1 figureJournal-ref: Ramanujan J. 65 (2024), 81-123Subjects: Number Theory (math.NT)
For $n \in \mathbb{N}$ let $\Pi[n]$ denote the set of partitions of $n$, i.e., the set of positive integer tuples $(x_1,x_2,\ldots,x_k)$ such that $x_1 \geq x_2 \geq \cdots \geq x_k$ and $x_1 + x_2 + \cdots + x_k = n$. Fixing $f:\mathbb{N}\to\{0,\pm 1\}$, for $\pi = (x_1,x_2,\ldots,x_k) \in \Pi[n]$ let $f(\pi) := f(x_1)f(x_2)\cdots f(x_k)$. In this way we define the {signed partition numbers} \[ p(n,f) = \sum_{\pi\in\Pi[n]} f(\pi). \] Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for quantities $p(n,\mu)$ and $p(n,\lambda)$, where $\mu$ and $\lambda$ denote the Möbius and Liouville functions from prime number theory, respectively. In addition we discuss how quantities $p(n,f)$ generalize the classical notion of restricted partitions.
- [227] arXiv:2311.10289 (replaced) [pdf, html, other]
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Title: Singular Trudinger--Moser inequality involving $L^{p}$ norm in bounded domainSubjects: Analysis of PDEs (math.AP)
In this paper, we use the method of blow-up analysis and capacity estimate to derive the singular Trudinger--Moser inequality involving $N$-Finsler--Laplacian and $L^{p}$ norm, precisely, for any $p>1$, $0\leq\gamma<\gamma_{1}:= \inf\limits_{u\in W^{1, N}_{0}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}F^{N}(\nabla u)dx}{\| u\|_p^N}$ and $0\leq\beta<N$, we have \begin{align} \sup_{u\in W_{0}^{1,N}(\Omega),\;\int_{\Omega}F^{N}(\nabla u)dx-\gamma\| u\|_p^N\leq1}\int_{\Omega}\frac{e^{\lambda_{N}(1-\frac{\beta}{N})\lvert u\rvert^{\frac{N}{N-1}}}}{F^{o}(x)^{\beta}}\;\mathrm{d}x<+\infty\notag, \end{align} where $\lambda_{N}=N^{\frac{N}{N-1}} \kappa_{N}^{\frac{1}{N-1}}$ and $\kappa_{N}$ is the volume of a unit Wulff ball in $\mathbb{R}^N$, moreover, extremal functions for the inequality are also obtained. When $F=\lvert\cdot\rvert$ and $p=N$, we can obtain the singular version of Tintarev type inequality by the obove inequality, namely, for any $0\leq\alpha<\alpha_{1}(\Omega):=\inf\limits_{u\in W^{1, N}_{0}(\Omega)\backslash \{0\}}\frac{\int_{\Omega}|\nabla u|^Ndx}{\| u\|_N^N}$ and $0\leq\beta<N$, it holds $$ \sup_{u\in W_{0}^{1,N}(\Omega),\;\int_{\Omega}\lvert\nabla u\rvert^{N}\;\mathrm{d}x-\alpha\|u\|_{N}^{N}\leq1}\int_{\Omega}\frac{e^{\alpha_{N}(1-\frac{\beta}{N})\lvert u\rvert^{\frac{N}{N-1}}}}{\lvert x\rvert^{\beta}}\;\mathrm{d}x<+\infty, $$ where $\alpha_{N}:=N^{\frac{N}{N-1}}\omega_{N}^{\frac{1}{N-1}}$ and $ \omega_{N}$ is the volume of unit ball in $\mathbb{R}^{N}$. Our results extend many well-known Trudinger--Moser type inequalities to more general setting.
- [228] arXiv:2311.11799 (replaced) [pdf, html, other]
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Title: A classification of Mengerian $4$-uniform hypergraphs derived from graphsComments: to appear in Ars CombinatoriaSubjects: Combinatorics (math.CO)
In this paper, we give a classification of all Mengerian $4$-uniform hypergraphs derived from graphs.
- [229] arXiv:2311.15900 (replaced) [pdf, html, other]
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Title: Asymptotic smoothness, concentration properties in Banach spaces and applicationsComments: 11 pages. arXiv admin note: text overlap with arXiv:2106.04297Subjects: Functional Analysis (math.FA); Metric Geometry (math.MG)
We prove an optimal result of stability under $\ell_p$-sums of some concentration properties for Lipschitz maps defined on Hamming graphs into Banach spaces. As an application, we give examples of spaces with Szlenk index arbitrarily high that admit nevertheless a concentration property. In particular, we get the very first examples of Banach spaces with concentration but without asymptotic smoothness property.
- [230] arXiv:2312.01730 (replaced) [pdf, html, other]
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Title: Set-valued stochastic integrals for convoluted L\'{e}vy processesComments: 28 pagesSubjects: Probability (math.PR); Mathematical Finance (q-fin.MF)
In this paper we study set-valued Volterra-type stochastic integrals driven by Lévy processes. Upon extending the classical definitions of set-valued stochastic integral functionals to convoluted integrals with square-integrable kernels, set-valued convoluted stochastic integrals are defined by taking the closed decomposable hull of the integral functionals for generic time. We show that, aside from well-established results for set-valued Itô integrals, while set-valued stochastic integrals with respect to a finite-variation Poisson random measure are guaranteed to be integrably bounded for bounded integrands, this is not true when the random measure is of infinite variation. For indefinite integrals, we prove that it is a mutual effect of kernel singularity and jumps that the set-valued convoluted integrals are possibly explosive and take extended vector values. These results have some important implications on how set-valued fractional dynamical systems are to be constructed in general. Two classes of set-monotone processes are studied for practical interests in economic and financial modeling.
- [231] arXiv:2312.10495 (replaced) [pdf, html, other]
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Title: Computing Optimal Joint Chance Constrained Control PoliciesSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
We consider the problem of optimally controlling stochastic, Markovian systems subject to joint chance constraints over a finite-time horizon. For such problems, standard Dynamic Programming is inapplicable due to the time correlation of the joint chance constraints, which calls for non-Markovian, and possibly stochastic, policies. Hence, despite the popularity of this problem, solution approaches capable of providing provably-optimal and easy-to-compute policies are still missing. We fill this gap by augmenting the dynamics via a binary state, allowing us to characterize the optimal policies and develop a Dynamic Programming based solution method.
- [232] arXiv:2312.10853 (replaced) [pdf, html, other]
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Title: Diophantine avoidance and small-height primitive elements in ideals of number fieldsComments: 17 pages; to appear in Combinatorics and Number TheorySubjects: Number Theory (math.NT)
Let $K$ be a number field of degree $d$. Then every ideal $I$ in the ring of integers ${\mathcal O}_K$ contains infinitely many primitive elements, i.e. elements of degree $d$. A bound on smallest height of such an element in $I$ follows from some recent developments in the direction of a 1998 conjecture of W. Ruppert. We prove a very explicit bound like this in the case of quadratic fields. Further, we consider primitive elements in an ideal outside of a finite union of other ideals and prove a bound on the height of a smallest such element. Our main tool is a result on points of small norm in a lattice outside of an algebraic hypersurface and a finite union of sublattices of finite index, which we prove by blending two previous Diophantine avoidance results. We also obtain an avoidance result like this for lattice points in the positive orthant in $\mathbb{R}^d$ and use it to obtain a small-height totally positive primitive element in an ideal of a totally real number field outside of a finite union of other ideals. Additionally, we use our avoidance method to prove a bound on the Mahler measure of a generating non-sparse polynomial for a given number field. Finally, we produce a bound on the height of a smallest primitive generator for a principal ideal in a quadratic number field.
- [233] arXiv:2401.01127 (replaced) [pdf, html, other]
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Title: Wireless 6G Connectivity for Massive Number of Devices and Critical ServicesAnders E. Kalør, Giuseppe Durisi, Sinem Coleri, Stefan Parkvall, Wei Yu, Andreas Mueller, Petar PopovskiComments: Accepted to Proceedings of the IEEE. 19 pages, 8 figuresSubjects: Information Theory (cs.IT)
Compared to the generations up to 4G, whose main focus was on broadband and coverage aspects, 5G has expanded the scope of wireless cellular systems towards embracing two new types of connectivity: massive machine-type communication (mMTC) and ultra-reliable low-latency communications (URLLC). This paper discusses the possible evolution of these two types of connectivity within the umbrella of 6G wireless systems. The paper consists of three parts. The first part deals with the connectivity for a massive number of devices. While mMTC research in 5G predominantly focuses on the problem of uncoordinated access in the uplink for a large number of devices, the traffic patterns in 6G may become more symmetric, leading to closed-loop massive connectivity. One of the drivers for this is distributed learning/inference. The second part of the paper discusses the evolution of wireless connectivity for critical services. While latency and reliability are tightly coupled in 5G, 6G will support a variety of safety critical control applications with different types of timing requirements, as evidenced by the emergence of metrics related to information freshness and information value. Additionally, ensuring ultra-high reliability for safety critical control applications requires modeling and estimation of the tail statistics of the wireless channel, queue length, and delay. The fulfillment of these stringent requirements calls for the development of novel AI-based techniques, incorporating optimization theory, explainable AI, generative AI and digital twins. The third part analyzes the coexistence of massive connectivity and critical services. We will consider scenarios in which a massive number of devices need to support traffic patterns of mixed criticality. This is followed by a discussion about the management of wireless resources shared by services with different criticality.
- [234] arXiv:2401.07557 (replaced) [pdf, other]
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Title: Conjugacy in finite classical groupsSubjects: Group Theory (math.GR)
Let $G$ be a classical group defined over a finite field. We consider the following fundamental problems concerning conjugacy in $G$:
1. List a representative for each conjugacy class of $G$.
2. Given $x \in G$, describe the centralizer of $x$ in $G$, by giving its group structure and a generating set.
3. Given $x,y \in G$, establish whether $x$ and $y$ are conjugate in $G$ and, if so, then find explicit $z \in G$ such that $z^{-1}xz = y$.
We present comprehensive theoretical solutions to all three problems, and use our solutions to formulate practical algorithms. In parallel to our theoretical work, we have developed in Magma complete implementations of our algorithms. They form a critical component of various general algorithms in computational group theory - for example, computing character tables and solving conjugacy problems in arbitrary finite groups. - [235] arXiv:2401.11557 (replaced) [pdf, html, other]
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Title: CAT(0) cube complexes and asymptotically rigid mapping class groupsSubjects: Group Theory (math.GR); Geometric Topology (math.GT)
In this article we study the asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. We present a family of CAT(0) cube complexes on which the latter groups act. Along the way, we determine in which cases the cube complexes introduced earlier by Genevois, Lonjou and Urech are CAT(0).
- [236] arXiv:2401.12598 (replaced) [pdf, other]
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Title: Asymptotic confidence interval for R2 in multiple linear regressionSubjects: Statistics Theory (math.ST)
Following White's approach of robust multiple linear regression, we give asymptotic confidence intervals for the multiple correlation coefficient R2 under minimal moment conditions. We also give the asymptotic joint distribution of the empirical estimators of the individual R2's. Through different sets of simulations, we show that the procedure is indeed robust (contrary to the procedure involving the near exact distribution of the empirical estimator of R2 is the multivariate Gaussian case) and can be also applied to count linear regression.
- [237] arXiv:2402.00836 (replaced) [pdf, html, other]
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Title: The Primitive Comparison Theorem in characteristic $p$Comments: This is the accepted versionSubjects: Algebraic Geometry (math.AG)
We prove an analogue of Scholze's Primitive Comparison Theorem for proper rigid spaces over an algebraically closed non-archimedean field $K$ of characteristic $p$. This implies a v-topological version of the Primitive Comparison Theorem for proper finite type morphisms $f:X\to Y$ of analytic adic spaces over $\mathbb Z_p$. We deduce new cases of the Proper Base Change Theorem for $p$-torsion coefficients and the Künneth formula in this setting.
- [238] arXiv:2402.05905 (replaced) [pdf, html, other]
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Title: Slices of Stable Polynomials and Connections to the Grace-Walsh-Szeg\H{o} theoremComments: 14 pagesSubjects: Algebraic Geometry (math.AG)
Univariate polynomials are called stable with respect to a circular region $\mathcal{A}$, if all of their roots are in $\mathcal{A}$. We consider the special case where $\mathcal{A}$ is a half-plane and investigate affine slices of the set of stable polynomials. In this setup, we show that an affine slice of codimension $k$ always contains a stable polynomial that possesses at most $2(k+2)$ distinct roots on the boundary and at most $(k+2)$ distinct roots in the interior of $\mathcal{A}$. This result also extends to affine slices of weakly Hurwitz polynomials. Subsequently, we apply these results to symmetric polynomials and varieties. Here we show that it is necessary and sufficient for a variety described by polynomials in few multiaffine polynomials to contain points in $\mathcal{A}^n$ with few distinct coordinates for its intersection with $\mathcal{A}^n$ being non-empty. This is at the same time a generalization of the degree principle to stable polynomials and a result similar to Grace-Walsh-Szegő's coincidence theorem on multiaffine symmetric polynomials.
- [239] arXiv:2402.10126 (replaced) [pdf, html, other]
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Title: Exchangeability, prediction and predictive modeling in Bayesian statisticsSubjects: Statistics Theory (math.ST)
There is currently a renewed interest in the Bayesian predictive approach to statistics. This paper offers a review on foundational concepts and focuses on predictive modeling, which by directly reasoning on prediction, bypasses inferential models or may characterize them. We detail predictive characterizations in exchangeable and partially exchangeable settings, for a large variety of data structures, and hint at new directions. The underlying concept is that Bayesian predictive rules are probabilistic learning rules, formalizing through conditional probability how we learn on future events given the available information. This concept has implications in any statistical problem and in inference, from classic contexts to less explored challenges, such as providing Bayesian uncertainty quantification to predictive algorithms in data science, as we show in the last part of the paper. The paper gives a historical overview, but also includes a few new results, presents some recent developments and poses some open questions.
- [240] arXiv:2402.13286 (replaced) [pdf, other]
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Title: Mass-energy Scattering Criterion For Double Power Schr{\"o}dinger EquationsThomas Duyckaerts (LAGA, DMA), Phan van Tin (LAGA)Subjects: Analysis of PDEs (math.AP)
We consider the nonlinear Schr{ö}dinger equation with double power nonlinearity. We extend the scattering result in [17] for all L 2-supercritical powers, specially, our results adapt to the cases of energy-supercritical nonlinearity.
- [241] arXiv:2402.16207 (replaced) [pdf, html, other]
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Title: Poset polytopes and pipe dreams: types C and BSubjects: Representation Theory (math.RT); Algebraic Geometry (math.AG); Combinatorics (math.CO)
The first part of this paper concerns type C. We present new explicitly defined families of algebro-combinatorial structures of three kinds: combinatorial bases in representations, Newton--Okounkov bodies of flag varieties and toric degenerations of flag varieties. All three families are parametrized by the same family of polytopes: the marked chain-order polytopes of Fang and Fourier which interpolate between the type C Gelfand--Tsetlin and FFLV polytopes. Thus, in each case the obtained structures interpolate between the well-known bases, Newton--Okounkov bodies or degenerations associated with the latter two polytopes. We then obtain similar results for type B after introducing a new family of poset polytopes to be considered in place of marked chain-order polytopes. In both types our constructions and proofs rely crucially on a combinatorial connection between poset polytopes and pipe dreams.
- [242] arXiv:2403.08884 (replaced) [pdf, html, other]
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Title: A note on spectral properties of random $S$-adic systemsComments: Revision after the referee report. To appear in the volume of Pure and Applied Functional Analysis dedicated to the memory of this http URLSubjects: Dynamical Systems (math.DS)
The paper is concerned with random $S$-adic systems arising from an i.i.d. sequence of unimodular substitutions. Using equidistribution results of Benoist and Quint, we show in Theorem 3.3 that, under some natural assumptions, if the Lyapunov exponent of the spectral cocycle is strictly less that 1/2 of the Lyapunov exponent of the random walk on $SL(2,\mathbb{R})$ driven by the sequence of substitution matrices, then almost surely the spectrum of the $S$-adic $\mathbb{Z}$-action is singular with respect to any (fixed in advance) continuous measure. Finally, the appendix discusses the weak-mixing property for random $S$-adic systems associated to the family of substitutions introduced in Example 4.2.
- [243] arXiv:2403.12324 (replaced) [pdf, html, other]
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Title: Towards a Theory of Pragmatic InformationComments: 15 pages, no figuresSubjects: Information Theory (cs.IT)
Standard information theory says nothing about how much meaning is conveyed by a message. We fill this gap with a rigorously justifiable, quantitative definition of ``pragmatic information'', the amount of meaning in a message relevant to a particular decision. We posit that such a message updates a random variable, $\omega$, that informs the decision. The pragmatic information of a single message is then defined as the Kulbach-Leibler divergence between the apriori and aposteriori probabilities of $\omega$; the pragmatic information of a message ensemble is the expected value of the pragmatic information of the ensemble's component messages. We justify these definitions by proving that the pragmatic information of a single message is the expected difference between the shortest binary encoding of $\omega$ under the a priori and a posteriori distributions, and that the average of the pragmatic values of individual messages, when sampled a large number of times from the ensemble, approaches its expected value.
Pragmatic information is non-negative and additive for independent decisions and ``pragmatically independent'' messages. Also, pragmatic information is the information analogue of free energy: just as free energy quantifies the part of a system's total energy available to do useful work, so pragmatic information quantifies the information actually used in making a decision.
We sketch 3 applications: the single play of a slot machine, a.k.a. a ``one armed bandit'', with an unknown payout probability; a characterization of the rate of biological evolution in the so-called ``quasi-species'' model; and a reformulation of the efficient market hypothesis of finance. We note the importance of the computational capacity of the receiver in each case. - [244] arXiv:2403.18727 (replaced) [pdf, html, other]
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Title: Modular representations of the Yangian $Y_2$Comments: 23 pages, final version, to appear in JLMSSubjects: Representation Theory (math.RT)
Let $Y_2$ be the Yangian associated to the general linear Lie algebra $\mathfrak{gl}_2$, defined over an algebraically closed field $\mathbbm{k}$ of characteristic $p > 0$. In this paper, we study the representation theory of the restricted Yangian $Y^{[p]}_2$. This leads to a description of the representations of $\mathfrak{gl}_{2n}$, whose $p$-character is nilpotent with Jordan type given by a two-row partition $(n, n)$.
- [245] arXiv:2403.19020 (replaced) [pdf, html, other]
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Title: The sticky particle dynamics of the 1D pressureless Euler-alignment system as a gradient flowComments: 34 pages, 6 figuresSubjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)
We show how the sticky dynamics for the one-dimensional pressureless Euler-alignment system can be obtained as an $L^2$-gradient flow of a convex functional. This is analogous to the Lagrangian evolution introduced by Natile and Savaré for the pressureless Euler system, and by Brenier et al. for the corresponding system with a self-interacting force field. Our Lagrangian evolution can be seen as the limit of sticky particle Cucker-Smale dynamics, similar to the solutions obtained by Leslie and Tan from a corresponding scalar balance law, and provides us with a uniquely determined distributional solution of the original system in the space of probability measures with quadratic moments and corresponding square-integrable velocities. Moreover, we show that the gradient flow also provides an entropy solution to the balance law of Leslie and Tan, and how their results on cluster formation follow naturally from (non-)monotonicity properties of the so-called natural velocity of the flow.
- [246] arXiv:2404.00824 (replaced) [pdf, html, other]
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Title: Identifying a piecewise affine signal from its nonlinear observation -- application to DNA replication analysisComments: 25 pages, 11 figuresSubjects: Optimization and Control (math.OC)
DNA replication stands as one of the fundamental biological processes crucial for cellular functioning. Recent experimental developments enable the study of replication dynamics at the single-molecule level for complete genomes, facilitating a deeper understanding of its main parameters. In these new data, replication dynamics is reported by the incorporation of an exogenous chemical, whose intra-cellular concentration follows a nonlinear function. The analysis of replication traces thus gives rise to a nonlinear inverse problem, presenting a nonconvex optimization challenge. We demonstrate that under noiseless conditions, the replication dynamics can be uniquely identified by the proposed model. Computing a global solution to this optimization problem is specially challenging because of its multiple local minima. We present the DNA-inverse optimization method that is capable of finding this global solution even in the presence of noise. Comparative analysis against state-of-the-art optimization methods highlights the superior computational efficiency of our approach. DNA-inverse enables the automatic recovery of all configurations of the replication dynamics, which was not possible with previous methods.
- [247] arXiv:2404.01756 (replaced) [pdf, html, other]
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Title: Antiparticles in non-relativistic quantum mechanicsComments: 18 pages; v3: some clarificationsSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Quantum Physics (quant-ph)
Non-relativistic quantum mechanics was originally formulated to describe particles. Using ideas from the geometric quantization approach, we show how the concept of antiparticles can and should be introduced in the non-relativistic case without appealing to quantum field theory. We discuss this in detail using the example of the one-dimensional harmonic oscillator.
- [248] arXiv:2404.05454 (replaced) [pdf, html, other]
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Title: B-ary Tree Push-Pull Method is Provably Efficient for Distributed Learning on Heterogeneous DataSubjects: Optimization and Control (math.OC)
This paper considers the distributed learning problem where a group of agents cooperatively minimizes the summation of their local cost functions based on peer-to-peer communication. Particularly, we propose a highly efficient algorithm, termed ``B-ary Tree Push-Pull'' (BTPP), that employs two B-ary spanning trees for distributing the information related to the parameters and stochastic gradients across the network. The simple method is efficient in communication since each agent interacts with at most $(B+1)$ neighbors per iteration. More importantly, BTPP achieves linear speedup for smooth nonconvex and strongly convex objective functions with only $\tilde{O}(n)$ and $\tilde{O}(1)$ transient iterations, respectively, significantly outperforming the state-of-the-art results to the best of our knowledge. Our code is available at this https URL.
- [249] arXiv:2404.11971 (replaced) [pdf, html, other]
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Title: Finite-zone PT-potentialsComments: 19 pagesSubjects: Spectral Theory (math.SP); Mathematical Physics (math-ph)
We give a description of finite-zone PT-potentials in terms of explicit theta functional formulas.
- [250] arXiv:2404.16583 (replaced) [pdf, html, other]
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Title: Fast Machine-Precision Spectral Likelihoods for Stationary Time SeriesSubjects: Numerical Analysis (math.NA); Computation (stat.CO); Methodology (stat.ME)
We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in $\mathcal{O}(n \log n)$ time with applications to fitting time series models. In particular, for a symmetric Toeplitz matrix $\mathbf{\Sigma}$ with values $\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$ where $S(\omega)$ is piecewise smooth, we give an approximation $\mathbf{\mathcal{F}} \mathbf{\Sigma} \mathbf{\mathcal{F}}^H \approx \mathbf{D} + \mathbf{U} \mathbf{V}^H$, where $\mathbf{\mathcal{F}}$ is the DFT matrix, $\mathbf{D}$ is diagonal, and the matrices $\mathbf{U}$ and $\mathbf{V}$ are in $\mathbb{C}^{n \times r}$ with $r \ll n$. Studying these matrices in the context of time series, we offer a theoretical explanation of this structure and connect it to existing spectral-domain approximation frameworks. We then give a complete discussion of the numerical method for assembling the approximation and demonstrate its efficiency for improving Whittle-type likelihood approximations, including dramatic examples where a correction of rank $r = 2$ to the standard Whittle approximation increases the accuracy of the log-likelihood approximation from $3$ to $14$ digits for a matrix $\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$. The method and analysis of this work applies well beyond time series analysis, providing an algorithm for extremely accurate solutions to linear systems with a wide variety of symmetric Toeplitz matrices whose entries are generated by a piecewise smooth $S(\omega)$. The analysis employed here largely depends on asymptotic expansions of oscillatory integrals, and also provides a new perspective on when existing spectral-domain approximation methods for Gaussian log-likelihoods can be particularly problematic.
- [251] arXiv:2404.17891 (replaced) [pdf, html, other]
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Title: Isotopy classification of Morse polynomials of degree 3 in ${\mathbb R}^3$Subjects: Algebraic Topology (math.AT)
We enumerate all isotopy classes of Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ of degree three with non-singular principal homogeneous part, in particular prove that there are exactly 37 of them. We also count all 2258 isotopy classes of {\em strictly} Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ of degree three with the maximal (equal to eight) number of real critical points.
One of main tools of this calculation is a combinatorial computer program formalizing Morse surgeries, local monodromy and the Picard-Lefschetz theory, which I hereby promote and recommend to the readers. - [252] arXiv:2405.04361 (replaced) [pdf, html, other]
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Title: On the Iwasawa theory of Cayley graphsComments: Version 2: 23 pages, minor corrections following referee reports. Example 2 added. Accepted for publication in Research in the Mathematical SciencesSubjects: Number Theory (math.NT); Combinatorics (math.CO); Group Theory (math.GR)
This paper explores Iwasawa theory from a graph theoretic perspective, focusing on the algebraic and combinatorial properties of Cayley graphs. Using representation theory, we analyze Iwasawa-theoretic invariants within $\mathbb{Z}_\ell$-towers of Cayley graphs, revealing connections between graph theory, number theory, and group theory. Key results include the factorization of associated Iwasawa polynomials and the decomposition of $\mu$- and $\lambda$-invariants. Additionally, we apply these insights to complete graphs, establishing conditions under which these invariants vanish.
- [253] arXiv:2405.05567 (replaced) [pdf, html, other]
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Title: Perfect Subset Privacy in Polynomial Computation via Reed-Muller Information Super-setsComments: Extension of ISIT 2024 publication; currently under reviewSubjects: Information Theory (cs.IT)
Delegating large-scale computations to service providers is a common practice which raises privacy concerns. This paper studies information-theoretic privacy-preserving delegation of data to a service provider, who may further delegate the computation to auxiliary worker nodes, in order to compute a polynomial over that data at a later point in time. We study techniques which are compatible with robust management of distributed computation systems, an area known as coded computing. Privacy in coded computing, however, has traditionally addressed the problem of colluding workers, and assumed that the server that administrates the computation is trusted. This viewpoint of privacy does not accurately reflect real-world privacy concerns, since normally, the service provider as a whole (i.e., the administrator and the worker nodes) form one cohesive entity which itself poses a privacy risk. This paper aims to shift the focus of privacy in coded computing to safeguarding the privacy of the user against the service provider as a whole, instead of merely against colluding workers inside the service provider. To this end, we leverage the recently defined notion of perfect subset privacy, which guarantees zero information leakage from all subsets of the data up to a certain size. Using known techniques from Reed-Muller decoding, we provide a scheme which enables polynomial computation with perfect subset privacy in straggler-free systems. Furthermore, by studying information super-sets in Reed-Muller codes, which may be of independent interest, we extend the previous scheme to tolerate straggling worker nodes inside the service provider.
- [254] arXiv:2405.05654 (replaced) [pdf, html, other]
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Title: Characterizing rational homogeneous spaces via $\mathbb{C}^*$-actionsComments: Updated version, 33 pages. To appear in Collectanea MathematicaSubjects: Algebraic Geometry (math.AG)
We study smooth varieties of Picard number one admitting a special dominating family of rational curves and an equalized $\mathbb{C}^*$-action. In particular we show that $X$ is a smooth variety of Picard number one with nef tangent bundle admitting an equalized $\mathbb{C}^*$-action with an isolated extremal fixed point if and only if $X$ is an irreducible Hermitian symmetric space.
- [255] arXiv:2405.05661 (replaced) [pdf, html, other]
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Title: Dynamics of a multilink wheeled vehicle: partial solutions and unbounded speedupJournal-ref: International Journal of Non-Linear Mechanics, 2024, vol. 165, 104774, 10 ppSubjects: Dynamical Systems (math.DS)
A mathematical model featuring the motion of a multilink wheeled vehicle is developed using a nonholonomic model. A detailed analysis of the inertial motion is made. Fixed points of the reduced system are identified, their stability is analyzed, and invariant manifolds are found. For the case of three platforms (links), a phase portrait for motion on an invariant manifold is shown and trajectories of the attachment points of the wheel pairs of the three-link vehicle are presented. In addition, an analysis is made of motion in the case where the leading platform has a rotor whose angular velocity is a periodic function of time. The existence of trajectories for which one of the velocity components increases without bound is established, and the asymptotics for it is found.
- [256] arXiv:2405.06375 (replaced) [pdf, html, other]
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Title: Accuracy and Stability of CUR decompositions with OversamplingComments: 31 pages, 4 figuresSubjects: Numerical Analysis (math.NA)
This work investigates the accuracy and numerical stability of CUR decompositions with oversampling. The CUR decomposition approximates a matrix using a subset of columns and rows of the matrix. When the number of columns and the rows are the same, the CUR decomposition can become unstable and less accurate due to the presence of the matrix inverse in the core matrix. Nevertheless, we demonstrate that the CUR decomposition can be implemented in a numerical stable manner and illustrate that oversampling, which increases either the number of columns or rows in the CUR decomposition, can enhance its accuracy and stability. Additionally, this work devises an algorithm for oversampling motivated by the theory of the CUR decomposition and the cosine-sine decomposition, whose competitiveness is illustrated through experiments.
- [257] arXiv:2405.06539 (replaced) [pdf, html, other]
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Title: Gradient Descent for Noisy OptimizationComments: 40 pages, 3 figuresSubjects: Optimization and Control (math.OC)
We study the use of gradient descent with backtracking line search (GD-BLS) to solve the noisy optimization problem $\theta_\star:=\mathrm{argmin}_{\theta\in\mathbb{R}^d} \mathbb{E}[f(\theta,Z)]$, imposing that the function $F(\theta):=\mathbb{E}[f(\theta,Z)]$ is strictly convex but not necessarily $L$-smooth. Assuming that $\mathbb{E}[\|\nabla_\theta f(\theta_\star,Z)\|^2]<\infty$, we first prove that sample average approximation based on GD-BLS allows to estimate $\theta_\star$ with an error of size $\mathcal{O}_{\mathbb{P}}(B^{-0.25})$, where $B$ is the available computational budget. We then show that we can improve upon this rate by stopping the optimization process earlier when the gradient of the objective function is sufficiently close to zero, and use the residual computational budget to optimize, again with GD-BLS, a finer approximation of $F$. By iteratively applying this strategy $J$ times, we establish that we can estimate $\theta_\star$ with an error of size $\mathcal{O}_{\mathbb{P}}(B^{-\frac{1}{2}(1-\delta^{J})})$, where $\delta\in(1/2,1)$ is a user-specified parameter. More generally, we show that if $\mathbb{E}[\|\nabla_\theta f(\theta_\star,Z)\|^{1+\alpha}]<\infty$ for some known $\alpha\in (0,1]$ then this approach, which can be seen as a retrospective approximation algorithm with a fixed computational budget, allows to learn $\theta_\star$ with an error of size $\mathcal{O}_{\mathbb{P}}(B^{-\frac{\alpha}{1+\alpha}(1-\delta^{J})})$, where $\delta\in(2\alpha/(1+3\alpha),1)$ is a tuning parameter. Beyond knowing $\alpha$, achieving the aforementioned convergence rates do not require to tune the algorithms parameters according to the specific functions $F$ and $f$ at hand, and we exhibit a simple noisy optimization problem for which stochastic gradient is not guaranteed to converge while the algorithms discussed in this work are.
- [258] arXiv:2405.09162 (replaced) [pdf, html, other]
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Title: Complete and Terminating Tableau Calculus for Undirected GraphComments: 19 pages, 6 figures, accepted as the conference 'AWPL 2024' post-proceedingSubjects: Logic (math.LO); Logic in Computer Science (cs.LO)
Hybrid logic is a modal logic with additional operators specifying nominals and is highly expressive. For example, there is no formula corresponding to the irreflexivity of Kripke frames in basic modal logic, but there is in hybrid logic. Irreflexivity is significant in that irreflexive and symmetric Kripke frames can be regarded as undirected graphs reviewed from a graph theoretic point of view. Thus, the study of the hybrid logic with axioms corresponding to irreflexivity and symmetry can help to elucidate the logical properties of undirected graphs. In this paper, we formulate the tableau method of the hybrid logic for undirected graphs. Our main result is to show the completeness theorem and the termination property of the tableau method, which leads us to prove the decidability.
- [259] arXiv:2405.09510 (replaced) [pdf, html, other]
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Title: The Instrumental Variable Model with Categorical Instrument, Treatment and Outcome: Characterization, Partial Identification, and Statistical InferenceComments: Added corollaries, statistical inference, and a real example compared to the previous versionSubjects: Statistics Theory (math.ST)
Instrumental variable (IV) analysis is a crucial tool in estimating causal relationships by addressing the issue of confounding variables that may bias the results. Among other work on IV models with binary exposure and outcomes, Richardson and Robins (2014) studied the instrumental variable model with binary exposure (X) and binary outcome (Y) with an instrument (Z) that takes Q states where Q>=2. However, IV models beyond binary X and Y have been less explored. In this work, we consider the instrumental variable model with categorical X, Y, Z taking values in {1, ..., K}, {1, ..., M}, and {1, ..., Q} respectively. We first give a simple closed-form characterization of the set of joint distributions of the potential outcomes P(Y(x=1), ..., Y(x=K)) compatible with a given observed probability distribution P(X, Y | Z). We further show the bounds we derived are necessary, sufficient, and non-redundant, and they hold under various versions of the independence assumptions that have been discussed in the literature. We also provide how a confidence region of any convex function of the joint counterfactual probability including the average causal effect (ATE) can be computed using an algorithm proposed by Guo and Richardson (2021) which is based on a new tail bound for the KL-divergence. We implement our bounds and provide practical recommendations through a real data example of a cash-incentive smoking cessation program.
- [260] arXiv:2406.00310 (replaced) [pdf, html, other]
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Title: $F$-Diophantine sets over finite fieldsComments: 7 pagesSubjects: Number Theory (math.NT)
Let $k \geq 2$, $q$ be an odd prime power, and $F \in \mathbb{F}_q[x_1, \ldots, x_k]$ be a polynomial. An $F$-Diophantine set over a finite field $\mathbb{F}_q$ is a set $A \subset \mathbb{F}_q^*$ such that $F(a_1, a_2, \ldots, a_k)$ is a square in $\mathbb{F}_q$ whenever $a_1, a_2, \ldots, a_k$ are distinct elements in $A$. In this paper, we provide a strategy to construct a large $F$-Diophantine set, provided that $F$ has a nice property in terms of its monomial expansion. In particular, when $F=x_1x_2\ldots x_k+1$, our construction gives a $k$-Diophantine tuple over $\mathbb{F}_q$ with size $\gg_k \log q$, significantly improving the $\Theta((\log q)^{1/(k-1)})$ lower bound in a recent paper by Hammonds-Kim-Miller-Nigam-Onghai-Saikia-Sharma.
- [261] arXiv:2406.12436 (replaced) [pdf, other]
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Title: Affordable mixed-integer Lagrangian methods: optimality conditions and convergence analysisComments: 18 pages, added motivating exampleSubjects: Optimization and Control (math.OC)
Necessary optimality conditions in Lagrangian form and the augmented Lagrangian framework are extended to mixed-integer nonlinear optimization, without any convexity assumptions. Building upon a recently developed notion of local optimality for problems with polyhedral and integrality constraints, a characterization of local minimizers and critical points is given for problems including also nonlinear constraints. This approach lays the foundations for developing affordable sequential minimization algorithms with convergence guarantees to critical points from arbitrary initializations. A primal-dual perspective, a local saddle point property, and the dual relationships with the proximal point algorithm are also advanced in the presence of integer variables.
- [262] arXiv:2406.15137 (replaced) [pdf, html, other]
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Title: A Tangent Category Perspective on Connections in Algebraic GeometryComments: Minor changes and typos fixes based on reviewer commentsSubjects: Category Theory (math.CT)
There is an abstract notion of connection in any tangent category. In this paper, we show that when applied to the tangent category of affine schemes, this recreates the classical notion of a connection on a module (and similarly, in the tangent category of schemes, this recreates the notion of connection on a quasi-coherent sheaf of modules). By contrast, we also show that in the tangent category of algebras, there are no non-trivial connections.
- [263] arXiv:2407.01240 (replaced) [pdf, html, other]
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Title: Self-shrinkers whose asymptotic cones fattenComments: Added additional references to recent related workSubjects: Differential Geometry (math.DG)
For each positive integer $g$ we use variational methods to construct a genus $g$ self-shrinker $\Sigma_g$ in $\mathbb{R}^3$ with entropy less than $2$ and prismatic symmetry group $\mathbb{D}_{g+1}\times\mathbb{Z}_2$. For $g$ sufficiently large, the self-shrinker $\Sigma_g$ has two graphical asymptotically conical ends and the sequence $\Sigma_g$ converges on compact subsets to a plane with multiplicity two as $g\to\infty$. Angenent-Chopp-Ilmanen conjectured the existence of such self-shrinkers in 1995 based on numerical experiments. Using these surfaces as initial conditions for large $g$, we obtain examples of mean curvature flows in $\mathbb{R}^3$ with smooth initial non-compact data that evolve non-uniquely after their first singular time.
- [264] arXiv:2407.03478 (replaced) [pdf, html, other]
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Title: Periodic gravity-capillary roll wave solutions to the inclined viscous shallow water equations in two dimensionsComments: Minor typos fixedSubjects: Analysis of PDEs (math.AP)
We study periodic, two-dimensional, gravity-capillary traveling wave solutions to a viscous shallow water system posed on an inclined plane. While thinking of the Reynolds and Bond numbers as fixed and finite, we vary the speed of the traveling frame and the degree of the incline and identify a set of the latter two parameters that classifies from which combinations nontrivial and small amplitude solution curves originate. Our principal technical tools are a combination of the implicit function theorem and a local multiparameter bifurcation theorem. To the best of the author's knowledge, this paper constitutes the first construction and mathematical study of properly two dimensional examples of viscous roll waves.
- [265] arXiv:2407.05819 (replaced) [pdf, html, other]
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Title: A characterization of quasi-homogeneous singularities of free and nearly free plane curvesComments: 20 pages, 2 figures. Accepted for publication in International Mathematics Research NoticesSubjects: Algebraic Geometry (math.AG)
The goal of this paper is to establish a new characterization of quasi-homogeneous isolated singularities of free curves and nearly free curves $C$ in $\mathbb{P}_\mathbb{C}^2$. The criterion will be in terms of a first syzygy matrix associated with the Jacobian ideal $J_f$ of $f$, where $f=0$ is the equation of the plane curve $C$.
- [266] arXiv:2407.10548 (replaced) [pdf, html, other]
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Title: Fluid Antenna Multiple Access Assisted Integrated Data and Energy Transfer: Outage and Multiplexing Gain AnalysisComments: submitted to IEEE journal for possible publicationSubjects: Information Theory (cs.IT)
Fluid antenna multiple access (FAMA) exploits the spatial opportunities in wireless channels to overcome multiuser interference by position (a.k.a.~port) switching, which can achieve better performance compared to traditional fixed multiple-input multiple-output (MIMO) systems. Additionally, integrated data and energy transfer (IDET) is capable of providing both wireless data transfer (WDT) and wireless energy transfer (WET) services towards low-power devices. In this paper, a FAMA-assisted IDET system is investigated, where a base station (BS) equipped with $N$ fixed antennas provides dedicated IDET services towards $N$ user equipments (UEs). Each UE is equipped with a single fluid antenna, while the power splitting (PS) approach is conceived for coordinating WDT and WET. The outage probabilities of both WDT and WET are derived and approximated into closed-forms, where the fluid antenna (FA) at each UE selects the optimal port to achieve the maximum signal-to-interference-plus-noise ratio (SINR) or the energy harvesting power (EHP). The IDET outage probabilities are defined and subsequently derived and approximated into closed-forms. Further, multiplexing gains of the proposed system are defined and analyzed to evaluate the performace. Numerical results validate the theoretical analysis, while also illustrate that the trade-off is achieved between WDT and WET performance by exploiting different port selection strategies. Furthermore, the number of UEs should be optimized to achieve better IDET performance of the system.
- [267] arXiv:2407.11841 (replaced) [pdf, other]
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Title: Asymptotic Analysis of Boundary Layers for Stokes Systems in Periodic HomogenizationSubjects: Analysis of PDEs (math.AP)
We investigate the asymptotics of boundary layers in periodic homogenization. The analysis is focused on a Stokes system with periodic coefficients and periodic Dirichlet data posed in the half-space $\{y\in \mathbb{R}^d: y\cdot n -s>0\}$. In particular, we establish the convergence of the velocity as $y\cdot n \rightarrow \infty$. We obtain this convergence for arbitrary normals $n\in \mathbb{S}^{d-1}$. Moreover, we build an asymptotic expansion of Poisson's kernel for the periodically oscillating Stokes operator in the half-space. The presence of the pressure and the incompressibility condition impose certain innovations. In particular, we provide a framework for the analysis of the boundary layers in homogenization that relies only on physical space techniques and not on techniques that rely on the quasiperiodic structure of the problem.
- [268] arXiv:2408.05561 (replaced) [pdf, html, other]
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Title: Normally torsion-freeness and normality criteria for monomial idealsComments: This paper has 20 pages and 3 figures. We corrected some typos. In addition, this paper will appear in "AMS Contemporary Mathematics"Subjects: Commutative Algebra (math.AC)
In this paper, we focus on the associated primes of powers of monomial ideals and asymptotic behavior properties such as normally torsion-freeness, normality, the strong persistence property, and the persistence property. In particular, we introduce the concept of monomial ideals of well-nearly normally torsion-free type, and show that these ideals are normal. After that, we present some results on the existence of embedded associated prime ideals in the associated primes set of powers of monomial ideals. Further, we employ them in investigating the edge and cover ideals of cones of graphs. Next, we present counterexamples to several questions concerning the relations between relevant algebraic properties of the edge ideals of clutters and complement clutters. We conclude by providing counterexamples to questions on the possible connections between normally torsion-freeness and normality of monomial ideals under polarization.
- [269] arXiv:2408.07400 (replaced) [pdf, html, other]
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Title: Polylogarithmic motivic Chabauty-Kim for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$: the geometric step via resultantsComments: v2: additional funding information added to the acknowledgements, no other changesSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
Given a finite set $S$ of distinct primes, we propose a method to construct polylogarithmic motivic Chabauty-Kim functions for $\mathbb{P}^1 \setminus \{ 0,1,\infty \}$ using resultants. For a prime $p\not\in S$, the vanishing loci of the images of such functions under the $p$-adic period map contain the solutions of the $S$-unit equation. In the case $\vert S\vert=2$, we explicitly construct a non-trivial motivic Chabauty-Kim function in depth 6 of degree 18, and prove that there do not exist any other Chabauty-Kim functions with smaller depth and degree. The method, inspired by work of Dan-Cohen and the first author, enhances the geometric step algorithm developed by Corwin and Dan-Cohen, providing a more efficient approach.
- [270] arXiv:2408.11575 (replaced) [pdf, html, other]
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Title: Contact Structure and Canonical Equations of Stochastic Vector BundlesComments: completely revisedSubjects: Mathematical Physics (math-ph)
This paper investigates the geometric structure of stochastic vector bundles. It finds that the probability space of stochastic vector bundles possesses an infinite-order jet structure, enabling a gemetrical analysis of stochastic processes. Furthermore, the paper demonstrates that stochastic vector bundles have a natural contact structure, leading to a decomposition of the tangent space and providing insights into the system's evolution and constraints. Finally, it derives a set of canonical equations for stochastic vector bundles, which resemble Hamilton's equations. These equations are connected to the principle of least action, showing the relation between geometric structure of stochastic system evolution and its tendency to minimize energy consumption. This study provides a valuable geometric framework for analyzing stochastic systems, with potential applications in various fields where probabilistic behavior is crucial.
- [271] arXiv:2408.14781 (replaced) [pdf, html, other]
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Title: A module-theoretic characterization of $S$-($w$-)Noetherian ringsComments: arXiv admin note: text overlap with arXiv:2307.10309; text overlap with arXiv:2201.07913 by other authorsSubjects: Commutative Algebra (math.AC)
Let $R$ be a ring and $S$ a multiplicative subset of $R$. In this paper, we obtain the ACC characterization, Cartan-Eilenberg-Bass theorem and the absolutely pure characterization for $S$-Noetherian rings. In details, we show that a ring $R$ is an $S$-Noetherian ring if and only if any ascending chain of ideals of $R$ is $S$-stationary, if and only if any direct sum of injective modules is $S$-injective, if and only if any direct limit of injective modules is $S$-injective. We also characterized $S$-$w$-Noetherian rings similarly.
- [272] arXiv:2408.17201 (replaced) [pdf, html, other]
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Title: Categorical quantization on K\"ahler manifoldsComments: 22 pagesSubjects: Symplectic Geometry (math.SG); Mathematical Physics (math-ph); Differential Geometry (math.DG)
Generalizing deformation quantizations with separation of variables of a Kähler manifold $M$, we adopt Fedosov's gluing argument to construct a category $\mathsf{DQ}$, enriched over sheaves of $\mathbb{C}[[\hbar]]$-modules on $M$, as a quantization of the category of Hermitian holomorphic vector bundles over $M$ with morphisms being smooth sections of hom-bundles.
We then define quantizable morphisms among objects in $\mathsf{DQ}$, generalizing Chan-Leung-Li's notion [4] of quantizable functions. Upon evaluation of quantizable morphisms at $\hbar = \tfrac{\sqrt{-1}}{k}$, we obtain an enriched category $\mathsf{DQ}_{\operatorname{qu}, k}$. We show that, when $M$ is prequantizable, $\mathsf{DQ}_{\operatorname{qu}, k}$ is equivalent to the category $\mathsf{GQ}$ of holomorphic vector bundles over $M$ with morphisms being holomorphic differential operators, via a functor obtained from Bargmann-Fock actions. - [273] arXiv:2409.02047 (replaced) [pdf, html, other]
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Title: On the Diophantine Equation $F_n = F_l^k (F_l^m-1)$Subjects: Number Theory (math.NT)
In this paper, we examine the Diophantine problem given by the equation $F_n = F_l^k (F_l^m - 1)$, where $n, l, m \geq 1$ and $k \geq 3$. Here, $\{ F_t \}_{t=0}^{\infty} $ denotes the Fibonacci numbers, defined by the recurrence relation $F_0 = 0$, $F_1 = 1$, and $F_t = F_{t-1} + F_{t-2}$ for $t \geq 2$. By applying Matveev's theorem, which provides lower bounds for linear forms in logarithms of algebraic numbers, along with a modified Baker-Davenport reduction method and a divisibility property of Fibonacci numbers, we show that $(n, l, k, m) = (6, 3, 3, 1)$ is the only positive integer quadruple that satisfies this equation.
- [274] arXiv:2409.10664 (replaced) [pdf, html, other]
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Title: Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and ApplicationsComments: Submitted to IEEE L-CSS and ACC, 7 pages, 1 figureSubjects: Optimization and Control (math.OC); Signal Processing (eess.SP); Systems and Control (eess.SY)
In this letter we study the proximal gradient dynamics. This recently-proposed continuous-time dynamics solves optimization problems whose cost functions are separable into a nonsmooth convex and a smooth component. First, we show that the cost function decreases monotonically along the trajectories of the proximal gradient dynamics. We then introduce a new condition that guarantees exponential convergence of the cost function to its optimal value, and show that this condition implies the proximal Polyak-Łojasiewicz condition. We also show that the proximal Polyak-Łojasiewicz condition guarantees exponential convergence of the cost function. Moreover, we extend these results to time-varying optimization problems, providing bounds for equilibrium tracking. Finally, we discuss applications of these findings, including the LASSO problem, certain matrix based problems and a numerical experiment on a feed-forward neural network.
- [275] arXiv:2409.10867 (replaced) [pdf, html, other]
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Title: Integral zeros of quadratic polynomials avoiding sublatticesComments: 8 pages; to appear in The Ramanujan JournalSubjects: Number Theory (math.NT)
Assuming an integral quadratic polynomial with nonsingular quadratic part has a nontrivial zero on an integer lattice outside of a union of finite-index sublattices, we prove that there exists such a zero of bounded norm and provide an explicit bound. This is a contribution related to the celebrated theorem of Cassels on small-height zeros of quadratic forms, which builds on some previous work in this area. We also demonstrate an application of these results to the problem of effective distribution of angles between vectors in the integer lattice.
- [276] arXiv:2409.16624 (replaced) [pdf, html, other]
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Title: Removable dynamics in the Nose-Hoover and Moore-Spiegel OscillatorsSubjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
We study the dynamics of the Nose-Hoover and Moore-Spiegel Oscillators, and in particular, their topological dynamics. We prove the dynamics of both these systems can be reduced to a flow on a solid torus, with at most a finite number of attracting periodic trajectories. As a consequence, we obtain that every periodic trajectory for the Nose-Hoover and the Moore-Spiegel Oscillators is a Torus knot.
- [277] arXiv:2409.17679 (replaced) [pdf, html, other]
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Title: Spectral Tur\'an problems for hypergraphs with bipartite or multipartite patternSubjects: Combinatorics (math.CO)
General criteria on spectral extremal problems for hypergraphs were developed by Keevash, Lenz, and Mubayi in their seminal work (SIAM J. Discrete Math., 2014), in which extremal results on \alpha-spectral radius of hypergraphs for \alpha>1 may be deduced from the corresponding hypergraph Turán problem which has the stability property and whose extremal construction satisfies some continuity assumptions. Using this criterion, we give two general spectral Turán results for hypergraphs with bipartite or mulitpartite pattern, transform corresponding the spectral Turán problems into pure combinatorial problems with respect to degree-stability of a nondegenerate k-graph family. As an application, we determine the maximum \alpha-spectral radius for some classes of hypergraphs and characterize the corresponding extremal hypergraphs, such as the expansion of complete graphs, the generalized Fans, the cancellative hypergraphs, the generalized triangles, and a special book hypergraph.
- [278] arXiv:2409.17921 (replaced) [pdf, html, other]
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Title: Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extensionComments: Accepted for publication in the Tohoku Math JSubjects: Number Theory (math.NT)
Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation $n=x^3+y^3$, where $x$ and $y$ belong to the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, consider the case when $n$ is not a sum of rational cubes. Then, we prove that $n$ cannot be a sum of two cubes in certain large families of prime cyclic extensions of $\mathbb{Q}$.
- [279] arXiv:2409.20416 (replaced) [pdf, html, other]
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Title: Solution of certain Diophantine equations in Gaussian integersComments: 11 pagesSubjects: Number Theory (math.NT)
In this article, we show that the quartic Diophantine equations $x^4 \pm pqy^4=\pm z^2$ and $ x^4 \pm pq y^4= \pm iz^2$ have only trivial solutions for some primes $p$ and $q$ satisfying conditions $ p \equiv 3 \pmod 8, ~ q \equiv 1 \pmod 8 ~\text{and}~ \displaystyle\legendre{p}{q} = -1$. Here we have found the torsion of the two families of elliptic curves to find the solutions of given Diophantine equations. Moreover, we also calculate the rank of these two families of elliptic curves over the Gaussian field $\mathbb{Q}(i)$.
- [280] arXiv:2410.00520 (replaced) [pdf, other]
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Title: Stretching of Polymers and turbulence: Fokker Planck equation, special stochastic scaling limit and stationary lawSubjects: Probability (math.PR); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
The aim of this work is understanding the stretching mechanism of stochastic models of turbulence acting on a simple model of dilute polymers. We consider a turbulent model that is white noise in time and activates frequencies in a shell $N\leq |k|\leq 2N$ and investigate the scaling limit as $N\rightarrow \infty$, under suitable intensity assumption, such that the stretching term has a finite limit covariance. The polymer density equation, initially an SPDE, converges weakly to a limit deterministic equation with a new term. Stationary solutions can be computed and show power law decay in the polymer length.
- [281] arXiv:2410.00553 (replaced) [pdf, other]
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Title: Semistable degenerations of double octicsComments: minor revisionsSubjects: Algebraic Geometry (math.AG)
We present an algorithm for computing semistable degeneration of double octic Calabi-Yau threefolds. Our method has a combinatorial representation by the means of double octic diagrams. The proposed algorithm is applicable both in classical context over a complex disk as well as in arithmetic setting over a spectrum of DVR. We illustrate algorithm's efficacy through three examples where we compute semistable degeneration and limiting mixed Hodge structure for explicit families of double octics.
- [282] arXiv:2410.00750 (replaced) [pdf, html, other]
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Title: Quasi-reversible bullet modelsComments: 27 pages, 13 figuresSubjects: Probability (math.PR)
We consider a large class of bullet models that contains, in particular, the colliding bullet model with creations and a new loop model. For this large class of bullet models, we give sufficient conditions on their parameter to be $\text{rot}(\pi)$-quasi-reversible and to be $\text{rot}(\pi/2)$-quasi-reversible. Moreover, those conditions assure them that one of their invariant measures is described by a Poisson Point Process. This result applied to the colliding bullet models with creations is a first step to study its property under its non-empty invariant measure.
- [283] arXiv:2410.03153 (replaced) [pdf, html, other]
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Title: Factorization of rational six vertex model partition functionsJournal-ref: Nuclear Physics B, 1009 (2024), 116743Subjects: Mathematical Physics (math-ph); Combinatorics (math.CO)
We show factorization formulas for a class of partition functions of rational six vertex model. First we show factorization formulas for partition functions under triangular boundary. Further, by combining the factorization formulas with the explicit forms of the generalized domain wall boundary partition functions by Belliard-Pimenta-Slavnov, we derive factorization formulas for partition functions under trapezoid boundary which can be viewed as a generalization of triangular boundary. We also discuss an application to emptiness formation probabilities under trapezoid boundary which admit determinant representations.
- [284] arXiv:2410.11676 (replaced) [pdf, html, other]
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Title: Global non-asymptotic super-linear convergence rates of regularized proximal quasi-Newton methods on non-smooth composite problemsSubjects: Optimization and Control (math.OC)
In this paper, we propose two regularized proximal quasi-Newton methods with symmetric rank-1 update of the metric (SR1 quasi-Newton) to solve non-smooth convex additive composite problems. Both algorithms avoid using line search or other trust region strategies. For each of them, we prove a super-linear convergence rate that is independent of the initialization of the algorithm. The cubic regularized method achieves a rate of order $\left(\frac{C}{N^{1/2}}\right)^{N/2}$, where $N$ is the number of iterations and $C$ is some constant, and the other gradient regularized method shows a rate of the order $\left(\frac{C}{N^{1/4}}\right)^{N/2}$. To the best of our knowledge, these are the first global non-asymptotic super-linear convergence rates for regularized quasi-Newton methods and regularized proximal quasi-Newton methods. The theoretical properties are also demonstrated in two applications from machine learning.
- [285] arXiv:2410.11754 (replaced) [pdf, html, other]
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Title: Measurable splittings and the measured group theoretic structure of wreath productsComments: 37 pages, 1 figure, minor revisions and correctionsSubjects: Group Theory (math.GR); Dynamical Systems (math.DS); Logic (math.LO); Operator Algebras (math.OA)
Let $\Gamma$ be a countable group that admits an essential measurable splitting (for instance, any group measure equivalent to a free product of nontrivial groups).
We show: (1) for any two nontrivial countable groups $B$ and $C$ that are measure equivalent, the wreath product groups $B\wr\Gamma$ and $C\wr\Gamma$ are measure equivalent (in fact, orbit equivalent) -- this is interesting even in the case when the groups $B$ and $C$ are finite; and (2) the groups $B\wr \Gamma$ and $(B\times\mathbf{Z})\wr\Gamma$ are measure equivalent (in fact, orbit equivalent) for every nontrivial countable group $B$.
On the other hand, we show that certain wreath product actions are not even stably orbit equivalent if $\Gamma$ is instead assumed to be a sofic icc group that is Bernoulli superrigid, and $B$ and $C$ have different cardinalities. - [286] arXiv:2410.17754 (replaced) [pdf, html, other]
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Title: Puncturing Quantum Stabilizer CodesComments: PreprintSubjects: Information Theory (cs.IT); Rings and Algebras (math.RA)
Classical coding theory contains several techniques to obtain new codes from other codes, including puncturing and shortening. For quantum codes, a form of puncturing is known, but its description is based on the code space rather than its generators. In this work, we generalize the puncturing procedure to allow more freedom in the choice of which coded states are kept and which are removed. We describe this puncturing by focusing on the stabilizer matrix containing the generators of the code. In this way, we are able to explicitly describe the stabilizer matrix of the punctured code given the stabilizer matrix of the original stabilizer code. The additional freedom in the procedure also opens up new ways to construct new codes from old, and we present several ways to utilize this for the search of codes with good or even optimal parameters. In particular, we use the construction to obtain codes whose parameters exceed the best previously known. Lastly, we generalize the proof of the Griesmer bound from the classical setting to stabilizer codes since the proof relies heavily on the puncturing technique.
- [287] arXiv:2410.18407 (replaced) [pdf, html, other]
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Title: On topological solutions to a generalized Chern-Simons equation on lattice graphsComments: 13 pagesSubjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA)
For $n \geq 2$, consider $\mathbb{Z}^n$ as a lattice graph. We explore a generalized Chern-Simons equation on $\mathbb{Z}^n$. Employing the method of exhaustion, we prove that there exists a global solution that also qualifies as a topological solution. Our results extend those of Hua et al. [arXiv:2310.13905] and complement the findings of Chao and Hou [J. Math. Anal. Appl. $\bf{519}$(1), 126787(2023)], as well as those of Hou and Qiao [J. Math. Phys. $\bf{65}$(8), 081503(2024)].
- [288] arXiv:2410.19181 (replaced) [pdf, html, other]
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Title: Stochastic dynamic programming under recursive Epstein-Zin preferencesSubjects: Optimization and Control (math.OC)
This paper investigates discrete-time Markov decision processes with recursive utilities (or payoffs) defined by the classic CES aggregator and the Kreps-Porteus certainty equivalent operator. According to the classification introduced by Marinacci and Montrucchio, the aggregators that we consider are Thompson. We focus on the existence and uniqueness of a solution to the Bellman equation. Since the per-period utilities can be unbounded, we work with the weighted supremum norm. Our paper shows three major points for such models. Firstly, we prove that the Bellman equation can be obtained by the Banach fixed point theorem for contraction mappings acting on a standard complete metric space. Secondly, we need not assume any boundary conditions, which are present when the Thompson metric or the Du's theorem are used. Thirdly, our results give better bounds for the geometric convergence of the value iteration algorithm than those obtained by Du's fixed point theorem. Moreover, our techniques allow to derive the Bellman equation for some values of parameters in the CES aggregator and the Kreps-Porteus certainty equivalent that cannot be solved by Du's theorem for increasing and convex or concave operators acting on an ordered Banach space.
- [289] arXiv:2410.20426 (replaced) [pdf, html, other]
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Title: Exact Temporal Variation for Fractional Stochastic Heat Equation Driven by Space-Time White NoiseSubjects: Probability (math.PR)
In this paper, we consider the exact fractional variation for the temporal process of the solution to the fractional stochastic heat equation on $\mathbb{R}$ driven by a space-time white noise, and as an application we give the estimate of drift parameter.
- [290] arXiv:2410.22196 (replaced) [pdf, html, other]
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Title: LLL Algorithm for Lattice Basis ReductionComments: 19 pagesSubjects: Number Theory (math.NT)
This is an expository paper intended to introduce the polynomial time lattice basis reduction algorithm first described by Arjen Lenstra, Hendrik Lenstra, and László Lovász in 1982. We begin by introducing the shortest vector problem, which motivates the underlying components of the LLL algorithm. Then, we introduce the details of the algorithm itself, followed by proofs of the correctness and runtime of the algorithm in complete detail, assuming only a basic linear algebra background and an understanding of big O notation. Finally, we apply the LLL algorithm to the shortest vector problem and explore other applications of the algorithm in various mathematical settings.
- [291] arXiv:2411.00250 (replaced) [pdf, html, other]
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Title: Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphsComments: 33 pages, 2 figures, and 2 tables. This version includes new results added in Section 6. Additionally, Section 7, which discusses the summary and further questions, has been introducedSubjects: Combinatorics (math.CO)
We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certain cycle(s) in a graph. A key result proves that every Johnson graph has a signed variant with exactly two distinct eigenvalues. We also explore applications to weighing matrices, linear ternary codes, tight frames, and compute the minimum rank of Johnson graphs. Further results involve the minimum number of distinct eigenvalues for graphs in association schemes, distance-regular graphs, and Hamming graphs. We also draw some connections with simplicial complexes and higher-order Laplacians.
- [292] arXiv:2411.00957 (replaced) [pdf, html, other]
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Title: Modularity of $d$-elliptic loci with level structureComments: minor changes, comments still welcomeSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We consider the generating series of special cycles on $\mathcal{A}_1(N)\times \mathcal{A}_g(N)$, with full level $N$ structure, valued in the cohomology of degree $2g$. The modularity theorem of Kudla-Millson for locally symmetric spaces implies that these series are modular. When $N=1$, the images of these loci in $\mathcal{A}_g$ are the $d$-elliptic Noether-Lefschetz loci, which are conjectured to be modular. In the appendix, it is shown that the resulting modular forms are nonzero for $g=2$ when $N\geq 11$ and $N\neq 12$.
- [293] arXiv:2411.01782 (replaced) [pdf, other]
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Title: The Tur\'an Density of 4-Uniform Tight CyclesComments: 45 pages + 6-page appendix, 8 figures. Comments welcomeSubjects: Combinatorics (math.CO)
For any uniformity $r$ and residue $k$ modulo $r$, we give an exact characterization of the $r$-uniform hypergraphs that homomorphically avoid tight cycles of length $k$ modulo $r$, in terms of colorings of $(r-1)$-tuples of vertices. This generalizes the result that a graph avoids all odd closed walks if and only if it is bipartite, as well as a result of Kam\v cev, Letzter, and Pokrovskiy in uniformity 3. In fact, our characterization applies to a much larger class of families than those of the form $\mathscr C_k^{(r)}=\{\text{$r$-uniform tight cycles of length $k$ modulo $r$}\}$.
We also outline a general strategy to prove that, if $\mathscr C$ is a family of tight-cycle-like hypergraphs (including but not limited to the families $\mathscr C_k^{(r)}$) for which the above characterization applies, then all sufficiently long $C\in \mathscr C$ will have the same Turán density. We demonstrate an application of this framework, proving that there exists an integer $L_0$ such that for every $L>L_0$ not divisible by 4, the tight cycle $C^{(4)}_L$ has Turán density $1/2$. - [294] arXiv:2411.02440 (replaced) [pdf, other]
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Title: Corrigendum to the equivalent statement of the Laplacian Spread ConjectureComments: The conclusion of the errant original article is correct, so there are errors in the erratum of this paper, I request to withdraw this paperSubjects: Combinatorics (math.CO)
For a graph $G,$ let $\alpha(G)$ denote its second smallest Laplacian eigenvalue. The Laplacian Spread Conjecture states that $\alpha(G)+\alpha(\overline{G}) \geq 1,$ where $\overline{G}$ is the complement of $G.$ In this paper, we have corrected two conclusions: First, the necessary and sufficient condition for $\alpha(G) + \alpha(\overline{G}) \geq 1$ is $\parallel \bigtriangledown_{x} - \bigtriangledown_{y} \parallel^{2} \geq 1$ rather than $\parallel \bigtriangledown_{x} - \bigtriangledown_{y} \parallel^{2} \geq 2$ which has been proved in \cite{BS} as demonstrated in our study. Second, we show that the Laplacian spread of balanced digraph $\Gamma$ satisfies $LS(\Gamma) \leq n - \frac{1}{2}$ but not $LS(\Gamma) \leq n - 1$ in \cite{BCEHK}, since inequality $\parallel \bigtriangledown_{x} - \bigtriangledown_{y} \parallel^{2} \geq 2$ does not hold.
- [295] arXiv:2411.02766 (replaced) [pdf, html, other]
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Title: Approximate controllability of impulsive semilinear evolution equations in Hilbert spacesSubjects: Optimization and Control (math.OC)
Several dynamical systems in fields such as engineering, chemistry, biology, and physics show impulsive behavior by reason of unexpected changes at specific times. These behaviors are described by differential systems under impulse effects. The current paper examines approximate controllability for semi-linear impulsive differential and neutral differential equations in Hilbert spaces. By applying a fixed-point method and semigroup theory, a new sufficient condition is provided for the ($\mathcal{A}$-controllability) approximate controllability of neutral and impulsive differential equations (IDEs). To demonstrate the value of the suggested consequences, three examples are presented, offering improvements over some recent findings.
- [296] arXiv:2411.04070 (replaced) [pdf, html, other]
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Title: Chow functions for partially ordered setsComments: 48 pages. Minor corrections and improvementsSubjects: Combinatorics (math.CO); Algebraic Geometry (math.AG)
Three decades ago, Stanley and Brenti initiated the study of the Kazhdan--Lusztig--Stanley (KLS) functions, putting on common ground several polynomials appearing in algebraic combinatorics, discrete geometry, and representation theory. In the present paper we develop a theory that parallels the KLS theory. To each kernel in a given poset, we associate a polynomial function that we call the \emph{Chow function}. The Chow function often exhibits remarkable properties, and sometimes encodes the graded dimensions of a cohomology or Chow ring. The framework of Chow functions provides natural polynomial analogs of graded module decompositions that appear in algebraic geometry, but that work for arbitrary posets, even when no graded module decomposition is known to exist. In this general framework, we prove a number of unimodality and positivity results without relying on versions of the Hard Lefschetz theorem. Our framework shows that there is an unexpected relation between positivity and real-rootedness conjectures about chains on face lattices of polytopes by Brenti and Welker, Hilbert--Poincaré series of matroid Chow rings by Ferroni and Schröter, and flag enumerations on Bruhat intervals of Coxeter groups by Billera and Brenti.
- [297] arXiv:2411.04477 (replaced) [pdf, html, other]
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Title: Medial quandle and detecting causalitySubjects: Geometric Topology (math.GT)
I investigated the capability of medial quandle, quandle whose operation satisfying that $(a_1*b_1)*(a_2*b_2)=(a_1*a_2)*(b_1*b_2)$, to detect causality in (2+1)-dimensional globally hyperbolic spacetime by determining if they can distinguished the connected sum of two Hopf links with an infinite series of relevant three-component links constructed by Allen and Swenberg in 2020, who suggested that any link invariant must be able to distinguish those links for them to detect causality in the given setting. I show that these quandles fail to do so as long as $a\sim b\Leftrightarrow a*b=a$ defines an equivalence relation. The Alexander quandles is an example that this result can apply to.
- [298] arXiv:2411.05304 (replaced) [pdf, html, other]
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Title: A sharp upper bound on the spectral radius of $\theta(1,3,3)$-free graphs with given sizeComments: 14pages, 1 figures. arXiv admin note: text overlap with arXiv:2410.07721 by other authorsSubjects: Combinatorics (math.CO)
A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\rho(G)$ be the spectral radius of a graph $G$. Let $\theta(1,p,q)$ denote the theta graph, which is obtained by connecting two distinct vertices with three internally disjoint paths with lengths $1, p, q$, where $p\leq q$. Let $S_{n,k}$ denote the graph obtained by joining every vertex of $K_{k}$ to $n-k$ isolated vertices and $S_{n,k}^{-}$ denote the graph obtained from $S_{n,k}$ by deleting an edge incident to a vertex of degree $k$, respectively. In this paper, we show that if $\rho(G)\geq\rho(S_{\frac{m+4}{2},2}^{-})$ for a graph $G$ with even size $m\geq 92$, then $G$ contains a $\theta(1,3,3)$ unless $G\cong S_{\frac{m+4}{2},2}^{-}$.
- [299] arXiv:2411.06043 (replaced) [pdf, html, other]
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Title: The subTuring degreesSubjects: Logic (math.LO); Logic in Computer Science (cs.LO)
In this article, we introduce a notion of reducibility for partial functions on the natural numbers, which we call subTuring reducibility. One important aspect is that the subTuring degrees correspond to the structure of the realizability subtoposes of the effective topos. We show that the subTuring degrees (that is, the realizability subtoposes of the effective topos) form a dense non-modular (thus, non-distributive) lattice. We also show that there is a nonzero join-irreducible subTuring degree (which implies that there is a realizability subtopos of the effective topos that cannot be decomposed into two smaller realizability subtoposes).
- [300] arXiv:2411.06571 (replaced) [pdf, html, other]
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Title: Moment-based approach for two erratic KPZ scaling limitsComments: v2: corrected/clarified some proofs, added references, added a few remarks (22 pages)Subjects: Probability (math.PR)
A recent paper of Tsai shows how the first few moments of a stochastic flow in the space of measures can completely determine its law. Here we give another proof of this result for the particular case of the one-dimensional multiplicative stochastic heat equation (mSHE), and then we investigate two corollaries. The first one recovers a recent result of Hairer on a ``variance blowup" problem related to the KPZ equation , albeit in a much weaker topology. The second one recovers a KPZ scaling limit result related to random walks in random environments, but in a weaker topology. In these two problems, we furthermore explain why it is hard to directly use the martingale characterization of the mSHE, the chaos expansion, or other known methods. Using the moment-based approach avoids technicalities, leading to a short proof.
- [301] arXiv:2411.06675 (replaced) [pdf, html, other]
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Title: FCA using the Concept Explorer in 2024Comments: 10 pages, 1 context, 9 figuresSubjects: Logic (math.LO); Logic in Computer Science (cs.LO)
In this note we give a very short introduction to Formal Concept Analysis, accompanied by an example in order to build concept lattices from a context. We build the lattice using the Java-based software Concept Explorer (ConExp) in a recent version of Linux. Installing an appropriate Java version is necessary, because ConExp was developed some time ago using a Sun Java version, which is not open-source. As a result, it has been observed that ConExp will not build a lattice when started with an open-source Java version. Therefore, we also sketch the procedure we followed to install an appropriate Java version which makes ConExp work again, i.e., to "build lattices again". We also show how to start ConExp with a 32 bit Java version, which requires a few additional libraries.
- [302] arXiv:2411.07947 (replaced) [pdf, html, other]
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Title: Approximation rates of entropic maps in semidiscrete optimal transportSubjects: Probability (math.PR); Optimization and Control (math.OC)
Entropic optimal transport offers a computationally tractable approximation to the classical problem. In this note, we study the approximation rate of the entropic optimal transport map (in approaching the Brenier map) when the regularization parameter $\varepsilon$ tends to zero in the semidiscrete setting, where the input measure is absolutely continuous while the output is finitely discrete. Previous work shows that the approximation rate is $O(\sqrt{\varepsilon})$ under the $L^2$-norm with respect to the input measure. In this work, we establish faster, $O(\varepsilon^2)$ rates up to polylogarithmic factors, under the dual Lipschitz norm, which is weaker than the $L^2$-norm. For the said dual norm, the $O(\varepsilon^2)$ rate is sharp. As a corollary, we derive a central limit theorem for the entropic estimator for the Brenier map in the dual Lipschitz space when the regularization parameter tends to zero as the sample size increases.
- [303] arXiv:2411.08941 (replaced) [pdf, html, other]
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Title: Generalized Cauchy-Riemann equations and relevant PDESubjects: Analysis of PDEs (math.AP); Complex Variables (math.CV)
Here we give a survey of consequences from the theory of the Beltrami equations in the complex plane $\mathbb C$ to generalized Cauchy-Riemann equations $\nabla v = B \nabla u$ in the real plane $\mathbb R^2$ and clarify the relationships of the latter to the $A-$harmonic equation ${\rm div} A\,{\rm grad}\, u = 0$ with matrix valued coefficients $A$ that is one of the main equations of the potential theory, namely, of the hydro\-mechanics (fluid mechanics) in anisotropic and inhomogeneous media. The survey includes various types of results as theorems on existence, representation and regularity of their solutions, in particular, for the main boundary value problems of Hilbert, Dirichlet, Neumann, Poincare and Riemann.
- [304] arXiv:2411.09324 (replaced) [pdf, html, other]
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Title: Riesz-Schur transformsComments: Minor improvementsSubjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
We investigate nontrigonometric forms of Riesz transforms in the context of Schur multipliers. This refines Grothendieck-Haagerup's endpoint criterion with a new condition for the Schatten p-boundedness of Schur multipliers and strengthens Potapov/Sukochev's solution of Arazy's conjecture. We recover as well dimension-free estimates for trigonometric Riesz transforms. Our discrete approach is much simpler than previous harmonic analysis and probabilistic approaches. As an application, we find a very simple proof of recent criteria for Schur multipliers of Hörmander-Mikhlin and Marcinkiewicz type.
- [305] arXiv:2411.10012 (replaced) [pdf, html, other]
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Title: Bilinear Strichartz estimates on rescaled waveguides with applicationsComments: 35pages, 1figureSubjects: Analysis of PDEs (math.AP)
We focus on the bilinear Strichartz estimates for free solutions to the Schrödinger equation on rescaled waveguides $\mathbb{R} \times \mathbb{T}_\lambda^n$, $\mathbb{T}_\lambda^n=(\lambda\mathbb{T})^n$ with $n\geq 1$ and their applications. First, we utilize a decoupling-type estimate originally from Fan-Staffilani-Wang-Wilson [Anal. PDE 11 (2018)] to establish a global-in-time bilinear Strichartz estimate with a `$N_2^\epsilon$' loss on $\mathbb{R} \times \mathbb{T}^n_\lambda$ when $n\geq1$, which generalize the local-in time estimate in Zhao-Zheng [SIAM J. Math. Anal. (2021)] and fills a gap left by the unresolved case in Deng et al. [J. Func. Anal. 287 (2024)]. Second, we prove the local-in-time angularly refined bilinear Strichartz estimates on the 2d rescaled waveguide $\mathbb{R} \times \mathbb{T}_\lambda$, which generalize the estimate obtained by Takaoka [J. Differ. Equa. 394 (2024)] with $\lambda=1$. As applications, we show the local well-posedness and small data scattering for nonlinear Schrödinger equations with algebraic nonlinearities in the critical space on $\mathbb R^m\times\mathbb{T}^n$ and the global well-posedness for cubic NLS on $\mathbb{R} \times \mathbb{T}$ in the lower regularity space $H^s$ with $s>\frac{1}{2}$.
- [306] arXiv:2411.10610 (replaced) [pdf, html, other]
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Title: Asymptotic expansion of the partition function for $\beta$-ensembles with complex potentialsComments: 64 pagesSubjects: Mathematical Physics (math-ph)
In this work we establish under certain hypotheses the $N \to +\infty$ asymptotic expansion of integrals of the form $$\mathcal{Z}_{N,\Gamma}[V] \, = \, \int_{\Gamma^N} \prod_{ a < b}^{N}(z_a - z_b)^\beta \, \prod_{k=1}^{N} \mathrm{e}^{ - N \beta V(z_k) } \, \mathrm{d}\mathbf{z}$$ where $V \in \mathbb{C}[X]$, $\beta \in 2 \mathbb{N}^*$ is an even integer and $\Gamma \subset \mathbb{C}$ is an unbounded contour such that the integral converges. For even degree, real valued $V$s and when $\Gamma = \mathbb{R}$, it is well known that the large-$N$ expansion is characterised by an equilibrium measure corresponding to the minimiser of an appropriate energy functional. This method bears a structural resemblance with the Laplace method. By contrast, in the complex valued setting we are considering, the analysis structurally resembles the classical steepest-descent method, and involves finding a critical point \textit{and} a steepest descent curve, the latter being a deformation of the original integration contour. More precisely, one minimises a curve-dependent energy functional with respect to measures on the curve and then maximises the energy over an appropriate space of curves. Our analysis deals with the one-cut regime of the associated equilibrium measure. We establish the existence of an all order asymptotic expansion for $\ln \mathcal{Z}_{N,\Gamma}[V]$ and explicitly identify the first few terms.
- [307] arXiv:2411.11594 (replaced) [pdf, other]
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Title: Interval Multiplicities of Persistence ModulesHideto Asashiba (1, 2 and 3), Enhao Liu (4) ((1) Department of Mathematics, Shizuoka University, (2) Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, (3) Institute for Advanced Study, Kyoto University, (4) Department of Mathematics, Kyoto University)Comments: 39 pages. We extended the definition of the multiplicity of an indecomposable module in the persistence module to that of a decomposable module, and slightly changed the definition of essential covering by removing the indecomposable assumption on $R(V_I)$ that appeared in the previous versionSubjects: Representation Theory (math.RT); Algebraic Topology (math.AT); Rings and Algebras (math.RA)
For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ in $\mathbf{P}$, we give a formula of the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of structure linear maps of the module $M$. This makes it possible to compute the maximal interval-decomposable direct summand of $M$, which gives us a way to decide whether $M$ is interval-decomposable or not. Moreover, the formula tells us essential morphisms of $\mathbf{P}$ that are necessary to compute the multiplicity $d_M(V_I)$. This suggests us some poset morphism $\zeta \colon Z \to \mathbf{P}$ such that the induced restriction functor $R \colon \operatorname{mod} \mathbf{P} \to \operatorname{mod} Z$ has the property that the multiplicity $d:= d_{R(M)}(R(V_I))$ is equal to $d_M(V_I)$. If $Z$ can be taken as a poset of Dynkin type $\mathbb{A}$ as in the bipath case, then the calculation of the multiplicity $d$ can be done more efficiently, starting from the filtration level of topological spaces. Thus this even makes it unnecessary to compute the structure linear maps of $M$.
- [308] arXiv:2411.12189 (replaced) [pdf, html, other]
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Title: Derrida-Retaux type models and related scaling limit theoremsComments: 28 pages, Version: 2024-11-21Subjects: Probability (math.PR)
We give characterizations of the transition semigroup and generator of a continuous-time Derrida--Retaux type process that generalizes the one introduced by Hu, Mallein and Pain (Commun. Math. Phys., 2020). It is shown that the process arises naturally as the scaling limit of the discrete-time max-type recursive models introduced by Hu and Shi (J. Stat. Phys., 2018).
- [309] arXiv:2411.12333 (replaced) [pdf, other]
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Title: Correspondences between codensity and coupling-based liftings, a practical approachSubjects: Category Theory (math.CT)
The Kantorovich distance is a widely used metric between probability distributions. The Kantorovich-Rubinstein duality states that it can be defined in two equivalent ways: as a supremum, based on non-expansive functions into [0, 1], and as an infimum, based on probabilistic couplings.</p><p>Orthogonally, there are categorical generalisations of both presentations proposed in the literature, in the form of codensity liftings and what we refer to as coupling-based liftings. Both lift endofunctors on the category Set of sets and functions to that of pseudometric spaces, and both are parameterised by modalities from coalgebraic modal logic. A generalisation of the Kantorovich-Rubinstein duality has been more nebulous-it is known not to work in some cases. In this paper we propose a compositional approach for obtaining such generalised dualities for a class of functors, which is closed under coproducts and products. Our approach is based on an explicit construction of modalities and also applies to and extends known cases such as that of the powerset functor.
- [310] arXiv:2411.12411 (replaced) [pdf, html, other]
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Title: Classification of the trajectories of uncharged particles in the Schwarzschild-Melvin metricComments: Corrected typo in titleJournal-ref: Physical Review D, 2024, vol. 110, 104031Subjects: Dynamical Systems (math.DS); General Relativity and Quantum Cosmology (gr-qc)
This paper investigates the trajectories of neutral particles in the Schwarzschild-Melvin spacetime. After reduction by cyclic coordinates this problem reduces to investigating a two-degree-of-freedom Hamiltonian system that has no additional integral. A classification of regions of possible motion of a particle is performed according to the values of the momentum and energy integrals. Bifurcations of periodic solutions of the reduced system are analyzed using a Poincare map.
- [311] arXiv:2411.12582 (replaced) [pdf, html, other]
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Title: Reconfiguration Using Generalized Token JumpingComments: To appear at WALCOM 2025Subjects: Combinatorics (math.CO); Data Structures and Algorithms (cs.DS)
In reconfiguration, we are given two solutions to a graph problem, such as Vertex Cover or Dominating Set, with each solu tion represented by a placement of tokens on vertices of the graph. Our task is to reconfigure one into the other using small steps while ensuring the intermediate configurations of tokens are also valid solutions. The two commonly studied settings are Token Jumping and Token Sliding, which allows moving a single token to an arbitrary or an adjacent vertex, respectively.
We introduce new rules that generalize Token Jumping, parameterized by the number of tokens allowed to move at once and by the maximum distance of each move. Our main contribution is identifying minimal rules that allow reconfiguring any possible given solution into any other for Independent Set, Vertex Cover, and Dominating Set. For each minimal rule, we also provide an efficient algorithm that finds a corresponding reconfiguration sequence.
We further focus on the rule that allows each token to move to an adjacent vertex in a single step. This natural variant turns out to be the minimal rule that guarantees reconfigurability for Vertex Cover. We determine the computational complexity of deciding whether a (shortest) reconfiguration sequence exists under this rule for the three studied problems. While reachability for Vertex Cover is shown to be in P, finding a shortest sequence is shown to be NP-complete. For Independent Set and Dominating Set, even reachability is shown to be PSPACE-complete. - [312] arXiv:2411.12868 (replaced) [pdf, html, other]
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Title: On the ill-posedness of kinetic wave equationsComments: 24 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
In this article we identify a sharp ill-posedness/well-posedness threshold for kinetic wave equations (KWE) derived from quasilinear Schrödinger models. We show well-posedness using a collisional averaging estimate proved in our earlier work \cite{AmLe}. Ill-posedness manifests as instantaneous loss of smoothness for well-chosen initial data. We also prove that both the gain-only and full equation share the same well-posedness threhold, thus legitimizing a gain-only approach to solving 4-wave kinetic equations.
- [313] arXiv:2411.13146 (replaced) [pdf, html, other]
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Title: An Analytical Exploration of the Erd\"os-Moser Equation $ \sum_{i=1}^{m-1} i^k = m^k $ Using Approximation MethodsSubjects: Number Theory (math.NT); Combinatorics (math.CO)
The Erdös-Moser equation $ \sum_{i=1}^{m - 1} i^k = m^k $ is a longstanding problem in number theory, with the only known solution in positive integers being $ (k, m) = (1, 3) $. This paper investigates the possibility of other solutions by employing approximation methods based on the Euler-MacLaurin formula to extend the discrete sum $ S(m - 1, k) $ to a continuous function $ S_{\mathbb{R}}(m - 1, k) $. Analyzing the approximate polynomial $ P_{\mathbb{R}}(m) = S_{\mathbb{R}}(m - 1, k) - m^k $, we apply the rational root theorem to search for potential integer solutions. Our investigation confirms that for $ k = 1 $, the only solution is $ m = 3 $. For $ k \geq 2 $, the approximation suggests that no additional positive integer solutions exist. However, we acknowledge the limitations of using approximation methods in the context of Diophantine equations, where exactness is crucial. The omission of correction terms in the approximation may overlook valid solutions. Despite these limitations, our work provides insights into the behavior of the Erdös-Moser equation and highlights the challenges in finding solutions using analytical methods. We discuss the implications of our findings and suggest directions for future research, emphasizing the need for exact analytical techniques to conclusively address the conjecture.
- [314] arXiv:2411.13199 (replaced) [pdf, html, other]
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Title: Sharp Bounds for Multiple Models in Matrix CompletionComments: 35 pages. Several typos have been corrected. All comments are warmly welcomedSubjects: Statistics Theory (math.ST)
In this paper, we demonstrate how a class of advanced matrix concentration inequalities, introduced in \cite{brailovskaya2024universality}, can be used to eliminate the dimensional factor in the convergence rate of matrix completion. This dimensional factor represents a significant gap between the upper bound and the minimax lower bound, especially in high dimension. Through a more precise spectral norm analysis, we remove the dimensional factors for five different estimators in various settings, thereby establishing their minimax rate optimality.
- [315] arXiv:2411.13204 (replaced) [pdf, html, other]
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Title: Preserving curvature lower bounds when Ricci flowing non-smooth initial dataComments: This survey paper appeared in "Surveys in Differential Geometry, Vol. 27, No. 1 (2022), pp. 147-187". Although the reference is 2022, the paper was first submitted/published in 2024. In this version, Version 2: Fixed up a formatting error. Otherwise no changes to Version 1Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Metric Geometry (math.MG)
In this paper we survey some results on Ricci flowing non-smooth initial data. Among other things, we give a non-exhaustive list of various weak initial data which can be evolved with the Ricci flow. We also survey results which show that various curvature lower bounds will, possibly up to a constant, be preserved, if we start with such possibly non-smooth initial data. Some proofs/proof sketches are given in certain cases. A list of some open problems related to these areas is given in the last section of the paper.
- [316] arXiv:2411.13255 (replaced) [pdf, html, other]
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Title: Note on the $a$-points of the Riemann zeta functionComments: 16 pages, 0 figuresSubjects: Number Theory (math.NT)
For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$ shown by Garunkštis and Steuding. We then deduce an asymptotic of the sum \[S_T(a,\delta)=\sum_{\tau<\gamma_a\leqslant T}\zeta'(\rho_a+i\delta)X^{\rho_a},\quad T\to\infty,\] where $0\ne\delta=\frac{2\pi\alpha}{\log\frac{T}{2\pi X}}\ll 1$, and $X>0$ and $\tau\geqslant|\delta|+1$ are fixed.
We also find the interesting varied behavior of $S_T(a,\delta)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump. - [317] arXiv:1806.07428 (replaced) [pdf, html, other]
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Title: Minimum Quadratic Helicity StatesComments: 15 pages, 3 figuresSubjects: Fluid Dynamics (physics.flu-dyn); Solar and Stellar Astrophysics (astro-ph.SR); Differential Geometry (math.DG)
Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have indeed minimum quadratic helicity.
- [318] arXiv:2301.10088 (replaced) [pdf, html, other]
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Title: Linear Arboreal CategoriesSubjects: Logic in Computer Science (cs.LO); Category Theory (math.CT); Logic (math.LO)
Arboreal categories, introduced by Abramsky and Reggio, axiomatise categories with tree-shaped objects. These categories provide a categorical language for formalising behavioural notions such as simulation, bisimulation, and resource-indexing. In this paper, we strengthen the axioms of an arboreal category to exclude `branching' behaviour, obtaining a notion of `linear arboreal category'. We then demonstrate that every arboreal category satisfying a linearisability condition has an associated linear arboreal subcategory related via an adjunction. This identifies the relationship between the pebble-relation comonad, of Montacute and Shah, and the pebbling comonad, of Abramsky, Dawar, and Wang, and generalises it further. As another outcome of this new framework, we obtain a linear variant of the arboreal category for modal logic. By doing so we recover different linear-time equivalences between transition systems as instances of their categorical definitions. We conclude with new preservation and characterisation theorems relating trace inclusion and trace equivalence with different linear fragments of modal logic.
- [319] arXiv:2306.13633 (replaced) [pdf, html, other]
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Title: Optimal Vaccination Policy to Prevent Endemicity: A Stochastic ModelComments: 51 pages, 7 figuresSubjects: Populations and Evolution (q-bio.PE); Probability (math.PR)
We examine here the effects of recurrent vaccination and waning immunity on the establishment of an endemic equilibrium in a population. An individual-based model that incorporates memory effects for transmission rate during infection and subsequent immunity is introduced, considering stochasticity at the individual level. By letting the population size going to infinity, we derive a set of equations describing the large scale behavior of the epidemic. The analysis of the model's equilibria reveals a criterion for the existence of an endemic equilibrium, which depends on the rate of immunity loss and the distribution of time between booster doses. The outcome of a vaccination policy in this context is influenced by the efficiency of the vaccine in blocking transmissions and the distribution pattern of booster doses within the population. Strategies with evenly spaced booster shots at the individual level prove to be more effective in preventing disease spread compared to irregularly spaced boosters, as longer intervals without vaccination increase susceptibility and facilitate more efficient disease transmission. We provide an expression for the critical fraction of the population required to adhere to the vaccination policy in order to eradicate the disease, that resembles a well-known threshold for preventing an outbreak with an imperfect vaccine. We also investigate the consequences of unequal vaccine access in a population and prove that, under reasonable assumptions, fair vaccine allocation is the optimal strategy to prevent endemicity.
- [320] arXiv:2309.14553 (replaced) [pdf, other]
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Title: Inverse non-linear problem of the long wave run-up on coastAlexei Rybkin, Efim Pelinovsky, Oleksandr Bobrovnikov, Noah Palmer, Ekaterina Pniushkova, Daniel AbramowiczComments: To appear in Journal of Ocean Engineering and Marine EnergySubjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph); Geophysics (physics.geo-ph)
The study of the process of catastrophic tsunami-type waves on the coast makes it possible to determine the destructive force of waves on the coast. In hydrodynamics, the one-dimensional theory of the run-up of non-linear waves on a flat slope has gained great popularity, within which rigorous analytical results have been obtained in the class of non-breaking waves. In general, the result depends on the characteristics of the wave approaching (or generated on) the slope, which is usually not known in the measurements. Here we describe a rigorous method for recovering the initial displacement in a source localised in an inclined power-shaped channel from the characteristics of a moving shoreline. The method uses the generalised Carrier-Greenspan transformation, which allows one-dimensional non-linear shallow-water equations to be reduced to linear ones. The solution is found in terms of Erdélyi-Kober integral operator. Numerical verification of our results is presented for the cases of a parabolic bay and an infinite plane beach.
- [321] arXiv:2311.13159 (replaced) [pdf, html, other]
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Title: Multi-Objective Optimization via Wasserstein-Fisher-Rao Gradient FlowYinuo Ren, Tesi Xiao, Tanmay Gangwani, Anshuka Rangi, Holakou Rahmanian, Lexing Ying, Subhajit SanyalSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
Multi-objective optimization (MOO) aims to optimize multiple, possibly conflicting objectives with widespread applications. We introduce a novel interacting particle method for MOO inspired by molecular dynamics simulations. Our approach combines overdamped Langevin and birth-death dynamics, incorporating a "dominance potential" to steer particles toward global Pareto optimality. In contrast to previous methods, our method is able to relocate dominated particles, making it particularly adept at managing Pareto fronts of complicated geometries. Our method is also theoretically grounded as a Wasserstein-Fisher-Rao gradient flow with convergence guarantees. Extensive experiments confirm that our approach outperforms state-of-the-art methods on challenging synthetic and real-world datasets.
- [322] arXiv:2401.08758 (replaced) [pdf, other]
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Title: DiscoTEX 1.0: Discontinuous collocation and implicit-turned-explicit (IMTEX) integration symplectic, symmetric numerical algorithms with higher order jumps for differential equations I: numerical black hole perturbation theory applicationsComments: 50 pages, 19 figures, 9 tables. Several typos corrected, higher-order results for the computation of energy and angular moment fluxes added. Includes overview of previous numerical methods implemented in the time-domain for the modelling of asymmetric mass ratio inspirals with suitability checks on Table 9. Now first of a series of papers. Comments are welcomeSubjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Astrophysical Phenomena (astro-ph.HE); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
Dirac $\delta-$ distributionally sourced differential equations emerge in many dynamical physical systems from machine learning, finance, neuroscience, and seismology to black hole perturbation theory. These systems lack exact analytical solutions and are thus best tackled numerically. We describe a generic numerical algorithm which constructs discontinuous spatial and temporal discretisations by operating on discontinuous Lagrange and Hermite interpolation formulae, respectively. By solving the distributionally sourced wave equation, possessing analytical solutions, we demonstrate that numerical weak-form solutions can be recovered to high-order accuracy by solving a first-order reduced system of ODEs. The method-of-lines framework is applied to the \texttt{DiscoTEX} algorithm i.e. through \underline{dis}continuous \underline{co}llocation with implicit\underline{-turned-explicit} integration methods which are symmetric and conserve symplectic structure. Furthermore, the main application of the algorithm is proved by calculating the amplitude at any desired location within the numerical grid, including at the position (and at its right and left limit) where the wave- (or wave-like) equation is discontinuous via interpolation using \texttt{DiscoTEX}. This is demonstrated, firstly by solving the wave- (or wave-like) equation and comparing the numerical weak-form solution to the exact solution. We further demonstrate how to reconstruct the gravitational metric perturbations from weak-form numerical solutions of a non-rotating black hole, which do not have known exact analytical solutions, and compare them against state-of-the-art frequency domain results. We conclude by motivating how \texttt{DiscoTEX}, and related numerical algorithms, both open a promising new alternative waveform generation route for modelling highly asymmetric binaries and complement current frequency domain methods.
- [323] arXiv:2403.14931 (replaced) [pdf, html, other]
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Title: Structured stability analysis of networked systems with uncertain linksSubjects: Systems and Control (eess.SY); Dynamical Systems (math.DS)
An input-output approach to stability analysis is explored for networked systems with uncertain link dynamics. The main result consists of a collection of integral quadratic constraints, which together imply robust stability of the uncertain networked system, under the assumption that stability is achieved with ideal links. The conditions are decentralized inasmuch as each involves only agent and uncertainty model parameters that are local to a corresponding link. This makes the main result, which imposes no restriction on network structure, suitable for the study of large-scale systems.
- [324] arXiv:2405.00663 (replaced) [pdf, html, other]
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Title: Quantum cryptographic protocols with dual messaging system via 2D alternate quantum walk of a genuine single-photon entangled stateComments: 13 pages (including appendix), two figures and one table, accepted for publication in Journal of Physics A: Mathematical and Theoretical as a letterJournal-ref: Journal of Physics A: Mathematical and Theoretical (2024)Subjects: Quantum Physics (quant-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn); Cryptography and Security (cs.CR); Quantum Algebra (math.QA); Optics (physics.optics)
A single-photon entangled state (or single-particle entangled state (SPES) in general) can offer a more secure way of encoding and processing quantum information than their multi-photon (or multi-particle) counterparts. The SPES generated via a 2D alternate quantum-walk setup from initially separable states can be either 3-way or 2-way entangled. This letter shows that the generated genuine three-way and nonlocal two-way SPES can be used as cryptographic keys to securely encode two distinct messages simultaneously. We detail the message encryption-decryption steps and show the resilience of the 3-way and 2-way SPES-based cryptographic protocols against eavesdropper attacks like intercept-and-resend and man-in-the-middle. We also detail the experimental realization of these protocols using a single photon, with the three degrees of freedom being OAM, path, and polarization. We have proved that the protocols have unconditional security for quantum communication tasks. The ability to simultaneously encode two distinct messages using the generated SPES showcases the versatility and efficiency of the proposed cryptographic protocol. This capability could significantly improve the throughput of quantum communication systems.
- [325] arXiv:2405.00776 (replaced) [pdf, html, other]
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Title: Higher spins and Finsler geometryComments: 36 pages. v2, published version: minor corrections, added referencesSubjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Differential Geometry (math.DG)
Finsler geometry is a natural generalization of (pseudo-)Riemannian geometry, where the line element is not the square root of a quadratic form but a more general homogeneous function. Parameterizing this in terms of symmetric tensors suggests a possible interpretation in terms of higher-spin fields. We will see here that, at linear level in these fields, the Finsler version of the Ricci tensor leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling. Nonlinear terms can also be systematically analyzed, suggesting a possible interacting structure. No particular choice of spacetime dimension is needed. The Stueckelberg mechanism breaks gauge transformations to a redundancy that does not change the geometry. This creates a serious issue: non-transverse modes are not eliminated, at least for the versions of Finsler dynamics examined in this paper.
- [326] arXiv:2405.01425 (replaced) [pdf, html, other]
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Title: In-and-Out: Algorithmic Diffusion for Sampling Convex BodiesComments: 33 pages. To appear in NeurIPS 2024 (spotlight). Improve Lemma 22 and 26Subjects: Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in Rényi divergence (which implies TV, $\mathcal{W}_2$, KL, $\chi^2$). The proof departs from known approaches for polytime algorithms for the problem -- we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the stationary density.
- [327] arXiv:2405.01715 (replaced) [pdf, html, other]
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Title: GRAMEP: an alignment-free method based on the Maximum Entropy Principle for identifying SNPsMatheus Henrique Pimenta-Zanon, André Yoshiaki Kashiwabara, André Luís Laforga Vanzela, Fabricio Martins LopesSubjects: Genomics (q-bio.GN); Information Theory (cs.IT); Applications (stat.AP)
Background: Advances in high throughput sequencing technologies provide a huge number of genomes to be analyzed. Thus, computational methods play a crucial role in analyzing and extracting knowledge from the data generated. Investigating genomic mutations is critical because of their impact on chromosomal evolution, genetic disorders, and diseases. It is common to adopt aligning sequences for analyzing genomic variations. However, this approach can be computationally expensive and restrictive in scenarios with large datasets. Results: We present a novel method for identifying single nucleotide polymorphisms (SNPs) in DNA sequences from assembled genomes. This study proposes GRAMEP, an alignment-free approach that adopts the principle of maximum entropy to discover the most informative k-mers specific to a genome or set of sequences under investigation. The informative k-mers enable the detection of variant-specific mutations in comparison to a reference genome or other set of sequences. In addition, our method offers the possibility of classifying novel sequences with no need for organism-specific information. GRAMEP demonstrated high accuracy in both in silico simulations and analyses of viral genomes, including Dengue, HIV, and SARS-CoV-2. Our approach maintained accurate SARS-CoV-2 variant identification while demonstrating a lower computational cost compared to methods with the same purpose. Conclusions: GRAMEP is an open and user-friendly software based on maximum entropy that provides an efficient alignment-free approach to identifying and classifying unique genomic subsequences and SNPs with high accuracy, offering advantages over comparative methods. The instructions for use, applicability, and usability of GRAMEP are open access at this https URL
- [328] arXiv:2406.02573 (replaced) [pdf, other]
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Title: On equivalence of gauge-invariant models for massive integer-spin fieldsComments: 39 pages. This work includes the main results of the unpublished manuscript arXiv:2310.00951; V2: references and comments added; V3: 46 pages, name of first author changed, comments and new appendix addedJournal-ref: Phys. Rev. D 110, 105014 (2024)Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
There are several approaches to formulate gauge-invariant models for massive integer-spin fields in $d$ dimensions including the following: (i) in terms of symmetric tensor fields $\phi_{\mu_1 \dots \mu_k} $, with $k = s, s-1, \dots , 0$, restricted to be double traceless for $k\geq 4$; and (ii) in terms of a quartet of $traceful$ symmetric tensor fields $\psi_{\mu_1 \dots \mu_k} $, of rank $k=s,s-1,s-2, s-3$. We demonstrate that these formulations in Minkowski space ${\mathbb M}^d$ are equivalent to the gauge-invariant theory for a massive integer-spin field proposed in 1989 by Pashnev. We also make use of the Klishevich-Zinoviev theory in ${\mathbb M}^d$ to derive a unique generalisation of the Singh-Hagen model for a massive integer-spin field in $d>4 $ dimensions.
- [329] arXiv:2406.05153 (replaced) [pdf, html, other]
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Title: Integrating Physics of the Problem into Data-Driven Methods to Enhance Elastic Full-Waveform Inversion with Uncertainty QuantificationSubjects: Geophysics (physics.geo-ph); Machine Learning (cs.LG); Numerical Analysis (math.NA)
Full-Waveform Inversion (FWI) is a nonlinear iterative seismic imaging technique that, by reducing the misfit between recorded and predicted seismic waveforms, can produce detailed estimates of subsurface geophysical properties. Nevertheless, the strong nonlinearity of FWI can trap the optimization in local minima. This issue arises due to factors such as improper initial values, the absence of low frequencies in the measurements, noise, and other related considerations. To address this challenge and with the advent of advanced machine-learning techniques, data-driven methods, such as deep learning, have attracted significantly increasing attention in the geophysical community. Furthermore, the elastic wave equation should be included in FWI to represent elastic effects accurately. The intersection of data-driven techniques and elastic scattering theories presents opportunities and challenges. In this paper, by using the knowledge of elastic scattering (physics of the problem) and integrating it with machine learning techniques, we propose methods for the solution of time-harmonic FWI to enhance accuracy compared to pure data-driven and physics-based approaches. Moreover, to address uncertainty quantification, by modifying the structure of the Variational Autoencoder, we introduce a probabilistic deep learning method based on the physics of the problem that enables us to explore the uncertainties of the solution. According to the limited availability of datasets in this field and to assess the performance and accuracy of the proposed methods, we create a comprehensive dataset close to reality and conduct a comparative analysis of the presented approaches to it.
- [330] arXiv:2406.16787 (replaced) [pdf, html, other]
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Title: A Parrondo paradox in susceptible-infectious-susceptible dynamics over periodic temporal networksComments: 10 figuresJournal-ref: Mathematical Biosciences, Volume 378, December 2024, 109336Subjects: Physics and Society (physics.soc-ph); Dynamical Systems (math.DS)
Many social and biological networks periodically change over time with daily, weekly, and other cycles. Thus motivated, we formulate and analyze susceptible-infectious-susceptible (SIS) epidemic models over temporal networks with periodic schedules. More specifically, we assume that the temporal network consists of a cycle of alternately used static networks, each with a given duration. We observe a phenomenon in which two static networks are individually above the epidemic threshold but the alternating network composed of them renders the dynamics below the epidemic threshold, which we refer to as a Parrondo paradox for epidemics. We find that network structure plays an important role in shaping this phenomenon, and we study its dependence on the connectivity between and number of subpopulations in the network. We associate such paradoxical behavior with anti-phase oscillatory dynamics of the number of infectious individuals in different subpopulations.
- [331] arXiv:2406.19430 (replaced) [pdf, other]
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Title: Invitation to Local AlgorithmsSubjects: Distributed, Parallel, and Cluster Computing (cs.DC); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
This text provides an introduction to distributed local algorithms -- an area at the intersection of theoretical computer science and discrete mathematics. We collect recent results in the area and demonstrate how they lead to a clean theory. We also discuss many connections of local algorithms to fields such as parallel, distributed, and sublinear algorithms, or descriptive combinatorics.
- [332] arXiv:2407.04756 (replaced) [pdf, html, other]
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Title: On Hamiltonian formulations of the Dirac systemComments: Annals of Physics, in press, 35 pages, new section addedSubjects: Quantum Physics (quant-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We extend a previously successful discussion of the constrained Schrödinger system through the Dirac--Bergmann algorithm to the case of the Dirac field. In order to follow the analogy, first we discuss the classical Dirac field as a spinorial variable, by introducing properly defined momenta and a suitably modified, factor ordered Poisson bracket. According to the Dirac--Bergmann algorithm two second class Hamiltonian constraints emerge, leading to a factor ordered Dirac bracket on the full phase space. This becomes the Poisson bracket on the reduced phase space in the canonical chart adapted to the shell. The Dirac equation is recovered both as consistency condition on the full phase space and as canonical equation on the reduced phase space. Alternatively, considering the Dirac field as odd Grassmann variable, we present the details of the Dirac--Bergmann algorithm (with either left and righ derivatives acting on Grassmann valued superfunctions and involving a different type of generalized Poisson and Dirac brackets). We propose a recipe for the canonical second quantization of all three versions of the generalized Dirac brackets, yielding the correct fundamental anticommutator.
- [333] arXiv:2408.05384 (replaced) [pdf, html, other]
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Title: Nonlinear Propagation of Non-Gaussian UncertaintiesSubjects: Space Physics (physics.space-ph); Symbolic Computation (cs.SC); Probability (math.PR); Chaotic Dynamics (nlin.CD)
This paper presents a novel approach for propagating uncertainties in dynamical systems building on high-order Taylor expansions of the flow and moment-generating functions (MGFs). Unlike prior methods that focus on Gaussian distributions, our approach leverages the relationship between MGFs and distribution moments to extend high-order uncertainty propagation techniques to non-Gaussian scenarios. This significantly broadens the applicability of these methods to a wider range of problems and uncertainty types. High-order moment computations are performed one-off and symbolically, reducing the computational burden of the technique to the calculation of Taylor series coefficients around a nominal trajectory, achieved by efficiently integrating the system's variational equations. Furthermore, the use of the proposed approach in combination with event transition tensors, allows for accurate propagation of uncertainties at specific events, such as the landing surface of a celestial body, the crossing of a predefined Poincaré section, or the trigger of an arbitrary event during the propagation. Via numerical simulations we demonstrate the effectiveness of our method in various astrodynamics applications, including the unperturbed and perturbed two-body problem, and the circular restricted three-body problem, showing that it accurately propagates non-Gaussian uncertainties both at future times and at event manifolds.
- [334] arXiv:2408.10001 (replaced) [pdf, html, other]
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Title: Coprime Bivariate Bicycle CodesSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT)
This work (1) proposes a novel numerical algorithm to accelerate the search process for good Bivariate Bicycle (BB) codes and (2) defines a new subclass of BB codes suitable for quantum error correction. The proposed acceleration search algorithm reduces the search space by excluding some equivalent codes from the search space, as well as setting thresholds to drop bad codes at an early stage. A number of new BB codes found by this algorithm are reported. The proposed subclass of BB codes employs coprimes to construct groups via polynomials as the basis for the BB code, rather than using the standard BB codes with unconstrained constructors. In contrast to vanilla BB codes, where parameters remain unknown prior to code discovery, the rate of the proposed code can be determined beforehand by specifying a factor polynomial as an input to the numerical search algorithm. Using this coprime BB construction, we found a number of surprisingly short to medium-length codes that were previously unknown.
- [335] arXiv:2408.12752 (replaced) [pdf, html, other]
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Title: High-distance codes with transversal Clifford and T-gatesComments: 2 tables, 3 figures. Updated version: Includes a family of triorthogonal codes with improved parameters. Includes a more in-depth discussion of T-gate code familiesSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Number Theory (math.NT)
The non-local interactions in several quantum devices allow for the realization of more compact quantum encodings while retaining the same degree of protection against noise. Anticipating that short to medium-length codes will soon be realizable, it is important to construct stabilizer codes that, for a given code distance, admit fault-tolerant implementations of logical gates with the fewest number of physical qubits. We extract high-distance doubly even codes from the quantum quadratic-residue code family that admit a transversal implementation of the single-qubit Clifford group and block transversal implementation of the full Clifford group. Applying a doubling procedure [arXiv:1509.03239] to such codes yields a family of high-distance weak triply even codes which admit a transversal implementation of the logical $\texttt{T}$-gate. Relaxing the triply even property, we also obtain a family of triorthogonal codes which requires an even lower overhead at the cost of additional Clifford gates to achieve the same logical operation. To our knowledge, our doubly even and triorthogonal families are the shortest qubit stabilizer codes of the same distance that can realize their respective gates.
- [336] arXiv:2409.04322 (replaced) [pdf, html, other]
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Title: Integrability of polynomial vector fields and a dual problemSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Classical Analysis and ODEs (math.CA); Dynamical Systems (math.DS)
We investigate the integrability of polynomial vector fields through the lens of duality in parameter spaces. We examine formal power series solutions annihilated by differential operators and explore the properties of the integrability variety in relation to the invariants of the associated Lie group. Our study extends to differential operators on affine algebraic varieties, highlighting the intrinsic connection between these operators and local analytic first integrals. To illustrate the duality the case of quadratic vector fields is considered in detail.
- [337] arXiv:2409.10549 (replaced) [pdf, html, other]
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Title: Confronting Conflicts to Yes: Untangling Wicked Problems with Open Design SystemsSubjects: Computers and Society (cs.CY); Optimization and Control (math.OC)
Current project development practices often fail to engage stakeholders early and effectively. Decision support is often non-inclusive, single-sided, and lacking in transparency, while complexity goes beyond human's comprehension. Additionally, many approaches focus primarily on technical system aspects, neglecting the integration of stakeholders' individual preferences. This often results in project impasses, leaving stakeholders unable to collaboratively achieve a "yes." There is a need for a purely associative, a-priori design approach that integrates system realities and stakeholder ideals within a joint socio-technical solution space. The state-of-the-art Preferendus, embedded in the proven Open Design Systems (Odesys) methodology, is a neutral tool for transforming complexity into success. Aiming for synthesis, Odesys' robust IMAP optimization method generates a single best-fit design solution. Here, Odesys is applied for a Dutch wind farm stalemate development, balancing multiple stakeholder preferences, wind farm performances, and project constraints. The success of this approach hinges on stakeholder trust and input. This article introduces a structured stakeholder assessment method using choice-based conjunctive analysis (CBCA), facilitating transparent determination of global and local stakeholder weights and preference functions. Modelling 'disputable' exogenous factors as endogenous design parameters, the application demonstrates how one can shift toward a collaborative "yes." For this, it is concluded that a zoomed-out solution space would enable the energy transition to be tackled with multiple options rather than a prescribed one. The Odesys approach fosters decision-making that aligns with the social threefold principles of freedom, equality, and fraternity, guiding projects toward genuine democratic outcomes rather than selecting from curated options.
- [338] arXiv:2409.13978 (replaced) [pdf, html, other]
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Title: FracGM: A Fast Fractional Programming Technique for Geman-McClure Robust EstimatorComments: 8 pages, 6 figuresJournal-ref: IEEE Robotics and Automation Letters, 9(12), 11666-11673, 2024Subjects: Computer Vision and Pattern Recognition (cs.CV); Robotics (cs.RO); Optimization and Control (math.OC)
Robust estimation is essential in computer vision, robotics, and navigation, aiming to minimize the impact of outlier measurements for improved accuracy. We present a fast algorithm for Geman-McClure robust estimation, FracGM, leveraging fractional programming techniques. This solver reformulates the original non-convex fractional problem to a convex dual problem and a linear equation system, iteratively solving them in an alternating optimization pattern. Compared to graduated non-convexity approaches, this strategy exhibits a faster convergence rate and better outlier rejection capability. In addition, the global optimality of the proposed solver can be guaranteed under given conditions. We demonstrate the proposed FracGM solver with Wahba's rotation problem and 3-D point-cloud registration along with relaxation pre-processing and projection post-processing. Compared to state-of-the-art algorithms, when the outlier rates increase from 20% to 80%, FracGM shows 53% and 88% lower rotation and translation increases. In real-world scenarios, FracGM achieves better results in 13 out of 18 outcomes, while having a 19.43% improvement in the computation time.
- [339] arXiv:2410.16402 (replaced) [pdf, html, other]
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Title: Universal time evolution of string order parameter in quantum critical systems with boundary invertible or non-invertible symmetry breakingComments: are welcome. 32 pages, many figures; v2: Refs addedSubjects: Strongly Correlated Electrons (cond-mat.str-el); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
The global symmetry, either invertible or non-invertible, has been extensively studied in two dimensional conformal field theories in recent years. When the theory is defined on a manifold with open boundaries, however, many interesting conformal boundary conditions will fully or partially break such global symmetry. In this work, we study the effect of symmetry-breaking boundaries or interfaces when the system is out of equilibrium. We show that the boundary or interface symmetry-breaking can be detected by the time evolution of string order parameters, which are constructed from the symmetry operators that implement the symmetry transformations. While the string order parameters are independent of time if the symmetry is preserved over the whole system, they evolve in time in a universal way if the boundary or interface breaks the symmetry. More explicitly, in the presence of boundary or interface symmetry-breaking, the string order parameters decay exponentially in time after a global quantum quench, and decay as a power-law in time after a local quantum quench. We also generalize our study to the case when the string order parameters are defined in a subsystem, which are related to the full counting statistics. It is found there are also universal features in the time evolution of string order parameters in this case. We verify our field theory results by studying the time evolution of these two different types of string order parameters in lattice models.
- [340] arXiv:2410.18044 (replaced) [pdf, html, other]
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Title: Time Evolution in Quantum Mechanics with a Minimal Time ScaleComments: 24 pages, 5 figuresJournal-ref: Symmetry 16, 1520 (2024)Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
The existence of a minimum measurable length scale was suggested by various theories of quantum gravity, string theory and black hole physics. Motivated by this, we examine a quantum theory exhibiting a minimum measurable time scale. We use the Page-Wootters formalism to describe time evolution of a quantum system with the modified commutation relations between the time and frequency operator. Such modification leads to a minimal uncertainty in the measurement of time. This causes breaking of the time-translation symmetry and results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We demonstrate the developed theory on a couple simple quantum systems.
- [341] arXiv:2410.18254 (replaced) [pdf, html, other]
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Title: Refining Ky Fan's majorization relation with linear programmingComments: 36 pages, 2 figures, error in version 1 correctedSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Rings and Algebras (math.RA)
A separable version of Ky Fan's majorization relation is proven for a sum of two operators that are each a tensor product of two positive semi-definite operators. In order to prove it, upper bounds are established for the relevant largest eigenvalue sums in terms of the optimal values of certain linear programs. The objective function of these linear programs is the dual of the direct sum of the spectra of the summands. The feasible sets are bounded polyhedra determined by positive numbers, called alignment terms, that quantify the overlaps between pairs of largest eigenvalue spaces of the summands. By appealing to geometric considerations, tight upper bounds are established on the alignment terms of tensor products of positive semi-definite operators. As an application, the spin alignment conjecture in quantum information theory is affirmatively resolved to the 2-letter level. Consequently, the coherent information of platypus channels is additive to the 2-letter level.
- [342] arXiv:2411.12700 (replaced) [pdf, other]
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Title: Learning multivariate Gaussians with imperfect adviceSubjects: Machine Learning (cs.LG); Data Structures and Algorithms (cs.DS); Information Theory (cs.IT); Machine Learning (stat.ML)
We revisit the problem of distribution learning within the framework of learning-augmented algorithms. In this setting, we explore the scenario where a probability distribution is provided as potentially inaccurate advice on the true, unknown distribution. Our objective is to develop learning algorithms whose sample complexity decreases as the quality of the advice improves, thereby surpassing standard learning lower bounds when the advice is sufficiently accurate.
Specifically, we demonstrate that this outcome is achievable for the problem of learning a multivariate Gaussian distribution $N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ in the PAC learning setting. Classically, in the advice-free setting, $\tilde{\Theta}(d^2/\varepsilon^2)$ samples are sufficient and worst case necessary to learn $d$-dimensional Gaussians up to TV distance $\varepsilon$ with constant probability. When we are additionally given a parameter $\tilde{\boldsymbol{\Sigma}}$ as advice, we show that $\tilde{O}(d^{2-\beta}/\varepsilon^2)$ samples suffices whenever $\| \tilde{\boldsymbol{\Sigma}}^{-1/2} \boldsymbol{\Sigma} \tilde{\boldsymbol{\Sigma}}^{-1/2} - \boldsymbol{I_d} \|_1 \leq \varepsilon d^{1-\beta}$ (where $\|\cdot\|_1$ denotes the entrywise $\ell_1$ norm) for any $\beta > 0$, yielding a polynomial improvement over the advice-free setting.