Mathematics
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- [1] arXiv:2407.03322 [pdf, other]
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Title: Negative Shaping Order K in Set Shaping Theory: A Comprehensive AnalysisSubjects: Information Theory (cs.IT)
This paper delves into an innovative aspect of the Set Shaping Theory, exploring the use of a negative shaping order K. Traditionally, the theory utilizes a positive K to extend the length of data strings, enhancing their testability and compressibility. We propose a paradigm shift by employing a negative K, which shortens data strings and potentially improves compression efficiency. However, this approach sacrifices the local testability of the data, a cornerstone in traditional Set Shaping Theory. We examine the theoretical implications, practical benefits, and challenges of this new methodology.
- [2] arXiv:2407.03323 [pdf, other]
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Title: FFT-acceleration and stabilization of the 3D Marching-on-in-Time Contrast Current Density Volume Integral Equation for scattering from high contrast dielectricsComments: Submitted to Progress In Electromagnetic ResearchSubjects: Numerical Analysis (math.NA)
An implicit causal space-time Galerkin scheme applied to the contrast current density volume integral equation gives rise to a marching-on-in-time scheme known as the MOT-JVIE, which is accelerated and stabilized via a fully embedded FIR filter to compute the electromagnetic scattering from high permittivity dielectric objects discretized with over a million voxels. A review of two different acceleration approaches previously developed for two-dimensional time-domain surface integral equations based on fast Fourier transforms (FFTs), leads to an understanding why these schemes obtain the same order of acceleration and the extension of this FFT-acceleration to the three-dimensional MOT-JVIE. The positive definite stability analysis (PDSA) for the MOT-JVIE shows that the number of voxels for a stable MOT-JVIE discretization is restricted by the finite precision of the matrix elements. The application of the PDSA provides the insight that stability can be enforced through regularization, at the cost of accuracy. To minimize the impact in accuracy, FIR-regularization is introduced, which is based on low group-delay linear-phase high-pass FIR-filters. We demonstrate the capabilities of the FFT-accelerated FIR-regularized MOT-JVIE for a number of numerical experiments with high permittivity dielectric scatterers.
- [3] arXiv:2407.03325 [pdf, html, other]
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Title: On the accuracy and efficiency of reduced order models: towards real-world applicationsSubjects: Numerical Analysis (math.NA)
This chapter provides an extended overview about Reduced Order Models (ROMs), with a focus on their features in terms of efficiency and accuracy. In particular, the aim is to browse the more common ROM frameworks, considering both intrusive and data-driven approaches. We present the validation of such techniques against several test cases. The first one is an academic benchmark, the thermal block problem, where a Poisson equation is considered. Here a classic intrusive ROM framework based on a Galerkin projection scheme is employed. The second and third test cases come from real-world applications, the one related to the investigation of the blood flow patterns in a patient specific coronary arteries configuration where the Navier Stokes equations are addressed and the other one concerning the granulation process within pharmaceutical industry where a fluid-particle system is considered. Here we employ two data-driven ROM approaches showing a very relevant trade-off between accuracy and efficiency. In the last part of the contribution, two novel technological platforms, ARGOS and ATLAS, are presented. They are designed to provide a user-friendly access to data-driven models for real-time predictions for complex biomedical and industrial problems.
- [4] arXiv:2407.03326 [pdf, html, other]
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Title: Universal Coefficients Formula for the Residual Power Series Method with General Integral TransformSubjects: General Mathematics (math.GM)
This paper introduces a novel approach to address inherent limitations in the Residual Power Series (RPS) method and its variants with Laplace-like transforms when applied to solving time-fractional differential equations. Existing methods, while successful, often require computationally expensive calculations for the coefficients of the series solution. To overcome this limitation, we propose a new framework for the RPS method that utilizes a general integral transform. This framework incorporates an explicit formula for calculating coefficients, thereby eliminating repetitive computations and streamlining the solution process. Moreover, it offers a universally applicable approach, remaining compatible with various RPS methods that employ Laplace-like transform variants.
- [5] arXiv:2407.03327 [pdf, html, other]
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Title: An optimal method for high order mixed derivatives of bivariate functionsComments: arXiv admin note: substantial text overlap with arXiv:2309.05425, arXiv:2309.09710, arXiv:2405.20020Subjects: Numerical Analysis (math.NA)
The problem of optimal recovering high-order mixed derivatives of bivariate functions with finite smoothness is studied. Based on the truncation method, an algorithm for numerical differentiation is constructed, which is order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. Numerical examples are provided to illustrate the fact that our approach can be implemented successfully.
- [6] arXiv:2407.03328 [pdf, html, other]
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Title: The kernel polynomial method based on Jacobi polynomialsSubjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
The kernel polynomial method based on Jacobi polynomials $P_n^{\alpha,\beta}(x)$ is proposed. The optimal-resolution positivity-preserving kernels and the corresponding damping factors are obtained. The results provide a generalization of the Jackson damping factors for arbitrary Jacobi polynomials. For $\alpha =\pm 1/2$, $\beta =\pm 1/2$ (Chebyshev polynomials of the first to fourth kinds), explicit trigonometric expressions for the damping factors are obtained. The resulting algorithm can be easily introduced into existing implementations of the kernel polynomial method.
- [7] arXiv:2407.03329 [pdf, html, other]
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Title: $L^{p}$-convergence of Kantorovich-type Max-Min Neural Network OperatorsComments: 23 pages, 6 figuresSubjects: Numerical Analysis (math.NA)
In this work, we study the Kantorovich variant of max-min neural network operators, in which the operator kernel is defined in terms of sigmoidal functions. Our main aim is to demonstrate the $L^{p}$-convergence of these nonlinear operators for $1\leq p<\infty$, which makes it possible to obtain approximation results for functions that are not necessarily continuous. In addition, we will derive quantitative estimates for the rate of approximation in the $L^{p}$-norm. We will provide some explicit examples, studying the approximation of discontinuous functions with the max-min operator, and varying additionally the underlying sigmoidal function of the kernel. Further, we numerically compare the $L^{p}$-approximation error with the respective error of the Kantorovich variants of other popular neural network operators. As a final application, we show that the Kantorovich variant has advantages compared to the sampling variant of the max-min operator and Kantorovich variant of the max-product operator when it comes to approximate noisy functions as for instance biomedical ECG signals.
- [8] arXiv:2407.03334 [pdf, other]
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Title: Linear model reduction using SPOD modesComments: 32 pages, 13 figuresSubjects: Numerical Analysis (math.NA)
The majority of model reduction approaches use an efficient representation of the state and then derive equations to temporally evolve the coefficients that encode the state in the representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval and solve for the coefficients that define the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, in particular, which possess properties that make them suitable for model reduction and are known to provide an accurate representation of trajectories. In fact, with the same number of total coefficients, the SPOD representation is substantially more accurate than any representation formed by specifying the coefficients in a spatial (e.g., POD) basis for the many time steps that make up the interval. We develop a method to solve for the SPOD coefficients that encode the trajectories in forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate representation of the trajectory. We apply the method to two examples, a linearized Ginzburg-Landau problem and an advection-diffusion problem. In both, the error of the proposed method is orders of magnitude lower than both POD-Galerkin and balanced truncation applied to the same problem, as well as the most accurate solution within the span of the POD modes. The method is also fast, with CPU time comparable to or lower than both benchmarks in the examples we present.
- [9] arXiv:2407.03335 [pdf, html, other]
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Title: Dual-Domain Deep D-bar Method for Solving Electrical Impedance TomographyComments: 15 pages, 7 figuresSubjects: Numerical Analysis (math.NA); Computer Vision and Pattern Recognition (cs.CV); Image and Video Processing (eess.IV)
The regularized D-bar method is one of the most prominent methods for solving Electrical Impedance Tomography (EIT) problems due to its efficiency and simplicity. It provides a direct approach by applying low-pass filtering to the scattering data in the non-linear Fourier domain, thereby yielding a smoothed conductivity approximation. However, D-bar images often present low contrast and low resolution due to the absence of accurate high-frequency information and ill-posedness of the problem. In this paper, we proposed a dual-domain neural network architecture to retrieve high-contrast D-bar image sequences from low-contrast D-bar images. To further accentuate the spatial features of the conductivity distribution, the widely adopted U-net has been tailored for conductivity image calibration from the predicted D-bar image sequences. We call such a hybrid approach by Dual-Domain Deep D-bar method due to the consideration of both scattering data and image information. Compared to the single-scale structure, our proposed multi-scale structure exhibits superior capabilities in reducing artifacts and refining conductivity approximation. Additionally, solving discrete D-bar systems using the GMRES algorithm entails significant computational complexity, which is extremely time-consuming on CPU-based devices. To remedy this, we designed a surrogate GPU-based Richardson iterative method to accelerate the data enhancement process by D-bar. Numerical results are presented for simulated EIT data from the KIT4 and ACT4 systems to demonstrate notable improvements in absolute EIT imaging quality when compared to existing methodologies.
- [10] arXiv:2407.03336 [pdf, other]
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Title: Efficient and Precise Calculation of the Confluent Hypergeometric FunctionComments: 22 pages, 8 figuresSubjects: Numerical Analysis (math.NA); Methodology (stat.ME)
Kummer's function, also known as the confluent hypergeometric function (CHF), is an important mathematical function, in particular due to its many special cases, which include the Bessel function, the incomplete Gamma function and the error function (erf). The CHF has no closed form expression, but instead is most commonly expressed as an infinite sum of ratios of rising factorials, which makes its precise and efficient calculation challenging. It is a function of three parameters, the first two being the rising factorial base of the numerator and denominator, and the third being a scale parameter. Accurate and efficient calculation for large values of the scale parameter is particularly challenging due to numeric underflow and overflow which easily occur when summing the underlying component terms. This work presents an elegant and precise mathematical algorithm for the calculation of the CHF, which is of particular advantage for large values of the scale parameter. This method massively reduces the number and range of component terms which need to be summed to achieve any required precision, thus obviating the need for the computationally intensive transformations needed by current algorithms.
- [11] arXiv:2407.03337 [pdf, html, other]
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Title: The Impact of Data Dependence, Convergence and Stability by $AT$ Iterative AlgorithmsComments: 17 pages, 2 figureSubjects: Classical Analysis and ODEs (math.CA); Numerical Analysis (math.NA)
This article aims to present the $AT$ algorithm, a novel two-step iterative approach for approximating fixed points of weak contractions within complete normed linear spaces. The article demonstrates the convergence of $AT$ algorithm towards fixed points of weak contractions. Notably, it establishes the algorithm's strong convergence properties, highlighting its faster convergence compared to established iterative methods such as $S$, normal-$S$, Varat, Mann, Ishikawa, $F^{*} $, and Picard algorithms. Additionally, the study explores the $AT$ algorithm's almost stable behavior for weak contractions. Emphasizing practical applicability, the paper offers data-dependent results through the $AT$ algorithm and substantiates findings with illustrative numerical examples
- [12] arXiv:2407.03338 [pdf, html, other]
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Title: Error Inhibiting Methods for Finite ElementsComments: arXiv admin note: substantial text overlap with arXiv:2011.14411Subjects: Numerical Analysis (math.NA)
Finite Difference methods (FD) are one of the oldest and simplest methods for solving partial differential equations (PDE). Block Finite Difference methods (BFD) are FD methods in which the domain is divided into blocks, or cells, containing two or more grid points, with a different scheme used for each grid point, unlike the standard FD method.
It was shown in recent works that BFD schemes might be one to three orders more accurate than their truncation errors. Due to these schemes' ability to inhibit the accumulation of truncation errors, these methods were called Error Inhibiting Schemes (EIS).
This manuscript shows that our BFD schemes can be viewed as a particular type of Discontinuous Galerkin (DG) method. Then, we prove the BFD scheme's stability using the standard DG procedure while using a Fourier-like analysis to establish its optimal convergence rate.
We present numerical examples in one and two dimensions to demonstrate the efficacy of these schemes. - [13] arXiv:2407.03339 [pdf, html, other]
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Title: Stabilized Time Series Expansions for High-Order Finite Element Solutions of Partial Differential EquationsSubjects: Numerical Analysis (math.NA)
Over the past decade, Finite Element Method (FEM) has served as a foundational numerical framework for approximating the terms of Time Series Expansion (TSE) as solutions to transient Partial Differential Equation (PDE). However, the application of high-order Finite Element (FE) to certain classes of PDEs, such as diffusion equations and the Navier-Stokes (NS) equations, often leads to numerical instabilities. These instabilities limit the number of valid terms in the series, though the efficiency of time series integration even when resummation techniques like the Borel-Padé-Laplace (BPL) integrators are employed. In this study, we introduce a novel variational formulation for computing the terms of a TSE associated with a given PDE using higher-order FEs. Our approach involves the incorporation of artificial diffusion terms on the left-hand side of the equations corresponding to each power in the series, serving as a stabilization technique. We demonstrate that this method can be interpreted as a minimization of an energy functional, wherein the total variations of the unknowns are considered. Furthermore, we establish that the coefficients of the artificial diffusion for each term in the series obey a recurrence relation, which can be determined by minimizing the condition number of the associated linear system. We highlight the link between the proposed technique and the Discrete Maximum Principle (DMP) of the heat equation. We show, via numerical experiments, how the proposed technique allows having additional valid terms of the series that will be substantial in enlarging the stability domain of the BPL integrators.
- [14] arXiv:2407.03343 [pdf, html, other]
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Title: Accurate close interactions of Stokes spheres using lubrication-adapted image systemsComments: 31 pages, 18 figuresSubjects: Numerical Analysis (math.NA)
Stokes flows with near-touching rigid particles induce near-singular lubrication forces under relative motion, making their accurate numerical treatment challenging. With the aim of controlling the accuracy with a computationally cheap method, we present a new technique that combines the method of fundamental solutions (MFS) with the method of images. For rigid spheres, we propose to represent the flow using Stokeslet proxy sources on interior spheres, augmented by lines of image sources adapted to each near-contact to resolve lubrication. Source strengths are found by a least-squares solve at contact-adapted boundary collocation nodes. We include extensive numerical tests, and validate against reference solutions from a well-resolved boundary integral formulation. With less than 60 image sources per particle per contact, we show controlled uniform accuracy to three relative digits in surface velocities, and up to five digits in particle forces and torques, for all separations down to a thousandth of the radius. In the special case of flows around fixed particles, the proxy sphere alone gives controlled accuracy. A one-body preconditioning strategy allows acceleration with the fast multipole method, hence close to linear scaling in the number of particles. This is demonstrated by solving problems of up to 2000 spheres on a workstation using only 700 proxy sources per particle.
- [15] arXiv:2407.03344 [pdf, html, other]
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Title: Factorial Series Representation of Stieltjes Series Converging FactorsSubjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph)
The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series, is nowadays beyond dispute. Weniger's transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better known resummation techniques, the most known being Padé approximants. However, our understanding of its theoretical features is still far from being satisfactory and particularly bad as far as the decoding of factorially divergent series is concerned.
Stieltjes series represent a class of power series of fundamental interest in mathematical physics. In the present paper, it is shown how the Stieltjes series converging factor of any order is expressible as an inverse factorial series, whose terms can be analytically retrieved through a simple recursive algorithm. A few examples of applications of our algorithm are presented, in order to show its effectiveness and implementation ease. We believe the results presented here could constitute an important, preliminary step for the development of a general convergence theory of Weniger's transformation on Stieltjes series. A rather ambitious project, but worthy of being pursued in the future. - [16] arXiv:2407.03346 [pdf, html, other]
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Title: Numerical solution to the Neumann problem in a Lipschitz domain, based on random walksSubjects: Probability (math.PR)
We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods.
- [17] arXiv:2407.03347 [pdf, html, other]
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Title: Chebyshev Spectral Neural Networks for Solving Partial Differential EquationsSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Mathematical Physics (math-ph)
The purpose of this study is to utilize the Chebyshev spectral method neural network(CSNN) model to solve differential equations. This approach employs a single-layer neural network wherein Chebyshev spectral methods are used to construct neurons satisfying boundary conditions. The study uses a feedforward neural network model and error backpropagation principles, utilizing automatic differentiation (AD) to compute the loss function. This method avoids the need to solve non-sparse linear systems, making it convenient for algorithm implementation and solving high-dimensional problems. The unique sampling method and neuron architecture significantly enhance the training efficiency and accuracy of the neural network. Furthermore, multiple networks enables the Chebyshev spectral method to handle equations on more complex domains. The numerical efficiency and accuracy of the CSNN model are investigated through testing on elliptic partial differential equations, and it is compared with the well-known Physics-Informed Neural Network(PINN) method.
- [18] arXiv:2407.03348 [pdf, html, other]
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Title: Jacobi Set Simplification for Tracking Topological Features in Time-Varying Scalar FieldsSubjects: Numerical Analysis (math.NA); Computational Geometry (cs.CG); Computer Vision and Pattern Recognition (cs.CV); Graphics (cs.GR)
The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features.
- [19] arXiv:2407.03349 [pdf, other]
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Title: Recursive construction of biorthogonal polynomials for handling polynomial regressionSubjects: Numerical Analysis (math.NA)
An adaptive procedure for constructing a series of biorthogonal polynomials to a basis of monomials spanning the same finite-dimensional inner product space is proposed. By taking advantage of the orthogonality of the original basis, our procedure circumvents the well-known instability problem arising from the matrix inversion involved in classical polynomial regression. Moreover, the recurrent generation of biorthogonal polynomials in our framework facilitates the upgrading of all polynomials to include one additional element in the set whilst also allowing for a natural downgrading of the polynomial regression approximation. This is achieved by the posterior removal of any basis element leading to a straightforward approach for reducing the approximation order. We illustrate the usefulness of this approach through a series of examples where we derive the resulting biorthogonal polynomials from Legendre, Laguerre, and Chebyshev orthogonal bases.
- [20] arXiv:2407.03351 [pdf, html, other]
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Title: A High-Order Perturbation of Envelopes (HOPE) Method for Vector Electromagnetic Scattering by Periodic Inhomogeneous Media: Analytic ContinuationComments: arXiv admin note: text overlap with arXiv:2307.15152Subjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
Electromagnetic waves interacting with three--dimensional periodic structures occur in many applications of great scientific and engineering interest. These three dimensional interactions are extremely complicated and subtle, so it is unsurprising that practitioners find their rapid, robust, and accurate numerical simulation to be of paramount interest. Among the wide array of possible numerical approaches, the High--Order Spectral algorithms are often preferred due to their surpassing fidelity with a moderate number of unknowns, and here we describe an algorithm that fits into this class. In addition, we take a perturbative approach to the problem which views the deviation of the permittivity from a reference value as the deformation and we conduct a regular perturbation theory. This work concludes a line of research on these methods which began with two-dimensional problems governed by the Helmholtz equation and moved to small perturbations in the fully three-dimensional vector Maxwell equations. We now extend these latter results to large (real) perturbations constituting a rigorous analytic continuation.
- [21] arXiv:2407.03353 [pdf, html, other]
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Title: Is there an optimal choice of configuration space for Lie group integration schemes applied to constrained MBS?Journal-ref: Proceedings of the ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, IDETC/CIE 2013, August 12-15, 2013, Portland, OR, USASubjects: Numerical Analysis (math.NA); Robotics (cs.RO)
Recently various numerical integration schemes have been proposed for numerically simulating the dynamics of constrained multibody systems (MBS) operating. These integration schemes operate directly on the MBS configuration space considered as a Lie group. For discrete spatial mechanical systems there are two Lie group that can be used as configuration space: $SE\left( 3\right) $ and $SO\left( 3\right) \times \mathbb{R}^{3}$. Since the performance of the numerical integration scheme clearly depends on the underlying configuration space it is important to analyze the effect of using either variant. For constrained MBS a crucial aspect is the constraint satisfaction. In this paper the constraint violation observed for the two variants are investigated. It is concluded that the $SE\left( 3\right) $ formulation outperforms the $SO\left( 3\right) \times \mathbb{R}^{3}$ formulation if the absolute motions of the rigid bodies, as part of a constrained MBS, belong to a motion subgroup. In all other cases both formulations are equivalent. In the latter cases the $SO\left( 3\right) \times \mathbb{R}^{3}$ formulation should be used since the $SE\left( 3\right) $ formulation is numerically more complex, however.
- [22] arXiv:2407.03354 [pdf, other]
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Title: Stable rationality of hypersurfaces of mock toric variety IIComments: 40 pagesSubjects: Algebraic Geometry (math.AG)
In recent years, there has been a development in approaching rationality problems through motivic methods (cf. [Kontsevich--Tschinkel'19], [Nicaise--Shinder'19], [Nicaise--Ottem'21]). This method requires the explicit construction of degeneration families of curves with favorable properties. While the specific construction is generally difficult, [Nicaise--Ottem'22] combines combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and mentions the stable rationality of a very general hypersurface in projective spaces. In this paper, we substitute mock toric varieties for toric varieties and we prove the following theorem from the motivic method: If a very general hypersurface of degree $d$ in $\mathbb{P}^{2n-5}_\mathbb{C}$ is not stably rational, then a very general hypersurface of degree $d$ in $\mathrm{Gr}_\mathbb{C}(2, n)$ is not stably rational.
- [23] arXiv:2407.03356 [pdf, other]
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Title: AI Driven Laser Parameter Search: Inverse Design of Photonic Surfaces using Greedy Surrogate-based OptimizationSubjects: Optimization and Control (math.OC); Computational Engineering, Finance, and Science (cs.CE); Machine Learning (cs.LG); Applied Physics (physics.app-ph); Optics (physics.optics)
Photonic surfaces designed with specific optical characteristics are becoming increasingly important for use in in various energy harvesting and storage systems. , In this study, we develop a surrogate-based optimization approach for designing such surfaces. The surrogate-based optimization framework employs the Random Forest algorithm and uses a greedy, prediction-based exploration strategy to identify the laser fabrication parameters that minimize the discrepancy relative to a user-defined target optical characteristics. We demonstrate the approach on two synthetic benchmarks and two specific cases of photonic surface inverse design targets. It exhibits superior performance when compared to other optimization algorithms across all benchmarks. Additionally, we demonstrate a technique of inverse design warm starting for changed target optical characteristics which enhances the performance of the introduced approach.
- [24] arXiv:2407.03357 [pdf, html, other]
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Title: Elementary Formulas for Greatest Common Divisors and Semiprime FactorsComments: Version 1 of this preprint was published to the Cryptology ePrint Archive on June 21, 2024. Updates in this version include: Additional GCD formulas, tightening of constraints on existing GCD formulas to handle an edge case, addition of a formula for Euler's totient function for semiprimes, a minor simplification of the factorial formula used in the semiprime factoring formulasSubjects: General Mathematics (math.GM)
We present new formulas for computing greatest common divisors (GCDs) and extracting the prime factors of semiprimes using only elementary arithmetic operations: addition, subtraction, multiplication, floored division, and exponentiation. Our GCD formula simplifies a result of Mazzanti, and is derived using Kronecker substitution techniques from our previous work. We utilize the GCD formula, along with recent developments on arithmetic terms for square roots and factorials, to derive explicit expressions for the prime factors of a semiprime $n=pq$.
- [25] arXiv:2407.03359 [pdf, html, other]
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Title: (Two-scale) $W^{1}L^{\Phi}$-gradient Young measures and homogenization of integral functionals in Orlicz-Sobolev spacesSubjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)
(Two-scale) gradient Young measures in Orlicz-Sobolev setting are introduced and characterized providing also an integral representation formula for non convex energies arising in homogenization problems with nonstandard growth.
- [26] arXiv:2407.03363 [pdf, html, other]
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Title: A novel direct imaging method for passive inverse obstacle scattering problemSubjects: Numerical Analysis (math.NA)
This paper investigates the inverse scattering problem of recovering a sound-soft obstacle using passive measurements taken from randomly distributed point sources. The randomness introduced by these sources poses significant challenges, leading to the failure of classical direct sampling methods that rely on scattered field measurements. To address this issue, we introduce the Doubly Cross-Correlating Method (DCM), a novel direct imaging scheme that consists of two major steps. Initially, DCM creates a cross-correlation between two passive measurements. This specially designed cross-correlation effectively handles the uncontrollability of incident sources and connects to the active scattering model via the Helmholtz-Kirchhoff identity. Subsequently, this cross-correlation is used to create a correlation-based imaging function that can qualitatively identify the obstacle. The stability and resolution of DCM are theoretically analyzed. Extensive numerical examples, including scenarios with two closely positioned obstacles and multiscale obstacles, demonstrate that DCM is computationally efficient, stable, and fast.
- [27] arXiv:2407.03366 [pdf, html, other]
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Title: Spherical Distributions on the De Sitter Space and their Spectral SingularitiesComments: 12 pages. arXiv admin note: text overlap with arXiv:2309.10685Subjects: Functional Analysis (math.FA)
A spherical distribution is an eigendistribution of the Laplace-Beltrami operator with certain invariance on the de Sitter space. Let G'=O(1,n;R) be the Lorentz group and H' = O(1,n-1;R) be its subgroup. The authors Olafsson and Sitiraju have constructed the spherical distributions, which are $H'$-invariant, as boundary values of some sesquiholomorphic kernels. In this survey article we will explore the connections of these kernels with reflection positivity and representations of the group G = SO(1,n;R)_e, which is the connected component of the Lorentz group. We will also discuss the singularities of spherical distributions in terms of their wavefront set.
- [28] arXiv:2407.03367 [pdf, html, other]
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Title: Classical orthogonal decomposition of a modular $\mathfrak{sl}_n$Comments: In this arXiv version, we correct the conclusion about the uniqueness of COD of $\mathfrak{sl}_3$ in Section 3 of the published version. Moreover, we provide the input for one part of Appendix A, making the Mathematica Code more accessible to readersJournal-ref: J. Pure Appl. Algebra 228 (2024) 107721Subjects: Rings and Algebras (math.RA)
An orthogonal decomposition problem of Lie algebras over the complex numbers has been studied since the 1980s. It has many applications and relations to other areas of mathematics and sciences. In this paper, we consider this decomposition problem over a field of prime characteristic. We define a classical orthogonal decomposition of a modular Lie algebra and construct it for $\mathfrak{sl}_n$ under certain sufficient conditions. Additionally, we provide more detailed analysis of the problem when $n = 2$ and $3$.
- [29] arXiv:2407.03373 [pdf, other]
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Title: Low-rank plus diagonal approximations for Riccati-like matrix differential equationsComments: SIAM Journal on Matrix Analysis and Applications, In pressSubjects: Numerical Analysis (math.NA)
We consider the problem of computing tractable approximations of time-dependent d x d large positive semi-definite (PSD) matrices defined as solutions of a matrix differential equation. We propose to use "low-rank plus diagonal" PSD matrices as approximations that can be stored with a memory cost being linear in the high dimension d. To constrain the solution of the differential equation to remain in that subset, we project the derivative at all times onto the tangent space to the subset, following the methodology of dynamical low-rank approximation. We derive a closed-form formula for the projection, and show that after some manipulations it can be computed with a numerical cost being linear in d, allowing for tractable implementation. Contrary to previous approaches based on pure low-rank approximations, the addition of the diagonal term allows for our approximations to be invertible matrices, that can moreover be inverted with linear cost in d. We apply the technique to Riccati-like equations, then to two particular problems. Firstly a low-rank approximation to our recent Wasserstein gradient flow for Gaussian approximation of posterior distributions in approximate Bayesian inference, and secondly a novel low-rank approximation of the Kalman filter for high-dimensional systems. Numerical simulations illustrate the results.
- [30] arXiv:2407.03382 [pdf, html, other]
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Title: Geometric statistics with subspace structure preservation for SPD matricesComments: arXiv admin note: substantial text overlap with arXiv:2304.07347Subjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Differential Geometry (math.DG); Computation (stat.CO)
We present a geometric framework for the processing of SPD-valued data that preserves subspace structures and is based on the efficient computation of extreme generalized eigenvalues. This is achieved through the use of the Thompson geometry of the semidefinite cone. We explore a particular geodesic space structure in detail and establish several properties associated with it. Finally, we review a novel inductive mean of SPD matrices based on this geometry.
- [31] arXiv:2407.03429 [pdf, html, other]
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Title: Fault-Ride-Through (FRT) Control of a grid-connected Fixed-Speed Wind Energy Conversion System using STATCOMComments: Accepted for Publication at IEEE IECON-2024Subjects: Optimization and Control (math.OC)
Wind energy conversion system (WECS) is stochastic in nature and has low inertia to grid voltage instability, poor reactive power compensation and most importantly fault susceptibility. Variable speed WECS such as a doubly-fed induction generators (DFIG) are well known to reach steady state quickly after fault occurrence without the need for an external reactive power source because of the presence of a back-to-back converter that provides independent control of the active and reactive power unlike in the fixed-speed squirrel cage induction generator (SCIG) counterpart that cant be stabilized unless ab external source of reactive power support is present. However, controlling DFIG is complicated and costly due to complete tripping unlike the fixed-speed generators which doesnt trip completely when fault occurs. Hence, in this work, a 48-pulse, 3-phase static synchronous compensator (STATCOM) is used to ensure reactive power compensation and fault-ride through (FRT) control of the SCIG against over-voltage emanating from fault occurrence in a grid-connected power system. The goal here is to guarantee voltage stability and fault-ride through control against injected faults within certain time ranges at the point of common coupling (PCC) between the AC sources, the load, and the fixed-speed WECS.
- [32] arXiv:2407.03435 [pdf, html, other]
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Title: Optimal synchronisation to a limit cycleComments: 13 pages, 6 figuresSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)
In the absence of external forcing, all trajectories on the phase plane of the van der Pol oscillator tend to a closed, periodic, trajectory -- the limit cycle -- after infinite time. Here, we drive the van der Pol oscillator with an external time-dependent force to reach the limit cycle in a given finite time. Specifically, we are interested in minimising the non-conservative contribution to the work when driving the system from a given initial point on the phase plane to any final point belonging to the limit cycle. There appears a speed limit inequality, which expresses a trade-off between the connection time and cost -- in terms of the non-conservative work. We show how the above results can be { generalized to the broader family of non-linear oscillators given by} the Liénard equation. Finally, we also look into the problem of minimising the total work done by the external force.
- [33] arXiv:2407.03444 [pdf, html, other]
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Title: Fault-Tolerant Decentralized Control for Large-scale Inverter-based Resources for Active Power TrackingComments: Accepted for Joint Submission MECC-ASME Letters in Dynamic Systems and ControlSubjects: Optimization and Control (math.OC)
Integration of Inverter Based Resources (IBRs) which lack the intrinsic characteristics such as the inertial response of the traditional synchronous-generator (SG) based sources presents a new challenge in the form of analyzing the grid stability under their presence. While the dynamic composition of IBRs differs from that of the SGs, the control objective remains similar in terms of tracking the desired active power. This letter presents a decentralized primal-dual-based fault-tolerant control framework for the power allocation in IBRs. Overall, a hierarchical control algorithm is developed with a lower level addressing the current control and the parameter estimation for the IBRs and the higher level acting as the reference power generator to the low level based on the desired active power profile. The decentralized network-based algorithm adaptively splits the desired power between the IBRs taking into consideration the health of the IBRs transmission lines. The proposed framework is tested through a simulation on the network of IBRs and the high-level controller performance is compared against the existing framework in the literature. The proposed algorithm shows significant performance improvement in the magnitude of power deviation and settling time to the nominal value under faulty conditions as compared to the algorithm in the literature.
- [34] arXiv:2407.03447 [pdf, html, other]
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Title: Division Algebras and Quadratic ReciprocitySubjects: Rings and Algebras (math.RA); Algebraic Geometry (math.AG)
The Grothendieck and Artin-Mumford exact sequences for the Brauer group of a function field in 1 or 2 variables are applied to derive reciprocity laws for $q$th power residues.
- [35] arXiv:2407.03458 [pdf, html, other]
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Title: Analysis of Iterative Deblurring: No Explicit NoiseSinethemba Neliswa Mamba (AIMS-RW), Pawel Danielewicz (Michigan State U and AIMS-RW)Comments: 17 pages, 9 figuresSubjects: Numerical Analysis (math.NA); High Energy Physics - Experiment (hep-ex); Nuclear Experiment (nucl-ex)
Iterative deblurring, notably the Richardson-Lucy algorithm with and without regularization, is analyzed in the context of nuclear and high-energy physics applications. In these applications, probability distributions may be discretized into a few bins, measurement statistics can be high, and instrument performance can be well understood. In such circumstances, it is essential to understand the deblurring first without any explicit noise considerations. We employ singular value decomposition for the blurring matrix in a low-count pixel system. A strong blurring may yield a null space for the blurring matrix. Yet, a nonnegativity constraint for images built into the deblurring may help restore null-space content in a high-contrast image with zero or low intensity for a sufficient number of pixels. For low-contrast images, the control over null-space content may be gained through regularization. When the regularization is applied, the blurred image is, in practice, restored to an image that is still blurred but less than the starting one.
- [36] arXiv:2407.03461 [pdf, html, other]
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Title: On the Zariski invariant of plane branchesSubjects: Algebraic Geometry (math.AG)
We show how to obtain the Zariski invariant of a plane branch employing the contact order or the intersection multiplicity with elements in a particular family of curves and we present some consequences of this result.
- [37] arXiv:2407.03464 [pdf, html, other]
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Title: Semiclassical limit of a non-polynomial $q$-Askey schemeSubjects: Classical Analysis and ODEs (math.CA)
We prove a semiclassical asymptotic formula for the two elements $\mathcal M$ and $\mathcal Q$ lying at the bottom of the recently constructed non-polynomial hyperbolic $q$-Askey scheme. We also prove that the corresponding exponent is a generating function of the canonical transformation between pairs of Darboux coordinates on the monodromy manifold of the Painlevé I and $\textrm{III}_3$ equations, respectively. Such pairs of coordinates characterize the asymptotics of the tau function of the corresponding Painlevé equation. We conjecture that the other members of the non-polynomial hyperbolic $q$-Askey scheme yield generating functions associated to the other Painlevé equations in the semiclassical limit.
- [38] arXiv:2407.03478 [pdf, html, other]
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Title: Periodic gravity-capillary roll wave solutions to the inclined viscous shallow water equations in two dimensionsComments: 21 pages, 2 figuresSubjects: 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.
- [39] arXiv:2407.03483 [pdf, html, other]
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Title: Construct accurate multi-continuum micromorphic homogenisations in multi-D space-time with computer algebraSubjects: Dynamical Systems (math.DS); Mathematical Software (cs.MS); Analysis of PDEs (math.AP)
Homogenisation empowers the efficient macroscale system level prediction of physical problems with intricate microscale structures. Here we develop an innovative powerful, rigorous and flexible framework for asymptotic homogenisation of dynamics at the finite scale separation of real physics, with proven results underpinned by modern dynamical systems theory. The novel systematic approach removes most of the usual assumptions, whether implicit or explicit, of other methodologies. By no longer assuming averages the methodology constructs so-called multi-continuum or micromorphic homogenisations systematically based upon the microscale physics. The developed framework and approach enables a user to straightforwardly choose and create such homogenisations with clear physical and theoretical support, and of highly controllable accuracy and fidelity.
- [40] arXiv:2407.03485 [pdf, html, other]
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Title: A strongly convergent inertial inexact proximal-point algorithm for monotone inclusions with applications to variational inequalitiesSubjects: Optimization and Control (math.OC)
We propose an inertial variant of the strongly convergent inexact proximal-point (PP) method of Solodov and Svaiter (2000) for monotone inclusions. We prove strong convergence of our main algorithm under less restrictive assumptions on the inertial parameters when compared to previous analysis of inertial PP-type algorithms, which makes our method of interest even in finite-dimensional settings. We also performed an iteration-complexity analysis and applied our main algorithm to variational inequalities for monotone operators, obtaining strongly convergent (inertial) variants of Korpolevich's extragradient, forward-backward and Tseng's modified forward-backward methods.
- [41] arXiv:2407.03488 [pdf, html, other]
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Title: Failures of Compositionality: A Short Note on Cohomology, Sheafification and Lavish PresheavesComments: 11 pagesSubjects: Commutative Algebra (math.AC); Category Theory (math.CT)
In many sciences one often builds large systems out of smaller constituent parts. Mathematically, to study these systems, one can attach data to the component pieces via a functor F. This is of great practical use if F admits a compositional structure which is compatible with that of the system under study (i.e. if the local data defined on the pieces can be combined into global data). However, sometimes this does not occur. Thus one can ask: (1) Does F fail to be compositional? (2) If so, can this failure be quantified? and (3) Are there general tools to fix failures of compositionality? The kind of compositionality we study in this paper is one in which one never fails to combine local data into global data. This is formalized via the understudied notion of what we call a lavish presheaf: one that satisfies the existence requirement of the sheaf condition, but not uniqueness. Adapting Čech cohomology to presheaves, we show that a presheaf has trivial zeroth presheaf-Čech cohomology if and only if it is lavish. In this light, cohomology is a measure of the failure of compositionality. The key contribution of this paper is to show that, in some instances, cohomology can itself display compositional structure. Formally, we show that, given any Abelian presheaf F : C^op --> A and any Grothendieck pretopology J, if F is flasque and separated, then the zeroth cohomology functor H^0(-,F) : C^op --> A is lavish. This follows from observation that, for separated presheaves, H^0(-,F) can be written as a cokernel of the unit of the adjunction given by sheafification. This last fact is of independent interest since it shows that cohomology is a measure of ``distance'' between separated presheaves and their closest sheaves (their sheafifications). On the other hand, the fact that H^0(-,F) is a lavish presheaf has unexpected algorithmic consequences.
- [42] arXiv:2407.03492 [pdf, html, other]
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Title: Compatible Forts and Maximum Nullity of a GraphSubjects: Combinatorics (math.CO)
We consider bounds on maximum nullity of a graph via transversal numbers of compatible collections of forts. Results include generalizations of theorems from symmetric to combinatorially symmetric matrices, special bases of matrix nullspaces derived from transversal sets, and examples of issues that arise when considering only minimal forts and how to avoid them. We also show an important difference between constructing symmetric and combinatorially symmetric matrices associated to a graph whose nullspaces are supported on collections of disjoint forts.
- [43] arXiv:2407.03494 [pdf, html, other]
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Title: Helicity is a topological invariant of massless particles: C=-2hComments: 13 pagesSubjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)
There is an elementary but indispensable relationship between the topology and geometry of massive particles. The geometric spin $s$ is related to the topological dimension of the internal space $V$ by $\dim V = 2s + 1$. This breaks down for massless particles, which are characterized by their helicity $h$, but all have 1D internal spaces. We show that a subtler relation exists between the topological and geometry of massless particles. Wave functions of massless particles are sections of nontrivial line bundles over the lightcone whose topology are completely characterized by their first Chern number $C$. We prove that in general $C = -2h$. In doing so, we also exhibit a method of generating all massless bundle representations via an abelian group structure of massless particles.
- [44] arXiv:2407.03499 [pdf, html, other]
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Title: An adaptive Newton-based free-boundary Grad-Shafranov solverSubjects: Numerical Analysis (math.NA)
Equilibriums in magnetic confinement devices result from force balancing between the Lorentz force and the plasma pressure gradient. In an axisymmetric configuration like a tokamak, such an equilibrium is described by an elliptic equation for the poloidal magnetic flux, commonly known as the Grad--Shafranov equation. It is challenging to develop a scalable and accurate free-boundary Grad--Shafranov solver, since it is a fully nonlinear optimization problem that simultaneouly solves for the magnetic field coil current outside the plasma to control the plasma shape. In this work, we develop a Newton-based free-boundary Grad--Shafranov solver using adaptive finite elements and preconditioning strategies. The free-boundary interaction leads to the evaluation of a domain-dependent nonlinear form of which its contribution to the Jacobian matrix is achieved through shape calculus. The optimization problem aims to minimize the distance between the plasma boundary and specified control points while satisfying two non-trivial constraints, which correspond to the nonlinear finite element discretization of the Grad--Shafranov equation and a constraint on the total plasma current involving a nonlocal coupling term. The linear system is solved by a block factorization, and AMG is called for subblock elliptic operators. The unique contributions of this work include the treatment of a global constraint, preconditioning strategies, nonlocal reformulation, and the implementation of adaptive finite elements. It is found that the resulting Newton solver is robust, successfully reducing the nonlinear residual to 1e-6 and lower in a small handful of iterations while addressing the challenging case to find a Taylor state equilibrium where conventional Picard-based solvers fail to converge.
- [45] arXiv:2407.03500 [pdf, html, other]
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Title: On the Moduli Space of Coherent Systems of Type $(2, c_1, c_2, 2)$ on Projective PlaneComments: 30 pages, no figuresSubjects: Algebraic Geometry (math.AG)
We study the moduli space of coherent systems in $P^2$ using the Segre invariant. We obtain necessary conditions for the existence of $\alpha$-semistable coherent systems $(E,V)$ of type $(2, c_1, c_2, k)$, with $k \geq 2$. Afterwards, we give numerical conditions to the nonemptiness of the moduli space and compute the critical values depending of the Chern classes. Finally, we give some topological properties of the flips.
- [46] arXiv:2407.03507 [pdf, html, other]
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Title: Improved Iteration Complexity in Black-Box Optimization Problems under Higher Order Smoothness Function ConditionSubjects: Optimization and Control (math.OC)
This paper is devoted to the study (common in many applications) of the black-box optimization problem, where the black-box represents a gradient-free oracle $\tilde{f} = f(x) + \xi$ providing the objective function value with some stochastic noise. Assuming that the objective function is $\mu$-strongly convex, and also not just $L$-smooth, but has a higher order of smoothness ($\beta \geq 2$) we provide a novel optimization method: Zero-Order Accelerated Batched Stochastic Gradient Descent, whose theoretical analysis closes the question regarding the iteration complexity, achieving optimal estimates. Moreover, we provide a thorough analysis of the maximum noise level, and show under which condition the maximum noise level will take into account information about batch size $B$ as well as information about the smoothness order of the function $\beta$.
- [47] arXiv:2407.03513 [pdf, html, other]
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Title: The chromatic number of 4-dimensional latticesSubjects: Combinatorics (math.CO)
The chromatic number of a lattice in n-dimensional Euclidean space is defined as the chromatic number of its Voronoi graph. The Voronoi graph is the Cayley graph on the lattice having the strict Voronoi vectors as generators. In this paper we determine the chromatic number of all 4-dimensional lattices. To achieve this we use the known classification of 52 parallelohedra in dimension 4. These 52 geometric types yield 16 combinatorial types of relevant Voronoi graphs. We discuss a systematic approach to checking for isomorphism of Cayley graphs of lattices. Lower bounds for the chromatic number are obtained from choosing appropriate small finite induced subgraphs of the Voronoi graphs. Matching upper bounds are derived from periodic colorings. To determine the chromatic numbers of these finite graphs, we employ a SAT solver.
- [48] arXiv:2407.03529 [pdf, html, other]
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Title: Geometric and Analytic Aspects of Simon-Lojasiewicz Inequalities on Vector BundlesComments: 21 pages, 4 figuresSubjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
In real analysis, the Lojasiewicz inequalities, revitalized by Leon Simon in his pioneering work on singularities of energy minimizing maps, have proven to be monumental in differential geometry, geometric measure theory, and variational problems. These inequalities provide specific growth and stability conditions for prescribed real-analytic functions, and have found applications to gradient flows, gradient systems, and as explicated in this paper, vector bundles over compact Riemannian manifolds. In this work, we outline the theory of functionals and variational problems over vector bundles, explore applications to arbitrary real-analytic functionals, and describe the energy functional on $S^{n-1}$ as a functional over a vector bundle.
- [49] arXiv:2407.03530 [pdf, html, other]
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Title: Comparative Prime Number Theory Problem ListSubjects: Number Theory (math.NT)
This is a list of problems that were collected from participants at the Comparative Prime Number Theory Symposium held at UBC from June 17 to June 21, 2024. Its goal is to stimulate research and future collaborations in this growing field. This event was part of the PIMS (Pacific Institute of Mathematical Sciences) Collaborative Research Group L-functions in Analytic Number Theory: 2022- 2025.
- [50] arXiv:2407.03534 [pdf, html, other]
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Title: Myers-Steenrod theorems for metric and singular Riemannian foliationsComments: 20 pages, we thank gold essen.trinken in Karlsruhe for its excellent working conditionsSubjects: Differential Geometry (math.DG)
We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space $X$, or a singular Riemannian foliation on a manifold $M$ is a closed subgroup of the isometry group of $X$ in the case of a metric foliation, or of the isometry group of $M$ for the case of a singular Riemannian foliation. We obtain a sharp upper bound for the dimension of these subgroups and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. Moreover, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces.
- [51] arXiv:2407.03546 [pdf, html, other]
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Title: Exponential Euler method for stiff SDEs driven by fractional Brownian motionSubjects: Probability (math.PR)
In a recent paper by Kamrani et al. (2024), exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise was discussed, and the convergence order close to the Hurst parameter H was proved. Utilizing the technique of Malliavin derivative, we prove the exponential Euler scheme and obtain a convergence order of one, which is the optimal rate in numerical simulation.
- [52] arXiv:2407.03547 [pdf, html, other]
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Title: Large Time Behavior of Solutions to Cauchy Problem for 1-D Compressible Isentropic Navier-Stokes/Allen-Cahn SystemComments: 26 pagesSubjects: Analysis of PDEs (math.AP)
This paper is concerned with the large time behavior of the solutions to the Cauchy problem for the one-dimensional compressible Navier-Stokes/Allen-Cahn system with the immiscible two-phase flow initially located near the phase separation state. Under the assumptions that the initial data is a small perturbation of the constant state, we prove the global existence and uniqueness of the solutions and establish the time decay rates of the solution as well as its higher-order spatial derivatives. Moreover, we derive that the solutions of the system are time asymptotically approximated by the solutions of the modified parabolic system and obtain decay rates in $L^2$ and $L^1$. Furthermore, we show that the solution of the system is time asymptotically approximated in $L^p (1 \leq p \leq+\infty)$ by the diffusion waves.
- [53] arXiv:2407.03553 [pdf, html, other]
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Title: On a generalized hexagonal dart board problemSubjects: Combinatorics (math.CO); Metric Geometry (math.MG)
Jung's theorem says that planar sets of diameter $1$ can be covered by a closed circular disc of radius $\frac 1{\sqrt3}$. In this paper we will restrict ourselves to finite point sets and consider a fractional Jung-type problem. Let $\mathcal{P}_n$ be the family of all finite sets of $n$ points of diameter $1$. Let the function value $N_n(r)$ ($0 < r \leq 1$) be the largest integer $k$ so that for every point set $P \in \mathcal{P}_n$ there is a circle of radius $r$ which covers at least $k$ points of $P$. We give many lower and upper bounds for $N_n(r)$ and determine intervals of $(0,1]$ where exact values of $N_n(r)$ can be determined. Among others we show the $n\leq N_{3n}(\frac{1}{2}) \leq n+1$ and $N_n(\frac{1}{4}) = \lceil \frac{n}{7} \rceil$ for all $n\neq 7$.
- [54] arXiv:2407.03554 [pdf, other]
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Title: Multi-phase high frequency solutions to Klein-Gordon-Maxwell equations in Lorenz gauge in (3+1) Minkowski spacetimeSubjects: Analysis of PDEs (math.AP)
We study a 1-parameter family (A{\lambda}, {\Phi}{\lambda}){\lambda} of multi-phase high frequency solutions to Klein-Gordon-Maxwell equations in Lorenz gauge in the (3+1)-dimensional Minkowski spacetime. This family is based on an initial ansatz. We prove that for {\lambda} small enough the family of solutions exists on an interval uniform in {\lambda} only function of the initial ansatz. These solutions are close to an approximate solution constructed by geometric optics. The initial ansatz needs to be regular enough, to satisfy a polarization condition and to satisfy the constraints for Maxwell null-transport in Lorenz gauge, but there is no need for smallness of any kind. The phases need to interact in a coherent way. We also observe that the limit (A0, {\Phi}0) is not solution to Klein-Gordon-Maxwell equations but to a Klein-Gordon-Maxwell null-transport type system.
- [55] arXiv:2407.03555 [pdf, html, other]
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Title: Adaptive Perturbation Enhanced SCL Decoder for Polar CodesSubjects: Information Theory (cs.IT)
For polar codes, successive cancellation list (SCL) decoding algorithm significantly improves finite-length performance compared to SC decoding. SCL-flip decoding can further enhance the performance but the gain diminishes as code length increases, due to the difficulty in locating the first error bit position. In this work, we introduce an SCL-perturbation decoding algorithm to address this issue. A basic version of the algorithm introduces small random perturbations to the received symbols before each SCL decoding attempt, and exhibits non-diminishing gain at large block lengths. Its enhanced version adaptively performs random perturbations or directional perturbation on each received symbol according to previous decoding results, and managed to correct more errors with fewer decoding attempts. Extensive simulation results demonstrate stable gains across various code rates, lengths and list sizes. To the best of our knowledge, this is the first SCL enhancement with non-diminishing gains as code length increases, and achieves unprecedented efficiency. With only one additional SCL-$L$ decoding attempt (in total two), the proposed algorithm achieves SCL-$2L$-equivalent performance. Since the gain is obtained without increasing list size, the algorithm is best suited for hardware implementation.
- [56] arXiv:2407.03559 [pdf, html, other]
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Title: On Finite Fields and Higher ReciprocityComments: 46 pagesSubjects: Number Theory (math.NT)
Cubic and biquadratic reciprocity have long since been referred to as "the forgotten reciprocity laws", largely since they provide special conditions that are widely considered to be unnecessary in the study of number theory. In this exposition of finite fields and higher reciprocity, we will begin by introducing concepts in abstract algebra and elementary number theory. This will motivate our approach toward understanding the structure and then existence of finite fields, especially with a focus on understanding the multiplicative group $\mathbb{F}^{*}$. While surveying finite fields we will provide another proof of quadratic reciprocity. We will proceed to investigate properties of the general multiplicative character, covering the concept of a general Gauss sum as well as basic notions of the Jacobi sum. From there we will begin laying the foundations for the cubic reciprocity law, beginning with a classification of the primes and units of the Eisenstein integers, denoted $\mathbb{Z}[\omega]$, and further investigations into the residue class ring $\mathbb{Z}[\omega]/\pi\mathbb{Z}[\omega]$ for $\pi$ prime, which is predominantly the world in which cubic reciprocity lies. We then define the general multiplicative character and state the full law of cubic reciprocity. We will finish the section on cubic reciprocity with a brief survey of the cubic residue character of the even prime $2$ and state a significant result due to Gauss that summarizes the conditions for $2$ to be a cubic residue. We conclude with a brief survey of the law of biquadratic reciprocity.
- [57] arXiv:2407.03560 [pdf, html, other]
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Title: Numerical semigroups from rational matrices I: power-integral matrices and nilpotent representationsComments: 13 pagesSubjects: Combinatorics (math.CO)
Our aim in this paper is to initiate the study of exponent semigroups for rational matrices. We prove that every numerical semigroup is the exponent semigroup of some rational matrix. We also obtain lower bounds on the size of such matrices and discuss the related class of power-integral matrices.
- [58] arXiv:2407.03561 [pdf, html, other]
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Title: Towards the Use of Anderson Acceleration in Coupled Transport-Gyrokinetic Turbulence SimulationsSubjects: Numerical Analysis (math.NA); Plasma Physics (physics.plasm-ph)
Predicting the behavior of a magnetically confined fusion plasma over long time periods requires methods that can bridge the difference between transport and turbulent time scales. The nonlinear transport solver, Tango, enables simulations of very long times, in particular to steady state, by advancing each process independently with different time step sizes and couples them through a relaxed iteration scheme. We examine the use of Anderson Acceleration (AA) to reduce the total number of coupling iterations required by interfacing Tango with the AA implementation, including several extensions to AA, provided by the KINSOL nonlinear solver package in SUNDIALS. The ability to easily enable and adjust algorithmic options through KINSOL allows for rapid experimentation to evaluate different approaches with minimal effort. Additionally, we leverage the GPTune library to automate the optimization of algorithmic parameters within KINSOL. We show that AA can enable faster convergence in stiff and very stiff tests cases without noise present and in all cases, including with noisy fluxes, increases robustness and reduces sensitivity to the choice of relaxation strength.
- [59] arXiv:2407.03565 [pdf, other]
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Title: Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schr\"odinger equationsComments: 52 pagesSubjects: Analysis of PDEs (math.AP)
We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic setting, the authors proved some well-posedness results, which contain the scaling critical case for $d\geq 2$. In the present paper, we prove the well-posedness of this system for the periodic setting. In particular, well-posedness is proved at the scaling critical regularity for $d\geq 3$ under some conditions for the coefficients of the Laplacian. We also prove some ill-posedness results. As long as we use an iteration argument, our well-posedness results are optimal except for some critical cases.
- [60] arXiv:2407.03566 [pdf, html, other]
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Title: Stacked Intelligent Metasurfaces for Wireless Sensing and Communication: Applications and ChallengesHao Liu, Jiancheng An, Xing Jia, Shining Lin, Xianghao Yao, Lu Gan, Bruno Clerckx, Chau Yuen, Mehdi Bennis, Mérouane DebbahComments: 8 pages, 5 figures, 1 tableSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
The rapid advancement of wireless communication technologies has precipitated an unprecedented demand for high data rates, extremely low latency, and ubiquitous connectivity. In order to achieve these goals, stacked intelligent metasurfaces (SIM) has been developed as a novel solution to perform advanced signal processing tasks directly in the electromagnetic wave domain, thus achieving ultra-fast computing speed and reducing hardware complexity. This article provides an overview of the SIM technology by discussing its hardware architectures, advantages, and potential applications for wireless sensing and communication. Specifically, we explore the utilization of SIMs in enabling wave-domain beamforming, channel modeling and estimation in SIM-assisted communication systems. Furthermore, we elaborate on the potential of utilizing a SIM to build a hybrid optical-electronic neural network (HOENN) and demonstrate its efficacy by examining two case studies: disaster monitoring and direction-of-arrival estimation. Finally, we identify key implementation challenges, including practical hardware imperfections, efficient SIM configuration for realizing wave-domain signal processing, and performance analysis to motivate future research on this important and far-reaching topic.
- [61] arXiv:2407.03571 [pdf, html, other]
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Title: A Fully Parameter-Free Second-Order Algorithm for Convex-Concave Minimax Problems with Optimal Iteration ComplexitySubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Machine Learning (stat.ML)
In this paper, we study second-order algorithms for the convex-concave minimax problem, which has attracted much attention in many fields such as machine learning in recent years. We propose a Lipschitz-free cubic regularization (LF-CR) algorithm for solving the convex-concave minimax optimization problem without knowing the Lipschitz constant. It can be shown that the iteration complexity of the LF-CR algorithm to obtain an $\epsilon$-optimal solution with respect to the restricted primal-dual gap is upper bounded by $\mathcal{O}(\frac{\rho\|z^0-z^*\|^3}{\epsilon})^{\frac{2}{3}}$, where $z^0=(x^0,y^0)$ is a pair of initial points, $z^*=(x^*,y^*)$ is a pair of optimal solutions, and $\rho$ is the Lipschitz constant. We further propose a fully parameter-free cubic regularization (FF-CR) algorithm that does not require any parameters of the problem, including the Lipschitz constant and the upper bound of the distance from the initial point to the optimal solution. We also prove that the iteration complexity of the FF-CR algorithm to obtain an $\epsilon$-optimal solution with respect to the gradient norm is upper bounded by $\mathcal{O}(\frac{\rho\|z^0-z^*\|^2}{\epsilon})^{\frac{2}{3}}$. Numerical experiments show the efficiency of both algorithms. To the best of our knowledge, the proposed FF-CR algorithm is the first completely parameter-free second-order algorithm for solving convex-concave minimax optimization problems, and its iteration complexity is consistent with the optimal iteration complexity lower bound of existing second-order algorithms with parameters for solving convex-concave minimax problems.
- [62] arXiv:2407.03578 [pdf, html, other]
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Title: Distributed online generalized Nash Equilibrium learning in multi-cluster games: A delay-tolerant algorithmSubjects: Optimization and Control (math.OC)
This paper addresses the problem of distributed online generalized Nash equilibrium (GNE) learning for multi-cluster games with delayed feedback information. Specifically, each agent in the game is assumed to be informed a sequence of local cost functions and constraint functions, which are known to the agent with time-varying delays subsequent to decision-making at each round. The objective of each agent within a cluster is to collaboratively optimize the cluster's cost function, subject to time-varying coupled inequality constraints and local feasible set constraints over time. Additionally, it is assumed that each agent is required to estimate the decisions of all other agents through interactions with its neighbors, rather than directly accessing the decisions of all agents, i.e., each agent needs to make decision under partial-decision information. To solve such a challenging problem, a novel distributed online delay-tolerant GNE learning algorithm is developed based upon the primal-dual algorithm with an aggregation gradient mechanism. The system-wise regret and the constraint violation are formulated to measure the performance of the algorithm, demonstrating sublinear growth with respect to time under certain conditions. Finally, numerical results are presented to verify the effectiveness of the proposed algorithm.
- [63] arXiv:2407.03579 [pdf, html, other]
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Title: A connection between Lipschitz and Kazhdan constants for groups of homeomorphisms of the real lineComments: 12 pages, 2 figures. Comments welcomeSubjects: Group Theory (math.GR); Dynamical Systems (math.DS); Functional Analysis (math.FA)
We exhibit an obstruction for groups with Property (T) to act on the real line by bi-Lipschitz homeomorphisms. This condition is expressed in terms of the Lipschitz constants and the Kazhdan constants associated to finite, generating subsets. As a corollary, we obtain an upper bound for the Kazhdan constants of orderable groups. Our main tool is the Koopman representation associated to the action $\operatorname{BiLip}_+(\mathbb{R})\curvearrowright\mathbb{R}$.
- [64] arXiv:2407.03592 [pdf, html, other]
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Title: Mixed type boundary value problem of elliptic equation in a thin domainSubjects: Analysis of PDEs (math.AP)
In this paper, we prove the a priori estimates for two-dimensional second order homogeneous linear elliptic equations in a narrow region. In a crescent-shaped area, part of the boundary is subject to an oblique derivative boundary condition, while the other part of the boundary is subject to a Dirichlet boundary condition. We show that, as the crescent-shaped area collapses into a segment under suitable conditions, the boundary value problem obeys uniform Schauder estimates and induces an asymptotic estimate.
- [65] arXiv:2407.03593 [pdf, html, other]
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Title: Green Multigrid NetworkSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG)
GreenLearning networks (GL) directly learn Green's function in physical space, making them an interpretable model for capturing unknown solution operators of partial differential equations (PDEs). For many PDEs, the corresponding Green's function exhibits asymptotic smoothness. In this paper, we propose a framework named Green Multigrid networks (GreenMGNet), an operator learning algorithm designed for a class of asymptotically smooth Green's functions.
Compared with the pioneering GL, the new framework presents itself with better accuracy and efficiency, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, Green's function is modeled as a piecewise function to take into account its singular behavior in some parts of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output(AugNN) so that it can capture singularity accurately. Second, the asymptotic smoothness property of Green's function is used to leverage the Multi-Level Multi-Integration (MLMI) algorithm for both the training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectiveness of the proposed method. On average, GreenMGNet achieves $3.8\%$ to $39.15\%$ accuracy improvement. To match the accuracy level of GL, GreenMGNet requires only about $10\%$ of the full grid data, resulting in a $55.9\%$ and $92.5\%$ reduction in training time and GPU memory cost for one-dimensional test problems, and a $37.7\%$ and $62.5\%$ reduction for two-dimensional test problems. - [66] arXiv:2407.03599 [pdf, html, other]
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Title: The exotic inverted Kloosterman sumSubjects: Number Theory (math.NT)
Let $B$ be a product of finitely many finite fields containing $\mathbb F_q$, $\psi:\mathbb F_q\to \overline{\mathbb Q}_\ell^*$ a nontrivial additive character, and $\chi: B^*\to \overline{\mathbb Q}_\ell^*$ a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum \begin{eqnarray*} \mathrm{EIK}(\mathbb F_q, a):=\sum_{\substack{x\in B^* \\ \mathrm{Tr}_{B/\mathbb F_q}(x)\not =0\\ \mathrm{N}_{B/\mathbb F_q}(x)=a}} \chi(x)\psi\Big(\frac{1}{\mathrm{Tr}_{B/\mathbb F_q}(x)}\Big), \ \ a\in \mathbb F_q^*. \end{eqnarray*} We estimate this sum using $\ell$-adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of weight at most $n$, where $n+1 = \dim_{\mathbb F_q}B$. Up to a trivial main term, this gives the expected square root cancellation.
- [67] arXiv:2407.03601 [pdf, html, other]
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Title: Online Non-Stationary Stochastic Quasar-Convex OptimizationSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
Recent research has shown that quasar-convexity can be found in applications such as identification of linear dynamical systems and generalized linear models. Such observations have in turn spurred exciting developments in design and analysis algorithms that exploit quasar-convexity. In this work, we study the online stochastic quasar-convex optimization problems in a dynamic environment. We establish regret bounds of online gradient descent in terms of cumulative path variation and cumulative gradient variance for losses satisfying quasar-convexity and strong quasar-convexity. We then apply the results to generalized linear models (GLM) when the underlying parameter is time-varying. We establish regret bounds of online gradient descent when applying to GLMs with leaky ReLU activation function, logistic activation function, and ReLU activation function. Numerical results are presented to corroborate our findings.
- [68] arXiv:2407.03605 [pdf, html, other]
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Title: Orthogonal Constrained Minimization with Tensor $\ell_{2,p}$ Regularization for HSI Denoising and DestripingSubjects: Optimization and Control (math.OC); Computer Vision and Pattern Recognition (cs.CV)
Hyperspectral images (HSIs) are often contaminated by a mixture of noises such as Gaussian noise, dead lines, stripes, and so on. In this paper, we propose a novel approach for HSI denoising and destriping, called NLTL2p, which consists of an orthogonal constrained minimization model and an iterative algorithm with convergence guarantees. The model of the proposed NLTL2p approach is built based on a new sparsity-enhanced Nonlocal Low-rank Tensor regularization and a tensor $\ell_{2,p}$ norm with $p\in(0,1)$. The low-rank constraints for HSI denoising utilize the spatial nonlocal self-similarity and spectral correlation of HSIs and are formulated based on independent higher-order singular value decomposition with sparsity enhancement on its core tensor to prompt more low-rankness. The tensor $\ell_{2,p}$ norm for HSI destriping is extended from the matrix $\ell_{2,p}$ norm. A proximal block coordinate descent algorithm is proposed in the NLTL2p approach to solve the resulting nonconvex nonsmooth minimization with orthogonal constraints. We show any accumulation point of the sequence generated by the proposed algorithm converges to a first-order stationary point, which is defined using three equalities of substationarity, symmetry, and feasibility for orthogonal constraints. In the numerical experiments, we compare the proposed method with state-of-the-art methods including a deep learning based method, and test the methods on both simulated and real HSI datasets. Our proposed NLTL2p method demonstrates outperformance in terms of metrics such as mean peak signal-to-noise ratio as well as visual quality.
- [69] arXiv:2407.03606 [pdf, html, other]
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Title: Decoding Analog Subspace Codes: Algorithms for Character-Polynomial CodesComments: ISIT 2024Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)
We propose efficient minimum-distance decoding and list-decoding algorithms for a certain class of analog subspace codes, referred to as character-polynomial (CP) codes, recently introduced by Soleymani and the second author. In particular, a CP code without its character can be viewed as a subcode of a Reed-Solomon (RS) code, where a certain subset of the coefficients of the message polynomial is set to zeros. We then demonstrate how classical decoding methods, including list decoders, for RS codes can be leveraged for decoding CP codes. For instance, it is shown that, in almost all cases, the list decoder behaves as a unique decoder. We also present a probabilistic analysis of the improvements in list decoding of CP codes when leveraging their certain structure as subcodes of RS codes.
- [70] arXiv:2407.03608 [pdf, html, other]
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Title: Gaussian process regression with log-linear scaling for common non-stationary kernelsSubjects: Numerical Analysis (math.NA); Computation (stat.CO)
We introduce a fast algorithm for Gaussian process regression in low dimensions, applicable to a widely-used family of non-stationary kernels. The non-stationarity of these kernels is induced by arbitrary spatially-varying vertical and horizontal scales. In particular, any stationary kernel can be accommodated as a special case, and we focus especially on the generalization of the standard Matérn kernel. Our subroutine for kernel matrix-vector multiplications scales almost optimally as $O(N\log N)$, where $N$ is the number of regression points. Like the recently developed equispaced Fourier Gaussian process (EFGP) methodology, which is applicable only to stationary kernels, our approach exploits non-uniform fast Fourier transforms (NUFFTs). We offer a complete analysis controlling the approximation error of our method, and we validate the method's practical performance with numerical experiments. In particular we demonstrate improved scalability compared to to state-of-the-art rank-structured approaches in spatial dimension $d>1$.
- [71] arXiv:2407.03613 [pdf, html, other]
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Title: Representation theory of the Reflection Equation Algebra II: Theory of shapesComments: 17 pagesSubjects: Quantum Algebra (math.QA); Representation Theory (math.RT)
We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the $R$-matrix associated to the standard $q$-deformation of $GL(N,\mathbb{C})$ for $0<q<1$. We consider the Poisson structure appearing as the classical limit of the $R$-matrix, and parametrize the symplectic leaves explicitly in terms of a type of matrix we call a shape matrix. We then introduce a quantized version of the shape matrix for the REA, and show that each irreducible representation of the REA has a unique shape.
- [72] arXiv:2407.03642 [pdf, html, other]
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Title: A Probabilistic Approach to Discounted Infinite Horizon and Invariant Mean Field GamesComments: 53 pagesSubjects: Optimization and Control (math.OC); Probability (math.PR)
This paper considers discounted infinite horizon mean field games by extending the probabilistic weak formulation of the game as introduced by Carmona and Lacker (2015). Under similar assumptions as in the finite horizon game, we prove existence and uniqueness of solutions for the extended infinite horizon game. The key idea is to construct local versions of the previously considered stable topologies. Further, we analyze how sequences of finite horizon games approximate the infinite horizon one. Under a weakened Lasry-Lions monotonicity condition, we can quantify the convergence rate of solutions for the finite horizon games to the one for the infinite horizon game using a novel stability result for mean field games. Lastly, applying our results allows to solve the invariant mean field game as well.
- [73] arXiv:2407.03643 [pdf, html, other]
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Title: Approximation of the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in general dimensionsComments: 23 pagesSubjects: Analysis of PDEs (math.AP)
We study the first Steklov-Dirichlet eigenvalue on eccentric spherical shells in $\mathbb{R}^{n+2}$ with $n\geq 1$, imposing the Steklov condition on the outer boundary sphere, denoted by $\Gamma_S$, and the Dirichlet condition on the inner boundary sphere. The first eigenfunction admits a Fourier--Gegenbauer series expansion via the bispherical coordinates, where the Dirichlet-to-Neumann operator on $\Gamma_S$ can be recursively expressed in terms of the expansion coefficients arXiv:2309.09587. In this paper, we develop a finite section approach for the Dirichlet-to-Neumann operator to approximate the first Steklov--Dirichlet eigenvalue on eccentric spherical shells. We prove the exponential convergence of this approach by using the variational characterization of the first eigenvalue. Furthermore, based on the convergence result, we propose a numerical computation scheme as an extension of the two-dimensional result in [Hong et al., Ann. Mat. Pura Appl., 2022] to general dimensions. We provide numerical examples of the first Steklov-Dirichlet eigenvalue on eccentric spherical shells with various geometric configurations.
- [74] arXiv:2407.03647 [pdf, html, other]
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Title: WANCO: Weak Adversarial Networks for Constrained Optimization problemsComments: 24 pages, 18 figuresSubjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI)
This paper focuses on integrating the networks and adversarial training into constrained optimization problems to develop a framework algorithm for constrained optimization problems. For such problems, we first transform them into minimax problems using the augmented Lagrangian method and then use two (or several) deep neural networks(DNNs) to represent the primal and dual variables respectively. The parameters in the neural networks are then trained by an adversarial process. The proposed architecture is relatively insensitive to the scale of values of different constraints when compared to penalty based deep learning methods. Through this type of training, the constraints are imposed better based on the augmented Lagrangian multipliers. Extensive examples for optimization problems with scalar constraints, nonlinear constraints, partial differential equation constraints, and inequality constraints are considered to show the capability and robustness of the proposed method, with applications ranging from Ginzburg--Landau energy minimization problems, partition problems, fluid-solid topology optimization, to obstacle problems.
- [75] arXiv:2407.03659 [pdf, html, other]
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Title: Strong Approximations in the Almost Sure Central Limit Theorem and Limit Behavior of the Center of MassSubjects: Probability (math.PR)
In this paper, we establish an almost sure central limit theorem for a general random sequence under a strong approximation condition. Additionally, we derive the law of the iterated logarithm for the center of mass corresponding to a random sequence under a different strong approximation condition. Applications to step-reinforced random walks are also discussed.
- [76] arXiv:2407.03660 [pdf, html, other]
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Title: A Number Field Analogue of Ramanujan's identity for $\zeta(2m+1)$Comments: 33 pages, comments are welcome!Subjects: Number Theory (math.NT)
Ramanujan's famous formula for $\zeta(2m+1)$ has captivated the attention of numerous mathematicians over the years. Grosswald, in 1972, found a simple extension of Ramanujan's formula which in turn gives transformation formula for Eisenstein series over the full modular group. Recently, Banerjee, Gupta and Kumar found a number field analogue of Ramanujan's formula. In this paper, we present a new number field analogue of the Ramanujan-Grosswald formula for $\zeta(2m+1)$ by obtaining a formula for Dedekind zeta function at odd arguments. We also obtain a number field analogue of an identity of Chandrasekharan and Narasimhan, which played a crucial role in proving our main identity. As an application, we generalize transformation formula for Eisenstein series $G_{2k}(z)$ and Dedekind eta function $\eta(z)$. A new formula for the class number of a totally real number field is also obtained, which provides a connection with the Kronceker's limit formula for the Dedekind zeta function.
- [77] arXiv:2407.03664 [pdf, html, other]
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Title: Deformation of the heat kernel and the Brownian motion from the perspective of the Ben Sa\"id--Kobayashi--{\O}rsted $(k,a)$-generalized Laguerre semigroup theoryComments: 58 pagesSubjects: Representation Theory (math.RT); Analysis of PDEs (math.AP); Probability (math.PR)
We deform the heat kernel and the Brownian motion on $\mathbb{R}^{N}$ from the perspective of "$(k,a)$-generalized Fourier analysis" with $k=0$. This is a new type of harmonic analysis proposed by S.Ben Saïd--T.Kobayashi--B.Ørsted from the representation theoretic viewpoint. In this paper, we construct the $a$-deformed heat kernel and $a$-deformed Brownian motion, and explore their some basic properties. We also prove that the $(k,a)$-generalized Fourier integral kernels are polynomial growth when $k=0$, for a justification of some discussions.
- [78] arXiv:2407.03675 [pdf, html, other]
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Title: Characterization of weakly porous sets via dyadic coveringsComments: 8 pagesSubjects: Functional Analysis (math.FA)
We consider the class of weakly porous sets in Euclidean spaces. As our main goal, we give a precise characterization in terms of dyadic coverings of these sets
- [79] arXiv:2407.03677 [pdf, html, other]
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Title: Nonlinear Model Reduction to Random Spectral Submanifolds in Random VibrationsComments: 26 pages, 15 figuresSubjects: Dynamical Systems (math.DS); Numerical Analysis (math.NA); Pattern Formation and Solitons (nlin.PS)
Dynamical systems in engineering and physics are often subject to irregular excitations that are best modeled as random. Monte Carlo simulations are routinely performed on such random models to obtain statistics on their long-term response. Such simulations, however, are prohibitively expensive and time consuming for high-dimensional nonlinear systems. Here we propose to decrease this numerical burden significantly by reducing the full system to very low-dimensional, attracting, random invariant manifolds in its phase space and performing the Monte Carlo simulations on that reduced dynamical system. The random spectral submanifolds (SSMs) we construct for this purpose generalize the concept of SSMs from deterministic systems under uniformly bounded random forcing. We illustrate the accuracy and speed of random SSM reduction by computing the SSM-reduced power spectral density of the randomly forced mechanical systems that range from simple oscillator chains to finite-element models of beams and plates.
- [80] arXiv:2407.03680 [pdf, html, other]
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Title: The condition for constructing a finite element from a supersplineComments: 22 pages, 4 figuresSubjects: Numerical Analysis (math.NA)
This paper addresses the sufficient and necessary conditions for constructing $C^r$ conforming finite element spaces from a superspline spaces on general simplicial triangulations. We introduce the concept of extendability for the pre-element spaces, which encompasses both the superspline space and the finite element space. By examining the extendability condition for both types of spaces, we provide an answer to the conditions regarding the construction. A corollary of our results is that constructing $C^r$ conforming elements in $d$ dimensions should in general require an extra $C^{2^{s}r}$ continuity on $s$-codimensional simplices, and the polynomial degree is at least $(2^d r + 1)$.
- [81] arXiv:2407.03692 [pdf, html, other]
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Title: A survey on embeddings of 3-manifolds in definite 4-manifoldsComments: 16 pages, 1 figureSubjects: Geometric Topology (math.GT)
This article presents a survey on the topic of embedding 3-manifolds in definite 4-manifolds, emphasizing the latest progress in the field. We will focus on the significant role played by Donaldson's diagonalization theorem and the combinatorics of integral lattices in understanding these embeddings. Additionally, the article introduces a new result concerning the embedding of amphichiral lens spaces in negative-definite manifolds.
- [82] arXiv:2407.03693 [pdf, html, other]
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Title: Multi-toric geometries with larger compact symmetryComments: 19 pagesSubjects: Differential Geometry (math.DG)
We study complete, simply-connected manifolds with special holonomy that are toric with respect to their multi-moment maps. We consider the cases where there is a connected non-Abelian symmetry group containing the torus. For $\mathrm{Spin}(7)$-manifolds, we show that the only possibility are structures with a cohomogeneity-two action of $T^{3} \times \mathrm{SU}(2)$. We then specialise the analysis to holonomy $G_{2}$, to Calabi-Yau geometries in real dimension six and to hyperKähler four-manifolds. Finally, we consider weakly coherent triples on $\mathbb{R} \times \mathrm{SU}(2)$, and their extensions over singular orbits, to give local examples in the $\mathrm{Spin}(7)$-case that have singular orbits where the stabiliser is of rank one.
- [83] arXiv:2407.03694 [pdf, html, other]
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Title: Holomorphic Functional Calculus approach to the Characteristic Function of Quantum ObservablesComments: Available at: this https URLJournal-ref: Journal of Stochastic Analysis: Vol. 5: No. 2, Article 1. (2024)Subjects: Mathematical Physics (math-ph); Functional Analysis (math.FA); Spectral Theory (math.SP)
We show how Cauchy's Integral Formula and the ideas of Dunford's Holomorphic Functional Calculus (for unbounded operators) can be used to compute the Vacuum Characteristic Function (Quantum Fourier Transform) of quantum random variables defined as self-adjoint operators on $L^2(\mathbb{R},\mathbb{C})$. We consider in detail several quantum observables defined in terms of the position and momentum operators $X$, $P$, respectively, on $L^2(\mathbb{R},\mathbb{C})$.
- [84] arXiv:2407.03706 [pdf, html, other]
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Title: Regularity of the (N-1)-particle electronic reduced density matrix for molecules with fixed nuclei and N electronsSubjects: Mathematical Physics (math-ph)
We consider an electronic bound state of the usual, non-relativistic, molecular Hamiltonian with Coulomb interactions, fixed nuclei, and N electrons (N>1). Near appropriate electronic collisions, we determine the regularity of the (N-1)-particle electronic reduced density matrix.
- [85] arXiv:2407.03707 [pdf, html, other]
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Title: An elementary approach based on variational inequalities for modelling a friction-based locomotion problemSubjects: Classical Analysis and ODEs (math.CA)
We propose an elementary proof based on a penalization technique to show the existence and uniqueness of the solution to a system of variational inequalities modelling the friction-based motion of a two-body crawling system. Here for each body, the static and dynamic friction coefficients are equal.
- [86] arXiv:2407.03710 [pdf, html, other]
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Title: Internal Controls for Stochastic Lattice DynamicsComments: 38 pages, 1 figureSubjects: Optimization and Control (math.OC)
In [6], we have designed impulsive and feed-back controls for harmonic chains with a point thermostat. In this work, a new type of control is proposed: internal control. We study the internal control for for stochastic lattice dynamics, with the goal of controlling the transition kernel of the kinetic equation in the limit.
- [87] arXiv:2407.03712 [pdf, other]
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Title: A second-order direct Eulerian GRP scheme for ten-moment Gaussian closure equations with source termsComments: 54 pages, 20 figures, 2tablesSubjects: Numerical Analysis (math.NA)
This paper proposes a second-order accurate direct Eulerian generalized Riemann problem (GRP) scheme for the ten-moment Gaussian closure equations with source terms. The generalized Riemann invariants associated with the rarefaction waves, the contact discontinuity and the shear waves are given, and the 1D exact Riemann solver is obtained. After that, the generalized Riemann invariants and the Rankine-Hugoniot jump conditions are directly used to resolve the left and right nonlinear waves (rarefaction wave and shock wave) of the local GRP in Eulerian formulation, and then the 1D direct Eulerian GRP scheme is derived. They are much more complicated, technical and nontrivial due to more physical variables and elementary waves. Some 1D and 2D numerical experiments are presented to check the accuracy and high resolution of the proposed GRP schemes, where the 2D direct Eulerian GRP scheme is given by using the Strang splitting method for simplicity. It should be emphasized that several examples of 2D Riemann problems are constructed for the first time.
- [88] arXiv:2407.03714 [pdf, html, other]
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Title: A distinction criterion for Iwahori-spherical representationsComments: Comments are welcomeSubjects: Representation Theory (math.RT)
Let $G/H$ be a Galois symmetric space for an unramified quadratic extension of a locally compact field $F$, where the group $H$ is semisimple, simply connected, defined and split over $F$. We prove that there exists a subgroup $\Gamma = \Gamma (G/H)$ of the group of invertible elements of the Iwahori-Hecke algebra $\mathcal H$ of $G$ such that an Iwahori-spherical representation of $G$ is $H$-distinguished if and only if the corresponding Iwahori-Hecke module is "$\Gamma$-distinguished".
- [89] arXiv:2407.03717 [pdf, html, other]
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Title: The existence of solutions for a Schrodinger equation with jumping nonlinearities crossing the essential spectrumSubjects: Analysis of PDEs (math.AP)
In this paper, we establish the existence of one solution for a Schrödinger equation with jumping nonlinearities: $-\Delta u+V(x)u=f(x,u)$, $x\in \mathbb {R}^N$, and $u(x)\to 0$, $|x|\to +\infty$, where $V$ is a potential function on which we make hypotheses, and in particular allow $V$ which is unbounded below, and $f(x,u)=au^-+bu^++g(x,u)$. No restriction on $b$ is required, which implies that $f(x,s)s^{-1}$ may interfere with the essential spectrum of $ -\Delta+V$ for $s\to +\infty$. Using the truncation method and the Morse theory, we can compute critical groups of the corresponding functional at zero and infinity, then prove the existence of one negative solution.
- [90] arXiv:2407.03722 [pdf, html, other]
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Title: More on the indivisibility of $\mathbb{Q}$Subjects: Logic (math.LO); Logic in Computer Science (cs.LO); Combinatorics (math.CO)
We study the complexity of the computational task ``Given a colouring $c : \mathbb{Q} \to \mathbf{k}$, find a monochromatic $S \subseteq \mathbb{Q}$ such that $(S,<) \cong (\mathbb{Q},<)$''. The framework is Weihrauch reducibility. Our results answer some open questions recently raised by Gill, and by Dzhafarov, Solomon and Valenti.
- [91] arXiv:2407.03731 [pdf, html, other]
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Title: Parameterizing Intersecting Surfaces via InvariantsComments: 30 pages, 12 figuresSubjects: Numerical Analysis (math.NA)
We introduce and analyze numerical companion matrix methods for the reconstruction of hypersurfaces with crossings from smooth interpolants given unordered or, without loss of generality, value-sorted data. The problem is motivated by the desire to machine learn potential energy surfaces arising in molecular excited state computational chemistry applications. We present simplified models which reproduce the analytically predicted convergence and stability behaviors as well as two application-oriented numerical experiments: the electronic excited states of Graphene featuring Dirac conical cusps and energy surfaces corresponding to a sulfur dioxide ($SO_2$) molecule in different configurations.
- [92] arXiv:2407.03741 [pdf, html, other]
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Title: A Unified Expression for Upper Bounds on the BLER of Spinal Codes over Fading ChannelsSubjects: Information Theory (cs.IT)
Performance evaluation of particular channel coding has been a significant topic in coding theory, often involving the use of bounding techniques. This paper focuses on the new family of capacity-achieving codes, Spinal codes, to provide a comprehensive analysis framework to tightly upper bound the block error rate (BLER) of Spinal codes in the finite block length (FBL) regime. First, we resort to a variant of the Gallager random coding bound to upper bound the BLER of Spinal codes over the fading channel. Then, this paper derives a new bound without resorting to the use of Gallager random coding bound, achieving provable tightness over the wide range of signal-to-noise ratios (SNR). The derived BLER upper bounds in this paper are generalized, facilitating the performance evaluations of Spinal codes over different types of fast fading channels. Over the Rayleigh, Nakagami-m, and Rician fading channels, this paper explicitly derived the BLER upper bounds on Spinal codes as case studies. Based on the bounds, we theoretically reveal that the tail transmission pattern (TTP) for ML-decoded Spinal codes remains optimal in terms of reliability performance. Simulations verify the tightness of the bounds and the insights obtained.
- [93] arXiv:2407.03746 [pdf, html, other]
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Title: Monolithic convex limiting and implicit pseudo-time stepping for calculating steady-state solutions of the Euler equationsSubjects: Numerical Analysis (math.NA)
In this work, we use the monolithic convex limiting (MCL) methodology to enforce relevant inequality constraints in implicit finite element discretizations of the compressible Euler equations. In this context, preservation of invariant domains follows from positivity preservation for intermediate states of the density and internal energy. To avoid spurious oscillations, we additionally impose local maximum principles on intermediate states of the density, velocity components, and specific total energy. For the backward Euler time stepping, we show the invariant domain preserving (IDP) property of the fully discrete MCL scheme by constructing a fixed-point iteration that is IDP and converges under a strong time step restriction. Our iterative solver for the nonlinear discrete problem employs a more efficient fixed-point iteration. The matrix of the associated linear system is a robust low-order Jacobian approximation that exploits the homogeneity property of the flux function. The limited antidiffusive terms are treated explicitly. We use positivity preservation as a stopping criterion for nonlinear iterations. The first iteration yields the solution of a linearized semi-implicit problem. This solution possesses the discrete conservation property but is generally not IDP. Further iterations are performed if any non-IDP states are detected. The existence of an IDP limit is guaranteed by our analysis. To facilitate convergence to steady-state solutions, we perform adaptive explicit underrelaxation at the end of each time step. The calculation of appropriate relaxation factors is based on an approximate minimization of nodal entropy residuals. The performance of proposed algorithms and alternative solution strategies is illustrated by the convergence history for standard two-dimensional test problems.
- [94] arXiv:2407.03747 [pdf, other]
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Title: An Example of Accurate Microlocal Tunneling in One DimensionAntide Duraffour (IRMAR, UR), Nicolas Raymond (UA, LAREMA)Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Spectral Theory (math.SP)
We investigate the spectral analysis of a class of pseudo-differential opera-tors in one dimension. Under symmetry assumptions, we prove an asymptotic formulafor the splitting of the first two eigenvalues. This article is a first example of extensionto pseudo-differential operators of the tunneling effect formulas known for the symmetricelectric Schr{ö}dinger operator.
- [95] arXiv:2407.03751 [pdf, html, other]
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Title: Equilibrium moderate deviations for occupation times of SSEP on regula treesSubjects: Probability (math.PR)
In this paper, we are concerned with the symmetric simple exclusion process on the regula tree $\mathbb{T}^d$ for $d\geq 2$. Our main result gives moderate deviation principles of occupation times of the process starting from an invariant product measure. Two replacement lemmas play key roles in the proof of our main result. To obtain these replacement lemmas, we utilize duality relationships between the symmetric exclusion process and two types of random walks on $\mathbb{T}^d$ and $\left(\mathbb{T}^d\right)^2$ respectively.
- [96] arXiv:2407.03752 [pdf, html, other]
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Title: The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equationsSubjects: Analysis of PDEs (math.AP)
In this paper, we derive the optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations. In particular, we prove that $\|u(t)\|_{\dot{B}^{\theta}_{p,1}({\mathop{\mathbb R\kern 0pt}\nolimits}^2)}={\mathcal O} (t^{\frac1p-\frac32-\frac{\theta}2})$ as $t\rightarrow\infty$ for any $p\in[2,\infty[,~\theta\in [0,2]$ if initially $\rho_0u_0\in \dot{B}^{-2}_{2,\infty}({\mathop{\mathbb R\kern 0pt}\nolimits}^2)$. This is optimal even for the classical homogeneous Navier-Stokes equations. Different with Schonbek and Wiegner's Fourier splitting device, our method here seems more direct, and can adapt to many other equations as well. Moreover, our method allows us to work in the $L^p$-based spaces.
- [97] arXiv:2407.03754 [pdf, other]
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Title: Genus theory, governing field, ramification and FrobeniusRoslan Ibara Ngiza Mfumu (University Marien Ngouabi of Brazzaville, FEMTO-ST), Christian Maire (FEMTO-ST)Subjects: Number Theory (math.NT)
In this work we develop, through a governing field, genus theory for a number field $\K$ with tame ramification in $T$ and splitting in $S$, where $T$ and $S$ are finite disjoint sets of primes of $\K$. This approach extends that initiated by the second author in the case of the class group. It allows expressing the $S$-$T$ genus number of a cyclic extension $Ł/\K$ of degree $p$ in terms of the rank of a matrix constructed from the Frobenius elements of the primes ramified in $Ł/\K$, in the Galois group of the underlying governing extension. For quadratic extensions $Ł/\Q$, the matrices in question are constructed from the Legendre symbols between the primes ramified in $Ł/\Q$ and the primes in $S$.
- [98] arXiv:2407.03758 [pdf, other]
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Title: Fisher information and continuity estimates for nonlinear but 1-homogeneous diffusive PDEs (via the JKO scheme)Thibault Caillet (MMCS), Filippo Santambrogio (MMCS)Subjects: Analysis of PDEs (math.AP)
In this short paper we prove, using the JKO scheme, that quantities such as the Fisher information stay bounded or decrease across time for a family of 1-homogeneous diffusive PDEs. As a corollary, we prove that moduli of continuity are conserved across time for the solutions of those PDEs.
- [99] arXiv:2407.03761 [pdf, other]
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Title: Universal piecewise polynomiality for counting curves in toric surfacesComments: 24 pages, 8 figures, 3 tables. arXiv admin note: text overlap with arXiv:1412.4563 by other authorsSubjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)
Inspired by piecewise polynomiality results of double Hurwitz numbers, Ardila and Brugallé introduced an enumerative problem which they call double Gromov--Witten invariants of Hirzebruch surfaces. These invariants serve as a two-dimensional analogue and satisfy a similar piecewise polynomial structure. More precisely, they introduced the enumeration of curves in Hirzebruch surfaces satisfying point conditions and tangency conditions on the two parallel toric boundaries. These conditions are stored in four partitions and the resulting invariants are piecewise polynomial in their entries. Moreover, they found that these expressions also behave polynomially with respect to the parameter determining the underlying Hirzebruch surfaces. Based on work of Ardila and Block, they proposed that such a polynomiality could also hold while changing between more general toric surfaces corresponding to $h$-transverse polygons. In this work, we answer this question affirmatively. Moreover, we express the resulting invariants for $h$-transverse polygons as matrix elements in the two-dimensional bosonic Fock space.
- [100] arXiv:2407.03769 [pdf, other]
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Title: Nonlinear compressive reduced basis approximation for multi-parameter elliptic problemSubjects: Numerical Analysis (math.NA)
Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space - when the parameters vary. This involves investigating the manifold and, in particular, understanding whether it is close to a low-dimensional affine space. This leads to the notion of Kolmogorov $N$-width that consists of evaluating to which extent the best choice of a vectorial space of dimension $N$ approximates ${\mathcal S}$ well enough. If a good approximation of elements in ${\mathcal S}$ can be done with some well-chosen vectorial space of dimension $N$ -- provided $N$ is not too large -- then a ``reduced'' basis can be proposed that leads to a Galerkin type method for the approximation of any element in ${\mathcal S}$. In many cases, however, the Kolmogorov $N$-width is not so small, even if the parameter set lies in a space of small dimension yielding a manifold with small dimension. In terms of complexity reduction, this gap between the small dimension of the manifold and the large Kolmogorov $N$-width can be explained by the fact that the Kolmogorov $N$-width is linear while, in contrast, the dependency in the parameter is, most often, non-linear. There have been many contributions aiming at reconciling these two statements, either based on deterministic or AI approaches. We investigate here further a new paradigm that, in some sense, merges these two aspects: the nonlinear compressive reduced basisapproximation. We focus on a simple multiparameter problem and illustrate rigorously that the complexity associated with the approximation of the solution to the parameter dependant PDE is directly related to the number of parameters rather than the Kolmogorov $N$-width.
- [101] arXiv:2407.03776 [pdf, html, other]
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Title: Energy-Efficient Probabilistic Semantic Communication over Space-Air-Ground Integrated NetworksSubjects: Information Theory (cs.IT)
Space-air-ground integrated networks (SAGINs) are emerging as a pivotal element in the evolution of future wireless networks. Despite their potential, the joint design of communication and computation within SAGINs remains a formidable challenge. In this paper, the problem of energy efficiency in SAGIN-enabled probabilistic semantic communication (PSC) system is investigated. In the considered model, a satellite needs to transmit data to multiple ground terminals (GTs) via an unmanned aerial vehicle (UAV) acting as a relay. During transmission, the satellite and the UAV can use PSC technique to compress the transmitting data, while the GTs can automatically recover the missing information. The PSC is underpinned by shared probability graphs that serve as a common knowledge base among the transceivers, allowing for resource-saving communication at the expense of increased computation resource. Through analysis, the computation overhead function in PSC is a piecewise function with respect to the semantic compression ratio. Therefore, it is important to make a balance between communication and computation to achieve optimal energy efficiency. The joint communication and computation problem is formulated as an optimization problem aiming to minimize the total communication and computation energy consumption of the network under latency, power, computation capacity, bandwidth, semantic compression ratio, and UAV location constraints. To solve this non-convex non-smooth problem, we propose an iterative algorithm where the closed-form solutions for computation capacity allocation and UAV altitude are obtained at each iteration. Numerical results show the effectiveness of the proposed algorithm.
- [102] arXiv:2407.03777 [pdf, html, other]
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Title: Semi and fully-discrete analysis of lowest-order nonstandard finite element methods for the biharmonic wave problemSubjects: Numerical Analysis (math.NA)
This paper discusses lowest-order nonstandard finite element methods for space discretization and explicit and implicit schemes for time discretization of the biharmonic wave equation with clamped boundary conditions. A modified Ritz projection operator defined on $H^2_0(\Omega)$ ensures error estimates under appropriate regularity assumptions on the solution. Stability results and error estimates of optimal order are established in suitable norms for the semidiscrete and explicit/implicit fully-discrete versions of the proposed schemes. Finally, we report on numerical experiments using explicit and implicit schemes for time discretization and Morley, discontinuous Galerkin, and {C$^0$ interior} penalty schemes for space discretization, that validate the theoretical error estimates.
- [103] arXiv:2407.03780 [pdf, html, other]
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Title: U-Gibbs measure rigidity for partially hyperbolic endomorphisms on surfacesSubjects: Dynamical Systems (math.DS)
We prove that, for a $C^2$ partially hyperbolic endomorphism of the 2-torus which is strongly transitive, given an ergodic $u$-Gibbs measure that has positive center Lyapunov exponent and has full support, then either the map is special (has only one unstable direction per point), or the measure is the unique absolutely continuous invariant measure. We can apply this result in many settings, in particular obtaining uniqueness of $u$-Gibbs measures for every non-special perturbation of irreducible linear expanding maps of the torus with simple spectrum. This gives new open sets of partially hyperbolic systems displaying a unique $u$-Gibbs measure.
- [104] arXiv:2407.03784 [pdf, html, other]
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Title: Eigenvalue backward errors of Rosenbrock systems and optimization of sums of Rayleigh quotientSubjects: Numerical Analysis (math.NA)
We address the problem of computing the eigenvalue backward error of the Rosenbrock system matrix under various types of block perturbations. We establish computable formulas for these backward errors using a class of minimization problems involving the Sum of Two generalized Rayleigh Quotients (SRQ2). For computational purposes and analysis, we reformulate such optimization problems as minimization of a rational function over the joint numerical range of three Hermitian matrices. This reformulation eliminates certain local minimizers of the original SRQ2 minimization and allows for convenient visualization of the solution. Furthermore, by exploiting the convexity within the joint numerical range, we derive a characterization of the optimal solution using a Nonlinear Eigenvalue Problem with Eigenvector dependency (NEPv). The NEPv characterization enables a more efficient solution of the SRQ2 minimization compared to traditional optimization techniques. Our numerical experiments demonstrate the benefits and effectiveness of the NEPv approach for SRQ2 minimization in computing eigenvalue backward errors of Rosenbrock systems.
- [105] arXiv:2407.03785 [pdf, html, other]
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Title: Impact of Channel Aging and Electromagnetic Interference on RIS-Assisted Cell-Free Massive MIMO SystemsComments: This paper contains 13 pages and 11 figures. This paper has been submitted to IEEE Journal for potential publication on 11th MaySubjects: Information Theory (cs.IT)
In this work, we investigate the impact of channel aging and electromagnetic interference (EMI) on spatially correlated reconfigurable intelligent surface (RIS) assisted cell-free massive multiple-input multiple-output (MIMO) systems. To effectively handle channel aging and EMI, we employ a novel two-phase channel estimation scheme with fractional power control-aided pilot assignment during the uplink channel estimation phase. This scheme provides improved channel estimates compared to existing approaches. The closed-form uplink and downlink spectral efficiency (SE) expressions incorporating fractional power control are derived to enable system performance evaluation. Additionally, we introduce the system's power consumption model to analyze energy efficiency (EE). Our numerical results illustrate the theoretical results and demonstrate the system performance with channel aging and EMI. Specifically, the proposed two-phase channel estimation scheme enhances estimation accuracy, compensating for performance degradation caused by channel aging and EMI. We find that increasing the number of access points (APs), RISs, antennas per AP, and elements per RIS can help to mitigate the SE performance degradation. We also find that an optimal number of APs can be selected to achieve energy efficiency (EE) maximization. However, in severe EMI environments, the benefits of deploying more RISs cannot be fully realized.
- [106] arXiv:2407.03793 [pdf, html, other]
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Title: A Preconditioned Discontinuous Galerkin Method for Biharmonic Equation with $C^0$-Reconstructed ApproximationSubjects: Numerical Analysis (math.NA)
In this paper, we present a high-order finite element method based on a reconstructed approximation to the biharmonic equation. In our construction, the space is reconstructed from nodal values by solving a local least squares fitting problem per element. It is shown that the space can achieve an arbitrarily high-order accuracy and share the same nodal degrees of freedom with the $C^0$ linear space. The interior penalty discontinuous Galerkin scheme can be directly applied to the reconstructed space for solving the biharmonic equation. We prove that the numerical solution converges with optimal orders under error measurements. More importantly, we establish a norm equivalence between the reconstructed space and the continuous linear space. This property allows us to precondition the linear system arising from the high-order space by the linear space on the same mesh. This preconditioner is shown to be optimal in the sense that the condition number of the preconditioned system admits a uniform upper bound independent of the mesh size. Numerical examples in two and three dimensions are provided to illustrate the accuracy of the scheme and the efficiency of the preconditioning method.
- [107] arXiv:2407.03798 [pdf, html, other]
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Title: Deligne-Knop tensor categories and functorialitySubjects: Representation Theory (math.RT); Category Theory (math.CT)
A general construction of Knop creates a symmetric monoidal category $\mathcal{T}(\mathcal{A},\delta)$ from any regular category $\mathcal{A}$ and a fixed degree function $\delta$. A special case of this construction are the Deligne categories $\underline{\operatorname{Rep}}(S_t)$ and $\underline{\operatorname{Rep}}(GL_t(\mathbb{F}_q))$. We discuss when a functor $F:\mathcal{A} \to \mathcal{A}'$ between regular categories induces a symmetric monoidal functor $\mathcal{T}(\mathcal{A},\delta) \to \mathcal{T}(\mathcal{A}',\delta')$. We then give a criterion when a pair of adjoint functors between two regular categories $\mathcal{A}, \ \mathcal{A}'$ lifts to a pair of adjoint functors between $\mathcal{T}(\mathcal{A},\delta)$ and $\mathcal{T}(\mathcal{A}',\delta')$.
- [108] arXiv:2407.03801 [pdf, html, other]
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Title: Solving the inverse source problem of the fractional Poisson equation by MC-fPINNsSubjects: Numerical Analysis (math.NA)
In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using MC-fPINNs. We construct two neural networks $ u_{NN}(x;\theta )$ and $f_{NN}(x;\psi)$ to approximate the solution $u^{*}(x)$ and the forcing term $f^{*}(x)$ of the fractional Poisson equation. To optimize these two neural networks, we use the Monte Carlo sampling method mentioned in MC-fPINNs and define a new loss function combining measurement data and the underlying physical model. Meanwhile, we present a comprehensive error analysis for this method, along with a prior rule to select the appropriate parameters of neural networks. Several numerical examples are given to demonstrate the great precision and robustness of this method in solving high-dimensional problems up to 10D, with various fractional order $\alpha$ and different noise levels of the measurement data ranging from 1$\%$ to 10$\%$.
- [109] arXiv:2407.03803 [pdf, html, other]
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Title: Extending Gromov's optimal systolic inequalityComments: 8 pages, published in Journal of GeometryJournal-ref: Journal of Geometry 114 (2023), article 23Subjects: Differential Geometry (math.DG)
The existence of nontrivial cup products or Massey products in the cohomology of a manifold leads to inequalities of systolic type, but in general such inequalities are not optimal (tight). Gromov proved an {optimal} systolic inequality for complex projective space. We provide a natural extension of Gromov's inequality to manifolds whose fundamental cohomology class is a cup product of 2-dimensional classes.
- [110] arXiv:2407.03807 [pdf, html, other]
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Title: Walecki tournaments with an arc that lies in a unique directed triangleComments: 14 pagesSubjects: Combinatorics (math.CO)
A Walecki tournament is any tournament that can be formed by choosing an orientation for each of the Hamilton cycles in the Walecki decomposition of a complete graph on an odd number of vertices. In this paper, we show that if some arc in a Walecki tournament on at least $7$ vertices lies in exactly one directed triangle, then there is a vertex of the tournament (the vertex typically labelled $*$ in the decomposition) that is fixed under every automorphism of the tournament. Furthermore, any isomorphism between such Walecki tournaments maps the vertex labelled $*$ in one to the vertex labelled $*$ in the other.
We also show that among Walecki tournaments with a signature of even length $2k$, of the $2^{2k}$ possible signatures, at least $2^k$ produce tournaments that have an arc that lies in a unique directed triangle (and therefore to which our result applies). - [111] arXiv:2407.03819 [pdf, html, other]
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Title: Words Avoiding TangramsComments: 15 pages, 2 figuresSubjects: Combinatorics (math.CO)
A \emph{tangram} is a word in which every letter occurs an even number of times. Such word can be cut into parts that can be arranged into two identical words. The minimum number of cuts needed is called the \emph{cut number} of a tangram. For example, the word $\mathtt{\color{red}{0102}\color{blue}{0102}}$ is a tangram with cut number one, while the word $\mathtt{\color{red}{01}\color{blue}{01023}\color{red}{023}}$ is a tangram with cut number two. Clearly, tangrams with cut number one coincide with the well known family of words, known as \emph{squares}, having the form $UU$ for some nonempty word $U$.
A word $W$ \emph{avoids} a word $T$ if it is not possible to write $W=ATB$, for any words $A$ and $B$ (possibly empty). The famous 1906 theorem of Thue asserts that there exist arbitrarily long words avoiding squares over alphabet with just \emph{three} letters. Given a fixed number $k\geqslant 1$, how many letters are needed to avoid tangrams with the cut number at most $k$? Let $t(k)$ denote the minimum size of an alphabet needed for that purpose. By Thue's result we have $t(1)=3$, which easily implies $t(2)=3$. Curiously, these are currently the only known exact values of this function.
In our main result we prove that $t(k)=\Theta(\log_2k)$. The proof uses \emph{entropy compression} argument and \emph{Zimin words}. By using a different method we prove that $t(k)\leqslant k+1$ for all $k\geqslant 4$, which gives more exact estimates for small values of $k$. The proof makes use of \emph{Dejean words} and a curious property of \emph{Gauss words}, which is perhaps of independent interest. - [112] arXiv:2407.03822 [pdf, html, other]
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Title: Some Diophantine equations involving arithmetic functions and Bhargava factorialsComments: 8 pages, comments welcome!Subjects: Number Theory (math.NT)
F. Luca proved for any fixed rational number $\alpha>0$ that the Diophantine equations of the form $\alpha\,m!=f(n!)$, where $f$ is either the Euler function or the divisor sum function or the function counting the number of divisors, have only finitely many integer solutions $(m,n)$. In this paper we generalize the mentioned result and show that Diophantine equations of the form $\alpha\,m_1!\cdots m_r!=f(n!)$ have finitely many integer solutions, too. In addition, we do so by including the case $f$ is the sum of $k$\textsuperscript{th} powers of divisors function. Moreover, we observe that the same holds by replacing some of the factorials with certain examples of Bhargava factorials.
- [113] arXiv:2407.03829 [pdf, html, other]
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Title: Recovering Initial States in Semilinear Parabolic Problems from Time-AveragesComments: 19 pagesSubjects: Analysis of PDEs (math.AP)
Well-posedness of certain semilinear parabolic problems with nonlocal initial conditions is shown in time-weighted spaces. The result is applied to recover the initial states in semilinear parabolic problems with nonlinearities of superlinear behavior near zero from small time-averages over arbitrary time periods.
- [114] arXiv:2407.03831 [pdf, html, other]
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Title: Exploring Algorithmic Solutions for the Independent Roman Domination Problem in GraphsSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
Given a graph $G=(V,E)$, a function $f:V\to \{0,1,2\}$ is said to be a \emph{Roman Dominating function} if for every $v\in V$ with $f(v)=0$, there exists a vertex $u\in N(v)$ such that $f(u)=2$. A Roman Dominating function $f$ is said to be an \emph{Independent Roman Dominating function} (or IRDF), if $V_1\cup V_2$ forms an independent set, where $V_i=\{v\in V~\vert~f(v)=i\}$, for $i\in \{0,1,2\}$. The total weight of $f$ is equal to $\sum_{v\in V} f(v)$, and is denoted as $w(f)$. The \emph{Independent Roman Domination Number} of $G$, denoted by $i_R(G)$, is defined as min$\{w(f)~\vert~f$ is an IRDF of $G\}$. For a given graph $G$, the problem of computing $i_R(G)$ is defined as the \emph{Minimum Independent Roman Domination problem}. The problem is already known to be NP-hard for bipartite graphs. In this paper, we further study the algorithmic complexity of the problem.
In this paper, we propose a polynomial-time algorithm to solve the Minimum Independent Roman Domination problem for distance-hereditary graphs, split graphs, and $P_4$-sparse graphs. - [115] arXiv:2407.03832 [pdf, html, other]
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Title: Homology of graph burningsComments: 22 pages, 10 figuresSubjects: Algebraic Topology (math.AT); Combinatorics (math.CO); Category Theory (math.CT)
In this paper we study graph burnings using methods of algebraic topology. We prove that the time function of a burning is a graph map to a path graph. Afterwards, we define a category whose objects are graph burnings and morphisms are graph maps which commute with the time functions of the burnings. In this category we study relations between burnings of different graphs and, in particular, between burnings of a graph and its subgraphs. For every graph, we define a simplicial complex, arising from the set of all the burnings, which we call a configuration space of the burnings. Further, simplicial structure of the configuration space gives burning homology of the graph. We describe properties of the configuration space and the burning homology theory. In particular, we prove that the one-dimensional skeleton of the configuration space of a graph $G$ coincides with the complement graph of $G$. The results are illustrated with numerous examples.
- [116] arXiv:2407.03837 [pdf, html, other]
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Title: Ponzi schemes on coarse spaces with uniform measureComments: 20pages.Comments are welcomeSubjects: General Topology (math.GN); Group Theory (math.GR); Metric Geometry (math.MG)
Ponzi schemes, defined by Block-Weinberger(1992) and Roe(2003), give a characterization of amenability from the viewpoint of coarse geometry. We consider measures in coarse spaces, and propose a reformulation of Ponzi schemes with measures.
- [117] arXiv:2407.03840 [pdf, html, other]
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Title: On the convergence of generalized kernel-based interpolation by greedy data selection algorithmsSubjects: Numerical Analysis (math.NA)
Besides standard Lagrange interpolation, i.e., interpolation of target functions from scattered point evaluations, positive definite kernel functions are well-suited for the solution of more general reconstruction problems. This is due to the intrinsic structure of the underlying reproducing kernel Hilbert space (RKHS). In fact, kernel-based interpolation has been applied to the reconstruction of bivariate functions from scattered Radon samples in computerized tomography (cf. Iske, 2018) and, moreover, to the numerical solution of elliptic PDEs (cf. Wenzel et al., 2022). As shown in various previous contributions, numerical algorithms and theoretical results from kernel-based Lagrange interpolation can be transferred to more general interpolation problems. In particular, greedy point selection methods were studied in (Wenzel et al., 2022), for the special case of Sobolev kernels. In this paper, we aim to develop and analyze more general kernel-based interpolation methods, for less restrictive settings. To this end, we first provide convergence results for generalized interpolation under minimalistic assumptions on both the selected kernel and the target function. Finally, we prove convergence of popular greedy data selection algorithms for totally bounded sets of functionals. Supporting numerical results are provided for illustration.
- [118] arXiv:2407.03844 [pdf, html, other]
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Title: Nonlocal-to-local convergence of the Cahn-Hilliard equation with degenerate mobility and the Flory-Huggins potentialComments: 18 pages + appendixSubjects: Analysis of PDEs (math.AP)
The Cahn-Hilliard equation is a fundamental model for phase separation phenomena. Its rigorous derivation from the nonlocal aggregation equation, motivated by the desire to link interacting particle systems and continuous descriptions, has received much attention in recent years. In the recent article, we showed how to treat the case of degenerate mobility for the first time. Here, we discuss how to adapt the exploited tools to the case of the mobility $m(u)=u\,(1-u)$ as in the original works of Giacomin-Lebowitz and Elliott-Garcke. The main additional information is the boundedness of $u$, implied by the form of mobility, which allows handling the nonlinear terms. We also discuss the case of (mildly) singular kernels and a model of cell-cell adhesion with the same mobility.
- [119] arXiv:2407.03849 [pdf, other]
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Title: A sequential multilinear Nystr\"om algorithm for streaming low-rank approximation of tensors in Tucker formatSubjects: Numerical Analysis (math.NA)
We present a sequential version of the multilinear Nyström algorithm which is suitable for the low-rank Tucker approximation of tensors given in a streaming format. Accessing the tensor $\mathcal{A}$ exclusively through random sketches of the original data, the algorithm effectively leverages structures in $\mathcal{A}$, such as low-rankness, and linear combinations. We present a deterministic analysis of the algorithm and demonstrate its superior speed and efficiency in numerical experiments including an application in video processing.
- [120] arXiv:2407.03853 [pdf, html, other]
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Title: Obstacles for Sobolev-homeomorphisms with low rank -- pointwise a.e. vs distributional JacobiansSubjects: Analysis of PDEs (math.AP); Algebraic Geometry (math.AG)
We show that for any $k$ and $s > \frac{k+1}{k+2}$ there exist neither $W^{s,\frac{k}{s}}$-Sobolev nor $C^s$-Hölder homeomorphisms from the disk $\mathbb{B}^n$ into $\mathbb{R}^N$ whose gradient has rank $< k$ in distributional sense. This complements known examples of such kind of homeomorphisms whose gradient has rank $<k$ almost everywhere.
- [121] arXiv:2407.03854 [pdf, html, other]
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Title: Risk Bounds on MDL Estimators for Linear Regression Models with Application to Simple ReLU Neural NetworksSubjects: Information Theory (cs.IT)
To investigate the theoretical foundations of deep learning from the viewpoint of the minimum description length (MDL) principle, we analyse risk bounds of MDL estimators based on two-stage codes for simple two-layers neural networks (NNs) with ReLU activation. For that purpose, we propose a method to design two-stage codes for linear regression models and establish an upper bound on the risk of the corresponding MDL estimators based on the theory on MDL estimators originated by Barron and Cover (1991). Then, we apply this result to the simple two-layers NNs with ReLU activation which consist of $d$ nodes in the input layer, $m$ nodes in the hidden layer and one output node. Since the object of estimation is only the $m$ weights from the hidden layer to the output node in our setting, this is an example of linear regression models. As a result, we show that the redundancy of the obtained two-stage codes is small owing to the fact that the eigenvalue distribution of the Fisher information matrix of the NNs is strongly biased, which was recently shown by Takeishi et al. (2023). That is, we establish a tight upper bound on the risk of our MDL estimators. Note that our risk bound, of which the leading term is $O(d^2 \log n /n)$, is independent of the number of parameters $m$.
- [122] arXiv:2407.03855 [pdf, html, other]
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Title: On Riemann curvature of spherically symmetric metricsComments: Accepted for publication in Mathematical Physics, Analysis and GeometrySubjects: Differential Geometry (math.DG)
In this paper, studying the inverse problem, we establish a curvature compatibility condition on a spherically symmetric Finsler metric. As an application, we characterize the spherically symmetric metrics of scalar curvature. We construct a Berwald frame for a spherically symmetric Finsler surface and calculate some associated geometric objects. Several examples are provided and discussed. Finally, we give a note on a certain general $(\alpha,\beta)$-metric that appears in the literature.
- [123] arXiv:2407.03870 [pdf, html, other]
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Title: A uniform-in-time nonlocal approximation of the standard Fokker-Planck equationComments: 46 pagesSubjects: Analysis of PDEs (math.AP); Probability (math.PR)
We study a nonlocal approximation of the Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which respects the local limit of the equation. This uniform estimate cannot be easily obtained with standard inequalities or entropy methods, but can be obtained through the use of Harris's theorem, finding interesting links to quantitative versions of the central limit theorem in probability. As a consequence one can prove that solutions of this nonlocal approximation converge to solutions of the usual Fokker-Planck equation uniformly in time -- hence we show the approximation is asymptotic-preserving in this sense. The associated equilibrium has some interesting tail and regularity properties, which we also study.
- [124] arXiv:2407.03873 [pdf, html, other]
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Title: Parallel-in-time solution of hyperbolic PDE systems via characteristic-variable block preconditioningSubjects: Numerical Analysis (math.NA)
We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that inter-variable coupling for characteristic variables is weak relative to intra-variable coupling, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small. For an $\ell$-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of $\ell$ scalar linear(ized)-advection-like problems, each being associated with a different characteristic wave-speed in the underlying linear(ized) PDE. We approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media, and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions.
- [125] arXiv:2407.03881 [pdf, html, other]
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Title: Generic nonexpansive Hilbert space mappingsComments: 18 pages, 2 figuresSubjects: Functional Analysis (math.FA)
We consider a closed convex set $C$ in a separable, infinite-dimensional Hilbert space and endow the set $\mathcal{N}(C)$ of nonexpansive self-mappings on $C$ with the topology of pointwise convergence. We introduce the notion of a somewhat bounded set and establish a strong connection between this property and the existence of fixed points for the generic $f\in\mathcal{N}(C)$, in the sense of Baire categories. Namely, if $C$ is somewhat bounded, the generic nonexpansive mapping on $C$ admits a fixed point, whereas if $C$ is not somewhat bounded, the generic nonexpansive mapping on $C$ does not have any fixed points. This results in a topological 0-1 law: the set of all $f\in\mathcal{N}(C)$ with a fixed point is either meager or residual. We further prove that, generically, there are no fixed points in the interior of $C$ and, under additional geometric assumptions, we show the uniqueness of such fixed points for the generic $f\in\mathcal{N}(C)$ and the convergence of the iterates of $f$ to its fixed point.
- [126] arXiv:2407.03888 [pdf, html, other]
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Title: Continuous-time q-Learning for Jump-Diffusion Models under Tsallis EntropySubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
This paper studies continuous-time reinforcement learning for controlled jump-diffusion models by featuring the q-function (the continuous-time counterpart of Q-function) and the q-learning algorithms under the Tsallis entropy regularization. Contrary to the conventional Shannon entropy, the general form of Tsallis entropy renders the optimal policy not necessary a Gibbs measure, where some Lagrange multiplier and KKT multiplier naturally arise from certain constraints to ensure the learnt policy to be a probability distribution. As a consequence,the relationship between the optimal policy and the q-function also involves the Lagrange multiplier. In response, we establish the martingale characterization of the q-function under Tsallis entropy and devise two q-learning algorithms depending on whether the Lagrange multiplier can be derived explicitly or not. In the latter case, we need to consider different parameterizations of the q-function and the policy and update them alternatively. Finally, we examine two financial applications, namely an optimal portfolio liquidation problem and a non-LQ control problem. It is interesting to see therein that the optimal policies under the Tsallis entropy regularization can be characterized explicitly, which are distributions concentrate on some compact support. The satisfactory performance of our q-learning algorithm is illustrated in both examples.
- [127] arXiv:2407.03898 [pdf, html, other]
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Title: Overflow-Avoiding Memory AMPSubjects: Information Theory (cs.IT)
Approximate Message Passing (AMP) type algorithms are widely used for signal recovery in high-dimensional noisy linear systems. Recently, a principle called Memory AMP (MAMP) was proposed. Leveraging this principle, the gradient descent MAMP (GD-MAMP) algorithm was designed, inheriting the strengths of AMP and OAMP/VAMP. In this paper, we first provide an overflow-avoiding GD-MAMP (OA-GD-MAMP) to address the overflow problem that arises from some intermediate variables exceeding the range of floating point numbers. Second, we develop a complexity-reduced GD-MAMP (CR-GD-MAMP) to reduce the number of matrix-vector products per iteration by 1/3 (from 3 to 2) with little to no impact on the convergence speed.
- [128] arXiv:2407.03899 [pdf, html, other]
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Title: Hybrid NOMA Assisted OFDMA Uplink TransmissionSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Hybrid non-orthogonal multiple access (NOMA) has recently received significant research interest due to its ability to efficiently use resources from different domains and also its compatibility with various orthogonal multiple access (OMA) based legacy networks. Unlike existing studies on hybrid NOMA that focus on combining NOMA with time-division multiple access (TDMA), this work considers hybrid NOMA assisted orthogonal frequency-division multiple access (OFDMA). In particular, the impact of a unique feature of hybrid NOMA assisted OFDMA, i.e., the availability of users' dynamic channel state information, on the system performance is analyzed from the following two perspectives. From the optimization perspective, analytical results are developed which show that with hybrid NOMA assisted OFDMA, the pure OMA mode is rarely adopted by the users, and the pure NOMA mode could be optimal for minimizing the users' energy consumption, which differs from the hybrid TDMA case. From the statistical perspective, two new performance metrics, namely the power outage probability and the power diversity gain, are developed to quantitatively measure the performance gain of hybrid NOMA over OMA. The developed analytical results also demonstrate the ability of hybrid NOMA to meet the users' diverse energy profiles.
- [129] arXiv:2407.03905 [pdf, html, other]
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Title: A network aggregation model for the dynamics and treatment of neurodegenerative diseases at the brain scaleSubjects: Dynamical Systems (math.DS); Biological Physics (physics.bio-ph); Medical Physics (physics.med-ph)
Neurodegenerative diseases are associated with the assembly of specific proteins into oligomers and fibrillar aggregates. At the brain scale, these protein assemblies can diffuse through the brain and seed other regions, creating an autocatalytic protein progression. The growth and transport of these assemblies depend on various mechanisms that can be targeted therapeutically. Here, we use spatially-extended nucleation-aggregation-fragmentation models for the dynamics of prion-like neurodegenerative protein-spreading in the brain to study the effect of different drugs on whole-brain Alzheimer's disease progression.
- [130] arXiv:2407.03908 [pdf, html, other]
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Title: Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifoldsComments: 30 pages, comments are welcomeSubjects: Analysis of PDEs (math.AP)
In this article, we investigate the global well-posedness for cubic nonlinear Schrödinger equation(NLS) $ i\partial_tu+\Delta_gu=|u|^2u$ posed on the three dimensional compact manifolds $(M,g)$ with initial data $u_0\in H^s(M)$ where $s>\frac{\sqrt{21}-1}{4}$ for Zoll manifold and $s>\frac{1+3\sqrt{5}}{8}$ for the product of spheres $\Bbb{S}^2\times\Bbb{S}^1$. We utilize the multilinear eigenfunction estimate on compact manifold to treat the interaction of different frequencies, which is more complicated compared to the case of flat torus [C. Fan, G. Staffilani, H. Wang, B. Wilson, Anal. PDE, 11 (2018), 919-944.] and waveguide manifold [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.]. Moreover, combining with the I-method adapted to the non-periodic case, bilinear Strichartz estimates along with the scale-invariant $L^p$ linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the $H^s$ norm of solution $u$.
- [131] arXiv:2407.03909 [pdf, html, other]
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Title: Wide stable neural networks: Sample regularity, functional convergence and Bayesian inverse problemsComments: 26 pages, 2 figuresSubjects: Statistics Theory (math.ST); Probability (math.PR)
We study the large-width asymptotics of random fully connected neural networks with weights drawn from $\alpha$-stable distributions, a family of heavy-tailed distributions arising as the limiting distributions in the Gnedenko-Kolmogorov heavy-tailed central limit theorem. We show that in an arbitrary bounded Euclidean domain $\mathcal{U}$ with smooth boundary, the random field at the infinite-width limit, characterized in previous literature in terms of finite-dimensional distributions, has sample functions in the fractional Sobolev-Slobodeckij-type quasi-Banach function space $W^{s,p}(\mathcal{U})$ for integrability indices $p < \alpha$ and suitable smoothness indices $s$ depending on the activation function of the neural network, and establish the functional convergence of the processes in $\mathcal{P}(W^{s,p}(\mathcal{U}))$. This convergence result is leveraged in the study of functional posteriors for edge-preserving Bayesian inverse problems with stable neural network priors.
- [132] arXiv:2407.03926 [pdf, html, other]
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Title: Rethinking the fundamental performance limits of integrated sensing and communication systemsSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Integrated sensing and communication (ISAC) has been recognized as a key enabler and feature of future wireless networks. In the existing works analyzing the performances of ISAC, discrete-time systems were commonly assumed, which, however, overlooked the impacts of temporal, spectral, and spatial properties. To address this issue, we establish a unified information model for the band-limited continuous-time ISAC systems. In the established information model, we employ a novel sensing performance metric, called the sensing mutual information (SMI). Through analysis, we show how the SMI can be utilized as a bridge between the mutual information domain and the mean squared error (MSE) domain. In addition, we illustrate the communication mutual information (CMI)-SMI and CMI-MSE regions to identify the performance bounds of ISAC systems in practical settings and reveal the trade-off between communication and sensing performances. Moreover, via analysis and numerical results, we provide two valuable insights into the design of novel ISAC-enabled systems: i) communication prefers the waveforms of random amplitude, sensing prefers the waveforms of constant amplitude, both communication and sensing favor the waveforms of low correlations with random phases; ii) There exists a linear positive proportional relationship between the allocated time-frequency resource and the achieved communication rate/sensing MSE.
- [133] arXiv:2407.03930 [pdf, html, other]
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Title: A Generalized Spiking Locally Competitive Algorithm for Multiple Optimization ProblemsComments: 26 pages, 6 figuresSubjects: Optimization and Control (math.OC)
We introduce a generalized Spiking Locally Competitive Algorithm (LCA) that is biologically plausible and exhibits adaptability to a large variety of neuron models and network connectivity structures. In addition, we provide theoretical evidence demonstrating the algorithm's convergence in optimization problems of signal recovery. Furthermore, our algorithm demonstrates superior performance over traditional optimization methods, such as FISTA, particularly by achieving faster early convergence in practical scenarios including signal denoising, seismic wave detection, and computed tomography reconstruction. Notably, our algorithm is compatible with neuromorphic chips, such as Loihi, facilitating efficient multitasking within the same chip architecture - a capability not present in existing algorithms. These advancements make our generalized Spiking LCA a promising solution for real-world applications, offering significant improvements in execution speed and flexibility for neuromorphic computing systems.
- [134] arXiv:2407.03932 [pdf, html, other]
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Title: Cohomology and K-theory of generalized Dold manifolds fibred by complex flag manifoldsSubjects: Algebraic Topology (math.AT)
Let $\nu=(n_1,\ldots, n_s), s\ge 2,$ be a sequence of positive integers and let $n=\sum_{1\le j\le s}n_j$. Let $\mathbb CG(\nu)=U(n)/(U(n_1)\times \cdots\times U(n_s))$ be the complex flag manifold. Denote by $P(m,\nu)=P(\mathbb S^m,\mathbb CG(\nu))$ the generalized Dold manifold $\mathbb S^m\times \mathbb CG(\nu)/\langle \theta\rangle $ where $\theta=\alpha\times \sigma$ with $\alpha:\mathbb S^m\to \mathbb S^m$ being the antipodal map and $\sigma:\mathbb CG(\nu)\to \mathbb CG(\nu)$, the complex conjugation. The manifold $P(m,\nu)$ has the structure of a smooth $\mathbb CG(\nu)$-bundle over the real projective space $\mathbb RP^m.$ We determine the additive structure of $H^*(P(m,\nu);R)$ when $R=\mathbb Z$ and its ring structure when $R$ is a commutative ring in which $2$ is invertible. As an application, we determine the additive structure of $K(P(m,\nu))$ almost completely and also obtain partial results on its ring structure. The results for the singular homology are obtained for generalized Dold spaces $P(S,X)=S\times X/\langle \theta\rangle$, where $\theta=\alpha\times \sigma$, $\alpha:S\to S$ is a fixed point free involution and $\sigma:X\to X$ is an involution with $\mathrm{Fix}(\sigma)\ne \emptyset,$ for a much wider class of spaces $S$ and $X$.
- [135] arXiv:2407.03936 [pdf, html, other]
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Title: Factorized binary polynomial optimizationSubjects: Optimization and Control (math.OC); Discrete Mathematics (cs.DM)
In binary polynomial optimization, the goal is to find a binary point maximizing a given polynomial function. In this paper, we propose a novel way of formulating this general optimization problem, which we call factorized binary polynomial optimization. In this formulation, we assume that the variables are partitioned into a fixed number of sets, and that the objective function is written as a sum of r products of linear functions, each one involving only variables in one set of the partition. Our main result is an algorithm that solves factorized binary polynomial optimization in strongly polynomial time, when r is fixed. This result provides a vast new class of tractable instances of binary polynomial optimization, and it even improves on the state-of-the-art for quadratic objective functions, both in terms of generality and running time. We demonstrate the applicability of our result through the binary tensor factorization problem, which arises in mining discrete patterns in data, and that contains as a special case the rank-1 Boolean tensor factorization problem. Our main result implies that these problems can be solved in strongly polynomial time, if the input tensor has fixed rank, and a rank factorization is given. For the rank-1 Boolean matrix factorization problem, we only require that the input matrix has fixed rank.
- [136] arXiv:2407.03938 [pdf, html, other]
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Title: Monochromatic Sumsets in Countable Colourings of Abelian GroupsSubjects: Combinatorics (math.CO)
Fernández-Bretón, Sarmiento and Vera showed that whenever a direct sum of sufficiently many copies of ${\mathbb Z}_4$, the cyclic group of order 4, is countably coloured there are arbitrarily large finite sets $X$ whose sumsets $X+X$ are monochromatic. They asked if the elements of order 4 are necessary, in the following strong sense: if $G$ is an abelian group having no elements of order 4, is it always the case there there is a countable colouring of $G$ for which there is not even a monochromatic sumset $X+X$ with $X$ of size 2? Our aim in this short note is to show that this is indeed the case.
- [137] arXiv:2407.03945 [pdf, html, other]
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Title: A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamicsSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG)
The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel operator learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.
- [138] arXiv:2407.03957 [pdf, other]
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Title: Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theoryComments: 31 pages, 7 figures, 3 tablesSubjects: Numerical Analysis (math.NA)
We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.
- [139] arXiv:2407.03975 [pdf, html, other]
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Title: Stacking faults in the limit of a discrete model for partial edge dislocationsSubjects: Analysis of PDEs (math.AP)
In the limit of vanishing lattice spacing we provide a rigorous variational coarse-graining result for a next-to-nearest neighbor lattice model of a simple crystal. We show that the $\Gamma$-limit of suitable scaled versions of the model leads to an energy describing a continuum mechanical model depending on partial dislocations and stacking faults. Our result highlights the necessary multiscale character of the energies setting the groundwork for more comprehensive models that can better explain and predict the mechanical behavior of materials with complex defect structures.
- [140] arXiv:2407.03983 [pdf, html, other]
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Title: Characterization of Lipschitz functions via commutators of maximal operators on slice spacesComments: 10 pagesSubjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA)
Let $0 \leq \alpha<n$, $M_{\alpha}$ be the fractional maximal operator, $M^{\sharp}$ be the sharp maximal operator and $b$ be the locally integrable function. Denote by $[b, M_{\alpha}]$ and $[b, M^{\sharp}]$ be the commutators of the fractional maximal operator $M_{\alpha}$ and the sharp maximal operator $M^{\sharp}$. In this paper, we show some necessary and sufficient conditions for the boundedness of the commutators $[b, M_{\alpha}]$ and $[b, M^{\sharp}]$ on slice spaces when the function $b$ is the Lipschitz function, by which some new characterizations of the non-negative Lipschitz function are obtained
- [141] arXiv:2407.03986 [pdf, html, other]
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Title: Potential trace inequalities via a Calder\'on-type theoremComments: 30 pagesSubjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)
We establish an approach to trace inequalities for potential-type operators based on an appropriate modification of an interpolation theorem due to Calderón. We develop a general theoretical tool for establishing boundedness of notoriously difficult operators (such as potentials) on certain specific types of rearrangement-invariant function spaces from analogous properties of operators that are easier to handle (such as fractional maximal operators). The key ingredient for the development of the theory is the initial pair of endpoint estimates for the easier operator whose pivotal example is based on a two-weight inequality of Sawyer. Among various applications we obtain a generalization of the celebrated trace inequality involving the Riesz potential and the Hausdorff content by Korobkov and Kristensen.
- [142] arXiv:2407.03988 [pdf, html, other]
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Title: Global well-posedness of 2D Navier-Stokes with Dirichlet boundary fractional noiseComments: arXiv admin note: text overlap with arXiv:2306.11081Subjects: Analysis of PDEs (math.AP); Probability (math.PR)
In this paper, we prove the global well-posedness and interior regularity for the 2D Navier-Stokes equations driven by a fractional noise acting as an inhomogeneous Dirichlet-type boundary condition. The model describes a vertical slice of the ocean with a relative motion between the two surfaces and can be thought of as a stochastic variant of the Couette flow. The relative motion of the surfaces is modeled by a Gaussian noise which is coloured in space and fractional in time with Hurst parameter greater than 3/4.
- [143] arXiv:2407.03989 [pdf, html, other]
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Title: Foliations transverse to a closed conformal vector fieldComments: 15 pages, 2 figuresSubjects: Differential Geometry (math.DG)
In this article, we study the geometric properties of codimension one foliations on Riemannian manifolds equipped with vector fields that are closed and conformal. Apart from its singularities, these vector fields define codimension one foliations with nice geometric features, which we call Montiel Foliations. We investigate conditions for which a foliation transverse to this structure has totally geodesic leaves, as well as how the ambient space and the geometry of the leaves forces a given foliation into a Montiel Foliation. Our main results concern minimal leaves and constant mean curvature foliations, having compact or complete noncompact leaves. Finally, we characterize totally geodesic foliations by means of its relation to a prior Montiel Foliation.
- [144] arXiv:2407.03991 [pdf, html, other]
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Title: Hamiltonian formalism for general variational problemsComments: 26 pagesSubjects: Mathematical Physics (math-ph); Differential Geometry (math.DG)
The present article introduces a generalization of the Hamiltonian field theory for a Lagrangian density, allowing the formulation of this kind of field theories for variational problem of more general nature than those associated to a classical variational problem. It is achieved by realizing that the usual construction of the Hamiltonian equations can be performed without the use of the so called Hamiltonian section, whose existence is problematic when general variational problems are considered. The developed formalism is applied to obtain an unified problem and a Hamiltonian field theory for the vakonomic Herglotz variational problem.
- [145] arXiv:2407.03999 [pdf, html, other]
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Title: A Consistent Sandpile Torsor Algorithm for Regular MatroidsSubjects: Combinatorics (math.CO)
Every regular matroid is associated with a sandpile group, which acts simply transitively on the set of bases in various ways. Ganguly and the second author introduced the notion of consistency to describe classes of actions that respect deletion-contraction in a precise sense, and proved the consistency of rotor-routing torsors (and uniqueness thereof) for plane graphs.
In this work, we prove that the class of actions introduced by Backman, Baker, and the fourth author, is consistent for regular matroids. More precisely, we prove the consistency of its generalization given by Backman, Santos and the fourth author, and independently by the first author. This extends the above existence assertion, as well as makes progress on the goal of classifying all consistent actions. - [146] arXiv:2407.04000 [pdf, html, other]
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Title: Smoothing low-dimensional algebraic cycles [after Koll\'ar and Voisin]Comments: 20 pages, séminaire Bourbaki, novembre 2024, exposé 1228Subjects: Algebraic Geometry (math.AG)
Let $X$ be a smooth projective complex algebraic variety. An old question of Borel and Haefliger asks whether any (possibly singular) algebraic subvariety of $X$ is homologically equivalent to a linear combination with integral coefficients of smooth algebraic subvarieties of $X$. In general, this question is too optimistic, and counterexamples have been known for a long time. The aim of this survey is to explain how János Kollár and Claire Voisin have provided a positive answer to Borel and Haefliger's question, for subvarieties of dimension less than half the dimension of $X$.
- [147] arXiv:2407.04005 [pdf, html, other]
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Title: Stochastic Processes: From Classical to QuantumComments: 52 pagesSubjects: Mathematical Physics (math-ph); Probability (math.PR); Quantum Physics (quant-ph)
The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems.
- [148] arXiv:2407.04006 [pdf, html, other]
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Title: Analysis and Optimization of RIS-Assisted Cell-Free Massive MIMO NOMA SystemsSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
We consider a reconfigurable intelligent surface (RIS) assisted cell-free massive multiple-input multiple-output non-orthogonal multiple access (NOMA) system, where each access point (AP) serves all the users with the aid of the RIS. We practically model the system by considering imperfect instantaneous channel state information (CSI) and employing imperfect successive interference cancellation at the users end. We first obtain the channel estimates using linear minimum mean square error approach considering the spatial correlation at the RIS and then derive a closed-form downlink spectral efficiency (SE) expression using the statistical CSI. We next formulate a joint optimization problem to maximize the sum SE of the system. We first introduce a novel successive Quadratic Transform (successive-QT) algorithm to optimize the transmit power coefficients using the concept of block optimization along with quadratic transform and then use the particle swarm optimization technique to design the RIS phase shifts. Note that most of the existing works on RIS-aided cell-free systems are specific instances of the general scenario studied in this work. We numerically show that i) the RIS-assisted link is more advantageous at lower transmit power regions where the direct link between AP and user is weak, ii) NOMA outperforms orthogonal multiple access schemes in terms of SE, and iii) the proposed joint optimization framework significantly improves the sum SE of the system.
- [149] arXiv:2407.04008 [pdf, other]
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Title: A Multi-Parameter Singular Perturbation Analysis of the Robertson ModelComments: 28 pages, 16 figuresSubjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)
The Robertson model describing a chemical reaction involving three reactants is one of the classical examples of stiffness in ODEs. The stiffness is caused by the occurrence of three reaction rates $k_1,\,k_2$, and $k_3$, with largely differing orders of magnitude, acting as parameters. The model has been widely used as a numerical test problem. Surprisingly, no asymptotic analysis of this multiscale problem seems to exist. In this paper we provide a full asymptotic analysis of the Robertson model under the assumption $k_1, k_3 \ll k_2$. We rewrite the equations as a two-parameter singular perturbation problem in the rescaled small parameters $(\varepsilon_1,\varepsilon_2):=(k_1/k_2,k_3/k_2)$, which we then analyze using geometric singular perturbation theory (GSPT). To deal with the multi-parameter singular structure, we perform blow-ups in parameter- and variable space. We identify four distinct regimes in a neighbourhood of the singular limit \mbox{$(\varepsilon_1,\varepsilon_2)= (0,0)$}. Within these four regimes we use GSPT and additional blow-ups to analyze the dynamics and the structure of solutions. Our asymptotic results are in excellent qualitative and quantitative agreement with the numerics.
- [150] arXiv:2407.04012 [pdf, other]
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Title: Transfer of homological objects in exact categories via adjoint triples. Applications to functor categoriesSubjects: Category Theory (math.CT); Representation Theory (math.RT)
For a given family $\{(\mathrm{q}_i, \mathrm{t}_i, \mathrm{p_i} )\}_{i \in I}$ of adjoint triples between exact categories $\mathcal{C}$ or $\mathcal{D}$, we show that any cotorsion pair in $\mathcal{C}$ and $\mathcal{D}$ yield two canonical cotorsion pairs providing a concrete description of objects without using any injectives/projectives object hypothesis. We firstly apply this result for the evaluation functor on the functor category $\operatorname{Add}(\mathcal{A}, R \mbox{-Mod})$ equipped with an exact structure $\mathcal{E}$. Under mild conditions on $\mathcal{A}$, we introduce the stalk functor at any object of $\mathcal{A}$, and subsequently, we investigate cotorsion pairs induced by stalk functors. Finally, we use them to present an intrinsic characterization of projective/injective objects in $(\mbox{Add}(\mathcal{A}, R\mbox{-Mod}); \mathcal{E})$.
- [151] arXiv:2407.04019 [pdf, html, other]
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Title: Cohomological field theories and generalized Seiberg-Witten equationsComments: 31 pages, 2 figuresSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)
We introduce a formalism for constructing cohomological field theories (CohFT) out of nonlinear PDEs based on the first author's previous work (arXiv:2202.12425). We apply the formalism to the generalized Seiberg-Witten equations and show that the obtained CohFT functionals agree with the existing ones proposed by physicists. This leads to a unified perspective from which to view the full supersymmetric functionals of the Donaldson-Witten, Seiberg-Witten, and Kapustin-Witten theories and understand the relations between them. We also outline a quantization program for our framework and discuss its potential to produce manifold invariants and quantum cohomologies.
- [152] arXiv:2407.04021 [pdf, other]
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Title: Total Lagrangian Smoothed Particle Hydrodynamics with An Improved Bond-Based Deformation Gradient for Large Strain Solid DynamicsSubjects: Numerical Analysis (math.NA)
Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH) is one variant of SPH where the variables are described using the fixed reference configuration and a Lagrangian smoothing kernel. TLSPH elevates the computational efficiency of the standard SPH when no topological change is involved, and it alleviates the stability of SPH scheme with respect to tensile loading. However, instabilities associated with spurious mode, or hourglass/zero-energy mode, persists and often affects the simulation of solids undergoing extremely large strain. This work proposes an alternative formulation to compute deformation gradient with improved accuracy and therefore minimising the possibility of encountering the zero-energy mode. Specifically, we leverage the local discrete computation of bond-based (or pairwise) deformation gradient smoothed by the kernel. Additionally, the bond of a particle with itself is considered to preserve the polynomial reproducibility imposed by the kernel correction scheme. We showcase the convergence of the approach using a two-dimensional benchmark example. Furthermore, the accuracy, robustness, and stability of the proposed approach are assessed in various two- and three-dimensional examples, highlighting on the stability improvement that allows for solid dynamic simulations with more extreme elongation.
- [153] arXiv:2407.04030 [pdf, html, other]
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Title: Dual Ramsey degrees for some classes of relational structuresSubjects: Combinatorics (math.CO)
In this paper we show that some natural classes of structures such as graphs, posets and metric spaces have both dual small and dual big Ramsey degrees with respect to some natural classes of morphisms such as quotient maps in case of graphs and posets, or non-expansive surjections in case of metric spaces. The only exception is the class of reflexive tournaments: they have neither dual small nor dual big Ramsey degrees. Our proof strategy is based on the categorical interpretation of structural Ramsey theory. Starting from a category we are interested in, we engineer an synthetic expansion to piggyback on a category where the dual Ramsey property has been established. We then use the additive properties of dual Ramsey degrees to get back to the original category of "natural" objects and morphisms and conclude that it has finite dual Ramsey degrees.
- [154] arXiv:2407.04032 [pdf, html, other]
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Title: Optimal planar immersions of prescribed winding number and Arnold invariantsComments: 38 pages, 7 figuresSubjects: Classical Analysis and ODEs (math.CA)
Vladimir Arnold defined three invariants for generic planar immersions, i.e. planar curves whose self-intersections are all transverse double points. We use a variational approach to study these invariants by investigating a suitably truncated knot energy, the tangent-point energy. We prove existence of energy minimizers for each truncation parameter ${\delta} > 0$ in a class of immersions with prescribed winding number and Arnold invariants, and establish Gamma convergence of the truncated tangent-point energies to a limiting renormalized tangent-point energy as ${\delta\to 0}$. Moreover, we show that any sequence of minimizers subconverges in ${C^1}$, and the corresponding limit curve has the same topological invariants, self-intersects exclusively at right angles, and minimizes the renormalized tangent-point energy among all curves with right self-intersection angles. In addition, the limit curve is an almost-minimizer for all of the original truncated tangent-point energies as long as the truncation parameter ${\delta}$ is sufficiently small. Therefore, this limit curve serves as an "optimal" curve in the class of generic planar immersions with prescribed winding number and Arnold invariants.
- [155] arXiv:2407.04035 [pdf, html, other]
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Title: A remark on the Whitney Broken Circuit TheoremSubjects: Combinatorics (math.CO)
In the present note we show, via the connection between chromatic polynomial and Potts model, that the Whitney Broken circuit theorem is in fact a special case of a more general identity relating the chromatic polynomial of a graph G=(V,E) to sums over forests of G associated to some partition scheme in G.
- [156] arXiv:2407.04038 [pdf, html, other]
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Title: Levinson FunctionsComments: 14 pages 6 figuresSubjects: Number Theory (math.NT)
Starting from some of Norman Levinson's results, we construct interesting examples of functions $f(s)$ such that for $s=\frac12+it$, we have $Z(t)=2\Re\{\pi^{-\frac{s}{2}}\Gamma(s/2)f(s)\}$. For example one such function is \[\begin{aligned}{\mathcal R }_{-3}(s)=\frac12&\int_{0\swarrow1}\frac{x^{-s}e^{3\pi ix^2}}{e^{\pi i x}-e^{-\pi i x}}\,dx\\&+\frac{1}{2\sqrt{3}}\int_{0\swarrow1}\frac{x^{-s}e^{\frac{\pi i}{3}x^2}}{e^{\pi i x}-e^{-\pi i x}}\Bigl(e^{\frac{\pi i}{2}}+2e^{-\frac{\pi i}{6}}\cos(\tfrac{2\pi x}{3})\Bigr)\,dx.\end{aligned}\]
- [157] arXiv:2407.04042 [pdf, html, other]
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Title: A simplified directional KeRF algorithmComments: 10 figures, 14 pagesSubjects: Statistics Theory (math.ST)
Random forest methods belong to the class of non-parametric machine learning algorithms. They were first introduced in 2001 by Breiman and they perform with accuracy in high dimensional settings. In this article, we consider, a simplified kernel-based random forest algorithm called simplified directional KeRF (Kernel Random Forest). We establish the asymptotic equivalence between simplified directional KeRF and centered KeRF, with additional numerical experiments supporting our theoretical results.
- [158] arXiv:2407.04045 [pdf, html, other]
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Title: Convergence rates for the Trotter-Kato splittingComments: 49 pages, 8 figures, comments welcomeSubjects: Mathematical Physics (math-ph); Functional Analysis (math.FA); Numerical Analysis (math.NA); Quantum Physics (quant-ph)
We study convergence rates of the Trotter-Kato splitting $e^{A+L} = \lim_{n \to \infty} (e^{L/n} e^{A/n})^n$ in the strong operator topology. In the first part, we use complex interpolation theory to treat generators $L$ and $A$ of contraction semigroups on Banach spaces, with $L$ relatively $A$-bounded. In the second part, we study unitary dynamics on Hilbert spaces and develop a new technique based on the concept of energy constraints. Our results provide a complete picture of the convergence rates for the Trotter splitting for all common types of Schrödinger and Dirac operators, including singular, confining and magnetic vector potentials, as well as molecular many-body Hamiltonians in dimension $d=3$. Using the Brezis-Mironescu inequality, we derive convergence rates for the Schrödinger operator with $V(x)=\pm |x|^{-a}$ potential. In each case, our conditions are fully explicit.
- [159] arXiv:2407.04054 [pdf, html, other]
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Title: Covering Numbers of Some Irreducible Characters of the Symmetric GroupComments: 33 pagesSubjects: Representation Theory (math.RT); Combinatorics (math.CO)
The covering number of a non-linear character $\chi$ of a finite group $G$ is the least positive integer $k$ such that every irreducible character of $G$ occurs in $\chi^k$. We determine the covering numbers of irreducible characters of the symmetric group $S_n$ indexed by certain two-row partitions (and their conjugates), namely $(n-2,2)$ and $((n+1)/2, (n-1)/2)$ when $n$ is odd. We also determine the covering numbers of irreducible characters indexed by certain hook-partitions (and their conjugates), namely $(n-2,1^2)$, the almost self-conjugate hooks $(n/2+1, 1^{n/2-1})$ when $n$ is even, and the self-conjugate hooks $((n+1)/2, 1^{(n-1)/2})$ when $n$ is odd.
- [160] arXiv:2407.04059 [pdf, html, other]
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Title: Precise large deviations through a uniform Tauberian theoremComments: 25 pagesSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main advantage of this method is that it can be easily applied to cases that are beyond the reach of the techniques currently used in the literature, such as random walks with long-ranged memory kernels or randomly stopped sums where the random time is not concentrated around its expectation. The method reveals the role of the characteristic function when Cramér's condition is violated and provides a unified approach within regular variation.
- [161] arXiv:2407.04081 [pdf, html, other]
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Title: Coincident Peak Prediction for Capacity and Transmission Charge ReductionComments: 24 pages, 17 figures, 7 tablesSubjects: Optimization and Control (math.OC)
Meeting the ever-growing needs of the power grid requires constant infrastructure enhancement. There are two important aspects for a grid ability to ensure continuous and reliable electricity delivery to consumers: capacity, the maximum amount the system can handle, and transmission, the infrastructure necessary to deliver electricity across the network. These capacity and transmission costs are then allocated to the end-users according to the cost causation principle. These charges are computed based on the customer demand on coincident peak (CP) events, time intervals when the system-wide electric load is highest. We tackle the problem of predicting CP events based on actual load and forecast data on the load of different jurisdictions. In particular, we identify two main use cases depending on the availability of a forecast. Our approach generates scenarios and formulates Monte-Carlo estimators for predicting CP-day and exact CP-hour events. Finally, we backtest the prediction performance of strategies with adaptive threshold for the prediction task. This analysis enables us to derive practical implications for load curtailment through Battery Energy Storage System (BESS) solutions.
- [162] arXiv:2407.04090 [pdf, html, other]
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Title: Global regularity of semi-critical case of anisotropic quasi-geostrophic equations in Sobolev spacesSubjects: Analysis of PDEs (math.AP)
In this paper, we consider the following anisotropic quasi-geostrophic equations \begin{equation}\tag*{$(AQG)_{\alpha,\beta}$} \partial_t\theta+ u_\theta.\nabla\theta +\mu|\partial_1|^{2\alpha}\theta+\nu |\partial_2|^{2\beta}\theta=0,\quad u_\theta=\mathcal{R}^{\perp}\theta, \end{equation} where $\min\{\alpha,\beta\}=\frac{1}{2}$ et $\max\{\alpha,\beta\}\in \left(\frac{1}{2},1\right)$. This equation is a particular case of the equation introduced by Ye (2019) in \cite{YZ}. In this paper, we prove that for any initial data $\theta^0$ in the Sobolev space $H^{s}(\mathbb{R}^2)$, $s >1$, the equation $(AQG)_{\alpha,\beta}$ has a global solution $\theta$ in $C_b(\mathbb{R}^+,H^s(\mathbb{R}^2)).$
- [163] arXiv:2407.04107 [pdf, other]
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Title: Properties of Besov-Lorentz spaces and application to Navier-Stokes equationsJournal-ref: SCIENCE CHINA Mathematics (2024)Subjects: Analysis of PDEs (math.AP)
Inspired by Caffarelli-Kohn-Nirenberg, Fefferman and Lin, we try to investigate how to control the set of large value points for the strong solution of Navier-Stokes equations. Besov-Lorentz spaces have multiple indices which can reflect complex changes of the set of the large value points. Hence we consider some properties of Gauss flow, paraproduct flow and couple flow related to the Besov-Lorentz spaces. When dealing with Lorentz index, we use wavelets and maximum norm to describe the decay situation in the binary time ring and to define time-frequency microlocal maximum norm space. We use maximum operator, $\alpha$-triangle inequality and Hölder inequality etc to get accurate estimates. As an application, we get a global wellposedness result of the Navier-Stokes equations where the solution can reflect how the size of the set of large value points changes.
- [164] arXiv:2407.04122 [pdf, html, other]
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Title: Linear Partial Differential Equations in Module of Copolynomials of Several Variables over a Commutative RingSubjects: Analysis of PDEs (math.AP)
We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from the ring of polynomials $K[x_1,...,x_n]$ into the commutative ring $K$.
We prove an existence and uniqueness theorem for a linear differential equation of infinite order which can be considered as an algebraic version of the classical Malgrange-Ehrenpreis theorem for the existence of the fundamental solution of a linear differential operator with constant coefficients. We find the fundamental solutions of linear differential operators of infinite order and show that the unique solution of the corresponding inhomogeneous equation can be represented as a convolution of the fundamental solution of this operator and the right-hand side. We also prove the existence and uniqueness theorem of the Cauchy problem for some linear differential equations in the module of formal power series with copolynomial coefficients. - [165] arXiv:2407.04124 [pdf, html, other]
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Title: Helson matrices induced by measuresComments: 20 pagesSubjects: Functional Analysis (math.FA)
We introduce a family of Helson matrices induced by the Laplace transform of a class of regular positive Borel measures $\mu,$ not necessarily finite, on $(0, \infty)$ and discuss their boundedness, Schatten-class properties and scattering theory. We also briefly discuss the class of Helson matrices induced by signed measures.
- [166] arXiv:2407.04131 [pdf, html, other]
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Title: Ends of (singular) Ricci shrinkersComments: 42 pagesSubjects: Differential Geometry (math.DG)
We estimate the number of ends of smooth and singular Ricci shrinkers focussing first on general ends and later on asymptotically conical ones. In particular, we obtain a variety of applications to sequences of Ricci shrinkers converging in a weak pointed sense to a possibly singular limit Ricci shrinker, for instance no new conical end can form in the limit.
- [167] arXiv:2407.04137 [pdf, html, other]
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Title: Autopolar conic bodies and polyhedraSubjects: Metric Geometry (math.MG)
An antinorm is a concave analogue of a norm. In contrast to norms, antinorms are not defined on the entire space $R^d$ but on a cone $K\subset R^d$. They are applied in the matrix analysis, optimal control, and dynamical systems. Their level sets are called conic bodies and (in case of piecewise-linear antinorms) conic polyhedra. The basic facts and notions of the "concave analysis" of antinorms such as separation theorems, duality, polars, Minkowski functionals, etc., are similar to those from the standard convex analysis. There are, however, some significant differences. One of them is the existence of many self-dual objects. We prove that there are infinitely many families of autopolar conic bodies and polyhedra in the cone $K=R^d_+$. For $d=2$, this gives a complete classification of self-dual antinorms, while for $d\ge 3$, there are counterexamples.
- [168] arXiv:2407.04148 [pdf, html, other]
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Title: Dynamic problem of a power-law graded half-plane and an associated Carleman problem for two functionsComments: 24 pages, 7 figuresSubjects: Complex Variables (math.CV); Mathematical Physics (math-ph)
A steady state plane problem of an inhomogeneous half-plane subjected to a load running along the boundary at subsonic speed is analyzed. The Lame coefficients and the density of the half-plane are assumed to be power functions of depth. The model is different from the standard static model have been used in contact mechanics since the Sixties and originated from the 1964 Rostovtsev exact solution of the Flamant problem of a power-law graded half-plane. To solve the governing dynamic equations with variable coefficients written in terms of the displacements, we propose a method that, by means of the Fourier and Mellin transforms, maps the model problem to a Carleman boundary value problem for two meromorphic functions in a strip with two shifts or, equivalently, to a system of two difference equations of the second order with variable coefficients. By partial factorization the Carleman problem is recast as a system of four singular integral equations on a segment with a fixed singularity and highly oscillating coefficients. A numerical method for its solution is proposed and tested. Numerical results for the displacement and stress fields are presented and discussed.
- [169] arXiv:2407.04150 [pdf, html, other]
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Title: Spectral Methods for Matrix Product FactorizationComments: Comments are welcomeSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
A graph $G$ is factored into graphs $H$ and $K$ via a matrix product if there exist adjacency matrices $A$, $B$, and $C$ of $G$, $H$, and $K$, respectively, such that $A = BC$. In this paper, we study the spectral aspects of the matrix product of graphs, including regularity, bipartiteness, and connectivity. We show that if a graph $G$ is factored into a connected graph $H$ and a graph $K$ with no isolated vertices, then certain properties hold. If $H$ is non-bipartite, then $G$ is connected. If $H$ is bipartite and $G$ is not connected, then $K$ is a regular bipartite graph, and consequently, $n$ is even. Furthermore, we show that trees are not factorizable, which answers a question posed by Maghsoudi et al.
- [170] arXiv:2407.04154 [pdf, html, other]
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Title: Liouville theorems and universal estimates for superlinear elliptic problems without scale invarianceComments: 56 pagesSubjects: Analysis of PDEs (math.AP)
We give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant elliptic problems, including Lane-Emden and Schrödinger type systems. This applies to various classes of nonlinearities with regular variation and possibly different behaviors at $0$ and $\infty$. To this end, we adapt the method from [72] to elliptic systems, which relies on a generalized rescaling technique and on doubling arguments from [55].
This is in particular facilitated by new Liouville type theorems in the whole space and in a half-space, for elliptic problems without scale invariance, that we obtain. Our results apply to some non-cooperative systems, for which maximum principle based techniques such as moving planes do not apply. To prove these Liouville type theorems, we employ two methods, respectively based on Pohozaev-type identities combined with functional inequalities on the unit sphere, and on reduction to a scalar equation by proportionality of components.
In turn we will survey the existing methods for proving Liouville-type theorems for superlinear elliptic equations and systems, and list some of the typical existing results for (Sobolev subcritical) systems.
In the case of scalar equations, we also revisit the classical Gidas-Spruck integral Bernstein method, providing some improvements which turn out to be efficient for certain nonlinearities, and we next compare the performances of various methods on a benchmark example. - [171] arXiv:2407.04161 [pdf, html, other]
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Title: On the Compatibility of Constructive Predicative Mathematics with Weyl's Classical PredicativitySubjects: Logic (math.LO)
It is well known that most foundations for Bishop's constructive mathematics are incompatible with a classical predicative development of analysis as put forward by Weyl in his $\textit{Das Kontinuum}$. Here, we first recall how this incompatibility arises from the possibility, present in most constructive foundations, to define sets by quantifying over (the exponentiation of) functional relations. This possibility is not allowed in modern formulations of Weyl's logical system. Then, we argue how a possible way out is offered by foundations, such as the Minimalist Foundation, where exponentiation is limited to a primitive notion of function defined by $\lambda$-terms as in dependent type theory. The price to pay is to renounce the so-called rule of unique choice identifying functional relations with $\lambda$-terms, and to number-theoretic choice principles, characteristic of foundations aimed to formalize Bishop's constructive analysis. This restriction calls for a point-free constructive development of topology as advocated by P. Martin-Löf and G. Sambin with the introduction of Formal Topology. Hence, we conclude that the Minimalist Foundation promises to be a natural crossroads between Bishop's constructivism and Weyl's classical predicativity provided that a point-free reformulation of classical analysis is viable.
- [172] arXiv:2407.04167 [pdf, html, other]
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Title: Non-uniform dependence on periodic initial data for the two-component Fornberg-Whitham system in Besov spacesSubjects: Analysis of PDEs (math.AP)
This paper establishes non-uniform continuity of the data-to-solution map in the periodic case, for the two-component Fornberg-Whitham system in Besov spaces $B^s_{p,r}(\mathbb{T}) \times B^{s-1}_{p,r}(\mathbb{T})$ for $s> \max\{2+\frac{1}{p}, \frac{5}{2}\}$. In particular, when $p=2$ and $r=2$, this proves the non-uniform dependence on initial data for the system in Sobolev spaces $H^s(\mathbb{T})\times H^{s-1}(\mathbb{T})$ for $s> \frac{5}{2}$.
- [173] arXiv:2407.04176 [pdf, html, other]
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Title: A Carath\'eodory's Extension Theorem for Families Simpler Than an Algebras of SetsSubjects: Probability (math.PR)
The Carathéodory's Extension Theorem is a powerful tool that allows us to generate a measure, over a sigma-algebra, from a pre-measure defined over an algebra of sets. However, although this result reduces our work to define a measure by only needing to define a pre-measure, it is not always easy to define the latter. The problem occurs when taking the smallest algebra that contains a family of targeted sets, it can be very complicated to consistently define the value of the pre-measure over its finite union of these sets - a union that is an element of the algebra. Thus, our objective in this article is to reproduce an extension theorem, just like the Carathéodory's Extension Theorem, but in the context of probability measures and replacing the need for a probability pre-measure defined over an algebra for now a quasi-measure defined over a refinement. The gain, then, is that the \textit{manual elaboration} of a quasi-measure is simpler than the elaboration of a pre-measure, since a refinement is a simpler structure than an algebra.
- [174] arXiv:2407.04178 [pdf, html, other]
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Title: Monomial web basis for the SL(N) skein algebra of the twice punctured sphereComments: 41 pages, 19 figuresSubjects: Geometric Topology (math.GT); Combinatorics (math.CO); Quantum Algebra (math.QA); Representation Theory (math.RT)
For any non-zero complex number $q$, excluding finitely many roots of unity of small order, a linear basis for the $\mathrm{SL}(n)$ skein algebra of the twice punctured sphere is constructed. In particular, the skein algebra is a commutative polynomial algebra in $n-1$ generators, where each generator is represented by an explicit $\mathrm{SL}(n)$ web, without crossings, on the surface. This includes the case $q=1$, where the skein algebra is identified with the coordinate ring of the $\mathrm{SL}(n)$ character variety of the twice punctured sphere. The proof of both the spanning and linear independence properties of the basis depends on the so-called $\mathrm{SL}(n)$ quantum trace map, due originally to Bonahon-Wong in the case $n=2$. A consequence of the proof is that the polynomial algebra sits as a distinguished subalgebra of the Lê-Sikora $\mathrm{SL}(n)$ stated skein algebra of the annulus. We end by discussing the relationship with Fock-Goncharov duality.
- [175] arXiv:2407.04193 [pdf, html, other]
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Title: Combinatorial Constructions of Optimal Quaternary Additive CodesComments: This work was mainly completed in the summer of 2023, and here we add some new developments. Everyone is welcome to discuss issues related to additional code with the first authorSubjects: Information Theory (cs.IT)
This paper aims to construct optimal quaternary additive codes with non-integer dimensions. Firstly, we propose combinatorial constructions of quaternary additive constant-weight codes, alongside additive anticode construction. Subsequently, we propose generalized Construction X, which facilitates the construction of non-integer dimensional optimal additive codes from linear codes. Then, we construct ten classes of optimal quaternary non-integer dimensional additive codes through these two methods. As an application, we also determine the optimal additive $[n,3.5,n-t]_4$ codes for all $t$ with variable $n$, except for $t=6,7,12$.
- [176] arXiv:2407.04194 [pdf, other]
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Title: Using Synthetic Data to Regularize Maximum Likelihood EstimationComments: 94 pagesSubjects: Statistics Theory (math.ST)
To overcome challenges in fitting complex models with small samples, catalytic priors have recently been proposed to stabilize the inference by supplementing observed data with synthetic data generated from simpler models. Based on a catalytic prior, the Maximum A Posteriori (MAP) estimator is a regularized estimator that maximizes the weighted likelihood of the combined data. This estimator is straightforward to compute, and its numerical performance is superior or comparable to other likelihood-based estimators. In this paper, we study several theoretical aspects regarding the MAP estimator in generalized linear models, with a particular focus on logistic regression. We first prove that under mild conditions, the MAP estimator exists and is stable against the randomness in synthetic data. We then establish the consistency of the MAP estimator when the dimension of covariates diverges slower than the sample size. Furthermore, we utilize the convex Gaussian min-max theorem to characterize the asymptotic behavior of the MAP estimator as the dimension grows linearly with the sample size. These theoretical results clarify the role of the tuning parameters in a catalytic prior, and provide insights in practical applications. We provide numerical studies to confirm the effective approximation of our asymptotic theory in finite samples and to illustrate adjusting inference based on the theory.
- [177] arXiv:2407.04201 [pdf, other]
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Title: A General Maximum Principle for Progressive Optimal Control of Fully Coupled Forward-Backward Stochastic Systems with JumpsComments: 32 pagesSubjects: Optimization and Control (math.OC)
This paper is concerned with a general maximum principle for the fully coupled forward-backward stochastic optimal control problem with jumps, where the control domain is not necessarily convex, within the progressively measurable framework. It is worth noting that not only the control variable enters into all the coefficients, but also the jump size "$e$" . We first proposed that the solution $Z$ of BSDEP also contains the variable "$e$", which is different from previous articles and we provide an explanation in Remark 2.1.
- [178] arXiv:2407.04206 [pdf, html, other]
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Title: Computational Graph Representation of Equations System Constructors in Hierarchical Circuit SimulationZichao Long, Lin Li, Lei Han, Xianglong Meng, Chongjun Ding, Ruiyan Li, Wu Jiang, Fuchen Ding, Jiaqing Yue, Zhichao Li, Yisheng Hu, Ding Li, Heng LiaoSubjects: Numerical Analysis (math.NA); Computational Engineering, Finance, and Science (cs.CE)
Equations system constructors of hierarchical circuits play a central role in device modeling, nonlinear equations solving, and circuit design automation. However, existing constructors present limitations in applications to different extents. For example, the costs of developing and reusing device models -- especially coarse-grained equivalent models of circuit modules -- remain high while parameter sensitivity analysis is complex and inefficient. Inspired by differentiable programming and leveraging the ecosystem benefits of open-source software, we propose an equations system constructor using the computational graph representation, along with its JSON format netlist, to address these limitations. This representation allows for runtime dependencies between signals and subcircuit/device parameters. The proposed method streamlines the model development process and facilitates end-to-end computation of gradients of equations remainders with respect to parameters. This paper discusses in detail the overarching concept of hierarchical subcircuit/device decomposition and nested invocation by drawing parallels to functions in programming languages, and introduces rules for parameters passing and gradient propagation across hierarchical circuit modules. The presented numerical examples, including (1) an uncoupled CMOS model representation using "equivalent circuit decomposition+dynamic parameters" and (2) operational amplifier (OpAmp) auto device sizing, have demonstrated that the proposed method supports circuit simulation and design and particularly subcircuit modeling with improved efficiency, simplicity, and decoupling compared to existing techniques.
- [179] arXiv:2407.04226 [pdf, html, other]
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Title: Dense sets of natural numbers with unusually large least common multiplesComments: 10 pages, no figuresSubjects: Number Theory (math.NT)
We construct a set $A \subset \mathbf{N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{1}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \frac{1}{(\sum_{n \in A: n \leq x} \frac{1}{n})^2} \sum_{n,m \in A: n < m \leq x} \frac{1}{\operatorname{lcm}(n,m)} \asymp 1$$ for sufficiently large $x$. The exponent $\frac{1}{2}$ can replaced by any other positive constant, but the growth rate is otherwise optimal. This answers in the negative a question of Erdős and Graham.
- [180] arXiv:2407.04228 [pdf, html, other]
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Title: On patched completed homology and a conjecture of VenkateshSubjects: Number Theory (math.NT)
Let $F$ be a CM field and $\Pi$ a regular algebraic cuspidal cohomological representation of $\mathbf{G}=\operatorname{PGL}_2/F$. A conjecture of Venkatesh describes the structure of the contribution of $\Pi$ to the homology of the locally symmetric spaces associated to $\mathbf{G}$. We investigate this conjecture in the setting of $p$-adic homology with $p$ a totally split prime. Along the way, we elaborate on the relations between Venkatesh's conjecture and completed homology, the Taylor-Wiles method and the $p$-adic local Langlands correspondence. Our main result is a `big $R=T$' theorem in characteristic 0, from which we deduce a variant of the $p$-adic realisation of Venkatesh's conjecture, conditional on various natural conjectures and technical assumptions.
- [181] arXiv:2407.04234 [pdf, html, other]
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Title: Subinvariant metric functionals for nonexpansive mappingsSubjects: Functional Analysis (math.FA); Optimization and Control (math.OC)
We investigate the existence of subinvariant metric functionals for commuting families of nonexpansive mappings in noncompact subsets of Banach spaces. Our findings underscore the practicality of metric functionals when searching for fixed points of nonexpansive mappings. To demonstrate this, we additionally investigate subsets of Banach spaces that have only nontrivial metric functionals. We particularly show that in certain cases every metric functional has a unique minimizer; thus, subinvariance implies the existence of a fixed point.
- [182] arXiv:2407.04235 [pdf, html, other]
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Title: Novel Optimization Techniques for Parameter EstimationSubjects: Optimization and Control (math.OC); Quantitative Methods (q-bio.QM)
In this paper, we introduce a new optimization algorithm that is well suited for solving parameter estimation problems. We call our new method cubic regularized Newton with affine scaling (CRNAS). In contrast to so-called first-order methods which rely solely on the gradient of the objective function, our method utilizes the Hessian of the objective. As a result it is able to focus on points satisfying the second-order optimality conditions, as opposed to first-order methods that simply converge to critical points. This is an important feature in parameter estimation problems where the objective function is often non-convex and as a result there can be many critical points making it is near impossible to identify the global minimum. An important feature of parameter estimation in mathematical models of biological systems is that the parameters are constrained by either physical constraints or prior knowledge. We use an affine scaling approach to handle a wide class of constraints. We establish that CRNAS identifies a point satisfying $\epsilon$-approximate second-order optimality conditions within $O(\epsilon^{-3/2})$ iterations. Finally, we compare CRNAS with MATLAB's optimization solver fmincon on three different test problems. These test problems all feature mixtures of heterogeneous populations, a problem setting that CRNAS is particularly well-suited for. Our numerical simulations show CRNAS has favorable performance, performing comparable if not better than fmincon in accuracy and computational cost for most of our examples.
- [183] arXiv:2407.04246 [pdf, html, other]
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Title: Conformally covariant probabilities, operator product expansions, and logarithmic correlations in two-dimensional critical percolationComments: 46 pagesSubjects: Mathematical Physics (math-ph); Probability (math.PR)
The large-scale behavior of two-dimensional critical percolation is expected to be described by a conformal field theory (CFT). Moreover, this putative percolation CFT is believed to be of the logarithmic type, exhibiting logarithmic corrections to the most commonly encountered and studied behavior of CFT correlations.
It was recently shown by the first author [Cam24] that critical connection probabilities, when appropriately rescaled, have a well-defined and conformally covariant scaling limit and therefore behave like CFT correlation functions. While constructing a full-fledged percolation CFT is still an open problem, in this paper we prove various CFT features of the scaling limit of two-dimensional critical percolation.
We identify several connectivity events, including arm-events and the events that a vertex is pivotal or belongs to the percolation backbone, whose probabilities have conformally covariant scaling limits and can be interpreted as CFT correlation functions.
For some of the probabilities mentioned above, we prove asymptotic expansions that can be interpreted as CFT operator product expansions (OPEs) and provide rigorous versions of CFT fusion rules.
In some of the probabilities mentioned above, we identify logarithmic singularities, providing the first rigorous confirmation of similar predictions made in the physics literature and establishing the logarithmic nature of the putative percolation CFT.
The latter result is particularly interesting because, while logarithmic CFTs are more complex and less studied than ordinary CFTs, they have attracted considerable attention in recent years due to their role in the analysis of important physical models and phenomena, such as, besides percolation, the Wess-Zumino-Witten (WZW) model, the quantum Hall effect, disordered critical systems, self-avoiding polymers, and the Fortuin-Kasteleyn (FK) model. - [184] arXiv:2407.04253 [pdf, html, other]
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Title: Delay differential equations with periodic coefficients: a numerical insightComments: In proceedings of the MATRIX Mathematical Research Institute workshop "Delay Differential Equations and Their Applications"Subjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)
Simple form scalar differential equation with delay and non-linear negative periodic feedback is considered. The existence of slowly oscillating periodic solutions with the same period as the feedback coefficient is shown numerically within the admissible range for the periods.
- [185] arXiv:2407.04259 [pdf, html, other]
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Title: Robust Q-Learning for finite ambiguity setsSubjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Probability (math.PR)
In this paper we propose a novel $Q$-learning algorithm allowing to solve distributionally robust Markov decision problems for which the ambiguity set of probability measures can be chosen arbitrarily as long as it comprises only a finite amount of measures. Therefore, our approach goes beyond the well-studied cases involving ambiguity sets of balls around some reference measure with the distance to reference measure being measured with respect to the Wasserstein distance or the Kullback--Leibler divergence. Hence, our approach allows the applicant to create ambiguity sets better tailored to her needs and to solve the associated robust Markov decision problem via a $Q$-learning algorithm whose convergence is guaranteed by our main result. Moreover, we showcase in several numerical experiments the tractability of our approach.
- [186] arXiv:2407.04276 [pdf, html, other]
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Title: On a continued fraction algorithm in finite extensions of $\Q_p$ and its metrical theoryComments: 20 pagesSubjects: Number Theory (math.NT)
We develop a continued fraction algorithm in finite extensions of $\Q_p$ generalising certain algorithms in $\Q_p$, and prove the finiteness property for certain small degree extensions. We also discuss the metrical properties of the associated continued fraction maps for our algorithms using subsequence ergodic theory and moving averages.
- [187] arXiv:2407.04278 [pdf, html, other]
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Title: Maximality of correspondence representationsComments: 26 pagesSubjects: Operator Algebras (math.OA); Functional Analysis (math.FA)
In this paper, we fully characterize maximal representations of a C*-correspondence. This strengthens several earlier results. We demonstrate the criterion with diverse examples. We also describe the noncommutative Choquet boundary and provide additional counterexamples to Arveson's hyperrigidity conjecture following the counterexample recently found by the author and Adam Dor-On.
- [188] arXiv:2407.04288 [pdf, html, other]
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Title: Lower gradient estimates for viscosity solutions to first-order Hamilton--Jacobi equations depending on the unknown functionSubjects: Analysis of PDEs (math.AP)
In this paper, we derive the lower bounds for the gradients of viscosity solutions to the Hamilton--Jacobi equation, where the convex Hamiltonian depends on the unknown function. We obtain gradient estimates using two different methods. First, we utilize the equivalence between viscosity solutions and Barron--Jensen solutions to study the properties of the inf-convolution. Second, we examine the Lie equation to understand how initial gradients propagate along its solutions.
- [189] arXiv:2407.04290 [pdf, html, other]
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Title: Onsager-Machlup functional for stochastic differential equations with time-varying noiseComments: 17 pages, 4 figuresSubjects: Probability (math.PR)
This paper is devoted to studying the Onsager-Machlup functional for stochastic differential equations with time-varying noise of the {\alpha}-Hölder, 0<{\alpha}<1/4,
dXt =f(t,Xt)dt+g(t)dWt.
Our study focuses on scenarios where the diffusion coefficient g(t) exhibits temporal variability, starkly contrasting the conventional assumption of a constant diffusion coefficient in the existing literature. This variance brings some complexity to the analysis. Through this investigation, we derive the Onsager-Machlup functional, which acts as the Lagrangian for mapping the most probable transition path between metastable states in stochastic processes affected by time-varying noise. This is done by introducing new measurable norms and applying an appropriate version of the Girsanov transformation. To illustrate our theoretical advancements, we provide numerical simulations, including cases of a one-dimensional SDE and a fast-slow SDE system, which demonstrate the application to multiscale stochastic volatility models, thereby highlighting the significant impact of time-varying diffusion coefficients. - [190] arXiv:2407.04298 [pdf, html, other]
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Title: Curvature of higher direct images of sheaves of twisted holomorphic formsComments: 50 pagesSubjects: Complex Variables (math.CV); Algebraic Geometry (math.AG); Differential Geometry (math.DG)
This paper investigates the curvature properties of higher direct images $ R^qf_*\Omega_{X/S}^p(E)$, where $f: X\rightarrow S$ is a family of compact Kähler manifolds equipped with a hermitian vector bundle $E \rightarrow X$. We derive a general curvature formula and explore several special cases, including those where $p + q = n$, $q = 0$, and $p = n$, with $E$ being a line bundle. Furthermore, the paper examines the curvature in the context of fiberwise hermitian flat cases, families of Hermite-Einstein vector bundles, and applications to moduli spaces and Weil-Petersson metrics, providing some insight into their geometric and analytical properties.
- [191] arXiv:2407.04301 [pdf, html, other]
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Title: Limits of limit sets in rank-one symmetric spacesSubjects: Geometric Topology (math.GT)
We consider the question of continuity of limit sets for sequences of geometrically finite subgroups of isometry groups of rank-one symmetric spaces, and prove analogues of classical (Kleinian) theorems in this context. In particular we show that, assuming strong convergence of the sequence of subgroups, the limit sets vary continuously with respect to Hausdorff distance, and if the sequence is weakly type-preserving, the sequence of Cannon-Thurston maps also converges uniformly to a limiting Cannon-Thurston map. Our approach uses the theory of extended geometrically finite representations, developed recently by the second author.
- [192] arXiv:2407.04306 [pdf, other]
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Title: Numerical stabilization method by switching time-delayKaïs Ammari, Stéphane Gerbi (LAMA)Subjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
In this paper, we propose a new numerical strategy for the stabilization of evolution systems. The method is based on the methodology given by Ammari, Nicaise and Pignotti in "Stabilization by switching time-delay, Asymptot. Anal., 83 (2013), 263--283". This method is then implemented in 1D by suitable numerical approximation techniques. Numerical experiments complete this study to confirm the theoretical announced results.
- [193] arXiv:2407.04309 [pdf, other]
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Title: Indirect stabilization of semilinear coupled wave systemsSubjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)
In this paper, we study the indirect stabilization problem for a system of two coupled semilinear wave equations with internal damping in a bounded domain in $\mathbb{R}^3$. The nonlinearity is assumed to be subcritical, defocusing and analytic. Under geometric control condition on both coupling and damping regions, we establish the exponential energy decay rate.
- [194] arXiv:2407.04310 [pdf, html, other]
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Title: Connection matrices on the Siegel-Jacobi upper half space and extended Siegel-Jacobi upper half spaceComments: 21 pages, Latex, amsart, AMS fontsSubjects: Differential Geometry (math.DG)
The inverse of the metric matrices on the Siegel-Jacobi upper half space ${\mathcal{X}}^J_n$, invariant to the restricted real Jacobi group $G^J_n(\mathbb{R})_0$ and extended Siegel-Jacobi $\tilde{\mathcal{X}}^J_n$ upper half space, invariant to the action of the real Jacobi $G^J_n(\mathbb{R})$, are presented. The results are relevant for Berezin quantization of the manifolds ${\mathcal{X}}^J_ n$ and $\tilde{\mathcal{X}}^J_n$. Explicit calculations in the case $n=2$ are given.
- [195] arXiv:2407.04312 [pdf, other]
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Title: Moments approaches for asymptotic inverse problems of depolymerisation and fragmentation systemsMarie Doumic (MERGE)Journal-ref: Journ{\'e}es Equations aux D{\'e}riv{\'e}es Partielles, Jun 2023, Aussois (France), FranceSubjects: Analysis of PDEs (math.AP)
Shrinkage of large particles, either through depolymerisation (i.e. progressive shortening) or through fragmentation (breakage into smaller pieces) may be modelled by discrete equations, of Becker-D\''oring type, or by continuous ones. In this note, we review two kinds of inverse problems: the first is the estimation of the initial size-distribution from moments measurements in a depolymerising system, in collaboration with Philippe Moireau and inspired by experiments carried out by Human Rezaei's team; the second is the inference of fragmentation characteristics from size distribution samples, in collaboration with Miguel Escobedo and Magali Tournus, based on biological questions and experiments of Wei-Feng Xue's team.
- [196] arXiv:2407.04313 [pdf, html, other]
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Title: Poisson stability of solutions for stochastic evolution equations driven by fractional Brownian motionComments: 25 pages, 1 figures. arXiv admin note: substantial text overlap with arXiv:1702.02718 by other authorsSubjects: Dynamical Systems (math.DS); Analysis of PDEs (math.AP); Probability (math.PR)
In this paper, we study the problem of Poisson stability of solutions for stochastic semi-linear evolution equation driven by fractional Brownian motion $$\mathrm{d} X(t)= \left( AX(t) + f(t,X(t)) \right) \mathrm{d}t + g\left(t,X(t)\right)\mathrm{d}B^H_{Q}(t),$$ where $A$ is an exponentially stable linear operator acting on a separable Hilbert space $\mathbb{H}$, coefficients $f$ and $g$ are Poisson stable in time, and $B^H_Q (t)$ is a $Q$-cylindrical fBm with Hurst index $H$. First, we establish the existence and uniqueness of the solution for this equation. Then, we prove that under the condition where the functions $f$ and $g$ are sufficiently "small", the equation admits a solution that exhibits the same character of recurrence as $f$ and $g$. The discussion is further extended to the asymptotic stability of these Poisson stable solutions. Finally, we include an example to validate our results.
- [197] arXiv:2407.04314 [pdf, html, other]
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Title: Beale--Kato--Majda-type continuation criteria for Hall- and electron-magnetohydrodynamicsSubjects: Analysis of PDEs (math.AP)
We show that regular solutions to electron-MHD with resistivity can be continued as long as the time integral of the supremum of the current gradient remains finite. This dimensionless continuation criterion is analogous to the celebrated result of Beale--Kato--Majda for the incompressible Euler and Navier--Stokes equations. A similar continuation criterion, formulated in terms of the time integral of the supremum of the vorticity, velocity gradient and current gradient, is established for the Hall-MHD with resistivity as well.
- [198] arXiv:2407.04318 [pdf, other]
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Title: Germs for scalar conservation laws: the Hamilton-Jacobi equation point of viewSubjects: Analysis of PDEs (math.AP)
We prove that the entropy solution to a scalar conservation law posed on the real line with a flux that is discontinuous at one point (in the space variable) coincides with the derivative of the solution to a Hamilton-Jacobi (HJ) equation whose Hamiltonian is discontinuous. Flux functions (Hamiltonians) are not assumed to be convex in the state (gradient) variable. The proof consists in proving the convergence of two numerical schemes. We rely on the theory developed by B.~Andreianov, K.~H.~Karlsen and N.~H.~Risebro (\textit{Arch. Ration. Mech. Anal.}, 2011) for such scalar conservation laws and on the viscosity solution theory developed by the authors (\textit{arxiv}, 2023) for the corresponding HJ equation. This study allows us to characterise certain germs introduced in the AKR theory (namely maximal and complete ones) and relaxation operators introduced in the viscosity solution framework.
- [199] arXiv:2407.04320 [pdf, other]
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Title: Asymptotic Analysis of a bi-monomeric nonlinear Becker-D{\"o}ring systemMarie Doumic (MERGE), Klemens Fellner (University of Graz), Mathieu Mezache (MaIAGE), Juan J L VelázquezSubjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS)
To provide a mechanistic explanation of sustained {then} damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-Döring system has been proposed in \cite{DFMR}. When all reaction rates are constant, the equations are the following: \begin{align*}\frac{dv}{dt} & =-vw+v\sum_{j=2}^{\infty}c_{j}, \qquad \frac{dw}{dt} =vw-w\sum_{j=1}^{\infty}c_{j}, \\ \frac{dc_{j}}{dt} & =J_{j-1}-J_{j}\ \ ,\ \ j\geq1\ \ ,\ \ \ J_{j}=wc_{j}-vc_{j+1}\ \ ,\ \ j\geq1\ \ ,\ J_{0}=0, \end{align*} where $v$ and $w$ are two distinct unit species, and $c_i$ represents the concentration of clusters containing $i$ units.
We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system. - [200] arXiv:2407.04321 [pdf, other]
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Title: A coupling strategy for Brownian motions at fixed time on Carnot groups using Legendre expansionSubjects: Probability (math.PR)
We propose a new simple construction of a coupling at a fixed time of two sub-Riemannian Brownian motions on the Heisenberg group and on the free step 2 Carnot groups. The construction is based on a Legendre expansion of the standard Brownian motion and of the L{é}vy area. We deduce sharp estimates for the decay in total variation distance between the laws of the Brownian motions. Using a change of probability method, we also obtain the log-Harnack inequality, a Bismut type integration by part formula and reverse Poincaré inequalities for the associated semi-group.
- [201] arXiv:2407.04329 [pdf, html, other]
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Title: Actual problems of the approximation theory in metrics of discrete spaces on sets of summable periodic and almost periodic functionsComments: in Ukrainian languageSubjects: Classical Analysis and ODEs (math.CA); Numerical Analysis (math.NA)
This review paper highlights the main aspects of the development of research related to the solution of extreme problems in the theory of approximation in the spaces ${\mathcal S}^p$ and $B{\mathcal S}^p$ of periodic and almost periodic summable functions, respectively, where the $l_p$-norms of the sequences of Fourier coefficients are finite. In particular, the review contains the results known so far concerning the best, best $n$-term approximations and widths of classes of functions of one and many variables defined by means of $\psi$-derivatives and generalized moduli of smoothness in the spaces ${\mathcal S}^p$ and $B{\mathcal S}^p$. Particular attention is paid to the development of studies related to the derivation of direct and inverse approximation theorems in these spaces.
- [202] arXiv:2407.04341 [pdf, html, other]
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Title: Flat sub-Lorentzian structures on Martinet distributionSubjects: Optimization and Control (math.OC); Differential Geometry (math.DG); Metric Geometry (math.MG)
Two flat sub-Lorentzian problems on the Martinet distribution are studied. For the first one, the attainable set has a nontrivial intersection with the Martinet plane, but for the second one it does not. Attainable sets, optimal trajectories, sub-Lorentzian distances and spheres are described.
- [203] arXiv:2407.04347 [pdf, html, other]
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Title: On a nonlinear nonlocal reaction-diffusion system applied to image restorationComments: 28 pages,7 figuresSubjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
This paper deals with a novel nonlinear coupled nonlocal reaction-diffusion system proposed for image restoration, characterized by the advantages of preserving low gray level features and textures.The gray level indicator in the proposed model is regularized using a new method based on porous media type equations, which is suitable for recovering noisy blurred images. The well-posedness, regularity, and other properties of the model are investigated, addressing the lack of theoretical analysis in those existing similar types of models. Numerical experiments conducted on texture and satellite images demonstrate the effectiveness of the proposed model in denoising and deblurring tasks.
- [204] arXiv:2407.04348 [pdf, html, other]
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Title: Representation Structure of the $SL(2, \mathbb{C})$ Acting in the Hilbert Space of the Quantum Coulomb FieldSubjects: Mathematical Physics (math-ph)
We give a complete description of the representation of $SL(2,\mathbb{C})$ acting in the Hilbert space of the quantum Coulomb field and a constructive consistency proof of the axioms of the quantum theory of the Coulomb field.
- [205] arXiv:2407.04354 [pdf, html, other]
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Title: Fluid-Limits of Fragmented Limit-Order MarketsSubjects: Probability (math.PR); Mathematical Finance (q-fin.MF)
Maglaras, Moallemi, and Zheng (2021) have introduced a flexible queueing model for fragmented limit-order markets, whose fluid limit remains remarkably tractable. In the present study we prove that, in the limit of small and frequent orders, the discrete system indeed converges to the fluid limit, which is characterized by a system of coupled nonlinear ODEs with singular coefficients at the origin. Moreover, we establish that the fluid system is asymptotically stable for an arbitrary number of limit order books in that, over time, it converges to the stationary equilibrium state studied by Maglaras et al. (2021).
- [206] arXiv:2407.04357 [pdf, html, other]
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Title: Chernoff's product formula: Semigroup approximations with non-uniform time intervalsComments: 9 pagesSubjects: Mathematical Physics (math-ph); Functional Analysis (math.FA)
Often, when we consider the time evolution of a system, we resort to approximation: Instead of calculating the exact orbit, we divide the time interval in question into uniform segments. Chernoff's results in this direction provide us with a general approximation scheme. There are situations when we need to break the interval into uneven pieces. In this paper, we explore alternative conditions to the one found by Smolyanov et. al such that Chernoff's original result can be extended to unevenly distributed time intervals. Two applications concerning the foundations of quantum mechanics and the central limit theorem are presented.
- [207] arXiv:2407.04358 [pdf, html, other]
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Title: An Adaptive Stochastic Gradient Method with Non-negative Gauss-Newton StepsizesSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
We consider the problem of minimizing the average of a large number of smooth but possibly non-convex functions. In the context of most machine learning applications, each loss function is non-negative and thus can be expressed as the composition of a square and its real-valued square root. This reformulation allows us to apply the Gauss-Newton method, or the Levenberg-Marquardt method when adding a quadratic regularization. The resulting algorithm, while being computationally as efficient as the vanilla stochastic gradient method, is highly adaptive and can automatically warmup and decay the effective stepsize while tracking the non-negative loss landscape. We provide a tight convergence analysis, leveraging new techniques, in the stochastic convex and non-convex settings. In particular, in the convex case, the method does not require access to the gradient Lipshitz constant for convergence, and is guaranteed to never diverge. The convergence rates and empirical evaluations compare favorably to the classical (stochastic) gradient method as well as to several other adaptive methods.
- [208] arXiv:2407.04361 [pdf, html, other]
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Title: Crouzeix-Raviart elements on simplicial meshes in $d$ dimensionsComments: 33 pagesSubjects: Numerical Analysis (math.NA)
In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions.
Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $\left( d-1\right) $ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does \textbf{not} exist in higher space dimension and $k>1$. - [209] arXiv:2407.04364 [pdf, html, other]
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Title: Robust Multiscale Methods for Helmholtz equations in high contrast heterogeneous mediaSubjects: Numerical Analysis (math.NA)
In this paper, we provide the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) to solve Helmholtz equations in heterogeneous medium. This novel multiscale method is specifically designed to overcome problems related to pollution effect, high-contrast coefficients, and the loss of hermiticity of operators. We establish the inf-sup stability and give an a priori error estimate for this method under a number of established assumptions and resolution conditions. The theoretical results are validated by a set of numerical tests, which further show that the multiscale technique can effectively capture pertinent physical phenomena.
- [210] arXiv:2407.04367 [pdf, html, other]
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Title: Reconfiguration of Independent TransversalsSubjects: Combinatorics (math.CO); Data Structures and Algorithms (cs.DS)
Given integers $\Delta\ge 2$ and $t\ge 2\Delta$, suppose there is a graph of maximum degree $\Delta$ and a partition of its vertices into blocks of size at least $t$. By a seminal result of Haxell, there must be some independent set of the graph that is transversal to the blocks, a so-called independent transversal. We show that, if moreover $t\ge2\Delta+1$, then every independent transversal can be transformed within the space of independent transversals to any other through a sequence of one-vertex modifications, showing connectivity of the so-called reconfigurability graph of independent transversals.
This is sharp in that for $t=2\Delta$ (and $\Delta\ge 2$) the connectivity conclusion can fail. In this case we show furthermore that in an essential sense it can only fail for the disjoint union of copies of the complete bipartite graph $K_{\Delta,\Delta}$. This constitutes a qualitative strengthening of Haxell's theorem. - [211] arXiv:2407.04373 [pdf, html, other]
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Title: A spring pair method of finding saddle points using the minimum energy path as a compassComments: 9 pages, 7 figuresSubjects: Mathematical Physics (math-ph); Computational Physics (physics.comp-ph)
Finding index-1 saddle points is crucial for understanding phase transitions. In this work, we propose a simple yet efficient approach, the spring pair method (SPM), to accurately locate saddle points. Without requiring Hessian information, SPM evolves a single pair of spring-coupled particles on the potential energy surface. By cleverly designing complementary drifting and climbing dynamics based on gradient decomposition, the spring pair converges onto the minimum energy path (MEP) and spontaneously aligns its orientation with the MEP tangent, providing a reliable ascent direction for efficient convergence to saddle points. SPM fundamentally differs from traditional surface walking methods, which rely on the eigenvectors of Hessian that may deviate from the MEP tangent, potentially leading to convergence failure or undesired saddle points. The efficiency of SPM for finding saddle points is verified by ample examples, including high-dimensional Lennard-Jones cluster rearrangement and the Landau energy functional involving quasicrystal phase transitions.
- [212] arXiv:2407.04374 [pdf, html, other]
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Title: Recollements for graded gentle algebras from spherical band objectsComments: 25 pagesSubjects: Representation Theory (math.RT)
In this paper we study the localization of a derived category of a graded gentle algebra by a subcategory generated by a spherical band object. This object corresponds to a simple closed curve under the equivalence between the perfect derived category of the graded gentle algebra and the partially wrapped Fukaya category of the associated graded marked surface, as established by Haiden, Katzarkov and Kontsevich.
We describe this localization as a recollement that involves the derived category of a new graded algebra given by quiver and relations. This leads us to the introduction of the class of graded pinched gentle algebras, a generalization of graded gentle algebras. We then show that these algebras are in bijection with graded marked surfaces with conical singularities. Moreover, under this correspondence the localization process amounts to the contraction of the closed curve. - [213] arXiv:2407.04375 [pdf, html, other]
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Title: An isomorphism between models of graphic arrangementsComments: 19 pages, 2 figuresSubjects: Algebraic Topology (math.AT)
This paper presents a bridge between the theories of wonderful models associated with toric arrangements and wonderful models associated with hyperplane arrangements. In a previous work, the same authors noticed that the model of the toric arrangement of type $A_{n-1}$ is isomorphic to the one of the hyperplane arrangement of type $A_{n}$; it is natural to ask if there exist similar isomorphisms between other families of arrangements. The aim of this paper is to study one such family, namely the family of arrangements defined by graphs. The main result states that there is indeed an isomorphism between the model of a toric arrangement defined by a graph $\Gamma$ and the model of a hyperplane arrangement defined by the cone of $\Gamma$, provided that a suitable building set is chosen.
- [214] arXiv:2407.04378 [pdf, html, other]
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Title: On pro-cdh descent on derived schemesComments: 28 pagesSubjects: K-Theory and Homology (math.KT); Algebraic Geometry (math.AG)
We prove a `pro-cdh descent' result for suitably connective localizing invariants and the cotangent complex on arbitrary qcqs derived schemes. As an application, we deduce a generalised Weibel vanishing for negative $K$-groups of non-Noetherian schemes.
- [215] arXiv:2407.04380 [pdf, html, other]
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Title: Gaps in the complex Farey sequence of an imaginary quadratic number fieldComments: 20 pages, 6 figuresSubjects: Number Theory (math.NT); Dynamical Systems (math.DS)
Given an imaginary quadratic number field $K$ with ring of integers $\mathcal{O}_K$, we are interested in the asymptotic \emph{distance to nearest neighbour} (or \emph{gap}) statistic of complex Farey fractions $\frac{p}{q}$, with $p,q \in \mathcal{O}_K$ and $0<|q|\leq T$, as $T \to \infty$. Reformulating this problem in a homogeneous dynamical setting, we follow the approach of J. Marklof for real Farey fractions with several variables (2013) and adapt a joint equidistribution result in the real $3$-dimensional hyperbolic space of J. Parkkonen and F. Paulin (2023) to derive the existence of a probability measure describing this asymptotic gap statistic. We obtain an integral formula for the associated cumulative distribution function, and use geometric arguments to find an explicit estimate for its tail distribution in the cases of Gaussian and Eisenstein fractions.
- [216] arXiv:2407.04387 [pdf, html, other]
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Title: The rigorous derivation of Vlasov equations with local alignments from moderately interacting particle systemsSubjects: Analysis of PDEs (math.AP); Probability (math.PR)
In this paper, we present a rigorous derivation of the mean-field limit for a moderately interacting particle system in $\R^d$ $(d\geq 2)$. For stochastic initial data, we demonstrate that the solution to the interacting particle model, with an appropriately applied cut-off, converges in probabilistic sense to the solution of the characteristics of the regularized Vlasov models featuring local alignments and Newtonian potential. Notably, the cutoff parameter for the singular potential is selected to scale polynomially with the number of particles, representing an improvement over the logarithmic cut-off obtained in [38].
- [217] arXiv:2407.04388 [pdf, html, other]
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Title: On a problem of Nathanson on non-minimal additive complementsComments: comments are welcomed!Subjects: Number Theory (math.NT)
Let $C$ and $W$ be two sets of integers. If $C+W=\mathbb{Z}$, then $C$ is called an additive complement to $W$. We further call $C$ a minimal additive complement to $W$ if no proper subset of $C$ is an additive complement to $W$. Answering a problem of Nathanson in part, we give sufficient conditions of $W$ which has no minimal additive complements. Our result also extends the prior result of Chen and Yang.
- [218] arXiv:2407.04395 [pdf, html, other]
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Title: On A Potential Contact Analogue Of Kirby Move Of Type 1Comments: 8 pages, 6 figuresSubjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)
In this note, we explore the possibility of the existence of Kirby move of type 1 for contact surgery diagrams. In particular, we give the necessary conditions on a contact surgery diagram to become a potential candidate for contact Kirby move of type 1. We prove that no other contact integral surgery diagram satisfies those conditions except for contact $(+2)$-surgery on Legendrian unknot with Thruston--Bennequin number $-1$.
- [219] arXiv:2407.04399 [pdf, html, other]
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Title: Theoretical analysis of a finite-volume scheme for a stochastic Allen-Cahn problem with constraintComments: arXiv admin note: text overlap with arXiv:2304.02259Subjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
The aim of this contribution is to address the convergence study of a time and space approximation scheme for an Allen-Cahn problem with constraint and perturbed by a multiplicative noise of Itô type. The problem is set in a bounded domain of $\mathbb{R}^d$ (with $d=2$ or $3$) and homogeneous Neumann boundary conditions are considered. The employed strategy consists in building a numerical scheme on a regularized version à la Moreau-Yosida of the constrained problem, and passing to the limit simultaneously with respect to the regularization parameter and the time and space steps, denoted respectively by $\epsilon$, $\Delta t$ and $h$. Combining a semi-implicit Euler-Maruyama time discretization with a Two-Point Flux Approximation (TPFA) scheme for the spatial variable, one is able to prove, under the assumption $\Delta t=\mathcal{O}(\epsilon^{2+\theta})$ for a positive $\theta$, the convergence of such a $(\epsilon, \Delta t, h)$ scheme towards the unique weak solution of the initial problem, \textit{ a priori} strongly in $L^2(\Omega;L^2(0,T;L^2(\Lambda)))$ and \textit{a posteriori} also strongly in $L^{p}(0,T; L^2(\Omega\times \Lambda))$ for any finite $p\geq 1$.
- [220] arXiv:2407.04401 [pdf, html, other]
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Title: High-order WENO finite-difference methods for hyperbolic nonconservative systems of Partial Differential EquationsSubjects: Numerical Analysis (math.NA)
This work aims to extend the well-known high-order WENO finite-difference methods for systems of conservation laws to nonconservative hyperbolic systems. The main difficulty of these systems both from the theoretical and the numerical points of view comes from the fact that the definition of weak solution is not unique: according to the theory developed by Dal Maso, LeFloch, and Murat in 1995, it depends on the choice of a family of paths. A general strategy is proposed here in which WENO operators are not only used to reconstruct fluxes but also the nonconservative products of the system. Moreover, if a Roe linearization is available, the nonconservative products can be computed through matrix-vector operations instead of path-integrals. The methods are extended to problems with source terms and two different strategies are introduced to obtain well-balanced schemes. These numerical schemes will be then applied to the two-layer shallow water equations in one- and two- dimensions to obtain high-order methods that preserve water-at-rest steady states.
- [221] arXiv:2407.04409 [pdf, html, other]
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Title: The Fibonacci-Fubini and Lucas-Fubini numbersComments: 18 pages, 3 figures, 3 tablesSubjects: Combinatorics (math.CO); Number Theory (math.NT)
Based on the combinatorial interpretation of the ordered Bell numbers, which count all the ordered partitions of the set $[n]=\{1,2,\dots,n\}$, we introduce the Fibonacci partition as a Fibonacci permutation of its blocks. Then we define the Fibonacci-Fubini numbers that count the total number of Fibonacci partitions of $[n]$. We study the classical properties of this sequence (generating function, explicit and Dobiński-like formula, etc.), we give combinatorial interpretation, and we extensively examine the Fibonacci-Fubini arithmetic triangle. We give some associate linear recurrence sequences, where in some sequences the Stirling numbers of the first and second kinds appear as well.
- [222] arXiv:2407.04410 [pdf, html, other]
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Title: On the Existence of an Extremal Function for the Delsarte Extremal ProblemComments: 9 pagesSubjects: Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
In the general setting of a locally compact Abelian group $G$, the Delsarte extremal problem asks for the supremum of integrals over the collection of continuous positive definite functions $f: G \to \mathbb{R}$ satisfying $f(0) = 1$ and having $\mathrm{supp} f_{+} \subset \Omega$ for some measurable subset $\Omega$ of finite measure. In this paper, we consider the question of the existence of an extremal function for the Delsarte extremal problem. In particular, we show that there exists an extremal function for the Delsarte problem when $\Omega$ is closed, extending previously known existence results to a larger class of functions.
- [223] arXiv:2407.04412 [pdf, html, other]
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Title: Projectivity of good moduli spaces of vector bundles on stacky curvesSubjects: Algebraic Geometry (math.AG)
Moduli of vector bundles on stacky curves behave similarly to moduli of vector bundles on curves, except there are additional numerical invariants giving many different notions of stability. We apply the existence criterion for good moduli spaces of stacks to show that the moduli stack of semistable vector bundles on a stacky curve has a proper good moduli space. We moduli-theoretically prove that a natural determinantal line bundle on this moduli space is ample, thus proving this moduli space is projective. Our methods give effective bounds for when a power of this line bundle is basepoint-free. As a special case, we obtain new and effective constructions of moduli spaces of parabolic bundles.
- [224] arXiv:2407.04414 [pdf, html, other]
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Title: Local multiplicity for fractional linear equations with Hardy potentialsSubjects: Analysis of PDEs (math.AP)
We exhibit existence of non-trivial solutions of some fractional linear Schrödinger equations which are radial and vanish at the origin. This is in stark contrast to what happens in the local case. We also prove analogous results in the presence of a Hardy potential.
- [225] arXiv:2407.04415 [pdf, other]
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Title: Quantifying redundancies and synergies with measures of inequalitySubjects: Information Theory (cs.IT)
Inequality measures provide a valuable tool for the analysis, comparison, and optimization based on system models. This work studies the relation between attributes or features of an individual to understand how redundant, unique, and synergetic interactions between attributes construct inequality. For this purpose, we define a family of inequality measures (f-inequality) from f-divergences. Special cases of this family are, among others, the Pietra index and the Generalized Entropy index. We present a decomposition for any f-inequality with intuitive set-theoretic behavior that enables studying the dynamics between attributes. Moreover, we use the Atkinson index as an example to demonstrate how the decomposition can be transformed to measures beyond f-inequality. The presented decomposition provides practical insights for system analyses and complements subgroup decompositions. Additionally, the results present an interesting interpretation of Shapley values and demonstrate the close relation between decomposing measures of inequality and information.
- [226] arXiv:2407.04421 [pdf, html, other]
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Title: On the classificarion of 3-dimensional spherical Sasakian manifoldsJournal-ref: Izv. Math. 85 (2021), no. 3, 518-528Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)
In this article we consider spherical hypersurfaces in $\mathbb C^2$ with a fixed Reeb vector field as 3-dimensional Sasakian manifolds. We establish the correspondence between three different sets of parameters, namely, those arising from representing the Reeb vector field as an automorphism of the Heisenberg sphere, the parameters used in Stanton's description of rigid spheres, and the parameters arising from the rigid normal forms. We also geometrically describe the moduli space for rigid spheres, and provide geometric distinction between Stanton's hypersurfaces and those found by Ezhov and Schmalz. Finally, we determine the Sasakian automorphism groups of the rigid spheres and detect the homogeneous Sasakian manifolds among them.
- [227] arXiv:2407.04426 [pdf, html, other]
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Title: Nonlinear chaotic Vlasov equationsSubjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS)
In this article, we study nonlinear Vlasov equations with a smooth interaction kernel on a compact manifold without boundary where the geodesic flow exhibits strong chaotic behavior, known as the Anosov property. We show that, for small initial data with finite regularity and supported away from the null section, there exist global solutions to the nonlinear Vlasov equations which weakly converge to an equilibrium of the free transport equation, and whose potential strongly converges to zero, both with exponential speed. Central to our approach are microlocal anisotropic Sobolev spaces, originally developed for studying Pollicott-Ruelle resonances, that we further refine to deal with the geometry of the full cotangent bundle, which paves the way to the analysis of nonlinear Vlasov equations.
- [228] arXiv:2407.04428 [pdf, other]
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Title: FEM-BEM coupling for the high-frequency Helmholtz problemSubjects: Numerical Analysis (math.NA)
We present a wavenumber-explicit analysis of FEM-BEM coupling methods for time-harmonic Helmholtz problems proposed in arXiv:2004.03523 for conforming discretizations and in arXiv:2105.06173 for discontinuous Galerkin (DG) volume discretizations. We show that the conditions that $kh/p$ be sufficiently small and that $\log(k) / p$ be bounded imply quasi-optimality of both conforming and DG-method, where $k$ is the wavenumber, $h$ the mesh size, and $p$ the approximation order. The analysis relies on a $k$-explicit regularity theory for a three-field coupling formulation.
- [229] arXiv:2407.04429 [pdf, html, other]
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Title: A combinatorial formula for the coefficients of multidimensional resultantsComments: 36 pagesSubjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG); Combinatorics (math.CO)
The classical multidimensional resultant can be defined as the, suitably normalized, generator of a projective elimination ideal in the ring of universal coefficients. This is the approach via the so-called inertia forms or Trägheitsformen. Using clever substitutions, Mertens and Hurwitz gave a criterion, for recognizing such inertia forms, which amounts to a linear system for their numerical coefficients. In this article we explicitly solve this linear system. We do so by identifying a subset of the available equations which forms a unitriangular system. The key notion we use is that of transversal, i.e., a selection of a monomial term in each of the homogeneous polynomials at hand. We need two such transversals which are disjoint and extremal, in the sense that they relate to extremizers of a, possibly new, determinantal inequality for differences of two substochastic matrices. Thanks to this notion of extremal pair of transversals, we derive an explicit formula for the coefficients of general multidimensional resultants, as a sum of terms made of a sign times a product of multinomial coefficients, thereby explicitly showing they are integer-valued.
- [230] arXiv:2407.04436 [pdf, html, other]
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Title: A Tunneling Method for Nonlinear Multi-objective Optimization ProblemsComments: 12 Pages; 2 Figures; 1 TableSubjects: Optimization and Control (math.OC)
In this paper, a tunneling method is developed for nonlinear multi-objective optimization problems. The proposed method is free from any kind of priori chosen parameter or ordering information of objective functions. Using this method, global Pareto front can be generated for non-convex problems with more than one local front. An algorithm is developed using some ideas of single objective tunneling method. Convergence of this algorithm is justified under some mild assumptions. In addition to this, some numerical examples are included to justify the theoretical results.
- [231] arXiv:2407.04456 [pdf, html, other]
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Title: $\beta$-dimensional sharp maximal function and applicationsComments: 28 pagesSubjects: Functional Analysis (math.FA)
In this paper, we study $\beta$-dimensional sharp maximal operator defined as \begin{align*} \mathcal{M}^{\#} _\beta f(x) := \sup_{Q} \inf_{c \in \mathbb{R}} \chi_{Q}(x) \frac{1}{\ell(Q)^\beta} \int_Q |f-c| \; d \mathcal{H}^{\beta}_\infty, \end{align*} where the supremum is taken over all cubes in $\mathbb{R}^d$ with sides pararell to the coordinate axes, $\ell(Q)$ is the length side of $Q$ and $\mathcal{H}^{\beta}_\infty$ is the Hausdorff content. In particular, we prove Fefferman-Stein inequality for $\mathcal{M}^{\#} _\beta f$ by giving a good lambda estimate for $\beta$-dimensional sharp maximal operator in the context of Hausdorff content. Additionally, we prove the Muckenhoupt-Wheeden inequality in this framework by establishing a good lambda inequality of independent interest.
- [232] arXiv:2407.04492 [pdf, html, other]
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Title: On the number of sets with small sumsetComments: 36 pages (including appendix)Subjects: Combinatorics (math.CO); Number Theory (math.NT)
We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and $|A + A| \leq m$ is at most \[2^{o(s)}\binom{\frac{m+\beta}{2}}{s},\] where $\beta$ is the size of the largest subgroup of $G$ of size at most $\left(1+o(1)\right)m$. This bound is sharp for $\mathbb{Z}$ and many other groups. Our result improves the one of Campos and nearly bridges the remaining gap in a conjecture of Alon, Balogh, Morris, and Samotij.
We also explore the behaviour of uniformly chosen random sets $A \subseteq \{1,\ldots,n\}$ with $|A| = s$ and $|A + A| \leq m$. Under the same assumption that $m \ll s^2/(\log n)^2$, we show that with high probability there exists an arithmetic progression $P \subseteq \mathbb{Z}$ of size at most $m/2 + o(m)$ containing all but $o(s)$ elements of $A$. Analogous results are obtained for asymmetric sumsets, improving results by Campos, Coulson, Serra, and Wötzel. The main tool behind our proofs is a graph container theorem combined with a variant of an asymmetric hypergraph container theorem. - [233] arXiv:2407.04497 [pdf, html, other]
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Title: Constructing Noncatenary Quasi-Excellent PrecompletionsComments: 16 pages, 3 figures. Comments welcomeSubjects: Commutative Algebra (math.AC)
Let $T$ be a local (Noetherian) ring and let $Q_1$ and $Q_2$ be prime ideals of $T$. We find sufficient conditions for there to exist a quasi-excellent local subring $B$ of $T$ satisfying the following conditions: (1) the completion of $B$ at its maximal ideal is isomorphic to the completion of $T$ at its maximal ideal, (2) $B \cap Q_1 = B \cap Q_2$, (3) the set of prime ideals of $T/(Q_1 \cap Q_2)$ of positive height is the same as the set of prime ideals of $B/(B \cap Q_1)$ of positive height when viewed as partially ordered sets, and (4) for $i = 1$ and for $i = 2$, there is a coheight preserving bijection between the minimal prime ideals of $T_{Q_i}$ and the minimal prime ideals of $B_{B \cap Q_1}$. Intuitively, this means that $T$ contains a quasi-excellent local subring in which $Q_1$ and $Q_2$ are "glued together" and such that both the completion and desirable properties of the prime spectrum are preserved. We use this result to show that certain complete local rings are the completion of a quasi-excellent local ring whose prime spectrum, when viewed as a partially ordered set, contains interesting noncatenary finite subsets.
- [234] arXiv:2407.04498 [pdf, html, other]
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Title: Global dynamics for the generalized chemotaxis-Navier-Stokes system in $\mathbb{R}^3$Comments: 39 pagesSubjects: Analysis of PDEs (math.AP)
We consider the Cauchy problem of the three-dimensional generalized chemotaxis-Navier-Stokes system
\begin{eqnarray*}
\begin{cases} \partial_t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi(c)n \nabla c),\\ \partial_t c+u \cdot \nabla c=\Delta c-nf(c),\\ \partial_t u +u \cdot \nabla u+\nabla P=-(-\Delta)^\alpha u-n\nabla \phi,\\ \nabla \cdot u=0.
\end{cases} \end{eqnarray*} First, we study the time extensibility criteria of strong solutions, including the Prodi-Serrin type criterion ($\alpha>\frac{3}{4}$) and the Beir${\rm\tilde{a}}$o da Veiga type criterion $(\alpha>\frac{1}{2})$. Furthermore, with Lions' dissipation exponent $\alpha\geq \frac{5}{4}$, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and oxygen concentration. These results reflect the influence of the generalized dissipation for the solutions of the coupled chemotaxis-fluid equations. Finally, in the scenario of weaker dissipation ($\frac{3}{4}<\alpha<\frac{5}{4}$), we establish uniform regularity estimates for global strong solutions and further obtain optimal time-decay rates under the mild condition that the initial $L^2$ energy is small. To our knowledge, this is the first result concerning the global existence and large-time behavior of strong solutions for the three-dimensional chemotaxis-Navier-Stokes equations with possibly large oscillations. - [235] arXiv:2407.04509 [pdf, html, other]
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Title: Analysis of SIR Reaction diffusion system with constant birth and death rateSubjects: Analysis of PDEs (math.AP)
This is a truncation of the second year group project at Imperial college london. In this paper, we consider a semilinear reaction diffusion system of SIR model which involves the birth rate and the death rate. We first prove the non-negativity and global existence theorem to ensure that the model makes sense. We prove the uniform convergence of the infection-free solution and study an example that separable solutions can be computed. We also focus on the steady state solution, which we prove the non-uniqueness of the solution and investigate the regularity of the general solution. In the end we also introduce an interesting phenomenon, which is called the Turing instability caused by the diffusion in the model.
- [236] arXiv:2407.04517 [pdf, html, other]
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Title: An identity involving $h$-polynomials of poset associahedra and type B Narayana polynomialsComments: arXiv admin note: text overlap with arXiv:2310.02512Subjects: Combinatorics (math.CO)
For any finite connected poset $P$, Galashin introduced a simple convex $(|P|-2)$-dimensional polytope $\mathscr{A}(P)$ called the poset associahedron. Let $P$ be a poset with a proper autonomous subposet $S$ that is a chain of size $n$. For $1\leq i \leq n$, let $P_i$ be the poset obtained from $P$ by replacing $S$ by an antichain of size $i$. We show that the $h$-polynomial of $\mathscr{A}(P)$ can be written in terms of the $h$-polynomials of $\mathscr{A}(P_i)$ and type B Narayana polynomials. We then use the identity to deduce several identities involving Narayana polynomials, Eulerian polynomials, and stack-sorting preimages.
- [237] arXiv:2407.04521 [pdf, other]
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Title: Unified continuous-time q-learning for mean-field game and mean-field control problemsSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Computational Finance (q-fin.CP)
This paper studies the continuous-time q-learning in the mean-field jump-diffusion models from the representative agent's perspective. To overcome the challenge when the population distribution may not be directly observable, we introduce the integrated q-function in decoupled form (decoupled Iq-function) and establish its martingale characterization together with the value function, which provides a unified policy evaluation rule for both mean-field game (MFG) and mean-field control (MFC) problems. Moreover, depending on the task to solve the MFG or MFC problem, we can employ the decoupled Iq-function by different means to learn the mean-field equilibrium policy or the mean-field optimal policy respectively. As a result, we devise a unified q-learning algorithm for both MFG and MFC problems by utilizing all test policies stemming from the mean-field interactions. For several examples in the jump-diffusion setting, within and beyond the LQ framework, we can obtain the exact parameterization of the decoupled Iq-functions and the value functions, and illustrate our algorithm from the representative agent's perspective with satisfactory performance.
- [238] arXiv:2407.04524 [pdf, html, other]
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Title: Energy-stable parametric finite element approximations for regularized solid-state dewetting in strongly anisotropic materialsSubjects: Numerical Analysis (math.NA)
In this work, we aim to develop energy-stable parametric finite element approximations for a sharp-interface model with strong surface energy anisotropy, which is derived from the first variation of an energy functional composed of film/vapor interfacial energy, substrate energy, and regularized Willmore energy. By introducing two geometric relations, we innovatively establish an equivalent regularized sharp-interface model and further construct an energy-stable parametric finite element algorithm for this equivalent model. We provide a detailed proof of the energy stability of the numerical scheme, addressing a gap in the relevant theory. Additionally, we develop another structure-preserving parametric finite element scheme that can preserve both area conservation and energy stability. Finally, we present several numerical simulations to show accuracy and efficiency as well as some structure-preserving properties of the proposed numerical methods. More importantly, extensive numerical simulations reveal that our schemes provide better mesh quality and are more suitable for long-term computations.
- [239] arXiv:2407.04539 [pdf, html, other]
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Title: Nijenhuis geometry of parallel tensorsComments: 20 pagesSubjects: Differential Geometry (math.DG)
A tensor -- meaning here a tensor field $\Theta$ of any type $(p,q)$ on a manifold -- may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of integrability for differential $q$-forms, $q=0,1,2,n-1,n$ (in dimension $n$), vectors, bivectors, symmetric $(2,0)$ and $(0,2)$ tensors, as well as complex-diagonalizable and nilpotent tensors of type $(1,1)$. In most cases, integrability is equivalent to algebraic constancy of $\Theta$ coupled with the vanishing of one or more suitably defined Nijenhuis-type tensors, depending on $\Theta$ via a quasilinear first-order differential operator. For $(p,q)=(1,1)$, they include the ordinary Nijenhuis tensor.
- [240] arXiv:2407.04546 [pdf, html, other]
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Title: Monotone heteroclinic solutions to semilinear PDEs in cylinders and applicationsComments: 16 pagesSubjects: Analysis of PDEs (math.AP)
In this paper we show the existence of strictly monotone heteroclinic type solutions of semilinear elliptic equations in cylinders. The motivation of this construction is twofold: first, it implies the existence of an entire bounded solution of a semilinear equation without critical points which is not one-dimensional. Second, this gives an example of a bounded stationary solution for the 2D Euler equations without stagnation points which is not a shear flow, completing previous results of Hamel and Nadirashvili. The proof uses a minimization technique together with a truncation argument, and a limit procedure.
- [241] arXiv:2407.04554 [pdf, html, other]
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Title: A Ramanujan bound for Drinfeld modular formsComments: 20 pages, comments welcome!Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
In this paper, we prove a Lefschetz trace formula for Böckle-Pink crystals on tame Deligne-Mumford stacks of finite type over $\mathbb{F}_q$ and apply it to the crystal associated to the universal Drinfeld module. Combined with the Eichler-Shimura theory developed by Böckle, this leads to a trace formula for Hecke operators on Drinfeld modular forms. As a corollary, we deduce a Ramanujan bound on the traces of Hecke operators.
- [242] arXiv:2407.04555 [pdf, html, other]
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Title: Traces of Hecke operators on Drinfeld modular forms for $\mathbb{F}_q[T]$Comments: 30 pages, 4 tables, 1 figure. Comments welcome!Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case $A = \mathbb{F}_q[T]$. We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree 1 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level $\Gamma_0(T)$ into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenvalue has multiplicity less than $p$.
- [243] arXiv:2407.04556 [pdf, html, other]
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Title: On certain determinants and the square root of some determinants involving Legendre SymbolsComments: 20 pagesSubjects: Number Theory (math.NT)
Let $p>3$ be a prime and $(\frac{.}{p})$ be the Legendre symbol. For any integer $d$ with $p\nmid d$ and any positive integer $m$, Sun introduced the determinants $$T_m(d,p)=\det\left[(i^2+dj^2)^m\left(\frac{i^2+dj^2}{p}\right)\right]_{1\leqslant i,j \leqslant (p-1)/2},$$ and $$D_p^{(m)}= \det\left[(i^2-j^2)^m\left(\frac{i^2-j^2}{p}\right)\right]_{1\leq i,j\leq (p-1)/2} .$$ In this paper, we obtain some properties of $T_m (d,p)$ and $ \sqrt{D_p^{(m)}}$ for some $m$. We also confirm some related conjectures posed by Zhi-Wei Sun.
- [244] arXiv:2407.04562 [pdf, html, other]
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Title: An SDE Perspective on Stochastic Inertial Gradient Dynamics with Time-Dependent Viscosity and Geometric DampingComments: 27 pages. arXiv admin note: text overlap with arXiv:2403.16775Subjects: Optimization and Control (math.OC)
Our approach is part of the close link between continuous dissipative dynamical systems and optimization algorithms. We aim to solve convex minimization problems by means of stochastic inertial differential equations which are driven by the gradient of the objective function. This will provide a general mathematical framework for analyzing fast optimization algorithms with stochastic gradient input. Our study is a natural extension of our previous work devoted to the first-order in time stochastic steepest descent. Our goal is to develop these results further by considering second-order stochastic differential equations in time, incorporating a viscous time-dependent damping and a Hessian-driven damping. To develop this program, we rely on stochastic Lyapunov analysis. Assuming a square-integrability condition on the diffusion term times a function dependant on the viscous damping, and that the Hessian-driven damping is a positive constant, our first main result shows that almost surely, there is convergence of the values, and states fast convergence of the values in expectation. Besides, in the case where the Hessian-driven damping is zero, we conclude with the fast convergence of the values in expectation and in almost sure sense, we also managed to prove almost sure weak convergence of the trajectory. We provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex and strongly convex case.
- [245] arXiv:2407.04569 [pdf, html, other]
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Title: Pencils of plane cubics with one base pointComments: 18 pages, 9 figures. Comments welcome!Subjects: Algebraic Geometry (math.AG)
We study pencils of plane cubics with only one base point and general member smooth, giving a complete classification. Under the additional hypothesis that all members are irreducible, we prove that there exists a unique non-isotrivial pencil with these properties up to projective transformation. We compare our construction with the classical approaches given by Gattazzo, Beauville and Miranda-Persson.
- [246] arXiv:2407.04570 [pdf, html, other]
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Title: A Degree Bound for Planar FunctionsSubjects: Combinatorics (math.CO)
Using Stickelberger's theorem on Gauss sums, we show that if $F$ is a planar function on a finite field $\mathbb{F}_q$, then for all non-zero functions $G : \mathbb{F}_q \to \mathbb{F}_q$, we have \begin{equation*} \mathrm{deg } \ G \circ F - \mathrm{deg } \ G \le \frac{n(p-1)}{2}\,, \end{equation*} where $q = p^n$ with $p$ a prime and $n$ a positive integer, and $\mathrm{deg } \ F$ is the algebraic degree of $F$, i.e., the degree of the corresponding multivariate polynomial over $\mathbb{F}_p$. This bound leads to a simpler proof of the classification of planar polynomials over $\mathbb{F}_p$ and planar monomials over $\mathbb{F}_{p^2}$. As a new result, using the same degree bound, we complete the classification of planar monomials for all $n = 2^k$ with $p>5$ and $k$ a non-negative integer. Finally, we state a conjecture on the sum of the base-$p$ digits of integers modulo $q-1$ that implies the complete classification of planar monomials over finite fields of characteristic $p>5$.
- [247] arXiv:2407.04571 [pdf, html, other]
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Title: Ultra-weak least squares discretizations for unique continuation and Cauchy problemsSubjects: Numerical Analysis (math.NA)
In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial- and test-spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the $L_2$-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of $C^1$-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound, or in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.
- [248] arXiv:2407.04582 [pdf, html, other]
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Title: Converse Techniques for Identification via ChannelsSubjects: Information Theory (cs.IT)
There is a growing interest in models that extend beyond Shannon's classical transmission scheme, renowned for its channel capacity formula $C$. One such promising direction is message identification via channels, introduced by Ahlswede and Dueck. Unlike in Shannon's classical model, where the receiver aims to determine which message was sent from a set of $M$ messages, message identification focuses solely on discerning whether a specific message $m$ was transmitted. The encoder can operate deterministically or through randomization, with substantial advantages observed particularly in the latter approach. While Shannon's model allows transmission of $M = 2^{nC}$ messages, Ahlswede and Dueck's model facilitates the identification of $M = 2^{2^{nC}}$ messages, exhibiting a double exponential growth in block length. In their seminal paper, Ahlswede and Dueck established the achievability and introduced a "soft" converse bound. Subsequent works have further refined this, culminating in a strong converse bound, applicable under specific conditions. Watanabe's contributions have notably enhanced the applicability of the converse bound. The aim of this survey is multifaceted: to grasp the formalism and proof techniques outlined in the aforementioned works, analyze Watanabe's converse, trace the evolution from earlier converses to Watanabe's, emphasizing key similarities and differences that underpin the enhancements. Furthermore, we explore the converse proof for message identification with feedback, also pioneered by Ahlswede and Dueck. By elucidating how their approaches were inspired by preceding proofs, we provide a comprehensive overview. This overview paper seeks to offer readers insights into diverse converse techniques for message identification, with a focal point on the seminal works of Hayashi, Watanabe, and, in the context of feedback, Ahlswede and Dueck.
- [249] arXiv:2407.04584 [pdf, other]
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Title: On partial derivatives of some summatory functionsSubjects: Number Theory (math.NT)
Let $f$ be a real arithmetic function and let $g:[1,\infty[\to{\mathbb R}$ be a smooth function. We describe two emblematic instances in which saddle-point estimates may be used to evaluate the frequency, on the set of integers $n\leqslant x$, of the event $\{f(n)\leqslant g(n)\}$ from those relevant to the event $\{f(n)\leqslant y\}$. The first example revisits Dickman's historical contribution to the theory of friable integers. The second is concerned with the distribution of the squarefree kernel of an integer.
- [250] arXiv:2407.04586 [pdf, html, other]
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Title: Global $C^{1,\beta}$ and $W^{2, p}$ regularity for some singular Monge-Amp\`ere equationsSubjects: Analysis of PDEs (math.AP)
We establish global $C^{1,\beta}$ and $W^{2, p}$ regularity for singular Monge-Ampère equations of the form \[\det D^2 u \sim \text{dist}^{-\alpha}(\cdot,\partial\Omega),\quad \alpha\in (0, 1),\] under suitable conditions on the boundary data and domains. Our results imply that the convex Aleksandrov solution to the singular Monge-Ampère equation \[\det D^2 u=|u|^{-\alpha}\quad \text{in}\quad\Omega,\quad u=0\quad \text{in}\quad \partial\Omega, \quad \alpha\in (0, 1),\] where $\Omega$ is a $C^3$, bounded, and uniformly convex domain, is globally $C^{1,\beta}$ and belongs to $W^{2, p}$ for all $p<1/\alpha$.
- [251] arXiv:2407.04588 [pdf, html, other]
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Title: Weak coloring numbers of minor-closed graph classesComments: 52 pages, 17 figuresSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
We study the growth rate of weak coloring numbers of graphs excluding a fixed graph as a minor. Van den Heuvel et al. (European J. of Combinatorics, 2017) showed that for a fixed graph $X$, the maximum $r$-th weak coloring number of $X$-minor-free graphs is polynomial in $r$. We determine this polynomial up to a factor of $\mathcal{O}(r \log r)$. Moreover, we tie the exponent of the polynomial to a structural property of $X$, namely, $2$-treedepth. As a result, for a fixed graph $X$ and an $X$-minor-free graph $G$, we show that $\mathrm{wcol}_r(G)= \mathcal{O}(r^{\mathrm{td}(X)-1}\mathrm{log}\ r)$, which improves on the bound $\mathrm{wcol}_r(G) = \mathcal{O}(r^{g(\mathrm{td}(X))})$ given by Dujmović et al. (SODA, 2024), where $g$ is an exponential function. In the case of planar graphs of bounded treewidth, we show that the maximum $r$-th weak coloring number is in $\mathcal{O}(r^2\mathrm{log}\ r$), which is best possible.
- [252] arXiv:2407.04602 [pdf, html, other]
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Title: Multiple stage stochastic linear programming with multiple objectives: flexible decision makingSubjects: Optimization and Control (math.OC); Probability (math.PR)
Optimization problems with random data have a wide range of applications. A typical feature of many such problems is that some variables have to be optimized before certain random coefficients have been realized and for other variables it is sufficient to decide on them afterwards. This leads to a multiple stage decision process. To optimize the variables in the first of two subsequent stages the stochastic problem is transformed into a deterministic program, called the (deterministic equivalent of the) recourse problem. In case of stochastic linear programs with finitely distributed random data this non-stochastic substitute is just a linear program. In the same way a multiple objective linear program is obtained if the original problem has multiple objective functions. In the first of the two stages, a decision maker usually would chose a feasible point out of the set of all Pareto-optimal points. This choice however has consequences to later stage decisions. We claim that the decision process in the earlier of the two stages is not fully transparent if a classical multi-objective decision process is applied: in addition to the original objectives of the problem a decision maker may have a preference for largest possible flexibility in later stage decisions. This additional objective is taken into account if the recourse problem in case of multiple objectives is taken to be a polyhedral convex set optimization problem instead of a multi-objective linear program only. We also discuss several surrogate problems to the recourse problem such as the wait-and-see problem and the expected valued problem for the multi-objective case. The new approach based on set optimization is illustrated by an example, the multi-objective newsvendor problem.
- [253] arXiv:2407.04608 [pdf, html, other]
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Title: A Multi-Player Potential Game Approach for Sensor Network Localization with Noisy MeasurementsSubjects: Optimization and Control (math.OC); Computer Science and Game Theory (cs.GT); Multiagent Systems (cs.MA)
Sensor network localization (SNL) is a challenging problem due to its inherent non-convexity and the effects of noise in inter-node ranging measurements and anchor node position. We formulate a non-convex SNL problem as a multi-player non-convex potential game and investigate the existence and uniqueness of a Nash equilibrium (NE) in both the ideal setting without measurement noise and the practical setting with measurement noise. We first show that the NE exists and is unique in the noiseless case, and corresponds to the precise network localization. Then, we study the SNL for the case with errors affecting the anchor node position and the inter-node distance measurements. Specifically, we establish that in case these errors are sufficiently small, the NE exists and is unique. It is shown that the NE is an approximate solution to the SNL problem, and that the position errors can be quantified accordingly. Based on these findings, we apply the results to case studies involving only inter-node distance measurement errors and only anchor position information inaccuracies.
- [254] arXiv:2407.04611 [pdf, html, other]
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Title: Unexpected phenomena in a one dimensional elliptic equation with a singular first order divergence termSubjects: Analysis of PDEs (math.AP)
We study existence of a weak solution for one-dimensional problems as \begin{equation}\label{intro}\tag{1} \begin{cases} \displaystyle -\frac{d}{dx}\left(a(x) \frac{d u}{dx}\right) = - \frac{d \phi (u) }{dx}- \frac{d g(x) }{dx}& \text{in}\;(0,L),\\
u(0)=u(L)=0\,, & \end{cases} \end{equation} where $a$ is a positive bounded function, $g\in L^2(0,L)$, and $\phi:\mathbb{R}\mapsto \mathbb{R}\cup \{+\infty\}$ is continuous as a function with values in $\mathbb{R}\cup \{+\infty\}$. Some relevant qualitative and quantitative facts concerning such problems and their solutions are described. In particular a precise characterization of the behaviour of suitable approximating solution is provided. Of particular (and independent) interest is the study of an associated ODE for which, we prove existence, uniqueness and comparison results. As a consequence of our arguments, a delicate stability result as well a quite unexpected multiplicity result is shown for problems as in \eqref{intro}. - [255] arXiv:2407.04612 [pdf, html, other]
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Title: Quasinormal modes for the Kerr black holeComments: 77 pages, 4 figuresSubjects: Analysis of PDEs (math.AP); General Relativity and Quantum Cosmology (gr-qc); Spectral Theory (math.SP)
We provide a rigorous definition of quasinormal modes for the Kerr black hole. They are obtained as the discrete set of poles of the meromorphically continued cutoff resolvent. The construction combines the method of complex scaling near asymptotically flat infinity with microlocal methods near the black hole horizon. We study the distribution of quasinormal modes in both the high and low energy regimes. We establish the existence of a high energy spectral gap and exclude the accumulation of quasinormal modes at zero energy.
- [256] arXiv:2407.04623 [pdf, html, other]
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Title: Balanced metrics and Gauduchon cone of locally conformally Kahler manifoldsComments: 17 pages, version 1.0Subjects: Differential Geometry (math.DG)
A complex Hermitian $n$-manifold $(M,I, \omega)$ is called locally conformally Kahler (LCK) if $d\omega=\theta\wedge\omega$, where $\theta$ is a closed 1-form, balanced if $\omega^{n-1}$ is closed, and SKT if $dId\omega=0$. We conjecture that any compact complex manifold admitting two of these three types of Hermitian forms (balanced, SKT, LCK) also admits a Kahler metric, and prove partial results towards this conjecture. We conjecture that the (1,1)-form $-d(I\theta)$ is Bott--Chern homologous to a positive (1,1)-current. This conjecture implies that $(M,I)$ does not admit a balanced Hermitian metric. We verify this conjecture for all known classes of LCK manifolds.
- [257] arXiv:2407.04634 [pdf, html, other]
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Title: A randomized small-block Lanczos method for large-scale null space computationsSubjects: Numerical Analysis (math.NA)
Computing the null space of a large sparse matrix $A$ is a challenging computational problem, especially if the nullity -- the dimension of the null space -- is large. When using a block Lanczos method for this purpose, conventional wisdom suggests to use a block size $d$ that is not smaller than the nullity. In this work, we show how randomness can be utilized to allow for smaller $d$ without sacrificing convergence or reliability. Even $d = 1$, corresponding to the standard single-vector Lanczos method, becomes a safe choice. This is achieved by using a small random diagonal perturbation, which moves the zero eigenvalues of $A^{\mathsf{T}} A$ away from each other, and a random initial guess. We analyze the effect of the perturbation on the attainable quality of the null space and derive convergence results that establish robust convergence for $d=1$. As demonstrated by our numerical experiments, a smaller block size combined with restarting and partial reorthogonalization results in reduced memory requirements and computational effort. It also allows for the incremental computation of the null space, without requiring a priori knowledge of the nullity.
- [258] arXiv:2407.04635 [pdf, html, other]
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Title: Hyperbolicity of the sub-Riemannian affine-additive groupComments: 17 pagesSubjects: Metric Geometry (math.MG); Differential Geometry (math.DG)
We consider the affine-additive group as a metric measure space with a canonical left-invariant measure and a left-invariant sub-Riemannian metric. We prove that this metric measure space is locally 4-Ahlfors regular and it is hyperbolic, meaning that it has a non-vanishing 4-capacity at infinity. This implies that the affine-additive group is not quasiconformally equivalent to the Heisenberg group or to the roto-translation group in contrast to the fact that both of these groups are globally contactomorphic to the affine-additive group. Moreover, each quasiregular map, from the Heisenberg group to the affine-additive group must be constant.
- [259] arXiv:2407.04637 [pdf, html, other]
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Title: Deep sections of the hypercubeComments: 46 pagesSubjects: Metric Geometry (math.MG); Combinatorics (math.CO); Functional Analysis (math.FA)
Consider a non-negative number $t$ and a hyperplane $H$ of $\mathbb{R}^d$ whose distance to the center of the hypercube $[0,1]^d$ is $t$. If $t$ is equal to $0$ and $H$ is orthogonal to a diagonal of $[0,1]^d$, it is known that the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ is a strictly increasing function of $d$ when $d$ is at least $3$. The study of the monotonicity of this volume is extended for $t$ up to above $1/2$ and, when $d$ is large enough, for every non-negative $t$. In particular, a range for $t$ is identified such that this volume is a strictly decreasing function of $d$ over the positive integers. The local extremality of the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ when $H$ is orthogonal to a diagonal of either $[0,1]^d$ or a lower dimensional face is also determined for the same values of $t$. It is shown for instance that when $t$ is above an explicit constant and $d$ is large enough, this volume is always strictly locally maximal when $H$ is orthogonal to a diagonal of $[0,1]^d$. A precise estimate for the convergence rate of the Eulerian numbers to their limit Gaussian behavior is provided along the way.
- [260] arXiv:2407.04640 [pdf, html, other]
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Title: Global Spectral Gap in Bosonic Molecular HamiltoniansComments: 35 pages, 2 figuresSubjects: Mathematical Physics (math-ph); Spectral Theory (math.SP)
We consider a neutral bosonic molecule in the Born-Oppenheimer approximation without spin and assume the physically obvious assertion that a neutral molecule prefers to break into smaller neutral clusters. We prove the existence of a global in space/uniform spectral gap between the ground state and first excited state energies. To do so, we improve upon previous results using a different tool, the time-independent Feshbach-Schur map.
- [261] arXiv:2407.04642 [pdf, html, other]
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Title: Some determinants involving binary formsComments: 13 pagesSubjects: Number Theory (math.NT)
In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This confirms a previous conjecture of the second author.
- [262] arXiv:2407.04644 [pdf, html, other]
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Title: Spectrality and monoidsSubjects: Rings and Algebras (math.RA); General Topology (math.GN)
We prove that the set of proper ideals of a monoid endowed with coarse lower topology is a spectral space.
- [263] arXiv:2407.04645 [pdf, html, other]
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Title: Small Hankel operator induced by measurable symbol acting on weighted Bergman spacesComments: arXiv admin note: text overlap with arXiv:2207.01086Subjects: Complex Variables (math.CV)
The boundedness of the small Hankel operator $h^\omega_{f}(g)=\overline{P_\omega}(fg)$ induced by a measurable symbol $f$ and the Bergman projection $P_\omega$ associated to a radial weight $\omega$ acting from the weighted Bergman space $A^p_\omega$ to its conjugate analytic counterpart $\overline{A^p_\omega}$ is characterized on the range $1<p<\infty$ when $\omega$ belongs to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. On the way to the proof a sharp integral estimate for certain modified Bergman kernels is obtained.
- [264] arXiv:2407.04646 [pdf, html, other]
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Title: Strongly consistent low-dissipation WENO schemes for finite elementsSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.
- [265] arXiv:2407.04654 [pdf, html, other]
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Title: Metastability of the contact process on slowly evolving scale-free networksComments: 88 pages, 6 tables 8 figuresSubjects: Probability (math.PR)
We investigate the contact process on scale-free networks evolving by a stationary dynamics whereby each vertex independently updates its connections with a rate depending on its power. This rate can be slowed down or speeded up by virtue of decreasing or increasing a parameter $\eta$, with $\eta\downarrow-\infty$ approaching the static and $\eta\uparrow\infty$ the mean-field case. We identify the regimes of slow, fast and ultra-fast extinction of the contact process. Slow extinction occurs in the form of metastability, when the contact process maintains a certain density of infected states for a time exponential in the network size. In our main result we identify the metastability exponents, which describe the decay of metastable densities as the infection rate goes to zero, in dependence on $\eta$ and the power-law exponent $\tau$. While the fast evolution cases have been treated in a companion paper, Jacob, Linker, Mörters (2019), the present paper looks at the significantly more difficult cases of slow network evolution. We describe various effects, like degradation, regeneration and depletion, which lead to a rich picture featuring numerous first-order phase transitions for the metastable exponents. To capture these effects in our upper bounds we develop a new martingale based proof technique combining a local and global analysis of the process.
- [266] arXiv:2407.04658 [pdf, other]
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Title: Thermodynamic Formalism for a family of cellular automata and duality with the shiftSubjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph); Probability (math.PR); Cellular Automata and Lattice Gases (nlin.CG)
We will consider a family of cellular automata $\Phi: \{1,2,...,r\}^\mathbb{N}\circlearrowright$ that are not of algebraic type. Our first goal is to determine conditions that result in the identification of probabilities that are at the same time $\sigma$-invariant and $\Phi$-invariant, where $\sigma$ is the full shift. Via the use of versions of the Ruelle operator $\mathcal{L}_{A,\sigma}$ and $\mathcal{L}_{B,\Phi}$ we will show that there is an abundant set of measures with this property; they will be equilibrium probabilities for different Lispchitz potentials $A,B$ and for the corresponding dynamics $\sigma$ and $\Phi$. Via the use of a version of the involution kernel $W$ for a $(\sigma,\Phi)$-mixed skew product $\hat{\Phi}: \{1,2,...,r\}^\mathbb{Z}\circlearrowright$, given $A$ one can determine $B$, in such way that the integral kernel $e^W$ produce a duality between eigenprobabilities $\rho_A$ for $(\mathcal{L}_{A,\sigma})^*$ and eigenfunctions $\psi_B$ for $\mathcal{L}_{B,\Phi}$. In another direction, considering the non-mixed extension $\hat{\Phi}_n : \{1,2,...,r\}^\mathbb{Z}\circlearrowright$ of $\Phi$, given a Lispchitz potential $\hat{A} : \{1,2,...,r\}^\mathbb{Z}\to \mathbb{R}$, we can identify a Lipschitz potential $A:\{1,2,...,r\}^\mathbb{N} \to \mathbb{R} $, in such away that relates the variational problem of $\hat{\Phi}_n$-Topological Pressure for $\hat{A}$ with the $\Phi$-Topological Pressure for $A$. We also present a version of Livsic's Theorem. Whether or not $\Phi$ (or $\hat{\Phi})$ can eventually be conjugated with another shift of finite type is irrelevant in our context.
- [267] arXiv:2407.04665 [pdf, html, other]
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Title: Lower spaces of multiplicative latticesComments: arXiv admin note: text overlap with arXiv:2308.06395Subjects: Rings and Algebras (math.RA)
We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower spaces, encompassing lower separation axioms and compactness. We characterize lower spaces that exhibit sobriety. Introducing the concept of strongly disconnected spaces, we establish a correlation between strongly disconnected lower spaces and the presence of nontrivial idempotent elements in the corresponding multiplicative lattices. Additionally, we provide a sufficient condition for a lower space to be connected. We prove that the lower space of proper elements is a spectral space, and we further explore continuous maps between lower spaces.
- [268] arXiv:2407.04683 [pdf, html, other]
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Title: Efficient Betti Matching Enables Topology-Aware 3D Segmentation via Persistent HomologySubjects: Algebraic Topology (math.AT); Computer Vision and Pattern Recognition (cs.CV)
In this work, we propose an efficient algorithm for the calculation of the Betti matching, which can be used as a loss function to train topology aware segmentation networks. Betti matching loss builds on techniques from topological data analysis, specifically persistent homology. A major challenge is the computational cost of computing persistence barcodes. In response to this challenge, we propose a new, highly optimized implementation of Betti matching, implemented in C++ together with a python interface, which achieves significant speedups compared to the state-of-the-art implementation Cubical Ripser. We use Betti matching 3D to train segmentation networks with the Betti matching loss and demonstrate improved topological correctness of predicted segmentations across several datasets. The source code is available at this https URL.
- [269] arXiv:2407.04696 [pdf, html, other]
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Title: Corks for exotic diffeomorphismsComments: 32 pages, 8 figuresSubjects: Geometric Topology (math.GT)
We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabilizing with one copy of $S^2 \times S^2$, is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold. For those that require more than one copy of $S^2 \times S^2$, we prove that the diffeomorphism can be isotoped to one that is supported in a submanifold homotopy equivalent to a wedge of 2-spheres, with null-homotopic inclusion map. We investigate the implications of these results by applying them to known exotic diffeomorphisms.
New submissions for Monday, 8 July 2024 (showing 269 of 269 entries )
- [270] arXiv:2401.07092 (cross-list from physics.class-ph) [pdf, html, other]
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Title: Flutter instability in solids and structures, with a view on biomechanics and metamaterialsDavide Bigoni, Francesco Dal Corso, Oleg N. Kirillov, Diego Misseroni, Giovanni Noselli, Andrea PiccolroazComments: 29 pages, 9 figuresJournal-ref: Proc. R. Soc. A. 479: 20230523 (2023)Subjects: Classical Physics (physics.class-ph); Materials Science (cond-mat.mtrl-sci); Dynamical Systems (math.DS); Instrumentation and Detectors (physics.ins-det)
The phenomenon of oscillatory instability called \lq flutter' was observed in aeroelasticity and rotor dynamics about a century ago. Driven by a series of applications involving nonconservative elasticity theory at different physical scales, ranging from nanomechanics to the mechanics of large space structures and including biomechanical problems of motility and growth, research on flutter is experiencing a new renaissance. A review is presented of the most notable applications and recent advances in fundamentals, both theoretical and experimental aspects, of flutter instability and Hopf bifurcation. Open problems, research gaps, and new perspectives for investigations are indicated.
- [271] arXiv:2407.01658 (cross-list from quant-ph) [pdf, html, other]
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Title: Fault-tolerant noise guessing decoding of quantum random codesSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT)
This work addresses the open question of implementing fault-tolerant QRLCs with feasible computational overhead. We present a new decoder for quantum random linear codes (QRLCs) capable of dealing with imperfect decoding operations. A first approach, introduced by Cruz et al., only considered channel errors, and perfect gates at the decoder. Here, we analyze the fault-tolerant characteristics of QRLCs with a new noise-guessing decoding technique, when considering preparation, measurement, and gate errors in the syndrome extraction procedure, while also accounting for error degeneracy. Our findings indicate a threshold error rate ($\pth$) of approximately $\pnum$ in the asymptotic limit, while considering realistic noise levels in the mentioned physical procedures.
- [272] arXiv:2407.03375 (cross-list from physics.soc-ph) [pdf, html, other]
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Title: Asymptotic and stability analysis of kinetic models for opinion formation on networks: an Allen-Cahn approachSubjects: Physics and Society (physics.soc-ph); Mathematical Physics (math-ph)
We present the analysis of the stationary equilibria and their stability in case of an opinion formation process in presence of binary opposite opinions evolving according to majority-like rules on social networks. The starting point is a kinetic Boltzmann-type model derived from microscopic interactions rules for the opinion exchange among individuals holding a certain degree of connectivity. The key idea is to derive from the kinetic model an Allen-Cahn type equation for the fraction of individuals holding one of the two opinions. The latter can be studied by means of a linear stability analysis and by exploiting integral operator analysis. While this is true for ternary interactions, for binary interactions the derived equation of interest is a linear scattering equation, that can be studied by means of General Relative Entropy tools and integral operators.
- [273] arXiv:2407.03378 (cross-list from quant-ph) [pdf, html, other]
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Title: Theory of Complex Particle without Extra DimensionsComments: 27 pagesSubjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Complex particle is a kind of bilocal particle having unexpected symmetry, which was proposed by the authour. In the present paper, we show that critical dimension of the complex particle in Minkowski spacetime is $D = 4$, while $D = 2, 4$ or $6$ are permitted in Euclid spacetime. The origin of the restriction to the dimension is the existence of tertiary constraint in the canonical theory, quantization of which leads to an eigenvalue equation having single-valued and bounded solutions only in particular dimension of spacetime. The derivation is based on a detailed analysis of Laplace-Beltrami operator on $S^{1,D-2}$ or $S^{D-1}$.
- [274] arXiv:2407.03395 (cross-list from gr-qc) [pdf, other]
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Title: Fluctuations and Correlations in Causal Set TheoryComments: 34 pagesSubjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We study the statistical fluctuations (such as the variance) of causal set quantities, with particular focus on the causal set action. To facilitate calculating such fluctuations, we develop tools to account for correlations between causal intervals with different cardinalities. We present a convenient decomposition of the fluctuations of the causal set action into contributions that depend on different kinds of correlations. This decomposition can be used in causal sets approximated by any spacetime manifold $\mathcal M$. Our work paves the way for investigating a number of interesting discreteness effects, such as certain aspects of the Everpresent $\Lambda$ cosmological model.
- [275] arXiv:2407.03454 (cross-list from cs.NE) [pdf, html, other]
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Title: Decomposition of Difficulties in Complex Optimization Problems Using a Bilevel ApproachComments: 9 pagesSubjects: Neural and Evolutionary Computing (cs.NE); Optimization and Control (math.OC)
Practical optimization problems may contain different kinds of difficulties that are often not tractable if one relies on a particular optimization method. Different optimization approaches offer different strengths that are good at tackling one or more difficulty in an optimization problem. For instance, evolutionary algorithms have a niche in handling complexities like discontinuity, non-differentiability, discreteness and non-convexity. However, evolutionary algorithms may get computationally expensive for mathematically well behaved problems with large number of variables for which classical mathematical programming approaches are better suited. In this paper, we demonstrate a decomposition strategy that allows us to synergistically apply two complementary approaches at the same time on a complex optimization problem. Evolutionary algorithms are useful in this context as their flexibility makes pairing with other solution approaches easy. The decomposition idea is a special case of bilevel optimization that separates the difficulties into two levels and assigns different approaches at each level that is better equipped at handling them. We demonstrate the benefits of the proposed decomposition idea on a wide range of test problems.
- [276] arXiv:2407.03574 (cross-list from stat.ML) [pdf, html, other]
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Title: An Axiomatic Definition of Hierarchical ClusteringSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
In this paper, we take an axiomatic approach to defining a population hierarchical clustering for piecewise constant densities, and in a similar manner to Lebesgue integration, extend this definition to more general densities. When the density satisfies some mild conditions, e.g., when it has connected support, is continuous, and vanishes only at infinity, or when the connected components of the density satisfy these conditions, our axiomatic definition results in Hartigan's definition of cluster tree.
- [277] arXiv:2407.03616 (cross-list from stat.ME) [pdf, other]
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Title: When can weak latent factors be statistically inferred?Subjects: Methodology (stat.ME); Econometrics (econ.EM); Statistics Theory (math.ST); Statistical Finance (q-fin.ST); Machine Learning (stat.ML)
This article establishes a new and comprehensive estimation and inference theory for principal component analysis (PCA) under the weak factor model that allow for cross-sectional dependent idiosyncratic components under nearly minimal the factor strength relative to the noise level or signal-to-noise ratio. Our theory is applicable regardless of the relative growth rate between the cross-sectional dimension $N$ and temporal dimension $T$. This more realistic assumption and noticeable result requires completely new technical device, as the commonly-used leave-one-out trick is no longer applicable to the case with cross-sectional dependence. Another notable advancement of our theory is on PCA inference $ - $ for example, under the regime where $N\asymp T$, we show that the asymptotic normality for the PCA-based estimator holds as long as the signal-to-noise ratio (SNR) grows faster than a polynomial rate of $\log N$. This finding significantly surpasses prior work that required a polynomial rate of $N$. Our theory is entirely non-asymptotic, offering finite-sample characterizations for both the estimation error and the uncertainty level of statistical inference. A notable technical innovation is our closed-form first-order approximation of PCA-based estimator, which paves the way for various statistical tests. Furthermore, we apply our theories to design easy-to-implement statistics for validating whether given factors fall in the linear spans of unknown latent factors, testing structural breaks in the factor loadings for an individual unit, checking whether two units have the same risk exposures, and constructing confidence intervals for systematic risks. Our empirical studies uncover insightful correlations between our test results and economic cycles.
- [278] arXiv:2407.03628 (cross-list from eess.SP) [pdf, html, other]
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Title: A Bistatic Sensing System in Space-Air-Ground Integrated NetworksSubjects: Signal Processing (eess.SP); Numerical Analysis (math.NA); Optimization and Control (math.OC)
Sensing is anticipated to have wider extensions in communication systems with the boom of non-terrestrial networks (NTNs) during the past years. In this paper, we study a bistatic sensing system by maximizing the signal-to-interference-plus-noise ration (SINR) from the target aircraft in the space-air-ground integrated network (SAGIN). We formulate a joint optimization problem for the transmit beamforming of low-earth orbit (LEO) satellite and the receive filtering of ground base station. To tackle this problem, we decompose the original problem into two sub-problems and use the alternating optimization to solve them iteratively. Using techniques of fractional programming and generalized Rayleigh quotient, the closed-form solution for each sub-problem is returned. Simulation results show that the proposed algorithm has good convergence performance.Moreover, the optimization of receive filtering dominates the optimality, especially when the satellite altitude becomes higher, which provides valuable network design insights.
- [279] arXiv:2407.03673 (cross-list from quant-ph) [pdf, other]
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Title: Hybrid Quantum-Classical Machine Learning with String DiagramsComments: 12 + 6 pagesSubjects: Quantum Physics (quant-ph); Category Theory (math.CT)
Central to near-term quantum machine learning is the use of hybrid quantum-classical algorithms. This paper develops a formal framework for describing these algorithms in terms of string diagrams: a key step towards integrating these hybrid algorithms into existing work using string diagrams for machine learning and differentiable programming. A notable feature of our string diagrams is the use of functor boxes, which correspond to a quantum-classical interfaces. The functor used is a lax monoidal functor embedding the quantum systems into classical, and the lax monoidality imposes restrictions on the string diagrams when extracting classical data from quantum systems via measurement. In this way, our framework provides initial steps toward a denotational semantics for hybrid quantum machine learning algorithms that captures important features of quantum-classical interactions.
- [280] arXiv:2407.03756 (cross-list from cs.NI) [pdf, other]
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Title: Perfect simulation of Markovian load balancing queueing networks in equilibriumCarl Graham (CMAP, ASCII)Subjects: Networking and Internet Architecture (cs.NI); Probability (math.PR)
We define a wide class of Markovian load balancing networks of identical single-server infinite-buffer queues. These networks may implement classic parallel server or work stealing load balancing policies, and may be asymmetric, for instance due to topological constraints. The invariant laws are usually not known even up to normalizing constant. We provide three perfect simulation algorithms enabling Monte Carlo estimation of quantities of interest in equilibrium. The state space is infinite, and the algorithms use a dominating process provided by the network with uniform routing, in a coupling preserving a preorder which is related to the increasing convex order. It constitutes an order up to permutation of the coordinates, strictly weaker than the product order. The use of a preorder is novel in this context. The first algorithm is in direct time and uses Palm theory and acceptance rejection. Its duration is finite, a.s., but has infinite expectation. The two other algorithms use dominated coupling from the past; one achieves coalescence by simulating the dominating process into the past until it reaches the empty state, the other, valid for exchangeable policies, is a back-off sandwiching method. Their durations have some exponential moments.
- [281] arXiv:2407.03826 (cross-list from cs.CE) [pdf, html, other]
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Title: Treatment of near-incompressibility and volumetric locking in higher order material point methodsJournal-ref: Computer Methods in Applied Mechanics and Engineering 395 (2022) 114985Subjects: Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
We propose a novel projection method to treat near-incompressibility and volumetric locking in small- and large-deformation elasticity and plasticity within the context of higher order material point methods. The material point method is well known to exhibit volumetric locking due to the presence of large numbers of material points per element that are used to decrease the quadrature error. Although there has been considerable research on the treatment of near-incompressibility in the traditional material point method, the issue has not been studied in depth for higher order material point methods. Using the Bbar and Fbar methods as our point of departure we develop an appropriate projection technique for material point methods that use higher order shape functions for the background discretization. The approach is based on the projection of the dilatational part of the appropriate strain rate measure onto a lower dimensional approximation space, according to the traditional Bbar and Fbar techniques, but tailored to the material point method. The presented numerical examples exhibit reduced stress oscillations and are free of volumetric locking and hourglassing phenomena.
- [282] arXiv:2407.03903 (cross-list from gr-qc) [pdf, html, other]
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Title: Characteristic Gluing with $\Lambda$: III. High-differentiability nonlinear gluingComments: 37 pages, 2 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Differential Geometry (math.DG)
We prove a nonlinear characteristic $C^k$-gluing theorem for vacuum gravitational fields in Bondi gauge for a class of characteristic hypersurfaces near static vacuum $n$-dimensional backgrounds, $n\ge 3$, with any finite $k$, with cosmological constant $ \Lambda \in \mathbb{R}$, near Birmingham-Kottler backgrounds. This generalises the $C^2$-gluing of Aretakis, Czimek and Rodnianski, carried-out near light cones in four-dimensional Minkowski spacetime.
- [283] arXiv:2407.03904 (cross-list from physics.soc-ph) [pdf, html, other]
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Title: Asymmetric Iterated Prisoner's Dilemma on BA Scale-Free NetworkComments: 25 pages, 17 figures, 35 referencesSubjects: Physics and Society (physics.soc-ph); Statistics Theory (math.ST)
In real-world scenarios, individuals often cooperate for mutual benefit. However, differences in wealth can lead to varying outcomes for similar actions. In complex social networks, individuals' choices are also influenced by their neighbors. To explore the evolution of strategies in realistic settings, we conducted repeated asymmetric prisoners dilemma experiments on a weighted BA scale-free network. Our analysis highlighted how the four components of memory-one strategies affect win rates, found two special strategies in the evolutionary process, and increased the cooperation levels among individuals. These findings offer practical insights for addressing real-world problems.
- [284] arXiv:2407.03924 (cross-list from cs.CE) [pdf, html, other]
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Title: TwinLab: a framework for data-efficient training of non-intrusive reduced-order models for digital twinsComments: Accepted version of the revised manuscript published in Engineering ComputationsSubjects: Computational Engineering, Finance, and Science (cs.CE); Artificial Intelligence (cs.AI); Systems and Control (eess.SY); Dynamical Systems (math.DS)
Purpose: Simulation-based digital twins represent an effort to provide high-accuracy real-time insights into operational physical processes. However, the computation time of many multi-physical simulation models is far from real-time. It might even exceed sensible time frames to produce sufficient data for training data-driven reduced-order models. This study presents TwinLab, a framework for data-efficient, yet accurate training of neural-ODE type reduced-order models with only two data sets. Design/methodology/approach: Correlations between test errors of reduced-order models and distinct features of corresponding training data are investigated. Having found the single best data sets for training, a second data set is sought with the help of similarity and error measures to enrich the training process effectively. Findings: Adding a suitable second training data set in the training process reduces the test error by up to 49% compared to the best base reduced-order model trained only with one data set. Such a second training data set should at least yield a good reduced-order model on its own and exhibit higher levels of dissimilarity to the base training data set regarding the respective excitation signal. Moreover, the base reduced-order model should have elevated test errors on the second data set. The relative error of the time series ranges from 0.18% to 0.49%. Prediction speed-ups of up to a factor of 36,000 are observed. Originality: The proposed computational framework facilitates the automated, data-efficient extraction of non-intrusive reduced-order models for digital twins from existing simulation models, independent of the simulation software.
- [285] arXiv:2407.03933 (cross-list from cond-mat.stat-mech) [pdf, html, other]
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Title: Composition of q-entropies and hyperbolic orthogonalityComments: 13 pages. No figures. Standard LaTeX2eSubjects: Statistical Mechanics (cond-mat.stat-mech); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Chaotic Dynamics (nlin.CD)
We point out that the q-entropy composition for independent events has exactly the same form as the Pythagorean theorem in hyperbolic geometry. We justify the formal relation of hyperbolic geometry with the q-entropy through the use of the $\kappa$-entropy, which is directly related to the hyperboloid model of hyperbolic space. We comment on the relation between orthogonality in this form of the Pythagorean theorem and the independence of the probability distributions appearing in the q-entropy composition through the use of the Dvoretzky-Rogers lemma.
- [286] arXiv:2407.03960 (cross-list from q-fin.MF) [pdf, html, other]
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Title: The second-order Esscher martingale densities for continuous-time market modelsSubjects: Mathematical Finance (q-fin.MF); Probability (math.PR)
In this paper, we introduce the second-order Esscher pricing notion for continuous-time models. Depending whether the stock price $S$ or its logarithm is the main driving noise/shock in the Esscher definition, we obtained two classes of second-order Esscher densities called linear class and exponential class respectively. Using the semimartingale characteristics to parametrize $S$, we characterize the second-order Esscher densities (exponential and linear) using pointwise equations. The role of the second order concept is highlighted in many manners and the relationship between the two classes is singled out for the one-dimensional case. Furthermore, when $S$ is a compound Poisson model, we show how both classes are related to the Delbaen-Haenzendonck's risk-neutral measure. Afterwards, we restrict our model $S$ to follow the jump-diffusion model, for simplicity only, and address the bounds of the stochastic Esscher pricing intervals. In particular, no matter what is the Esscher class, we prove that both bounds (upper and lower) are solutions to the same linear backward stochastic differential equation (BSDE hereafter for short) but with two different constraints. This shows that BSDEs with constraints appear also in a setting beyond the classical cases of constraints on gain-processes or constraints on portfolios. We prove that our resulting constrained BSDEs have solutions in our framework for a large class of claims' payoffs including any bounded claim, in contrast to the literature, and we single out the monotonic sequence of BSDEs that ``naturally" approximate it as well.
- [287] arXiv:2407.03973 (cross-list from quant-ph) [pdf, html, other]
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Title: Logical Operators and Fold-Transversal Gates of Bivariate Bicycle CodesComments: 21 pages, comments welcomeSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT)
Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware. The question of how to implement logical gates fault-tolerantly for these codes is still open. We present new examples of high-rate bivariate bicycle (BB) codes with enhanced symmetry properties. These codes feature explicit nice bases of logical operators (similar to toric codes) and support fold-transversal Clifford gates without overhead. As examples, we construct $[[98,6,12]]$ and $[[162, 8, 12]]$ BB codes which admit interesting fault-tolerant Clifford gates. Our work also lays the mathematical foundations for explicit bases of logical operators and fold-transversal gates in quantum two-block and group algebra codes, which might be of independent interest.
- [288] arXiv:2407.03977 (cross-list from q-bio.PE) [pdf, html, other]
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Title: Statistics for Phylogenetic Trees in the Presence of StickinessComments: 37 pages, 16 figuresSubjects: Populations and Evolution (q-bio.PE); Statistics Theory (math.ST)
Samples of phylogenetic trees arise in a variety of evolutionary and biomedical applications, and the Fréchet mean in Billera-Holmes-Vogtmann tree space is a summary tree shown to have advantages over other mean or consensus trees. However, use of the Fréchet mean raises computational and statistical issues which we explore in this paper. The Fréchet sample mean is known often to contain fewer internal edges than the trees in the sample, and in this circumstance calculating the mean by iterative schemes can be problematic due to slow convergence. We present new methods for identifying edges which must lie in the Fréchet sample mean and apply these to a data set of gene trees relating organisms from the apicomplexa which cause a variety of parasitic infections. When a sample of trees contains a significant level of heterogeneity in the branching patterns, or topologies, displayed by the trees then the Fréchet mean is often a star tree, lacking any internal edges. Not only in this situation, the population Fréchet mean is affected by a non-Euclidean phenomenon called stickness which impacts upon asymptotics, and we examine two data sets for which the mean tree is a star tree. The first consists of trees representing the physical shape of artery structures in a sample of medical images of human brains in which the branching patterns are very diverse. The second consists of gene trees from a population of baboons in which there is evidence of substantial hybridization. We develop hypothesis tests which work in the presence of stickiness. The first is a test for the presence of a given edge in the Fréchet population mean; the second is a two-sample test for differences in two distributions which share the same sticky population mean.
- [289] arXiv:2407.03984 (cross-list from eess.SY) [pdf, html, other]
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Title: Forward Reachability for Discrete-Time Nonlinear Stochastic Systems via Mixed-Monotonicity and Stochastic OrderComments: 8 pages, 3 figures, submitted to the 63rd IEEE Conference on Decision and ControlSubjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
We present a method to overapproximate forward stochastic reach sets of discrete-time, stochastic nonlinear systems with interval geometry. This is made possible by extending the theory of mixed-monotone systems to incorporate stochastic orders, and a concentration inequality result that lower-bounds the probability the state resides within an interval through a monotone mapping. Then, we present an algorithm to compute the overapproximations of forward reachable set and the probability the state resides within it. We present our approach on two aerospace examples to show its efficacy.
- [290] arXiv:2407.04028 (cross-list from hep-th) [pdf, html, other]
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Title: Celestial $Lw_{1+\infty}$ charges from a twistor actionComments: 44 pagesSubjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
The celestial $Lw_{1+\infty}$ symmetries in asymptotically flat spacetimes have a natural geometric interpretation on twistor space in terms of Poisson diffeomorphisms. Using this framework, we provide a first-principle derivation of the canonical generators associated with these symmetries starting from the Poisson BF twistor action for self-dual gravity. We express these charges as surface integrals over the celestial sphere in terms of spacetime data at null infinity. The connection between twistor space and spacetime expressions at $\mathscr{I}$ is achieved via an integral formula for the asymptotic Bianchi identities due to Bramson and Tod. Finally, we clarify how $Lw_{1+\infty}$ transformations are symmetries of gravity from a phase space perspective by showing the invariance of the asymptotic Bianchi identities.
- [291] arXiv:2407.04043 (cross-list from gr-qc) [pdf, other]
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Title: Geometric relational framework for general-relativistic gauge field theoriesComments: 81 pagesSubjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We remind how relationality arises as the core insight of general-relativistic gauge field theories from the articulation of the generalised hole and point-coincidence arguments. Hence, a compelling case for a manifestly relational framework ensues naturally. We propose our formulation for such a framework, based on a significant development of the dressing field method of symmetry reduction.
We first develop a version for the group $\text{Aut}(P)$ of automorphisms of a principal bundle $P$ over a manifold $M$, as it is the most natural and elegant, and as $P$ hosts all the mathematical structures relevant to general-relativistic gauge field theory. Yet, as the standard formulation is local, on $M$, we then develop the relational framework for local field theory. It manifestly implements the generalised point-coincidence argument, whereby the physical field-theoretical degrees of freedoms co-define each other and define, coordinatise, the physical spacetime itself. Applying the framework to General Relativity, we obtain relational Einstein equations, encompassing various notions of "scalar coordinatisation" à la Kretschmann-Komar and Brown-Kuchař. - [292] arXiv:2407.04098 (cross-list from gr-qc) [pdf, other]
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Title: Quasinormal modes on Kerr spacetimesComments: 86 pages, 4 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
We introduce a rigorous framework for defining quasinormal modes on stationary, asymptotically flat spacetimes as isolated eigenvalues of the infinitesimal generator of time translations. We consider time functions corresponding to a foliation of asymptotically hyperboloidal hypersurfaces and restrict to suitable Hilbert spaces of functions. These functions have finite Sobolev regularity in bounded regions, but need to be Gevrey-regular at null infinity. This framework is developed in the context of sub-extremal Kerr spacetimes, but also gives uniform-in-$\Lambda$ resolvent estimates on Kerr--de Sitter spacetimes with a small cosmological constant $\Lambda$. As a corollary, we also construct the meromorphic continuation (in a sector of the complex plane) of the cut-off resolvent in Kerr that is associated to the standard Boyer--Lindquist time function. The framework introduced in this paper bridges different notions of quasinormal modes found in the literature. As further applications of our methods, we prove stability of quasinormal frequencies in a sector of the complex plane, with respect to suitably small perturbations and establish convergence properties for Kerr--de Sitter quasinormal frequencies when the cosmological constant approaches zero.
- [293] arXiv:2407.04133 (cross-list from cs.SC) [pdf, html, other]
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Title: Computing Clipped ProductsComments: To appear in Computer Algebra in Scientific Computing (CASC 2024)Subjects: Symbolic Computation (cs.SC); Numerical Analysis (math.NA)
Sometimes only some digits of a numerical product or some terms of a polynomial or series product are required. Frequently these constitute the most significant or least significant part of the value, for example when computing initial values or refinement steps in iterative approximation schemes. Other situations require the middle portion. In this paper we provide algorithms for the general problem of computing a given span of coefficients within a product, that is the terms within a range of degrees for univariate polynomials or range digits of an integer. This generalizes the "middle product" concept of Hanrot, Quercia and Zimmerman. We are primarily interested in problems of modest size where constant speed up factors can improve overall system performance, and therefore focus the discussion on classical and Karatsuba multiplication and how methods may be combined.
- [294] arXiv:2407.04189 (cross-list from cs.LG) [pdf, html, other]
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Title: Meta-Learning and representation learner: A short theoretical noteSubjects: Machine Learning (cs.LG); Statistics Theory (math.ST)
Meta-learning, or "learning to learn," is a subfield of machine learning where the goal is to develop models and algorithms that can learn from various tasks and improve their learning process over time. Unlike traditional machine learning methods focusing on learning a specific task, meta-learning aims to leverage experience from previous tasks to enhance future learning. This approach is particularly beneficial in scenarios where the available data for a new task is limited, but there exists abundant data from related tasks. By extracting and utilizing the underlying structure and patterns across these tasks, meta-learning algorithms can achieve faster convergence and better performance with fewer data. The following notes are mainly inspired from \cite{vanschoren2018meta}, \cite{baxter2019learning}, and \cite{maurer2005algorithmic}.
- [295] arXiv:2407.04264 (cross-list from cs.LG) [pdf, other]
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Title: Langevin Dynamics: A Unified Perspective on Optimization via Lyapunov PotentialsSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
We study the problem of non-convex optimization using Stochastic Gradient Langevin Dynamics (SGLD). SGLD is a natural and popular variation of stochastic gradient descent where at each step, appropriately scaled Gaussian noise is added. To our knowledge, the only strategy for showing global convergence of SGLD on the loss function is to show that SGLD can sample from a stationary distribution which assigns larger mass when the function is small (the Gibbs measure), and then to convert these guarantees to optimization results.
We employ a new strategy to analyze the convergence of SGLD to global minima, based on Lyapunov potentials and optimization. We convert the same mild conditions from previous works on SGLD into geometric properties based on Lyapunov potentials. This adapts well to the case with a stochastic gradient oracle, which is natural for machine learning applications where one wants to minimize population loss but only has access to stochastic gradients via minibatch training samples. Here we provide 1) improved rates in the setting of previous works studying SGLD for optimization, 2) the first finite gradient complexity guarantee for SGLD where the function is Lipschitz and the Gibbs measure defined by the function satisfies a Poincaré Inequality, and 3) prove if continuous-time Langevin Dynamics succeeds for optimization, then discrete-time SGLD succeeds under mild regularity assumptions. - [296] arXiv:2407.04360 (cross-list from cs.CV) [pdf, html, other]
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Title: Shape Prior Segmentation Guided by Harmonic Beltrami SignatureComments: 34 pages, 15 figuresSubjects: Computer Vision and Pattern Recognition (cs.CV); Complex Variables (math.CV)
This paper presents a novel shape prior segmentation method guided by the Harmonic Beltrami Signature (HBS). The HBS is a shape representation fully capturing 2D simply connected shapes, exhibiting resilience against perturbations and invariance to translation, rotation, and scaling. The proposed method integrates the HBS within a quasi-conformal topology preserving segmentation framework, leveraging shape prior knowledge to significantly enhance segmentation performance, especially for low-quality or occluded images. The key innovation lies in the bifurcation of the optimization process into two iterative stages: 1) The computation of a quasi-conformal deformation map, which transforms the unit disk into the targeted segmentation area, driven by image data and other regularization terms; 2) The subsequent refinement of this map is contingent upon minimizing the $L_2$ distance between its Beltrami coefficient and the reference HBS. This shape-constrained refinement ensures that the segmentation adheres to the reference shape(s) by exploiting the inherent invariance, robustness, and discerning shape discriminative capabilities afforded by the HBS. Extensive experiments on synthetic and real-world images validate the method's ability to improve segmentation accuracy over baselines, eliminate preprocessing requirements, resist noise corruption, and flexibly acquire and apply shape priors. Overall, the HBS segmentation framework offers an efficient strategy to robustly incorporate the shape prior knowledge, thereby advancing critical low-level vision tasks.
- [297] arXiv:2407.04366 (cross-list from econ.TH) [pdf, html, other]
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Title: Nash epidemicsSubjects: Theoretical Economics (econ.TH); Systems and Control (eess.SY); Optimization and Control (math.OC); Physics and Society (physics.soc-ph)
Faced with a dangerous epidemic humans will spontaneously social distance to reduce their risk of infection at a socio-economic cost. Compartmentalised epidemic models have been extended to include this endogenous decision making: Individuals choose their behaviour to optimise a utility function, self-consistently giving rise to population behaviour. Here we study the properties of the resulting Nash equilibria, in which no member of the population can gain an advantage by unilaterally adopting different behaviour. We leverage a new analytic solution to obtain, (1) a simple relationship between rational social distancing behaviour and the current number of infections; (2) new scaling results for how the infection peak and number of total cases depend on the cost of contracting the disease; (3) characteristic infection costs that divide regimes of strong and weak behavioural response and depend only on the basic reproduction number of the disease; (4) a closed form expression for the value of the utility. We discuss how these analytic results provide a deep and intuitive understanding into the disease dynamics, useful for both individuals and policymakers. In particular the relationship between social distancing and infections represents a heuristic that could be communicated to the population to encourage, or "bootstrap", rational behaviour.
- [298] arXiv:2407.04406 (cross-list from cs.LG) [pdf, other]
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Title: On Quantum Channel LearningMikhail Gennadievich Belov, Victor Victorovich Dubov, Alexey Vladimirovich Filimonov, Vladislav Gennadievich MalyshkinComments: The unitary learning from arXiv:2405.10263 is generalized to density matrices and quantum channelsSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA); Quantum Physics (quant-ph)
The problem of an optimal mapping between Hilbert spaces $IN$ and $OUT$, based on a series of density matrix mapping measurements $\rho^{(l)} \to \varrho^{(l)}$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}=\sum_{l=1}^{M} \omega^{(l)} F\left(\varrho^{(l)},\sum_s B_s \rho^{(l)} B^{\dagger}_s\right)$ subject to probability preservation constraints on Kraus operators $B_s$. For $F(\varrho,\sigma)$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}=\sum_s\left\langle B_s\middle|S\middle| B_s \right\rangle$ (either exactly or as an approximation) an iterative algorithm is developed to find the global maximum. The result comprises in $N_s$ operators $B_s$ that collectively form an $IN$ to $OUT$ quantum channel $A^{OUT}=\sum_s B_s A^{IN} B_s^{\dagger}$. The work introduces two important generalizations of unitary learning: 1. $IN$/$OUT$ states are represented as density matrices. 2. The mapping itself is formulated as a general quantum channel. This marks a crucial advancement from the commonly studied unitary mapping of pure states $\phi_l=\mathcal{U} \psi_l$ to a general quantum channel, what allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping $\varrho^{(l)}=\mathcal{U} \rho^{(l)} \mathcal{U}^{\dagger}$, in this case a quadratic on $\mathcal{U}$ fidelity can be constructed by considering $\sqrt{\rho^{(l)}} \to \sqrt{\varrho^{(l)}}$ mapping, and on a general quantum channel of Kraus rank $N_s$, where quadratic on $B_s$ fidelity is an approximation -- a quantum channel is then built as a hierarchy of unitary mappings. The approach can be applied to study decoherence effects, spontaneous coherence, synchronizing, etc.
- [299] arXiv:2407.04455 (cross-list from gr-qc) [pdf, html, other]
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Title: Mean Curvature, Singularities and Time Functions in CosmologyComments: 15 pages, 2 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Differential Geometry (math.DG)
In this contribution, we study spacetimes of cosmological interest, without making any symmetry assumptions. We prove a rigid Hawking singularity theorem for positive cosmological constant, which sharpens known results. In particular, it implies that any spacetime with $\operatorname{Ric} \geq -ng$ in timelike directions and containing a compact Cauchy hypersurface with mean curvature $H \geq n$ is timelike incomplete. We also study the properties of cosmological time and volume functions, addressing questions such as: When do they satisfy the regularity condition? When are the level sets Cauchy hypersurfaces? What can one say about the mean curvature of the level sets? This naturally leads to consideration of Hawking type singularity theorems for Cauchy surfaces satisfying mean curvature inequalities in a certain weak sense.
- [300] arXiv:2407.04478 (cross-list from quant-ph) [pdf, html, other]
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Title: Minimal eigenvalue estimates for self-adjoint trace-class operatorsComments: 20 pages, 3 figuresSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Spectral Theory (math.SP)
Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics, and arguably the most important one is being positive semidefinite. For each self-adjoint, trace-class operator $O$ we construct a monotone increasing sequence $q_n$ which tends to the minimal eigenvalue $\lambda_{\min}$ if $O$ is not positive semidefinite, and to $0$ otherwise. This sequence only depends on the moments of $O$ and a concrete upper estimate of its $1$-norm; we also demonstrate that it can be effectively calculated for a large class of physically relevant operators. As a by-product, we obtain computable estimates for the $1$-norm of $O$, too.
First assume that $O$ is positive semidefinite. Unfortunately, positivity tests fail to prove this in finitely many steps. However, $q_n$ gives a rigorous, monotone increasing lower estimate for all eigenvalues, providing a quantitative way of measuring positivity. In this case the speed of convergence is $q_n\approx -\frac cn$.
Now suppose that $O$ is not positive semidefinite. Then $q_n$ monotonically converges to $\lambda_{\min}$ with super-exponential speed. Hence if $q_n$ stabilizes at a negative value, we obtain a strong indication that $O$ is in fact not positive semidefinite. We also construct an easier computable sequence $q_{n,0}$ which fails to be monotone, but converges to $\lambda_{\min}<0$ faster, providing an even better indicator of non-positivity. - [301] arXiv:2407.04480 (cross-list from cs.LG) [pdf, other]
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Title: LoCo: Low-Bit Communication Adaptor for Large-scale Model TrainingSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
To efficiently train large-scale models, low-bit gradient communication compresses full-precision gradients on local GPU nodes into low-precision ones for higher gradient synchronization efficiency among GPU nodes. However, it often degrades training quality due to compression information loss. To address this, we propose the Low-bit Communication Adaptor (LoCo), which compensates gradients on local GPU nodes before compression, ensuring efficient synchronization without compromising training quality. Specifically, LoCo designs a moving average of historical compensation errors to stably estimate concurrent compression error and then adopts it to compensate for the concurrent gradient compression, yielding a less lossless compression. This mechanism allows it to be compatible with general optimizers like Adam and sharding strategies like FSDP. Theoretical analysis shows that integrating LoCo into full-precision optimizers like Adam and SGD does not impair their convergence speed on nonconvex problems. Experimental results show that across large-scale model training frameworks like Megatron-LM and PyTorch's FSDP, LoCo significantly improves communication efficiency, e.g., improving Adam's training speed by 14% to 40% without performance degradation on large language models like LLAMAs and MoE.
- [302] arXiv:2407.04483 (cross-list from hep-th) [pdf, other]
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Title: A new construction of $c=1$ Virasoro blocksComments: 66 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
We introduce a nonabelianization map for conformal blocks, which relates $c=1$ Virasoro blocks on a Riemann surface $C$ to Heisenberg blocks on a branched double cover $\widetilde{C}$ of $C$. The nonabelianization map uses the datum of a spectral network on $C$. It gives new formulas for Virasoro blocks as regularized Fredholm determinants of integral operators, with kernel given by an appropriate free-fermion two-point function on $\widetilde{C}$. The nonabelianization map also intertwines with the action of Verlinde loop operators, and can be used to construct eigenblocks. This leads to new Kyiv-type formulas and Fredholm determinant formulas for $\tau$-functions.
- [303] arXiv:2407.04516 (cross-list from cs.LG) [pdf, html, other]
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Title: G-Adaptive mesh refinement -- leveraging graph neural networks and differentiable finite element solversJames Rowbottom, Georg Maierhofer, Teo Deveney, Katharina Schratz, Pietro Liò, Carola-Bibiane Schönlieb, Chris BuddSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
We present a novel, and effective, approach to the long-standing problem of mesh adaptivity in finite element methods (FEM). FE solvers are powerful tools for solving partial differential equations (PDEs), but their cost and accuracy are critically dependent on the choice of mesh points. To keep computational costs low, mesh relocation (r-adaptivity) seeks to optimise the position of a fixed number of mesh points to obtain the best FE solution accuracy. Classical approaches to this problem require the solution of a separate nonlinear "meshing" PDE to find the mesh point locations. This incurs significant cost at remeshing and relies on certain a-priori assumptions and guiding heuristics for optimal mesh point location. Recent machine learning approaches to r-adaptivity have mainly focused on the construction of fast surrogates for such classical methods. Our new approach combines a graph neural network (GNN) powered architecture, with training based on direct minimisation of the FE solution error with respect to the mesh point locations. The GNN employs graph neural diffusion (GRAND), closely aligning the mesh solution space to that of classical meshing methodologies, thus replacing heuristics with a learnable strategy, and providing a strong inductive bias. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. This method outperforms classical and prior ML approaches to r-adaptive meshing on the test problems we consider, in particular achieving lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work.
- [304] arXiv:2407.04535 (cross-list from cs.LO) [pdf, other]
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Title: Characterisation of Lawvere-Tierney Topologies on Simplicial Sets, Bicolored Graphs, and Fuzzy SetsSubjects: Logic in Computer Science (cs.LO); Category Theory (math.CT)
Simplicial sets generalize many categories of graphs. In this paper, we give a complete characterization of the Lawvere-Tierney topologies on (semi-)simplicial sets, on bicolored graphs, and on fuzzy sets. We apply our results to establish that 'partially simple' simplicial sets and 'partially simple' graphs form quasitoposes.
- [305] arXiv:2407.04558 (cross-list from quant-ph) [pdf, html, other]
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Title: Convex roofs witnessing Kirkwood-Dirac nonpositivityComments: 16 pages, 2 figuresSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Given two observables $A$ and $B$, one can associate to every quantum state a Kirkwood-Dirac (KD) quasiprobability distribution. KD distributions are like joint classical probabilities except that they can have negative or nonreal values, which are associated to nonclassical features of the state. In the last decade, KD distributions have come to the forefront as a versatile tool to investigate and construct quantum advantages and nonclassical phenomena. KD distributions are also used to determine quantum-classical boundaries. To do so, one must have witnesses for when a state is KD nonpositive. Previous works have established a relation between the uncertainty of a pure state with respect to the eigenbases of $A$ and $B$ and KD positivity. If this $\textit{support uncertainty}$ is large, the state cannot be KD positive. Here, we construct two witnesses for KD nonpositivity for general mixed states. Our first witness is the convex roof of the support uncertainty; it is not faithful, but it extends to the convex hull of pure KD-positive states the relation between KD positivity and small support uncertainty. Our other witness is the convex roof of the total KD nonpositivity, which provides a faithful witness for the convex hull of the pure KD-positive states. This implies that the convex roof of the total nonpositivity captures the nonpositive nature of the KD distribution at the underlying pure state level.
- [306] arXiv:2407.04576 (cross-list from cs.DM) [pdf, other]
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Title: Optimal Mixing for Randomly Sampling Edge Colorings on Trees Down to the Max DegreeSubjects: Discrete Mathematics (cs.DM); Data Structures and Algorithms (cs.DS); Probability (math.PR)
We address the convergence rate of Markov chains for randomly generating an edge coloring of a given tree. Our focus is on the Glauber dynamics which updates the color at a randomly chosen edge in each step. For a tree $T$ with $n$ vertices and maximum degree $\Delta$, when the number of colors $q$ satisfies $q\geq\Delta+2$ then we prove that the Glauber dynamics has an optimal relaxation time of $O(n)$, where the relaxation time is the inverse of the spectral gap. This is optimal in the range of $q$ in terms of $\Delta$ as Dyer, Goldberg, and Jerrum (2006) showed that the relaxation time is $\Omega(n^3)$ when $q=\Delta+1$. For the case $q=\Delta+1$, we show that an alternative Markov chain which updates a pair of neighboring edges has relaxation time $O(n)$. Moreover, for the $\Delta$-regular complete tree we prove $O(n\log^2{n})$ mixing time bounds for the respective Markov chain. Our proofs establish approximate tensorization of variance via a novel inductive approach, where the base case is a tree of height $\ell=O(\Delta^2\log^2{\Delta})$, which we analyze using a canonical paths argument.
- [307] arXiv:2407.04591 (cross-list from cs.LG) [pdf, html, other]
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Title: Proximal Point Method for Online Saddle Point ProblemSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
This paper focuses on the online saddle point problem, which involves a sequence of two-player time-varying convex-concave games. Considering the nonstationarity of the environment, we adopt the duality gap and the dynamic Nash equilibrium regret as performance metrics for algorithm design. We present three variants of the proximal point method: the Online Proximal Point Method~(OPPM), the Optimistic OPPM~(OptOPPM), and the OptOPPM with multiple predictors. Each algorithm guarantees upper bounds for both the duality gap and dynamic Nash equilibrium regret, achieving near-optimality when measured against the duality gap. Specifically, in certain benign environments, such as sequences of stationary payoff functions, these algorithms maintain a nearly constant metric bound. Experimental results further validate the effectiveness of these algorithms. Lastly, this paper discusses potential reliability concerns associated with using dynamic Nash equilibrium regret as a performance metric.
- [308] arXiv:2407.04605 (cross-list from stat.ML) [pdf, html, other]
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Title: Linear causal disentanglement via higher-order cumulantsSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Algebraic Geometry (math.AG); Combinatorics (math.CO); Statistics Theory (math.ST)
Linear causal disentanglement is a recent method in causal representation learning to describe a collection of observed variables via latent variables with causal dependencies between them. It can be viewed as a generalization of both independent component analysis and linear structural equation models. We study the identifiability of linear causal disentanglement, assuming access to data under multiple contexts, each given by an intervention on a latent variable. We show that one perfect intervention on each latent variable is sufficient and in the worst case necessary to recover parameters under perfect interventions, generalizing previous work to allow more latent than observed variables. We give a constructive proof that computes parameters via a coupled tensor decomposition. For soft interventions, we find the equivalence class of latent graphs and parameters that are consistent with observed data, via the study of a system of polynomial equations. Our results hold assuming the existence of non-zero higher-order cumulants, which implies non-Gaussianity of variables.
- [309] arXiv:2407.04626 (cross-list from cs.CC) [pdf, html, other]
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Title: Determination Problems for Orbit Closures and Matrix GroupsComments: 22 pagesSubjects: Computational Complexity (cs.CC); Algebraic Geometry (math.AG)
Computational problems concerning the orbit of a point under the action of a matrix group occur in numerous subfields of computer science, including complexity theory, program analysis, quantum computation, and automata theory. In many cases the focus extends beyond orbits proper to orbit closures under a suitable topology. Typically one starts from a group and several points and asks questions about the orbit closure of the points under the action of the group, e.g., whether two given orbit closures intersect.
In this paper we consider a collection of what we call determination problems concerning groups and orbit closures. These problems begin with a given variety and seek to understand whether and how it arises either as an algebraic group or as an orbit closure. The how question asks whether the underlying group is $s$-generated, meaning it is topologically generated by $s$ matrices for a given number $s$. Among other applications, problems of this type have recently been studied in the context of synthesising loops subject to certain specified invariants on program variables.
Our main result is a polynomial-space procedure that inputs a variety $V$ and a number $s$ and determines whether $V$ arises as an orbit closure of a point under an $s$-generated commutative matrix group. The main tools in our approach are rooted in structural properties of commutative algebraic matrix groups and lattice theory. We leave open the question of determining whether a variety is an orbit closure of a point under an algebraic matrix group (without the requirement of commutativity). In this regard, we note that a recent paper by Nosan et al. [NPSHW2021] gives an elementary procedure to compute the orbit closure of a point under finitely many matrices. - [310] arXiv:2407.04669 (cross-list from physics.plasm-ph) [pdf, html, other]
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Title: Two methods to analyse radial diffusion ensembles: the peril of space- and time- dependent diffusionSubjects: Plasma Physics (physics.plasm-ph); Earth and Planetary Astrophysics (astro-ph.EP); Mathematical Physics (math-ph); Geophysics (physics.geo-ph); Space Physics (physics.space-ph)
Particle dynamics in Earth's outer radiation belt can be modelled using a diffusion framework, where large-scale electron movements are captured by a diffusion equation across a single adiabatic invariant, $L^{*}$ $``(L)"$. While ensemble models are promoted to represent physical uncertainty, as yet there is no validated method to analyse radiation belt ensembles. Comparisons are complicated by the domain dependent diffusion, since diffusion coefficient $D_{LL}$ is dependent on $L$. We derive two tools to analyse ensemble members: time to monotonicity $t_m$ and mass/energy moment quantities $\mathcal{N}, \mathcal{E}$. We find that the Jacobian ($1/L^2$) is necessary for radiation belt error metrics. Components of $\partial\mathcal{E}/\partial t$ are explicitly calculated to compare the effects of outer and inner boundary conditions, and loss, on the ongoing diffusion. Using $t_m$, $\mathcal{N}$ and $\mathcal{E}$, we find that: (a) different physically motivated choices of outer boundary condition and location result in different final states and different rates of evolution; (b) the gradients of the particle distribution affect evolution more significantly than $D_{LL}$; (c) the enhancement location, and the amount of initial background particles, are both significant factors determining system evolution; (d) loss from pitch-angle scattering is generally dominant; it mitigates but does not remove the influence of both initial conditions and outer boundary settings, which are due to the $L$-dependence of $D_{LL}$. We anticipate this study will promote renewed focus on the distribution gradients, on the location and nature of the outer boundary in radiation belt modelling, and provide a foundation for systematic ensemble modelling.
- [311] arXiv:2407.04672 (cross-list from cs.DS) [pdf, other]
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Title: Rapid Mixing via Coupling Independence for Spin Systems with Unbounded DegreeSubjects: Data Structures and Algorithms (cs.DS); Probability (math.PR)
We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both $2$-spin and multi-spin systems. As applications for this approach:
$\bullet$ We prove the optimal $O(n)$ relaxation time for the Glauber dynamics of random $q$-list-coloring on an $n$-vertices triangle-tree graph with maximum degree $\Delta$ such that $q/\Delta > \alpha^\star$, where $\alpha^\star \approx 1.763$ is the unique positive solution of the equation $\alpha = \exp(1/\alpha)$. This improves the $n^{1+o(1)}$ relaxation time for Glauber dynamics obtained by the previous work of Jain, Pham, and Vuong (2022). Besides, our framework can also give a near-linear time sampling algorithm under the same condition.
$\bullet$ We prove the optimal $O(n)$ relaxation time and near-optimal $\widetilde{O}(n)$ mixing time for the Glauber dynamics on hardcore models with parameter $\lambda$ in $\textit{balanced}$ bipartite graphs such that $\lambda < \lambda_c(\Delta_L)$ for the max degree $\Delta_L$ in left part and the max degree $\Delta_R$ of right part satisfies $\Delta_R = O(\Delta_L)$. This improves the previous result by Chen, Liu, and Yin (2023).
At the heart of our proof is the notion of $\textit{coupling independence}$ which allows us to consider multiple vertices as a huge single vertex with exponentially large domain and do a "coarse-grained" local-to-global argument on spin systems. The technique works for general (multi) spin systems and helps us obtain some new comparison results for Glauber dynamics. - [312] arXiv:2407.04686 (cross-list from cs.DS) [pdf, other]
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Title: Near-optimal hierarchical matrix approximation from matrix-vector productsSubjects: Data Structures and Algorithms (cs.DS); Numerical Analysis (math.NA)
We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an $n\times n$ matrix $\mathbf{A}$, accessible only though matrix-vector products with $\mathbf{A}$ and $\mathbf{A}^{\mathsf{T}}$. We prove that, for the rank-$k$ HODLR approximation problem, our method achieves a $(1+\beta)^{\log(n)}$-optimal approximation in expected Frobenius norm using $O(k\log(n)/\beta^3)$ matrix-vector products. In particular, the algorithm obtains a $(1+\varepsilon)$-optimal approximation with $O(k\log^4(n)/\varepsilon^3)$ matrix-vector products, and for any constant $c$, an $n^c$-optimal approximation with $O(k \log(n))$ matrix-vector products. Apart from matrix-vector products, the additional computational cost of our method is just $O(n \operatorname{poly}(\log(n), k, \beta))$. We complement the upper bound with a lower bound, which shows that any matrix-vector query algorithm requires at least $\Omega(k\log(n) + k/\varepsilon)$ queries to obtain a $(1+\varepsilon)$-optimal approximation.
Our algorithm can be viewed as a robust version of widely used "peeling" methods for recovering HODLR matrices and is, to the best of our knowledge, the first matrix-vector query algorithm to enjoy theoretical worst-case guarantees for approximation by any hierarchical matrix class. To control the propagation of error between levels of hierarchical approximation, we introduce a new perturbation bound for low-rank approximation, which shows that the widely used Generalized Nyström method enjoys inherent stability when implemented with noisy matrix-vector products. We also introduced a novel randomly perforated matrix sketching method to further control the error in the peeling algorithm.
Cross submissions for Monday, 8 July 2024 (showing 43 of 43 entries )
- [313] arXiv:1003.5705 (replaced) [pdf, html, other]
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Title: Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $S^1$Comments: 56 pages, Revised version; Accepted, Differential and Integral EquationsJournal-ref: Differential and Integral Equations, Volume 24, Numbers 7-8 (2011), 653-718Subjects: Analysis of PDEs (math.AP)
We consider Nonlinear Schrodinger type equations on $S^1$. In this paper, we obtain polynomial bounds on the growth in time of high Sobolev norms of their solutions. The key is to derive an iteration bound based on a frequency decomposition of the solution. This iteration bound is different than the one used earlier in the work of Bourgain, and is less dependent on the structure of the nonlinearity. We first look at the defocusing NLS equation with nonlinearity of degree $\geq 5$. For the quintic NLS, Bourgain derives stronger bounds using different techniques. However, our approach works for higher nonlinearities, where the techniques of Bourgain don't seem to apply. Furthermore, we study variants of the defocusing cubic NLS in which the complete integrability is broken. Among this class of equations, we consider in particular the Hartree Equation, with sufficiently regular convolution potential. For most of the equations that come from modifying the defocusing cubic NLS, we obtain better bounds than for the other equations due to the fact that we can use higher modified energies as in the work of the I-Team.
- [314] arXiv:1003.5707 (replaced) [pdf, html, other]
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Title: Bounds on the growth of high Sobolev norms of solutions to Nonlinear Schrodinger Equations on $\mathbb{R}$Comments: 35 pages, Revised versionJournal-ref: Indiana University Mathematics Journal, Vol 60, Number 5 (2011), 1487-1516Subjects: Analysis of PDEs (math.AP)
In this paper, we consider the cubic nonlinear Schrodinger equation, and the Hartree equation, with sufficiently regular convolution potential, both on the real line. We are interested in bounding the growth of high Sobolev norms of solutions to these equations. Since the cubic NLS is completely integrable, it makes sense to bound only the fractional Sobolev norms of solutions, whose initial data is of restricted smoothness. For the Hartree equation, we consider all Sobolev norms. For both equations, we derive our results by using an appropriate frequency decomposition. In the case of the cubic NLS, this method allows us to recover uniform bounds on the integral Sobolev norms, up to a factor of $t^{0+}$. For the Hartree equation, we use the same method as in our previous work on $S^1$, and the improved Strichartz estimate to obtain a better bound than we previously obtained in the periodic setting.
- [315] arXiv:1505.05000 (replaced) [pdf, other]
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Title: An asymptotic shape theorem for random linear growth modelsAurelia Deshayes (LAMA), Pierrick Siest (IECL)Subjects: Probability (math.PR)
In this paper, we define a class of random growth models whose growth is at least and at most linear and prove an asymptotic shape theorem for these models. This proof generalizes already known proofs for the classical contact process or some of its variants and allows us to obtain conjectured asymptotic shape theorems for several models: the contact process in a randomly evolving environment, the oriented percolation with hostile immigration, the bounded modified contact process, Richardson's model with stirring and the contact process with stirring
- [316] arXiv:1807.00326 (replaced) [pdf, html, other]
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Title: On non-elliptic symplectic manifoldsComments: 72 pages, Comments are welcomeSubjects: Symplectic Geometry (math.SG); Differential Geometry (math.DG)
Let $M$ be a closed symplectic manifold of dimension $2n$ with non-ellipticity. We can define an almost Kähler structure on $M$ by using the given symplectic form. Hence, we have a $\G=\pi_1(M)$-invariant almost Kähler structure on the universal covering, $\ti M$, of $M$. Using Darboux coordinate charts, we globally deform the given almost Kähler structure on $\ti M$ off a Lebesgue measure zero subset to obtain a $\G$-invariant Lipschitz Kähler flat structure on $\ti M$ which is $\G$-homotopy equivalent to the given almost Kähler structure. Analogous to Teleman's $L^2$-Hodge decomposition on PL manifolds or Lipschitz Riemannian manifolds, we give a $L^2$-Hodge decomposition theorem on $\ti M$ with respect to the Lipschitz Kähler flat metric. Using an argument of Gromov, we give a vanishing theorem for $L^2$ harmonic $p$-forms, $p\not=n$ (resp. a non-vanishing theorem for $L^2$ harmonic $n$-forms) on $\ti M$, then the signed Euler characteristic satisfies $(-1)^n\chi(M)\geq0$ (resp. $(-1)^n\chi(M)>0$). Similarly, for any closed even dimensional Riemannian manifold $(M, g)$, we can construct a $\G$-invariant Lipschitz Kähler flat structure on the universal covering, $(\ti M, \ti g)$, of $(M, g)$ which is $\G$-homotopy equivalent to and quasi-isometric to the metric $\ti g$. As an application, using Gromov's method we show that the Chern-Hopf conjecture holds true in closed even dimensional Riemannian manifolds with nonpositive curvature (resp. strictly negative curvature), it gives a positive answer to a Yau's problem due to S. S. Chern and H. Hopf.
- [317] arXiv:1901.03207 (replaced) [pdf, html, other]
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Title: Unique continuation for many-body Schr\"odinger operators and the Hohenberg-Kohn theorem. II. The Pauli HamiltonianJournal-ref: Doc. Math. 25, 869-898 (2020)Subjects: Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Quantum Physics (quant-ph)
We prove the strong unique continuation property for many-body Pauli operators with external potentials, interaction potentials and magnetic fields in $L^p\loc(\R^d)$, and with magnetic potentials in ${L^{q}\loc(\R^d)}$, where ${p > \max(2d/3,2)}$ and ${q > 2d}$. For this purpose, we prove a singular Carleman estimate involving fractional Laplacian operators.
- [318] arXiv:1908.03648 (replaced) [pdf, html, other]
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Title: Symmetry, Unimodality, and Lefschetz Properties for Graded ModulesComments: 29 pages. SubmittedSubjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
We investigate the Weak Lefschetz Properties for modules whose minimal free resolutions are given by generalized Kosuzl complexes in dimension three through a careful study of their Betti numbers and the symmetry and unimodality of their Hilbert functions. We also study the non-Lefschetz locus for finite length modules in arbitrary dimension, and are able to generalize several previous results on the non-Lefschetz locus in this setting. Along the way, we find several connections with a Gorenstein analogue for finite length modules and Artin level modules that are both interesting and useful throughout this paper.
- [319] arXiv:2005.05583 (replaced) [pdf, html, other]
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Title: Modular affine Hecke category and regular unipotent centralizerComments: v1: 52 pages; v2: 51 pages, minor change of title, corrections and changes in notation for consistency with later papers arXiv:2206.03738 and arXiv:2402.08281Subjects: Representation Theory (math.RT)
In this paper we provide, under some mild explicit assumptions, a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the Langlands dual affine flag variety. This equivalence is suggested and motivated by the "geometric Langlands" philosophy, and is used in later work to construct equivalences of categories relating various geometric incarnations of the affine Hecke algebra of the given reductive group.
- [320] arXiv:2005.09038 (replaced) [pdf, other]
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Title: Gorenstein objects in the n-Trivial extensions of abelian categoriesComments: Conflict between authorsSubjects: K-Theory and Homology (math.KT); Category Theory (math.CT)
Given an abelian category, we introduce a categorical concept of (strongly) Gorenstein projective (resp., injective) objects, by defining a new special class of objects. Then we study the transfer of these properties when passing to an abelian category and its n-trivial extension category and also give a characterization of Gorenstein object over it. We give, at the end, applications of this study on the category of modules over an associative ring and triangular matrix rings.
- [321] arXiv:2005.09897 (replaced) [pdf, html, other]
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Title: Fragile minor-monotone parameters under random edge perturbationComments: 20 pagesSubjects: Combinatorics (math.CO)
We conduct a quantitative analysis on the number of random edges required to be added to a base graph~$H$ to significantly increase natural minor-monotone graph parameters in the resulting graph~$R$. Specifically, we show that if $R$ is obtained from a connected graph $H$ by adding only a few random edges, the tree-width, genus, and Hadwiger number of $R$ become very large, irrespective of the structure of~$H$.
- [322] arXiv:2007.02544 (replaced) [pdf, html, other]
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Title: On the Cauchy problem for Friedrichs systems on globally hyperbolic manifolds with timelike boundaryNicolas Ginoux (IECL), Simone Murro (UNITN)Comments: 34 pages -- accepted in Advances in Differential EquationsJournal-ref: Adv. Differential Equations 27(7/8): 497-542 (2022)Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Differential Geometry (math.DG)
In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided.
- [323] arXiv:2007.14821 (replaced) [pdf, html, other]
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Title: Group measure space construction, ergodicity and $W^\ast$-rigidity for stable random fieldsComments: Minor typos have been fixed. Remark 6.6 (a list of problems/conjectures that have been resolved recently) has been added. 30 pages, 1 figureSubjects: Probability (math.PR); Dynamical Systems (math.DS); Operator Algebras (math.OA)
This work discovers a novel link between probability theory (of stable random fields) and von Neumann algebras. It is established that the group measure space construction corresponding to a minimal representation is an invariant of a stationary symmetric $\alpha$-stable (S$\alpha$S) random field indexed by any countable group $G$. When $G=\mathbb{Z}^d$, we characterize ergodicity (and also absolute non-ergodicity) of stationary S$\alpha$S fields in terms of the central decomposition of this crossed product von Neumann algebra coming from any (not necessarily minimal) Rosinski representation. This shows that ergodicity (or the complete absence of it) is a $W^\ast$-rigid property (in a suitable sense) for this class of fields. All our results have analogues for stationary max-stable random fields as well.
- [324] arXiv:2010.02622 (replaced) [pdf, other]
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Title: On the relative Gersten conjecture for Milnor K-theory in the smooth caseComments: Corrected, published versionJournal-ref: J. Pure Appl. Algebra 228(11),(2024)Subjects: Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)
We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the the generic point of the special fiber. This complements results of Gillet and Levine for K-theory, Geisser for motivic cohomology and Schmidt and Strunk and the author for étale cohomology.
- [325] arXiv:2105.06892 (replaced) [pdf, html, other]
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Title: Birational description of moduli spaces of rank 2 logarithmic connectionsComments: 21 pages. To appear in Kyoto Journal of MathematicsSubjects: Algebraic Geometry (math.AG)
In this paper, we provide an explicit description of the Zariski-open subset of the moduli space of rank 2 parabolic logarithmic connections in the case $g\geq 2$. Our approach is to analyze the underlying parabolic bundles and the apparent singularities of the parabolic connections. We prove that a Zariski-open subset of the product of a projective space and the moduli space of parabolic bundles gives a Darboux coordinate for the moduli space of parabolic connections.
- [326] arXiv:2107.01981 (replaced) [pdf, html, other]
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Title: Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curvesComments: 59 pages, 6 figures; v3: revision after referee comments, added comparison to toric mirror construction (section 2.2) and clarified proof of A_infty relations (section 3.3)Subjects: Symplectic Geometry (math.SG); Algebraic Geometry (math.AG)
Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry.
- [327] arXiv:2111.02812 (replaced) [pdf, html, other]
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Title: Reductive quotients of klt singularitiesComments: v3: Final version. To appear in Inventiones MathematicaeSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Differential Geometry (math.DG)
We prove that the quotient of a klt type singularity by a reductive group is of klt type. In particular, given a klt variety $X$ endowed with the action of a reductive group $G$ and admitting a quasi-projective good quotient $X\rightarrow X/\!/G$, we can find a boundary $B$ on $X/\!/G$ so that the pair $(X/\!/G,B)$ is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler $G$-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing $n$-dimensional K-polystable Fano manifolds of volume $v$ has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.
- [328] arXiv:2112.00280 (replaced) [pdf, html, other]
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Title: On the Mordell-Weil Ranks of supersingular abelian varieties over $\mathbb{Z}_p^2$-extensionsSubjects: Number Theory (math.NT); Representation Theory (math.RT)
Let $p$ be a fixed odd prime and let $K$ be an imaginary quadratic field in which $p$ splits. Let $A$ be an abelian variety defined over $K$ with supersingular reduction at both primes above $p$ in $K$. Under certain assumptions, we give a growth estimate for the Mordell--Weil rank of $A$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$. In the last section, written by Chris Williams, he includes some speculative remarks on the $p$-adic $L$-functions for $\mathrm{GSp}(4)$ corresponding to the multi-signed Selmer groups constructed in this paper.
- [329] arXiv:2201.04568 (replaced) [pdf, html, other]
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Title: Vector bundles on quantum conjugacy classesComments: 42 pages, no figures. A revised version. The main changes: a dense open set of admissible deformation parameter values is indicatedSubjects: Quantum Algebra (math.QA); Representation Theory (math.RT)
Let $\mathfrak{g}$ be a simple complex Lie algebra of a classical type and $U_q(\mathfrak{g})$ the corresponding Drinfeld-Jimbo quantum group at $q$ not a root of unity. With every point $t$ of the fixed maximal torus $T$ of an algebraic group $G$ with Lie algebra $\mathfrak{g}$ we associate an additive category $\mathcal{O}_q(t)$ of $U_q(\mathfrak{g})$-modules that is stable under tensor product with finite-dimensional quasi-classical $U_q(\mathfrak{g})$-modules. We prove that $\mathcal{O}_q(t)$ is essentially semi-simple and use it to explicitly quantize equivariant vector bundles on the conjugacy class of $t$.
- [330] arXiv:2202.10258 (replaced) [pdf, other]
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Title: Brownian continuum random tree conditioned to be largeSubjects: Probability (math.PR)
We consider a Feller diffusion (Zs, s $\ge$ 0) (with diffusion coefficient $\sqrt$ 2$\beta$ and drift $\theta$ $\in$ R) that we condition on {Zt = at}, where at is a deterministic function, and we study the limit in distribution of the conditioned process and of its genealogical tree as t $\rightarrow$ +$\infty$. When at does not increase too rapidly, we recover the standard size-biased process (and the associated genealogical tree given by the Kesten's tree). When at behaves as $\alpha$$\beta$ 2 t 2 when $\theta$ = 0 or as $\alpha$ e 2$\beta$|$\theta$|t when $\theta$ = 0, we obtain a new process whose distribution is described by a Girsanov transformation and equivalently by a SDE with a Poissonian immigration. Its associated genealogical tree is described by an infinite discrete skeleton (which does not satisfy the branching property) decorated with Brownian continuum random trees given by a Poisson point measure. As a by-product of this study, we introduce several sets of trees endowed with a Gromovtype distance which are of independent interest and which allow here to define in a formal and measurable way the decoration of a backbone with a family of continuum random trees.
- [331] arXiv:2203.10559 (replaced) [pdf, html, other]
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Title: Envelopes of Bisection Lines of PolygonsComments: 20 pages, 11 figuresSubjects: Differential Geometry (math.DG)
A bisection line divides a convex planar curve into two parts with equal areas. It is natural to study the envelope of these lines, which in general present singularities. The polygonal case is particularly inte\-resting, since there are several different notions of a discrete envelope. In this paper, we study three different notions of discrete envelopes of bisection lines and the connections between them.
- [332] arXiv:2204.00158 (replaced) [pdf, html, other]
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Title: Some 2-adic conjectures concerning polyomino tilings of Aztec diamondsComments: Note: The proof given in the second-to-last paragraph of section 5 is incorrect. See the published version of the paper for a correct proofJournal-ref: Published in Integers volume 23 (2023), article #A30: https://math.colgate.edu/~integers/x30/x30.pdfSubjects: Combinatorics (math.CO)
For various sets of tiles, we count the ways to tile an Aztec diamond of order $n$ using tiles from that set. The resulting function $f(n)$ often has interesting behavior when one looks at $n$ and $f(n)$ modulo powers of 2.
- [333] arXiv:2204.07101 (replaced) [pdf, html, other]
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Title: Walsh Diffusions as Time Changed Multi-parameter ProcessesComments: Typos corrected and some lemmata whose are added for additional clarificationSubjects: Probability (math.PR)
Inspired by allocation strategies in multi-armed bandit model, we propose a pathwise construction of Walsh diffusions. For any infinitesimal generator on a star shaped graph, there exists a unique time change associated with a multi-parameter process such that the time change of this multi-parameter process is the desired diffusion. The time change has an interpretation of time allocation of the process on each edge, and it can be derived explicitly from a family of equations.
- [334] arXiv:2204.07277 (replaced) [pdf, html, other]
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Title: P\'{o}lya-type inequalities on spheres and hemispheresComments: 54 pages. Some misprints corrected. To appear in the Annales de l'Institut FourierSubjects: Spectral Theory (math.SP); Differential Geometry (math.DG)
Given an eigenvalue $\lambda$ of the Laplace-Beltrami operator on $n-$spheres or $-$hemispheres, with multiplicity $m$ such that $\lambda=\lambda_{k}=\dots = \lambda_{k+m-1}$, we characterise the lowest and highest orders in the set $\left\{k,\dots,k+m-1\right\}$ for which Pólya's conjecture holds and fails. In particular, we show that Pólya's conjecture holds for hemispheres in the Neumann case, but not in the Dirichlet case when $n$ is greater than two. We further derive Pólya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. We also obtain direct and reversed Li-Yau inequalities for $\mathbb{S}^2$ and $\mathbb{S}^4$, respectively.
- [335] arXiv:2204.08045 (replaced) [pdf, html, other]
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Title: Counting divisorial contractions with centre a $cA_n$-singularityComments: 18 pages, to appear in Publications of the Research Institute for Mathematical Sciences, Kyoto University. Update the theorem numbering of the citation [Pae21]Subjects: Algebraic Geometry (math.AG)
First, we simplify the existing classification due to Kawakita and Yamamoto of 3-dimensional divisorial contractions with centre a $cA_n$-singularity, also called compound $A_n$ singularity. Next, we describe the global algebraic divisorial contractions corresponding to a given local analytic equivalence class of divisorial contractions with centre a point. Finally, we consider divisorial contractions of discrepancy at least 2 to a fixed variety with centre a $cA_n$-singularity. We show that if there exists one such divisorial contraction, then there exist uncountably many such divisorial contractions.
- [336] arXiv:2206.03738 (replaced) [pdf, other]
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Title: Modular affine Hecke category and regular centralizerComments: v1: 164 pages, this paper is a sequel to arXiv:2005.05583 (whose title will be updated); v2: 148 pages, minor revision, some changes of notation for consistency with later paper arXiv:2402.08281, removed Section 13 (which will be treated in a separate paper)Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG)
In this paper we provide a "combinatorial" description of the category of tilting perverse sheaves on the affine flag variety of a reductive algebraic group, and its free-monodromic variant, with coefficients in a field of positive characteristic. This provides a replacement for the familiar "Soergel theory" for characteristic-0 coefficients, and the second step in our project towards the construction of an equivalence of categories relating the two natural geometric realizations of the associated affine Hecke algebra in the case of positive-characteristic coefficients.
- [337] arXiv:2206.05583 (replaced) [pdf, other]
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Title: Hamiltonicity of covering graphs of treesComments: 18 pages, minor revisions, Thm 3.10 fixedSubjects: Combinatorics (math.CO)
In this paper, we consider covering graphs obtained by lifting a tree with a loop at each vertex as a voltage graph over a cyclic group. We generalize a tool of Hell, Nishiyama, and Stacho, known as the billiard strategy, for constructing Hamiltonian cycles in the covering graphs of paths. We show that our extended tool can be used to provide new sufficient conditions for the Hamiltonicity of covering graphs of trees that are similar to those of Batagelj and Pisanski and of Hell, Nishiyama, and Stacho. Next, we focus specifically on covering graphs obtained from trees lifted as voltage graphs over cyclic groups $\mathbb Z_p$ of large prime order $p$. We prove that for a given reflexive tree $T$ whose edge labels are assigned uniformly at random from a finite set, the corresponding lift is almost surely Hamiltonian for a large enough prime-ordered cyclic group $\mathbb Z_p$. Finally, we show that if a reflexive tree $T$ is lifted over a group $\mathbb Z_p$ of a large prime order, then for any assignment of nonzero elements of $\mathbb Z_p$ to the edges of $T$, the corresponding cover of $T$ has a large circumference.
- [338] arXiv:2206.11770 (replaced) [pdf, html, other]
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Title: Asymptotic flocking for the Cucker-Smale model with time variable time delaysSubjects: Optimization and Control (math.OC)
In this paper, we investigate a Cucker-Smale flocking model with varying time delay. We establish exponential asymptotic flocking without requiring smallness assumptions on the time delay size and the monotonicity of the influence function.
- [339] arXiv:2207.03972 (replaced) [pdf, html, other]
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Title: Weakly bounded cohomology classes and a counterexample to a conjecture of GromovComments: 21 pages, 9 figures. This version of the article has been accepted for publication, after peer review, in GAFA, but is not the Version of Record and does not reflect post-acceptance improvements, or any correctionsJournal-ref: Geom. Funct. Anal. 34 (2024), 631-658Subjects: Group Theory (math.GR)
We exhibit a finitely presented group whose second cohomology contains a weakly bounded, but not bounded, class. As an application, we disprove a long-standing conjecture of Gromov about bounded primitives of differential forms on universal covers of closed manifolds.
- [340] arXiv:2208.04655 (replaced) [pdf, html, other]
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Title: Integral models of moduli spaces of shtukas with deep Bruhat-Tits level structuresComments: 36 pages. Minor corrections, to appear in IMRNSubjects: Algebraic Geometry (math.AG)
We construct integral models for moduli spaces of shtukas with deep Bruhat-Tits level structures. We embed the moduli space of global shtukas for a deep Bruhat-Tits group scheme into the limit of the moduli spaces of shtukas for all associated parahoric group schemes. Its schematic image defines an integral model of the moduli space of shtukas with deep Bruhat-Tits level with favourable properties: They admit proper, surjective and generically étale level maps as well as a natural Newton stratification. In the Drinfeld case, this general construction of integral models recovers the moduli space of Drinfeld shtukas with Drinfeld level structures.
- [341] arXiv:2208.12871 (replaced) [pdf, html, other]
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Title: Quantitative limit theorems and bootstrap approximations for empirical spectral projectorsSubjects: Probability (math.PR); Statistics Theory (math.ST)
Given finite i.i.d.~samples in a Hilbert space with zero mean and trace-class covariance operator $\Sigma$, the problem of recovering the spectral projectors of $\Sigma$ naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator $\hat \Sigma$, and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of $\Sigma$. In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject only to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.
- [342] arXiv:2208.14331 (replaced) [pdf, html, other]
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Title: Integration on the SurrealsComments: This substantially revised version of the paper is forthcoming in Advances in MathematicsSubjects: Logic (math.LO)
Conway's real closed field $\mathbf{No}$ of surreal numbers is a sweeping generalization of the real numbers and the ordinals to which a number of elementary functions such as log and exponentiation have been shown to extend. The problems of identifying significant classes of functions that can be so extended and of defining integration for them have proven to be formidable. In this paper we address this and related unresolved issues by showing that extensions to $\mathbf{No}$, and thereby integrals, exist for most functions arising in practical applications. In particular, we show they exist for a large subclass of the \emph{resurgent functions}, a subclass that contains the functions that at $\infty$ are semi-algebraic, semi-analytic, analytic, meromorphic, and Borel summable as well as solutions to nonresonant linear and nonlinear meromorphic systems of ODEs or of difference equations. By suitable changes of variables we deal with arbitrarily located singular points. We further establish a sufficient condition for the theory to carry over to ordered exponential subfields of $\mathbf{No}$ more generally and illustrate the result with structures familiar from the surreal literature. The extensions of functions and integrals that concern us are constructive in nature, which permits us to work in NBG less the Axiom of Choice (for both sets and proper classes). Following the completion of the positive portion of the paper, it is shown that the existence of such constructive extensions and integrals of substantially more general types of functions (e.g. smooth functions) is obstructed by considerations from the foundations of mathematics.
- [343] arXiv:2210.08886 (replaced) [pdf, html, other]
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Title: Learning Decentralized Linear Quadratic Regulators with $\sqrt{T}$ RegretComments: 50 pages, 3 figuresSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Systems and Control (eess.SY)
We propose an online learning algorithm that adaptively designs a decentralized linear quadratic regulator when the system model is unknown a priori and new data samples from a single system trajectory become progressively available. The algorithm uses a disturbance-feedback representation of state-feedback controllers coupled with online convex optimization with memory and delayed feedback. Under the assumption that the system is stable or given a known stabilizing controller, we show that our controller enjoys an expected regret that scales as $\sqrt{T}$ with the time horizon $T$ for the case of partially nested information pattern. For more general information patterns, the optimal controller is unknown even if the system model is known. In this case, the regret of our controller is shown with respect to a linear sub-optimal controller. We validate our theoretical findings using numerical experiments.
- [344] arXiv:2210.10089 (replaced) [pdf, html, other]
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Title: Unknotting number 21 knots are slice in K3Comments: 13 pages, 7 figures. Final version accepted for publicationSubjects: Geometric Topology (math.GT)
We prove that all knots with unknotting number at most 21 are smoothly slice in the K3 surface. We also prove a more general statement for 4-manifolds that contain a plumbing tree of spheres. Our strategy is based on a flexible method to remove double points of immersed surfaces in 4-manifolds by tubing over neighbourhoods of embedded trees. As a byproduct, we recover a classical result of Norman and Suzuki that every knot is smoothly slice in $S^2 \times S^2$ and in $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$.
- [345] arXiv:2210.14720 (replaced) [pdf, html, other]
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Title: Convergence of fractional Fourier series on the torus and applicationsSubjects: Functional Analysis (math.FA)
In this paper, we introduce the fractional Fourier series on the fractional torus and study some basic facts of fractional Fourier series, such as fractional convolution and fractional approximation. Meanwhile, fractional Fourier inversion and Poisson summation formula are also given. We further discuss the relationship between the decay of fractional Fourier coefficients and the smoothness of a function. Using the properties of fractional Fejer kernel, the pointwise convergence of fractional Fourier series can be established. Finally, we present the applications of fractional Fourier series to fractional partial differential equations with periodic boundary condition. Moreover, we apply approximation methods on the fractional torus to recover the non-stationary signals.
- [346] arXiv:2211.05835 (replaced) [pdf, html, other]
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Title: Optimal stopping of Gauss-Markov bridgesComments: 32 pages, 2 figuresSubjects: Probability (math.PR); Mathematical Finance (q-fin.MF)
We solve the non-discounted, finite-horizon optimal stopping problem of a Gauss-Markov bridge by using a time-space transformation approach. The associated optimal stopping boundary is proved to be Lipschitz continuous on any closed interval that excludes the horizon, and it is characterized by the unique solution of an integral equation. A Picard iteration algorithm is discussed and implemented to exemplify the numerical computation and geometry of the optimal stopping boundary for some illustrative cases.
- [347] arXiv:2211.06357 (replaced) [pdf, html, other]
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Title: Parity of ranks of Jacobians of curvesComments: Major changes. Parts of this paper have been split out into separate submissions, which will appear shortly ("On Galois covers of curves and arithmetic of Jacobians" and "A note on the parity conjecture and base change"). Added a section on the decomposition of L-functions of curves with automorphisms. Added a new proof of the parity conjecture for elliptic curves, assuming the finiteness of ShaSubjects: Number Theory (math.NT)
We investigate Selmer groups of Jacobians of curves that admit an action of a non-trivial group of automorphisms, and give applications to the study of the parity of Selmer ranks. Under the Shafarevich--Tate conjecture, we give an expression for the parity of the Mordell--Weil rank of an arbitrary Jacobian in terms of purely local invariants; the latter can be seen as an arithmetic analogue of local root numbers, which, under the Birch--Swinnerton-Dyer conjecture, similarly control parities of ranks of abelian varieties. As an application, we give a new proof of the parity conjecture for elliptic curves. The core of the paper is devoted to developing the arithmetic theory of Jacobians for Galois covers of curves, including decomposition of their L-functions, and the interplay between Brauer relations and Selmer groups.
- [348] arXiv:2211.07523 (replaced) [pdf, html, other]
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Title: Closed string mirrors of symplectic cluster manifoldsComments: revised and expanded based on referee commentsSubjects: Symplectic Geometry (math.SG)
For the base $B$ of a Maslov $0$ Lagrangian torus fibration with singularities consider the sheaf assigning to each $P\subset B$ the relative symplectic cohomology in degree $0$ of its pre-image. We compute this sheaf for nodal Lagrangian torus fibrations on four dimensional symplectic cluster manifolds. We show that it is the pushforward of the structure sheaf of a certain rigid analytic space under a non-archimedean torus fibration. The rigid analytic space is constructed in a canonical way from the relative SH sheaf and is referred as the \emph{closed string mirror}. The construction relies on computing relative SH for local models by applying general axiomatic properties rather than ad hoc analysis of holomorphic curves. These axiomatic properties include previously established ones such as the Mayer-Vietoris property and locality for complete embeddings; and new ones such as the Hartogs property and the holomorphic volume form preservation property of wall crossing in relative $SH$. We indicate some higher dimensional settings where the same techniques apply.
- [349] arXiv:2212.09037 (replaced) [pdf, html, other]
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Title: Collinearity of points on Poincar\'e unit disk and Riemann sphereComments: 26 pages, 7 figuresSubjects: Metric Geometry (math.MG)
We study certain points significant for the hyperbolic geometry of the unit disk. We give explicit formulas for the intersection points of the Euclidean lines and the stereographic projections of great circles of the Riemann sphere passing through these points. We prove several results related to collinearity of these intersection points, offer new ways to find the hyperbolic midpoint, and represent a formula for the chordal midpoint. The proofs utilize Gröbner bases from computer algebra for the solution of polynomial equations.
- [350] arXiv:2212.12925 (replaced) [pdf, other]
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Title: Jacobson program for Lie algebrasComments: There are some flaws and mistakesSubjects: Rings and Algebras (math.RA)
Following the structure theory approach for rings, the aim of this paper is to study some distinguished classes of Lie algebras. We introduce the notion of a Lie-module and discuss some relations of it with various classes of ideals of a Lie algebra. We give several Lie-module-related characterizations of primitive Lie algebras and prove a representation theorem of a Lie algebra in terms of primitive Lie algebras. We prove a density theorem. Finally, we highlight on the importance of nuclear ideals of prime and semiprime Lie algebras in determining them.
- [351] arXiv:2301.04797 (replaced) [pdf, html, other]
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Title: Faithful tropicalization and Skeleton of $\overline{M}_{0,n}$Subjects: Algebraic Geometry (math.AG)
We propose a comparison between the Berkovich skeleton of Berkovich analytification of $(\overline{\textsf{M}}_{0,n},{\overline{\textsf{M}}_{0,n} \setminus \textsf{M}_{0,n}})$ and faithful tropicalization of $\textsf{M}_{0,n}$ over a complete discrete valued field. In particular, we proved the two combinatorial structures are the same in terms of valuation in $\overline{\textsf{M}}^{\textsf{an}}_{0,n}$.
- [352] arXiv:2301.08684 (replaced) [pdf, other]
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Title: Smoothed Moreau-Yosida Tensor Train Approximation of State-constrained Optimization Problems under UncertaintyComments: 28 pagesSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
We propose an algorithm to solve optimization problems constrained by partial (ordinary) differential equations under uncertainty, with almost sure constraints on the state variable. To alleviate the computational burden of high-dimensional random variables, we approximate all random fields by the tensor-train decomposition. To enable efficient tensor-train approximation of the state constraints, the latter are handled using the Moreau-Yosida penalty, with an additional smoothing of the positive part (plus/ReLU) function by a softplus function. In a special case of a quadratic cost minimization constrained by linear elliptic partial differential equations, and some additional constraint qualification, we prove strong convergence of the regularized solution to the optimal control. This result also proposes a practical recipe for selecting the smoothing parameter as a function of the penalty parameter. We develop a second order Newton type method with a fast matrix-free action of the approximate Hessian to solve the smoothed Moreau-Yosida problem. This algorithm is tested on benchmark elliptic problems with random coefficients, optimization problems constrained by random elliptic variational inequalities, and a real-world epidemiological model with 20 random variables. These examples demonstrate mild (at most polynomial) scaling with respect to the dimension and regularization parameters.
- [353] arXiv:2301.09073 (replaced) [pdf, html, other]
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Title: The $\mu$-invariant change for abelian varieties over finite $p$-extensions of global fieldsComments: 38 pages, refined the main results of the previous version and corrected a few errorsSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $\mu$-invariants, with respect to a finite Galois p-extension $K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension of global fields $L/K$ that ramifies at a finite number of places at which $A$ has ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for the size $\delta_v$ of the local Galois cohomology of the Mordell-Weil group of $A$ with respect to a $p$-extension ramified at a supersingular place $v$. Next, in all characteristics, we describe the asymptotic growth of $\delta_v$ along a multiple $\mathbb{Z}_p$-extension $L/K$ and provide a lower bound for the change of $\mu$-invariants of $A$ from the tower $L/K$ to the tower $LK'/K'$. Finally, we present numerical evidence supporting these results.
- [354] arXiv:2302.00934 (replaced) [pdf, other]
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Title: High-dimensional variable clustering based on maxima of a weakly dependent random processComments: 47 pages, 6 figuresSubjects: Statistics Theory (math.ST); Methodology (stat.ME); Machine Learning (stat.ML)
We propose a new class of models for variable clustering called Asymptotic Independent block (AI-block) models, which defines population-level clusters based on the independence of the maxima of a multivariate stationary mixing random process among clusters. This class of models is identifiable, meaning that there exists a maximal element with a partial order between partitions, allowing for statistical inference. We also present an algorithm depending on a tuning parameter that recovers the clusters of variables without specifying the number of clusters \emph{a priori}. Our work provides some theoretical insights into the consistency of our algorithm, demonstrating that under certain conditions it can effectively identify clusters in the data with a computational complexity that is polynomial in the dimension. A data-driven selection method for the tuning parameter is also proposed. To further illustrate the significance of our work, we applied our method to neuroscience and environmental real-datasets. These applications highlight the potential and versatility of the proposed approach.
- [355] arXiv:2302.02085 (replaced) [pdf, html, other]
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Title: Semicontinuous maps on module varietiesComments: 17 pages. V2: small corrections and slightly reorganized after a referee report. Final version, to appear in CrelleSubjects: Representation Theory (math.RT)
We study semicontinuous maps on varieties of modules over finite-dimensional algebras. We prove that truncated Euler maps are upper or lower semicontinuous. This implies that $g$-vectors and $E$-invariants of modules are upper semicontinuous. We also discuss inequalities of generic values of some upper semicontinuous maps.
- [356] arXiv:2302.05303 (replaced) [pdf, html, other]
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Title: A Syntax for Strictly Associative and Unital $\infty$-CategoriesComments: 38 pages, 9 figuresSubjects: Category Theory (math.CT); Logic in Computer Science (cs.LO)
We present the first definition of strictly associative and unital $\infty$-category. Our proposal takes the form of a type theory whose terms describe the operations of such structures, and whose definitional equality relation enforces desired strictness conditions. The key technical device is a new computation rule in the definitional equality of the theory, which we call insertion, defined in terms of a universal property. On terms for which it is defined, this operation "inserts" one of the arguments of a substituted coherence into the coherence itself, appropriately modifying the pasting diagram and result type, and simplifying the syntax in the process. We generate an equational theory from this reduction relation and we study its properties in detail, showing that it yields a decision procedure for equality.
Expressed as a type theory, our model is well-adapted for generating and verifying efficient proofs of higher categorical statements. We illustrate this via an OCaml implementation, and give a number of examples, including a short encoding of the syllepsis, a 5-dimensional homotopy that plays an important role in the homotopy groups of spheres. - [357] arXiv:2303.00724 (replaced) [pdf, other]
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Title: Cluster-size decay in supercritical kernel-based spatial random graphsComments: 73 pages, 3 figuresSubjects: Probability (math.PR)
We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent $\zeta\in(0,1)$ of the decay of the cluster-size distribution. That is, with $|\mathcal{C}(0)|$ denoting the number of vertices in the component of the vertex at $0\in \mathbb{R}^d$, we prove \[
\mathbb{P}(k< |\mathcal{C}(0)|<\infty)=\exp\big(-\Theta(k^{\zeta})\big), \qquad \text{as }k\to\infty.
\] The value of $\zeta$ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension $d$, the power-law tail exponent $\tau$ of the degree distribution and a long-range parameter $\alpha$ governing the presence of long edges in Euclidean space. In this paper we present the proof for the region in the phase diagram where the model is a generalization of continuum scale-free percolation and/or hyperbolic random graphs: $\zeta$ in this regime depends both on $\tau,\alpha$. We also prove that the second-largest component in a box of volume $n$ is of size $\Theta((\log n)^{1/\zeta})$ with high probability. We develop a deterministic algorithm, the cover expansion, as new methodology. This algorithm enables us to prevent too large components that may be de-localized or locally dense in space. - [358] arXiv:2303.01889 (replaced) [pdf, html, other]
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Title: Scale invariant bounds for mixing in the Rayleigh-Taylor instabilitySubjects: Analysis of PDEs (math.AP)
We study the Rayleigh-Taylor instability for two miscible, incompressible, inviscid fluids. Scale-invariant estimates for the size of the mixing zone and coarsening of internal structures in the fully nonlinear regime are established following techniques introduced for the Saffman-Taylor instability in [OttoMenon, 2004]. These bounds provide optimal scaling laws and reveal the strong role of dissipation in slowing down mixing.
- [359] arXiv:2303.05337 (replaced) [pdf, html, other]
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Title: Proximity Operators of Perspective Functions with Nonlinear ScalingSubjects: Optimization and Control (math.OC)
A perspective function is a construction which combines a base function defined on a given space with a nonlinear scaling function defined on another space and which yields a lower semicontinuous convex function on the product space. Since perspective functions are typically nonsmooth, their use in first-order algorithms necessitates the computation of their proximity operator. This paper establishes closed-form expressions for the proximity operator of a perspective function defined on a Hilbert space in terms of a proximity operator involving its base function and one involving its scaling function.
- [360] arXiv:2303.07060 (replaced) [pdf, html, other]
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Title: Parametric Estimation of Tempered Stable LawsSubjects: Statistics Theory (math.ST); Probability (math.PR)
Tempered stable distributions are frequently used in financial applications (e.g., for option pricing) in which the tails of stable distributions would be too heavy. Given the non-explicit form of the probability density function, estimation relies on numerical algorithms which typically are time-consuming. We compare several parametric estimation methods such as the maximum likelihood method and different generalized method of moment approaches. We study large sample properties and derive consistency, asymptotic normality, and asymptotic efficiency results for our estimators. Additionally, we conduct simulation studies to analyze finite sample properties measured by the empirical bias, precision, and asymptotic confidence interval coverage rates and compare computational costs. We cover relevant subclasses of tempered stable distributions such as the classical tempered stable distribution and the tempered stable subordinator. Moreover, we discuss the normal tempered stable distribution which arises by subordinating a Brownian motion with a tempered stable subordinator. Our financial applications to log returns of asset indices and to energy spot prices illustrate the benefits of tempered stable models.
- [361] arXiv:2303.10798 (replaced) [pdf, other]
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Title: An aperiodic monotileComments: 91 pages, 57 figures. Copyedited journal version of article. Significant editing of the exposition throughout, but particularly in Section 3. Overall the same results appear in the same orderSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Metric Geometry (math.MG)
A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.
- [362] arXiv:2303.10994 (replaced) [pdf, other]
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Title: Novikov algebras and their primitive idealsComments: There are some flaws and mistakesSubjects: Rings and Algebras (math.RA)
The aim of this paper is to study the primitive ideals of Novikov algebras. In terms of modular maximal right ideals, a characterization of the primitive ideals of a Novikov algebra has been obtained. We prove a Chevalley-Jacobson density-type theorem for primitive Novikov algebras. We obtain some equivalences between prime, simple, and primitive Novikov algebras. We describe a subalgebra of a Novikov algebra as a Novikov algebra of endomorphisms.
- [363] arXiv:2303.13598 (replaced) [pdf, html, other]
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Title: Bootstrap-Assisted Inference for Generalized Grenander-type EstimatorsSubjects: Statistics Theory (math.ST); Econometrics (econ.EM); Methodology (stat.ME)
Westling and Carone (2020) proposed a framework for studying the large sample distributional properties of generalized Grenander-type estimators, a versatile class of nonparametric estimators of monotone functions. The limiting distribution of those estimators is representable as the left derivative of the greatest convex minorant of a Gaussian process whose monomial mean can be of unknown order (when the degree of flatness of the function of interest is unknown). The standard nonparametric bootstrap is unable to consistently approximate the large sample distribution of the generalized Grenander-type estimators even if the monomial order of the mean is known, making statistical inference a challenging endeavour in applications. To address this inferential problem, we present a bootstrap-assisted inference procedure for generalized Grenander-type estimators. The procedure relies on a carefully crafted, yet automatic, transformation of the estimator. Moreover, our proposed method can be made ``flatness robust'' in the sense that it can be made adaptive to the (possibly unknown) degree of flatness of the function of interest. The method requires only the consistent estimation of a single scalar quantity, for which we propose an automatic procedure based on numerical derivative estimation and the generalized jackknife. Under random sampling, our inference method can be implemented using a computationally attractive exchangeable bootstrap procedure. We illustrate our methods with examples and we also provide a small simulation study. The development of formal results is made possible by some technical results that may be of independent interest.
- [364] arXiv:2303.14085 (replaced) [pdf, html, other]
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Title: Optimal transport and Wasserstein distances for causal modelsSubjects: Statistics Theory (math.ST); Optimization and Control (math.OC); Probability (math.PR)
In this paper, we introduce a variant of optimal transport adapted to the causal structure given by an underlying directed graph $G$. Different graph structures lead to different specifications of the optimal transport problem. For instance, a fully connected graph yields standard optimal transport, a linear graph structure corresponds to causal optimal transport between the distributions of two discrete-time stochastic processes, and an empty graph leads to a notion of optimal transport related to CO-OT, Gromov-Wasserstein distances and factored OT. We derive different characterizations of $G$-causal transport plans and introduce Wasserstein distances between causal models that respect the underlying graph structure. We show that average treatment effects are continuous with respect to $G$-causal Wasserstein distances and small perturbations of structural causal models lead to small deviations in $G$-causal Wasserstein distance. We also introduce an interpolation between causal models based on $G$-causal Wasserstein distance and compare it to standard Wasserstein interpolation.
- [365] arXiv:2304.02338 (replaced) [pdf, html, other]
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Title: Solving decision problems with endogenous uncertainty and conditional information revelation using influence diagramsComments: 27 pages, 10 figures, 6 tablesSubjects: Optimization and Control (math.OC)
Despite methodological advances for modeling decision problems under uncertainty, representing endogenous uncertainty still proves challenging both in terms of modeling capabilities and computational requirements. A novel reformulation based on rooted junction trees (RJTs) provides an approach for solving such decision problems using off-the-shelf mathematical optimization solvers. This is made possible by using an influence diagram to represent a given decision problem and reformulating it as an RJT, which is then represented as a mixed-integer linear programming model. In this paper, we focus on the type of endogenous uncertainty that received less attention in the rooted junction tree approach: conditionally observed information. Multi-stage stochastic programming models use conditional non-anticipativity constraints to represent such uncertainties, and we show how such constraints can be incorporated into RJT formulations. This allows us to consider the two main types of endogenous uncertainty simultaneously, namely decision-dependent information structure and decision-dependent probability distribution. Finally, the extended framework is illustrated with a large-scale cost-benefit problem regarding climate change mitigation.
- [366] arXiv:2304.03163 (replaced) [pdf, html, other]
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Title: Compact K\"ahler 3-folds with nef anti-canonical bundleComments: v2; 39 pages; An erroneous argument in Case 1 of Subsection 4.3 has been replaced with a correct alternative proof.; to appear in Mathematische AnnalenSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Differential Geometry (math.DG)
In this paper, we prove that a non-projective compact Kähler $3$-fold with nef anti-canonical bundle is, up to a finite étale cover, one of the following: a manifold with vanishing first Chern class; the product of a K3 surface and the projective line; or a projective space bundle over a $2$-dimensional torus. This result extends Cao-Höring's structure theorem for projective manifolds to compact Kähler manifolds in dimension $3$. For the proof, we investigate the Minimal Model Program for compact Kähler $3$-folds with nef anti-canonical bundles by using the positivity of direct image sheaves, $\mathbb{Q}$-conic bundles, and orbifold vector bundles.
- [367] arXiv:2304.04517 (replaced) [pdf, html, other]
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Title: Approximating branchwidth on parametric extensions of planarityComments: Accepted to WG 2024Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
The branchwidth of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an extension of this algorithm to minor-closed graph classes, further than planar graphs, as follows: Let $H_{1}$ be a graph embeddable in the torus and $H_{2}$ be a graph embeddable in the projective plane. We prove that every $\{H_{1},H_{2}\}$-minor free graph $G$ contains a subgraph $G'$ where the difference between the branchwidth of $G$ and the branchwidth of $G'$ is bounded by some constant, depending only on $H_{1}$ and $H_{2}$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving a constant-additive approximation for branchwidth: For $\{H_{1},H_{2}\}$-minor free graphs, there is a constant $c$ (depending on $H_{1}$ and $H_{2}$) and an $\Ocal(|V(G)|^{3})$-time algorithm that, given a graph $G$, outputs a value $b$ such that the branchwidth of $G$ is between $b$ and $b+c$.
- [368] arXiv:2304.06318 (replaced) [pdf, html, other]
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Title: On the connected blocks polytopeComments: to be published in Discrete & Computational GeometrySubjects: Combinatorics (math.CO)
In this paper, we study the connected blocks polytope, which, apart from its own merits, can be seen as the generalization of certain connectivity based or Eulerian subgraph polytopes. We provide a complete facet description of this polytope, characterize its edges and show that it is Hirsch. We also show that connected blocks polytopes admit a regular unimodular triangulation by constructing a squarefree Gröbner basis. In addition, we prove that the polytope is Gorenstein of index $2$ and that its $h^\ast$-vector is unimodal.
- [369] arXiv:2304.09496 (replaced) [pdf, html, other]
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Title: Weak Convergence Of Tamed Exponential Integrators for Stochastic Differential EquationsComments: 24 pages, 3 figuresSubjects: Numerical Analysis (math.NA)
We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschtiz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the stepsize unlike the GBM tamed method.
- [370] arXiv:2304.10802 (replaced) [pdf, html, other]
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Title: An extended Merton problem with relaxed benchmark trackingComments: Keywords: Benchmark tracking, capital injection, expected largest shortfall, consumption and portfolio choice, Neumann boundary conditionSubjects: Optimization and Control (math.OC); Portfolio Management (q-fin.PM)
This paper studies a Merton's optimal consumption problem in an extended formulation incorporating the tracking of a benchmark process described by a geometric Brownian motion. We consider a relaxed tracking formulation such that the wealth process compensated by a fictitious capital injection outperforms the benchmark at all times. The fund manager aims to maximize the expected utility of consumption deducted by the cost of the capital injection, where the latter term can also be regarded as the expected largest shortfall of the wealth with reference to the benchmark. By introducing an auxiliary state process with reflection, we formulate and tackle an equivalent stochastic control problem by means of the dual transform and probabilistic representation, where the dual PDE can be solved explicitly. On the strength of the closed-form results, we can derive and verify the optimal feedback control for the primal control problem, allowing us to discuss some new and interesting financial implications induced by the additional risk-taking from the capital injection and the goal of tracking.
- [371] arXiv:2304.11945 (replaced) [pdf, html, other]
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Title: On the Viability and Invariance of Proper Sets under Continuity Inclusions in Wasserstein SpacesComments: 44 pagesJournal-ref: SIAM Journal on Mathematical Analysis 56 (3), 2863-2914 (2024)Subjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP)
In this article, we derive conditions for the existence of solutions to state-constrained continuity inclusions in Wasserstein spaces whose right-hand sides may be discontinuous in time. These latter are based on a fine investigation of the infinitesimal behaviour of the underlying reachable sets, through which we show that up to a negligible set of times, every admissible velocity of a continuity inclusion can be approximately realised as the metric derivative of a solution of the dynamics, and vice versa. Building on these results, we are able to establish necessary and sufficient geometric conditions for the viability and invariance of stationary and time-dependent constraint sets which involve a suitable notion of contingent cones in Wasserstein spaces, and presented in ascending order of generality. We then close the article by exhibiting two prototypical examples of constraints sets appearing in applications for which one can compute relevant subfamilies of contingent directions.
- [372] arXiv:2305.00091 (replaced) [pdf, html, other]
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Title: A Theory of the NEPv Approach for Optimization On the Stiefel ManifoldComments: 86 pages, 0 figuresSubjects: Optimization and Control (math.OC)
The NEPv approach has been increasingly used lately for optimization on the Stiefel manifold arising from machine learning. General speaking, the approach first turns the first order optimality condition, also known as the KKT condition, into a nonlinear eigenvalue problem with eigenvector dependency (NEPv) or a nonlinear polar decomposition with orthogonal factor dependency (NPDo) and then solve the nonlinear problem via some variations of the self-consistent-field (SCF) iteration. The difficulty, however, lies in designing a proper SCF iteration so that a maximizer is found at the end. Currently, each use of the approach is very much individualized, especially in its convergence analysis to show that the approach does work or otherwise. In this paper, a unifying framework is established. The framework is built upon some basic assumptions. If the basic assumptions are satisfied, globally convergence is guaranteed to a stationary point and during the SCF iterative process that leads to the stationary point, the objective function increases monotonically. Also a notion of atomic functions is proposed, which include commonly used matrix traces of linear and quadratic forms as special ones. It is shown that the basic assumptions are satisfied by atomic functions and by convex compositions of atomic functions. Together they provide a large collection of objectives for which the NEPv/NPDo approach is guaranteed to work.
- [373] arXiv:2305.05863 (replaced) [pdf, html, other]
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Title: Hybrid hyperinterpolation over general regionsComments: 17 pages, 5 figuresSubjects: Numerical Analysis (math.NA)
We present an $\ell^2_2+\ell_1$-regularized discrete least squares approximation over general regions under assumptions of hyperinterpolation, named hybrid hyperinterpolation. Hybrid hyperinterpolation, using a soft thresholding operator and a filter function to shrink the Fourier coefficients approximated by a high-order quadrature rule of a given continuous function with respect to some orthonormal basis, is a combination of Lasso and filtered hyperinterpolations. Hybrid hyperinterpolation inherits features of them to deal with noisy data once the regularization parameter and the filter function are chosen well. We derive $L_2$ errors in theoretical analysis for hybrid hyperinterpolation to approximate continuous functions with noise data on sampling points. Numerical examples illustrate the theoretical results and show that well chosen regularization parameters can enhance the approximation quality over the unit-sphere and the union of disks.
- [374] arXiv:2305.12456 (replaced) [pdf, html, other]
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Title: Recursions and characteristic polynomials of the Rows of the Circuit ArrayEmily J. Evans (1), Russell J. Hendel (2) ((1) Brigham Young University, (2) Towson University)Comments: Still undergoing review by the Fibonacci Quarterly. This latest version incorporates suggestions by an anonymous referee and completely describes the factorizations of the factors of the numerator and denominator of the polynomial circuit arraySubjects: Combinatorics (math.CO)
The 2022 Fibonacci Conference held in Sarajevo introduced the Circuit Array, a two-dimensional array associated with the resistance labels of electrical circuits whose underlying graph when embedded in the Cartesian plane has the form of a triangular n-grid. The presentation used row reduction, a computational method alternative to the traditional method using the Laplacian. The presentation explored reconsidering the Circuit Array in terms of polynomials instead of numbers as a means to facilitate finding patterns. The present paper shows the fruitfulness of this approach. The main result of the current paper states that the characteristic polynomials corresponding to the recursions of single or multi-variable polynomial formulations of the Circuit Array exclusively have powers of 9 as roots. The proof methodology uses the approach of annihilators. Several initial cases and one major sub-case are proven.
- [375] arXiv:2305.12509 (replaced) [pdf, html, other]
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Title: Concerning Keisler Measures over ultraproductsComments: 16 pagesSubjects: Logic (math.LO)
As consequence of the VC theorem, any pseudo-finite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that any finitely approximable measure over an ultraproduct is itself pseudo-finite (even without the NIP assumption). We also analyze the connection between the Morley product and the pseudo-finite product. In particular, we show that if $\mu$ is definable and both $\mu$ and $\nu$ are pseudo-finite, then the Morley product of $\mu$ and $\nu$ agrees with the pseudo-finite product of $\mu$ and $\nu$. Using this observation, we construct generically stable idempotent measures on pseudo-finite NIP groups.
- [376] arXiv:2305.19225 (replaced) [pdf, html, other]
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Title: Learning Decision-Focused Uncertainty Sets in Robust OptimizationSubjects: Optimization and Control (math.OC)
We propose a data-driven technique to automatically learn the uncertainty sets in robust optimization. Our method reshapes the uncertainty sets by minimizing the expected performance across a family of problems subject to guaranteeing constraint satisfaction. Our approach is very flexible and can learn a wide variety of uncertainty sets while preserving tractability. We solve the constrained learning problem using a stochastic augmented Lagrangian method that relies on differentiating the solutions of the robust optimization problems with respect to the parameters of the uncertainty set. Due to the nonsmooth and nonconvex nature of the augmented Lagrangian function, we apply the nonsmooth conservative implicit function theorem to establish convergence to a critical point, which is a feasible solution of the constrained problem under mild assumptions. Using empirical process theory, we show finite-sample probabilistic guarantees of constraint satisfaction for the resulting solutions. Numerical experiments show that our method outperforms traditional approaches in robust and distributionally robust optimization in terms of out-of-sample performance and constraint satisfaction guarantees.
- [377] arXiv:2306.10203 (replaced) [pdf, html, other]
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Title: On a sharper bound on the stability of non-autonomous Schr\"odinger equations and applications to quantum controlComments: arXiv admin note: text overlap with arXiv:2108.00495Subjects: Mathematical Physics (math-ph); Functional Analysis (math.FA); Optimization and Control (math.OC)
We study the stability of the Schrödinger equation generated by time-dependent Hamiltonians with constant form domain. That is, we bound the difference between solutions of the Schrödinger equation by the difference of their Hamiltonians. The stability theorem obtained in this article provides a sharper bound than those previously obtained in the literature. This makes it a potentially useful tool for time-dependent problems in Quantum Physics, in particular for Quantum Control. We apply this result to prove two theorems about global approximate controllability of infinite-dimensional quantum systems. These results improve and generalise existing results on infinite-dimensional quantum control.
- [378] arXiv:2306.14394 (replaced) [pdf, html, other]
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Title: Revisiting $L_q(0\leq q<1)$ Norm Regularized OptimizationSubjects: Optimization and Control (math.OC)
Sparse optimization has seen its advances in recent decades. For scenarios where the true sparsity is unknown, regularization turns out to be a promising solution. Two popular non-convex regularizations are the so-called $L_0$ norm and $L_q$ norm with $q\in(0,1)$, giving rise to extensive research on their induced optimization. However, the majority of these work centered around the main function that is twice continuously differentiable and the best convergence rate for an algorithm solving the optimization with $q\in(0,1)$ is superlinear. This paper explores the $L_q$ norm regularized optimization in a unified way for any $q\in[0,1)$, where the main function has a semismooth gradient. In particular, we establish the first-order and the second-order optimality conditions under mild assumptions and then integrate the proximal operator and semismooth Newton method to develop a proximal semismooth Newton pursuit algorithm. Under the second sufficient condition, the whole sequence generated by the algorithm converges to a unique local minimizer. Moreover, the convergence is superlinear and quadratic if the gradient of the main function is semismooth and strongly semismooth at the local minimizer, respectively. Hence, this paper accomplishes the quadratic rate for an algorithm designed to solve the $L_q$ norm regularization problem for any $q\in(0,1)$. Finally, some numerical experiments have showcased its nice performance when compared with several existing solvers.
- [379] arXiv:2307.00358 (replaced) [pdf, html, other]
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Title: The Error in Multivariate Linear Extrapolation with Applications to Derivative-Free OptimizationComments: 28 pages, 5 figures. arXiv admin note: text overlap with arXiv:2209.12606Subjects: Optimization and Control (math.OC)
We study in this paper the function approximation error of multivariate linear extrapolation. While the sharp error bound of linear interpolation already exists in the literature, linear extrapolation is used far more often in applications such as derivative-free optimization, and its error is not well-studied. A method to numerically compute the sharp error bound is introduced, and several analytical bounds are presented along with the conditions under which they are sharp. The approximation error achievable by quadratic functions and the error bound for the bivariate case are analyzed in depth. Additionally, we provide the convergence theories regarding the simplex derivative-free optimization method as a demonstration of the utility of the derived bounds. All results are under the assumptions that the function being interpolated has Lipschitz continuous gradient and is interpolated on an affinely independent sample set.
- [380] arXiv:2307.05828 (replaced) [pdf, html, other]
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Title: List Privacy Under Function RecoverabilitySubjects: Information Theory (cs.IT)
For a given function of user data, a querier must recover with at least a prescribed probability, the value of the function based on a user-provided query response. Subject to this requirement, the user forms the query response so as to minimize the likelihood of the querier guessing a list of prescribed size to which the data value belongs based on the query response. We obtain a general converse upper bound for maximum list privacy. This bound is shown to be tight for the case of a binary-valued function through an explicit achievability scheme that involves an add-noise query response.
- [381] arXiv:2307.11366 (replaced) [pdf, html, other]
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Title: Many equiprojective polytopesComments: 25 pages, 5 figuresSubjects: Metric Geometry (math.MG); Combinatorics (math.CO)
A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective polytopes. It has been shown recently that the number of combinatorial types of $k$-equiprojective polytopes is at least linear as a function of $k$. Here, it is shown that there are at least $k^{3k/2+o(k)}$ such combinatorial types as $k$ goes to infinity. This relies on the Goodman--Pollack lower bound on the number of order types and on new constructions of equiprojective polytopes via Minkowski sums.
- [382] arXiv:2308.03744 (replaced) [pdf, html, other]
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Title: Lie reductions and exact solutions of dispersionless Nizhnik equationComments: extended version, 43 pages, 1 table, application of results of arXiv:2211.09759Subjects: Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Exactly Solvable and Integrable Systems (nlin.SI)
We exhaustively classify the Lie reductions of the real dispersionless Nizhnik equation to partial differential equations in two independent variables and to ordinary differential equations. Lie and point symmetries of reduced equations are comprehensively studied, including the analysis of which of them correspond to hidden symmetries of the original equation. If necessary, associated Lie reductions of a nonlinear Lax representation of the dispersionless Nizhnik equation are carried out as well. As a result, we construct wide families of new invariant solutions of this equation in explicit form in terms of elementary, Lambert and hypergeometric functions as well as in parametric or implicit form. We show that Lie reductions to algebraic equations lead to no new solutions of this equation in addition to the constructed ones. Multiplicative separation of variables is used for illustrative construction of non-invariant solutions.
- [383] arXiv:2308.06104 (replaced) [pdf, other]
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Title: Morse homology with DG coefficientsComments: 256 pages, v2: Minor modifications and additions to address referee's comments. Change to book formatSubjects: Symplectic Geometry (math.SG); Algebraic Topology (math.AT); Geometric Topology (math.GT)
We develop a theory of Morse homology and cohomology with coefficients in a derived local system, for manifolds and also more generally for colimits of spaces that have the homotopy type of manifolds, with a view towards Floer theory. The model that we adopt for derived, or differential graded (DG) local systems is that of DG modules over chains on the based loop space of a manifold. These encompass both classical (non DG) local systems and chains on fibers of Hurewicz fibrations. We prove that the Morse homology and cohomology groups that we construct are isomorphic to DG Tor and Ext functors. The key ingredient in the definition is a notion of twisting cocycle obtained by evaluating into based loops a coherent system of representatives for the fundamental classes of the moduli spaces of Morse trajectories of arbitrary dimensions. From this perspective, our construction sits midway between classical Morse homology with twisted coefficients and more refined invariants of Floer homotopical flavor. The construction of the twisting cocycle is originally due to Barraud and Cornea with Z/2-coefficients. We show that the twisting cocycle with integer coefficients is equivalent to Brown's universal twisting cocycle. We prove that Morse homology with coefficients in chains on the fiber of a Hurewicz fibration recovers the homology of the total space of the fibration. We study several structural properties of the theory: invariance, functoriality, and Poincaré duality, also in the nonorientable case.
- [384] arXiv:2308.11038 (replaced) [pdf, html, other]
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Title: Logistics Hub Location Optimization: A K-Means and P-Median Model Hybrid Approach Using Road Network DistancesSubjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)
Logistic hubs play a pivotal role in the last-mile delivery distance; even a slight increment in distance negatively impacts the business of the e-commerce industry while also increasing its carbon footprint. The growth of this industry, particularly after Covid-19, has further intensified the need for optimized allocation of resources in an urban environment. In this study, we use a hybrid approach to optimize the placement of logistic hubs. The approach sequentially employs different techniques. Initially, delivery points are clustered using K-Means in relation to their spatial locations. The clustering method utilizes road network distances as opposed to Euclidean distances. Non-road network-based approaches have been avoided since they lead to erroneous and misleading results. Finally, hubs are located using the P-Median method. The P-Median method also incorporates the number of deliveries and population as weights. Real-world delivery data from Muller and Phipps (M&P) is used to demonstrate the effectiveness of the approach. Serving deliveries from the optimal hub locations results in the saving of 815 (10%) meters per delivery.
- [385] arXiv:2308.13395 (replaced) [pdf, html, other]
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Title: A Gr\"obner Approach to Dual-Containing Cyclic Left Module $(\theta,\delta)$-Codes over Finite Commutative Frobenius RingsComments: 20 pages, 11 tablesSubjects: Information Theory (cs.IT); Discrete Mathematics (cs.DM); Quantum Algebra (math.QA); Rings and Algebras (math.RA)
For a skew polynomial ring $R=A[X;\theta,\delta]$ where $A$ is a commutative Frobenius ring, $\theta$ an endomorphism of $A$ and $\delta$ a $\theta$-derivation of $A$, we consider cyclic left module codes $\mathcal{C}=Rg/Rf\subset R/Rf$ where $g$ is a left and right divisor of $f$ in $R$. In this paper, we derive a parity check matrix when $A$ is a finite commutative Frobenius ring using only the framework of skew polynomial rings. We consider rings $A=B[a_1,\ldots,a_s]$ which are free $B$-modules where the restriction of $\delta$ and $\theta$ to $B$ are polynomial maps. If a Gröbner basis can be computed over $B$, then we show that all Euclidean and Hermitian dual-containing codes $\mathcal{C}=Rg/Rf\subset R/Rf$ can be computed using a Gröbner basis. We also give an algorithm to test if the dual code is again a cyclic left module code. We illustrate our approach for rings of order $4$ with non-trivial endomorphism and the Galois ring of characteristic $4$.
- [386] arXiv:2308.13913 (replaced) [pdf, html, other]
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Title: Spectral Theory of Isogeny GraphsComments: Main result improvedSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
We consider finite graphs whose vertexes are supersingular elliptic curves, possibly with level structure, and edges are isogenies. They can be applied to the study of modular forms and to isogeny based cryptography. The main result of this paper is an upper bound on the modules of the eigenvalues of their adjacency matrices, which in particular implies that these graphs are Ramanujan. We also study the asymptotic distribution of the eigenvalues of the adjacency matrices, the number of connected components, the automorphisms of the graphs, and the connection between the graphs and modular forms.
- [387] arXiv:2309.00961 (replaced) [pdf, html, other]
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Title: Asymptotes of macroscopic observables in Gibbs measures of general interacting particle systemsComments: Showed that hypotheses are satisfied for Coulomb, Riesz, interactions, and more. Wrote concentration estimates in terms of more usual normsSubjects: Probability (math.PR)
This paper studies the Gibbs measure of an interacting particle system with a general interaction kernel at various temperature regimes. We are particularly interested in fine features of the convergence to the mean-field density as the number of particles tends to infinity. Our main results are concentration bounds, and estimates on the Laplace transform of fluctuations. The main technique is a regularization procedure for general interaction kernels, based on an associated parabolic flow.
- [388] arXiv:2309.15398 (replaced) [pdf, html, other]
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Title: Finite convergence of Moment-SOS relaxations with non-real radical idealsSubjects: Optimization and Control (math.OC)
We consider the linear conic optimization problem with the cone of nonnegative polynomials. Its dual optimization problem is the generalized moment problem. Moment-SOS relaxations are powerful for solving them. This paper studies finite convergence of the Moment-SOS hierarchy when the constraining set is defined by equations whose ideal may not be real radical. Under the archimedeanness, we show that the Moment-SOS hierarchy has finite convergence if some classical optimality conditions hold at every minimizer of the optimal nonnegative polynomial for the linear conic optimization problem. When the archimedeanness fails (this is the case for unbounded sets), we propose a homogenized Moment-SOS hierarchy and prove its finite convergence under similar assumptions. Furthermore, we also prove the finite convergence of the Moment-SOS hierarchy with denominators. In particular, this paper resolves a conjecture posed in the earlier work.
- [389] arXiv:2309.16660 (replaced) [pdf, other]
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Title: Folding QQ-relations and transfer matrix eigenvalues: towards a unified approach to Bethe ansatz for super spin chainsComments: 101 pages, v2: an appendix added; v3: small improvements; v4 misprints correctedJournal-ref: Nuclear Physics B 1005 (2024) 116607Subjects: Mathematical Physics (math-ph)
Extending the method proposed in [arXiv:1109.5524], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras $U_{q}(gl(2r+1|2s)^{(2)})$, $U_{q}(gl(2r|2s+1)^{(2)})$, $U_{q}(gl(2r|2s)^{(2)})$, $U_{q}(osp(2r|2s)^{(2)})$ and the untwisted quantum affine orthosymplectic superalgebras $U_{q}(osp(2r+1|2s)^{(1)})$ and $U_{q}(osp(2r|2s)^{(1)})$ (and their Yangian counterparts, $Y(osp(2r+1|2s))$ and $Y(osp(2r|2s))$) as reductions (a kind of folding) of those associated with $U_{q}(gl(M|N)^{(1)})$. In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [arXiv:0911.5393,arXiv:0911.5390], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of $U_{q}(gl(M|N)^{(1)})$.
- [390] arXiv:2310.10990 (replaced) [pdf, other]
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Title: A second-order exponential integration constraint energy minimizing generalized multiscale method for parabolic problemsSubjects: Numerical Analysis (math.NA)
This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method.
- [391] arXiv:2310.14911 (replaced) [pdf, html, other]
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Title: Joint Fronthaul Load Balancing and Computation Resource Allocation in Cell-Free User-Centric Massive MIMO NetworksComments: 15 pages, 7 figures, accepted to be published in IEEE Transactions on Wireless CommunicationsSubjects: Information Theory (cs.IT)
We consider scalable cell-free massive multiple-input multiple-output networks under an open radio access network paradigm comprising user equipments (UEs), radio units (RUs), and decentralized processing units (DUs). UEs are served by dynamically allocated user-centric clusters of RUs. The corresponding cluster processors (implementing the physical layer for each user) are hosted by the DUs as software-defined virtual network functions. Unlike the current literature, mainly focused on the characterization of the user rates under unrestricted fronthaul communication and computation, in this work we explicitly take into account the fronthaul topology, the limited fronthaul communication capacity, and computation constraints at the DUs. In particular, we systematically address the new problem of joint fronthaul load balancing and allocation of the computation resource. As a consequence of our new optimization framework, we present representative numerical results highlighting the existence of an optimal number of quantization bits in the analog-to-digital conversion at the RUs.
- [392] arXiv:2310.15619 (replaced) [pdf, html, other]
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Title: Spanning trees in $\mathbb{Z}$-covers of a finite graph and Mahler measuresComments: Final version. To appear in the Journal of the Australian Mathematical SocietySubjects: Number Theory (math.NT); Combinatorics (math.CO)
Using the special value at $u=1$ of Artin-Ihara $L$-functions, we associate to every $\mathbb{Z}$-cover of a finite connected graph a polynomial which we call the \emph{Ihara polynomial}. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce-Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specializing to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and $I$-graphs (including the generalized Petersen graphs). We also express the $p$-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the $p$-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb{Z}$-cover, our result gives us back Lengyel's calculation of the $p$-adic valuations of Fibonacci numbers.
- [393] arXiv:2310.16442 (replaced) [pdf, html, other]
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Title: Right preconditioned GMRES for arbitrary singular systemsComments: 25 pages, 15 figuresSubjects: Numerical Analysis (math.NA)
Brown and Walker (1997) showed that GMRES determines a least squares solution of $ A x = b $ where $ A \in {\bf R}^{n \times n} $ without breakdown for arbitrary $ b, x_0 \in {\bf R}^n $ if and only if $A$ is range-symmetric, i.e. $ {\cal R} (A^{\rm T}) = {\cal R} (A) $, where $ A $ may be singular and $ b $ may not be in the range space ${\cal R} A)$ of $A$. In this paper, we propose applying GMRES to $ A C A^{\rm T} z = b $, where $ C \in {\bf R}^{n \times n} $ is symmetric positive definite. This determines a least squares solution $ x = CA^{\rm T} z $ of $ A x = b $ without breakdown for arbitrary (singular) matrix $A \in {\bf R}^{n \times n}$ and $ b \in {\bf R}^n $. To make the method numerically stable, we propose using the pseudoinverse with an appropriate threshold parameter to suppress the influence of tiny singular values when solving the severely ill-conditioned Hessenberg systems which arise in the Arnoldi process of GMRES when solving inconsistent range-symmetric systems. Numerical experiments show that the method taking $C$ to be the identity matrix and the inverse matrix of the diagonal matrix whose diagonal elements are the diagonal of $A A^{\rm T}$ gives a least squares solution even when $A$ is not range-symmetric, including the case when $ {\rm index}(A) >1$.
- [394] arXiv:2310.20394 (replaced) [pdf, html, other]
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Title: The miniversal deformation of certain complete intersection monomial curvesComments: v2: 30 pages, minor typos fixed; v1: 29 pages, comments are very welcomeSubjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Combinatorics (math.CO)
The aim of this paper is to provide an explicit basis of the miniversal deformation of a monomial curve defined by a free semigroup -- these curves make up a notable family of complete intersection monomial curves. First, we dispense a general decomposition result of a basis of the miniversal deformation of any complete intersection monomial curve. As a consequence, we explicitly calculate this basis in the particular case of a monomial curve defined from a free semigroup. This explicit computation yields some estimates for the dimension of the moduli space of these family of curves.
- [395] arXiv:2311.05372 (replaced) [pdf, html, other]
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Title: Joint Angle and Delay Cram\'{e}r-Rao Bound Optimization for ISACSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
In this paper, we study a multi-input multi-output (MIMO) beamforming design in an integrated sensing and communication (ISAC) system, in which an ISAC base station (BS) is used to communicate with multiple downlink users and simultaneously the communication signals are reused for sensing multiple targets. Our interested sensing parameters are the angle and delay information of the targets, which can be used to locate these targets. Under this consideration, we first derive the Cramér-Rao bound (CRB) for joint angle and delay estimation. Then, we optimize the transmit beamforming at the BS to minimize the CRB, subject to the communication rate requirement and the maximum transmit power constraint. In particular, we obtain the closed-form optimal solution in the case of single-target and single-user, and in the case of multi-target and multi-user scenario, the sparsity of the optimal solution is proven, leading to a reduction in computational complexity during optimization. The numerical results demonstrate that the optimized beamforming yields excellent positioning performance and effectively reduces the requirement for a large number of antennas at the BS.
- [396] arXiv:2311.05901 (replaced) [pdf, html, other]
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Title: Chromatic numbers for facet colouring of some generalised associahedraComments: This is a new, fully rewritten version of the paper, 16 pagesSubjects: Combinatorics (math.CO)
The chromatic number related to a colouring of facets of certain classes of generalised associahedra is studied. The exact values are obtained for permutohedra, associahedra and simple permutoassociahedra, while lower and upper bounds are established for cyclohedra and stellohedra. The asymptotic values of the chromatic numbers for associahedra, cyclohedra and simple permutoassociahedra are given.
- [397] arXiv:2311.09738 (replaced) [pdf, html, other]
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Title: A Projection-Free Method for Solving Convex Bilevel Optimization ProblemsSubjects: Optimization and Control (math.OC)
When faced with multiple minima of an "inner-level" convex optimization problem, the convex bilevel optimization problem selects an optimal solution which also minimizes an auxiliary "outer-level" convex objective of interest. Bilevel optimization requires a different approach compared to single-level optimization problems since the set of minimizers for the inner-level objective is not given explicitly. In this paper, we propose a new projection-free method for convex bilevel optimization which require only a linear optimization oracle over the base domain. We establish $O(t^{-1/2})$ convergence rate guarantees for our method in terms of both inner- and outer-level objectives, and demonstrate how additional assumptions such as quadratic growth and strong convexity result in accelerated rates of up to $O(t^{-1})$ and $O(t^{-2/3})$ for inner- and outer-levels respectively. Lastly, we conduct a numerical study to demonstrate the performance of our method.
- [398] arXiv:2311.10071 (replaced) [pdf, other]
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Title: Moduli space of rank three logarithmic connections on the projective line with three polesComments: 54 pages. To appear in Annales de la Faculté des sciences de ToulouseSubjects: Algebraic Geometry (math.AG)
In this paper, we describe the moduli space of rank three parabolic logarithmic connections on the projective line with three poles for any local exponents. In particular, we show that the family of moduli spaces of rank three parabolic $\phi$-connections on the projective line with three poles is isomorphic to the family of $A^{(1)*}_2$-surfaces in Sakai's classification of Painlevé equations. Through this description, we investigate the relation between the apparent singularities and underlying parabolic bundles.
- [399] arXiv:2311.10527 (replaced) [pdf, html, other]
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Title: Functional degrees and arithmetic applications III: Beyond Prime ExponentComments: 27 pagesSubjects: Number Theory (math.NT); Group Theory (math.GR)
Continuing our work on group-theoretic generalizations of the prime Ax-Katz Theorem, we give a lower bound on the $p$-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\ldots,f_r)$ of $r$ maps $f_j:A\rightarrow B_j$ between arbitrary finite commutative groups $A$ and $B_j$ in terms of the invariant factors of $A, B_1,B_2,\dotsc,B_r$ and the \emph{functional degrees} of the maps $f_1,f_2,\dotsc,f_r$.
- [400] arXiv:2311.14023 (replaced) [pdf, html, other]
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Title: Algorithm-agnostic low-rank approximation of operator monotone matrix functionsSubjects: Numerical Analysis (math.NA); Data Structures and Algorithms (cs.DS)
Low-rank approximation of a matrix function, $f(A)$, is an important task in computational mathematics. Most methods require direct access to $f(A)$, which is often considerably more expensive than accessing $A$. Persson and Kressner (SIMAX 2023) avoid this issue for symmetric positive semidefinite matrices by proposing funNyström, which first constructs a Nyström approximation to $A$ using subspace iteration, and then uses the approximation to directly obtain a low-rank approximation for $f(A)$. They prove that the method yields a near-optimal approximation whenever $f$ is a continuous operator monotone function with $f(0) = 0$.
We significantly generalize the results of Persson and Kressner beyond subspace iteration. We show that if $\widehat{A}$ is a near-optimal low-rank Nyström approximation to $A$ then $f(\widehat{A})$ is a near-optimal low-rank approximation to $f(A)$, independently of how $\widehat{A}$ is computed. Further, we show sufficient conditions for a basis $Q$ to produce a near-optimal Nyström approximation $\widehat{A} = AQ(Q^T AQ)^{\dagger} Q^T A$. We use these results to establish that many common low-rank approximation methods produce near-optimal Nyström approximations to $A$ and therefore to $f(A)$. - [401] arXiv:2311.14074 (replaced) [pdf, html, other]
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Title: A special class of $k$-harmonic maps inducing calibrated fibrationsComments: 25 pages, no figures. Version 2: Various minor improvements based on referee's report, including strengthening results showing that the two formulations of the Smith immersion/submersion equations, in terms of calibration forms or in terms of cross products, are always equivalent. Final version, to appear in Mathematical Research LettersSubjects: Differential Geometry (math.DG)
We consider two special classes of $k$-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map $u \colon (L^k, g) \to (M^n, h)$ where $k \leq n$ and $\alpha$ is a calibration $k$-form on $M$. Away from the critical set, the image is an $\alpha$-calibrated submanifold of $M$. These were previously studied by Cheng-Karigiannis-Madnick when $\alpha$ was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map $u \colon (M^n, h) \to (L^k, g)$ where $n \geq k$ and $\alpha$ is a calibration $(n-k)$-form on $M$. Away from the critical set, the fibres $u^{-1} \{ u(x) \}$ are $\alpha$-calibrated submanifolds of $M$.
We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where $(M, h)$ are the Bryant-Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger-Yau-Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the $\mathrm{G}_2$ version by Gukov-Yau-Zaslow in terms of coassociative fibrations; and we present several open questions for future study. - [402] arXiv:2311.14862 (replaced) [pdf, html, other]
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Title: Probabilistic Galois Theory in Function FieldsComments: v2: changed conditional results to depend on a more standard version of Chowla's conjecture. Added references acknowledgments. Fixed small errors, typos and style issues. v3: fixed a few small errorsSubjects: Number Theory (math.NT)
We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial $f=y^n+\sum_{i=0}^{n-1}a_i(x)y^i\in\mathbb F_q[x][y]$ with i.i.d coefficients $a_i$ taking values in the set $\{a(x)\in\mathbb{F}_q[x]: \mathrm{deg}\, a\leq d\}$ with uniform probability, is irreducible with probability tending to $1-\frac{1}{q^d}$ as $n\to\infty$, where $d$ and $q$ are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group $A_n$. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over $\mathbb{F}_q[x]$, then the Galois group of this polynomial is actually equal to the symmetric group $S_n$ with probability tending to $1-\frac{1}{q^d}$. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with $n$ fixed and $d\to\infty$.
- [403] arXiv:2311.16600 (replaced) [pdf, html, other]
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Title: Morphisms of Cuntz-Pimsner algebras from completely positive mapsComments: 32 pagesSubjects: Operator Algebras (math.OA)
We introduce positive correspondences as right C*-modules with left actions given by completely positive maps. Positive correspondences form a semi-category that contains the C*-correspondence (Enchilada) category as a "retract". Kasparov's KSGNS construction provides a semi-functor from this semi-category onto the C*-correspondence category. The need for left actions by completely positive maps appears naturally when we consider morphisms between Cuntz-Pimsner algebras, and we describe classes of examples arising from projections on C*-correspondences and Fock spaces, as well as examples from conjugation by bi-Hilbertian bimodules of finite index.
- [404] arXiv:2311.18132 (replaced) [pdf, other]
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Title: The Brauer Group of $\mathscr{Y}_0(2)$Comments: Comments welcome, 34 pagesSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We determine the Brauer group of the Deligne-Mumford stack $\mathscr{Y}_0(2)$, the moduli space of elliptic curves with a marked $2$-torsion subgroup over bases of arithmetic interest. Antieau and Meier determine the Brauer group for $\mathscr{M}_{1,1}$, the moduli stack of elliptic curves by exploiting the fact it is covered by the Legendre family and using the Hochschild-Serre spectral sequence. Over an algebraically closed field, Shin uses the coarse space map to determine the Brauer group of $\mathscr{M}_{1,1}$. We combine techniques from both papers to determine the Brauer group of $\mathscr{Y}_0(2)$.
- [405] arXiv:2312.03092 (replaced) [pdf, html, other]
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Title: Coloring GroupsSubjects: Combinatorics (math.CO); Group Theory (math.GR)
We introduce coloring groups, which are permutation groups obtained from a proper edge coloring of a graph. These groups generalize the generalized toggle groups of Striker (which themselves generalize the toggle groups introduced by Cameron and Fon-der-Flaass). We present some general results connecting the structure of a coloring group to the structure of its graph coloring, providing graph-theoretic characterizations of the centralizer and primitivity of a coloring group. We apply these results particularly to generalized toggle groups arising from trees as well as coloring groups arising from the independence posets introduced by Thomas and Williams.
- [406] arXiv:2312.04741 (replaced) [pdf, html, other]
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Title: Computability in infinite Galois theory and algorithmically random algebraic fieldsSubjects: Logic (math.LO); Number Theory (math.NT)
We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of $\mathbb{Q}$ that are random according to our definition. We show that there are noncomputable algebraic fields which are not random. We also partially characterize the index set, relative to an oracle, of the set of random algebraic fields computable relative to that oracle.
In order to carry out this investigation of randomness for fields, we develop computability in the context of infinite Galois theory (where the relevant Galois groups are uncountable), including definitions of computable and computably enumerable Galois groups and computability of Haar measure on the Galois groups. - [407] arXiv:2312.04776 (replaced) [pdf, html, other]
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Title: Asymptotic convergence of restarted Anderson acceleration for certain normal linear systemsComments: This version is a significant update of previous versions, including changes to the title and many other major changes throughout the documentSubjects: Numerical Analysis (math.NA)
Anderson acceleration (AA) is widely used for accelerating the convergence of an underlying fixed-point iteration $\bm{x}_{k+1} = \bm{q}( \bm{x}_{k} )$, $k = 0, 1, \ldots$, with $\bm{x}_k \in \mathbb{R}^n$, $\bm{q} \colon \mathbb{R}^n \to \mathbb{R}^n$. Despite AA's widespread use, relatively little is understood theoretically about the extent to which it may accelerate the underlying fixed-point iteration. To this end, we analyze a restarted variant of AA with a restart size of one, a method closely related to GMRES(1). We consider the case of $\bm{q}( \bm{x} ) = M \bm{x} + \bm{b}$ with matrix $M \in \mathbb{R}^{n \times n}$ either symmetric or skew-symmetric. For both classes of $M$ we compute the worst-case root-average asymptotic convergence factor of the AA method, partially relying on conjecture in the symmetric setting, proving that it is strictly smaller than that of the underlying fixed-point iteration. For symmetric $M$, we show that the AA residual iteration corresponds to a fixed-point iteration for solving an eigenvector-dependent nonlinear eigenvalue problem (NEPv), and we show how this can result in the convergence factor strongly depending on the initial iterate, which we quantify exactly in certain special cases. Conversely, for skew-symmetric $M$ we show that the AA residual iteration is closely related to a power iteration for $M$, and how this results in the convergence factor being independent of the initial iterate. Supporting numerical results are given, which also indicate the theory is applicable to the more general setting of nonlinear $\bm{q}$ with Jacobian at the fixed point that is symmetric or skew symmetric.
- [408] arXiv:2312.07230 (replaced) [pdf, html, other]
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Title: Operadic structure of boundary conditions for two-dimensional Markov Gaussian random fields on the latticeComments: 43 pages, shortened version for publication requirements (see version 1 for more content about operads)Subjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
The theory of Markov processes on the square lattice has been given recently by the second author a new algebraic description in terms of operads. In particular, this new approach allows for a nice description of invariant boundary conditions and infinite-volume Gibbs measures. This theory comes with new algebraic objects which have not been constructed on any non trivial model yet. In this article, the main objective is to exhibit and understand these structures in the particular case of Gaussian Markov fields on the two-dimensional square lattice. This article, in the Gaussian framework, is the first time where all the operadic constructions -- products and eigen-elements up to morphisms -- introduced by the second author are defined rigorously. We also relate these constructions to more classical approach such as the transfer matrix of statistical mechanics and the Fourier transform. The description of half-strips and corners is new and requires the introduction of new operations such as folding. From the probabilistic point of view, we also show that the operadic products on the boundaries are not easily defined and most operations are lifted to the level of parameter spaces, here quadratic forms through Schur complements.
- [409] arXiv:2312.08280 (replaced) [pdf, html, other]
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Title: New High-Order Numerical Methods for Hyperbolic Systems of Nonlinear PDEs with UncertaintiesAlina Chertock, Michael Herty, Arsen S. Iskhakov, Safa Janajra, Alexander Kurganov, Maria Lukacova-MedvidovaSubjects: Numerical Analysis (math.NA)
In this paper, we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations (PDEs) with uncertainties. The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory (WENO) interpolations in (multidimensional) random space combined with second-order piecewise linear reconstruction in physical space. Compared with spectral approximations in the random space, the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy. The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations. In the latter case, the methods are also proven to be well-balanced and positivity-preserving.
- [410] arXiv:2312.08516 (replaced) [pdf, html, other]
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Title: A shooting-Newton procedure for solving fractional terminal value problemsComments: 24 pages, 4 figuresSubjects: Numerical Analysis (math.NA)
In this paper we consider the numerical solution of fractional terminal value problems (FDE-TVPs). In particular, the proposed procedure uses a Newton-type iteration which is particularly efficient when coupled with a recently-introduced step-by-step procedure for solving fractional initial value problems (FDE-IVPs), able to produce spectrally accurate solutions of FDE problems. Some numerical tests are reported to make evidence of its effectiveness.
- [411] arXiv:2312.13231 (replaced) [pdf, html, other]
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Title: Random Matrices and the Free Energy of Ising-Like Models with DisorderComments: 31 pages, 12 figuresSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)
We consider an Ising model with quenched surface disorder, the disorder average of the free energy is the main object of interest. Explicit expressions for the free energy distribution are difficult to obtain if the quenched surface spins take values of $\pm 1$. Thus, we choose a different approach and model the surface disorder by Gaussian random matrices. The distribution of the free energy is calculated. We chose skew-circulant random matrices and analytically compute the characteristic function of the free energy distribution. From the characteristic function we numerically calculate the distribution and show that it becomes log-normal for sufficiently large dimensions of the disorder matrices, and in the limit of infinitely large matrices tends to a Gaussian. Furthermore, we establish a connection to the central limit theorem.
- [412] arXiv:2312.16655 (replaced) [pdf, html, other]
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Title: Deformation of Fuchsian representations and proper affine actionsComments: 30 pages, statement of the main theorem improved, abstract changed, minor errors correctedSubjects: Geometric Topology (math.GT); Differential Geometry (math.DG)
The main goal of this article is to generalize Mess' work and using results from Labourie--Wentworth, Potrie--Sambarino and Smilga, to show that inside Hitchin representations, infinitesimal deformations of Fuchsian representations of a cocompact surface group do not act properly along the directions corresponding to the sum of a mixed odd differential and a $2m$-differential for any $1\leq m \leq \lfloor\frac{n}{2}\rfloor$.
In the process, we introduce affine versions of cross ratios and triple ratios. We introduce Margulis invariants and relate them with affine crossratios and infinitesimal Jordan projections. We obtain a general equivalent criterion for existence of proper affine actions in terms of the structure of the Margulis invariant spectra. Also, using a stability argument we show the existence of proper affine actions of non-abelian free groups whose linear part is a Hitchin representation. - [413] arXiv:2312.17405 (replaced) [pdf, html, other]
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Title: Renormalization of Translated Cone Exchange TransformationsComments: 32 pages, 12 figuresSubjects: Dynamical Systems (math.DS)
In this paper, we investigate a class of non-invertible piecewise isometries on the upper half-plane known as Translated Cone Exchanges. These maps include a simple interval exchange on a boundary we call the baseline. We provide a geometric construction for the first return map to a neighbourhood of the vertex of the middle cone for a large class of parameters, then we show a recurrence in the first return map tied to Diophantine properties of the parameters, and subsequently prove the infinite renormalizability of the first return map for these parameters.
- [414] arXiv:2401.03460 (replaced) [pdf, html, other]
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Title: Five tori in $S^4$Comments: 30 pages, 22 figuresSubjects: Geometric Topology (math.GT); Differential Geometry (math.DG)
Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic complement. We describe $L$ explicitly and study its properties, and discover that $L$ is in many aspects similar to the Borromean rings in $S^3$. In particular the following hold: (1) Any two tori in $L$ are unlinked, but three are not; (2) The complement $M=S^4\setminus L$ is integral arithmetic hyperbolic; (3) The symmetry group of $L$ acts $k$-transitively on its components for all $k$; (4) The double branched covering over $L$ has geometry $\mathbb H^2 \times \mathbb H^2$; (5) The fundamental group of $M$ has a nice presentation via commutators; (6) The Alexander ideal is explicit; (7) Every class $x\in H^1(M, \mathbb Z)=\mathbb Z^5$ with $x_i\neq 0$ is represented by a perfect circle-valued Morse function; (8) By longitudinal Dehn surgery along $L$ we get a closed 4-manifold with fundamental group $\mathbb Z^5$; (9) The link $L$ can be put in perfect position.
This leads also to the first descriptions of a cusped hyperbolic 4-manifold as a complement of tori in $\mathbb R \mathbb P^4$ and of some explicit Lagrangian tori in the product of two surfaces of genus two. - [415] arXiv:2401.03684 (replaced) [pdf, html, other]
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Title: The Two Lives of the GrassmannianComments: 19 pages. Theorem 3.8 and Remark 5.11 are newSubjects: Algebraic Geometry (math.AG); Combinatorics (math.CO); Optimization and Control (math.OC)
The real Grassmannian is both a projective variety (via Plücker coordinates) and an affine variety (via orthogonal projections). We connect these two representations, and we develop the commutative algebra of the latter variety. We introduce the squared Grassmannian, and we study applications to determinantal point processes in statistics.
- [416] arXiv:2401.07146 (replaced) [pdf, html, other]
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Title: The spectrum of the Vladimirov sub-Laplacian on the compact Heisenberg groupSubjects: Representation Theory (math.RT)
Let $p>2$ be a prime number. In this short note, we calculate explicitly the unitary dual and the matrix coefficients of the Heisenberg group over the $p$-adic integers. As an application, we consider directional Vladimirov-Taibleson derivatives, and some polynomials in these operators. In particular, we calculate explicitly the spectrum of the Vladimirov sub-Laplacian, and show how it provides a non-trivial example of a sub-elliptic operator on compact graded $p$-adic Lie groups.
- [417] arXiv:2401.07823 (replaced) [pdf, html, other]
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Title: A three-grid high-order immersed finite element method for the analysis of CAD modelsSubjects: Numerical Analysis (math.NA)
The automated finite element analysis of complex CAD models using boundary-fitted meshes is rife with difficulties. Immersed finite element methods are intrinsically more robust but usually less accurate. In this work, we introduce an efficient, robust, high-order immersed finite element method for complex CAD models. Our approach relies on three adaptive structured grids: a geometry grid for representing the implicit geometry, a finite element grid for discretising physical fields and a quadrature grid for evaluating the finite element integrals. The geometry grid is a sparse VDB (Volumetric Dynamic B+ tree) grid that is highly refined close to physical domain boundaries. The finite element grid consists of a forest of octree grids distributed over several processors, and the quadrature grid in each finite element cell is an octree grid constructed in a bottom-up fashion. We discretise physical fields on the finite element grid using high-order Lagrange basis functions. The resolution of the quadrature grid ensures that finite element integrals are evaluated with sufficient accuracy and that any sub-grid geometric features, like small holes or corners, are resolved up to a desired resolution. The conceptual simplicity and modularity of our approach make it possible to reuse open-source libraries, i.e. openVDB and p4est for implementing the geometry and finite element grids, respectively, and BDDCML for iteratively solving the discrete systems of equations in parallel using domain decomposition. We demonstrate the efficiency and robustness of the proposed approach by solving the Poisson equation on domains given by complex CAD models and discretised with tens of millions of degrees of freedom.
- [418] arXiv:2401.09863 (replaced) [pdf, html, other]
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Title: Well-posedness results for general reaction-diffusion transport of oxygen in encapsulated cellsSubjects: Analysis of PDEs (math.AP)
In this paper, we provide well-posedness results for nonlinear parabolic PDEs given by reaction-diffusion equations describing the concentration of oxygen in encapsulated cells. The cells are described in terms of a core and a shell, which introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. In addition, the cells are subject to general nonlinear consumption of oxygen. As no monotonicity condition is imposed on the consumption monotone operator theory cannot be used. Moreover, the discontinuity in the diffusion coefficient bars us to apply classical results. However, by directly applying a Galerkin method we obtain uniqueness and existence of the strong form solution. These results will provide the basis to study the dynamics of cells in critical states.
- [419] arXiv:2401.10923 (replaced) [pdf, other]
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Title: Online estimation of the inverse of the Hessian for stochastic optimization with application to universal stochastic Newton algorithmsSubjects: Optimization and Control (math.OC); Machine Learning (stat.ML)
This paper addresses second-order stochastic optimization for estimating the minimizer of a convex function written as an expectation. A direct recursive estimation technique for the inverse Hessian matrix using a Robbins-Monro procedure is introduced. This approach enables to drastically reduces computational complexity. Above all, it allows to develop universal stochastic Newton methods and investigate the asymptotic efficiency of the proposed approach. This work so expands the application scope of secondorder algorithms in stochastic optimization.
- [420] arXiv:2401.11622 (replaced) [pdf, html, other]
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Title: The Markov-Chain Polytope with ApplicationsComments: v2 corrects some typos that were present in v1Subjects: Information Theory (cs.IT)
This paper addresses the problem of finding a minimum-cost $m$-state Markov chain $(S_0,\ldots,S_{m-1})$ in a large set of chains. The chains studied have a reward associated with each state. The cost of a chain is its "gain", i.e., its average reward under its stationary distribution.
Specifically, for each $k=0,\ldots,m-1$ there is a known set ${\mathbb S}_k$ of type-$k$ states. A permissible Markov chain contains exactly one state of each type; the problem is to find a minimum-cost permissible chain.
The original motivation was to find a cheapest binary AIFV-$m$ lossless code on a source alphabet of size $n$. Such a code is an $m$-tuple of trees, in which each tree can be viewed as a Markov Chain state. This formulation was then used to address other problems in lossless compression. The known solution techniques for finding minimum-cost Markov chains were iterative and ran in exponential time.
This paper shows how to map every possible type-$k$ state into a type-$k$ hyperplane and then define a "Markov Chain Polytope" as the lower envelope of all such hyperplanes. Finding a minimum-cost Markov chain can then be shown to be equivalent to finding a "highest" point on this polytope.
The local optimization procedures used in the previous iterative algorithms are shown to be separation oracles for this polytope. Since these were often polynomial time, an application of the Ellipsoid method immediately leads to polynomial time algorithms for these problems. - [421] arXiv:2401.13232 (replaced) [pdf, html, other]
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Title: Distributed Source Coding Using Constrained-Random-Number GeneratorsComments: 25 pages, this is the extended version of the paper submitted to ISIT2024, appendices are the revision of arXiv:2206.00792, (v2) add Refs. [10][31], (v3) fix typosSubjects: Information Theory (cs.IT)
This paper investigates the general distributed lossless/lossy source coding formulated by Jana and Blahut. Their multi-letter rate-distortion region, an alternative to the region derived by Yang and Qin, is characterized by entropy functions for arbitrary general correlated sources. Achievability is shown by constructing a code based on constrained-random number generators.
- [422] arXiv:2401.18033 (replaced) [pdf, html, other]
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Title: Optimal boundary regularity and a Hopf-type lemma for Dirichlet problems involving the logarithmic LaplacianComments: Revised version, 30 pagesSubjects: Analysis of PDEs (math.AP)
We study the optimal boundary regularity of solutions to Dirichlet problems involving the logarithmic Laplacian. Our proofs are based on the construction of suitable barriers via the Kelvin transform and direct computations. As applications of our results, we show a Hopf-type Lemma for nonnegative weak solutions and the uniqueness of solutions to some nonlinear problems.
- [423] arXiv:2402.01609 (replaced) [pdf, other]
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Title: Estimating Max-Stable Random Vectors with Discrete Spectral Measure using Model-Based ClusteringComments: 42 pages, 6 figuresSubjects: Statistics Theory (math.ST)
This study introduces a novel estimation method for the entries and structure of a matrix $A$ in the linear factor model $\textbf{X} = A\textbf{Z} + \textbf{E}$. This is applied to an observable vector $\textbf{X} \in \mathbb{R}^d$ with $\textbf{Z} \in \mathbb{R}^K$, a vector composed of independently regularly varying random variables, and independent lighter tail noise $\textbf{E} \in \mathbb{R}^d$. This leads to max-linear models treated in classical multivariate extreme value theory. The spectral of the limit distribution is subsequently discrete and completely characterised by the matrix $A$. Every max-stable random vector with discrete spectral measure can be written as a max-linear model. Each row of the matrix $A$ is supposed to be both scaled and sparse. Additionally, the value of $K$ is not known a priori. The problem of identifying the matrix $A$ from its matrix of pairwise extremal correlation is addressed. In the presence of pure variables, which are elements of $\textbf{X}$ linked, through $A$, to a single latent factor, the matrix $A$ can be reconstructed from the extremal correlation matrix. Our proofs of identifiability are constructive and pave the way for our innovative estimation for determining the number of factors $K$ and the matrix $A$ from $n$ weakly dependent observations on $\textbf{X}$. We apply the suggested method to weekly maxima rainfall and wildfires to illustrate its applicability.
- [424] arXiv:2402.04407 (replaced) [pdf, html, other]
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Title: Sharp Lower Bounds on the Manifold Widths of Sobolev and Besov SpacesSubjects: Numerical Analysis (math.NA)
We consider the problem of determining the manifold $n$-widths of Sobolev and Besov spaces with error measured in the $L_p$-norm. The manifold widths control how efficiently these spaces can be approximated by general non-linear parametric methods with the restriction that the parameter selection and parameterization maps must be continuous. Existing upper and lower bounds only match when the Sobolev or Besov smoothness index $q$ satisfies $q\leq p$ or $1 \leq p \leq 2$. We close this gap and obtain sharp lower bounds for all $1 \leq p,q \leq \infty$ for which a compact embedding holds. A key part of our analysis is to determine the exact value of the manifold widths of finite dimensional $\ell^M_q$-balls in the $\ell_p$-norm when $p\leq q$. Although this result is not new, we provide a new proof and apply it to lower bounding the manifold widths of Sobolev and Besov spaces. Our results show that the Bernstein widths, which are typically used to lower bound the manifold widths, decay asymptotically faster than the manifold widths in many cases.
- [425] arXiv:2402.05470 (replaced) [pdf, html, other]
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Title: Complete immersions with constant scalar curvature and flat normal bundleComments: A normalisation error corrected. Missing references added. Introduction rewrittenSubjects: Differential Geometry (math.DG)
Let $M^n, n\geq 3,$ be a complete Riemannian manifold of constant scalar curvature $R$ and $f: M^n\rightarrow M^{n+k}(c)$ be an isometric immersion into a space form with flat normal bundle. Assume that $f$ admits a principal normal vector field of multiplicity $n-1$ at each point. Our main result is of a global character and states that (i) $R\geq 0$ if $c= 0;$\ (ii) $ R> (n-1)(n-2)c$ if $c> 0;$ and (iii) $R\geq n(n-1)c$ if $c< 0.$ The inequalities are optimal. Our second result is of a local character and states that if in addition we assume that $c$ is non-negative and the mean curvature field of $f$ is parallel then $M^n$ has non-negative sectional curvature.
- [426] arXiv:2402.05638 (replaced) [pdf, other]
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Title: Interval maps with dense periodicityComments: The main change is an improved corollary 5.10Subjects: Dynamical Systems (math.DS)
We consider the class of interval maps with dense set of periodic points CP and its closure Cl(CP) equipped with the metric of uniform convergence. Besides studying basic topological properties and density results in the spaces CP and Cl(CP) we prove that Cl(CP) is dynamically characterized as the set of interval maps for which every point is chain-recurrent. Furthermore, we prove that a strong topological expansion property called topological exactness (or leo property) is attained on the open dense set of maps in CP and on a residual set in Cl(CP). Moreover, we show that every second category set in CP and Cl(CP) is rich in a sense that it contains uncountably many conjugacy classes. An analogous conclusion also holds in the setting of interval maps preserving any fixed non-atomic probability measure with full support. Finally, we give a detailed description of the structure of periodic points of generic maps in CP and Cl(CP) and show that generic maps in CP and Cl(CP) satisfy the shadowing property.
- [427] arXiv:2402.06271 (replaced) [pdf, html, other]
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Title: Adaptive proximal gradient methods are universal without approximationSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
We show that adaptive proximal gradient methods for convex problems are not restricted to traditional Lipschitzian assumptions. Our analysis reveals that a class of linesearch-free methods is still convergent under mere local Hölder gradient continuity, covering in particular continuously differentiable semi-algebraic functions. To mitigate the lack of local Lipschitz continuity, popular approaches revolve around $\varepsilon$-oracles and/or linesearch procedures. In contrast, we exploit plain Hölder inequalities not entailing any approximation, all while retaining the linesearch-free nature of adaptive schemes. Furthermore, we prove full sequence convergence without prior knowledge of local Hölder constants nor of the order of Hölder continuity. Numerical experiments make comparisons with baseline methods on diverse tasks from machine learning covering both the locally and the globally Hölder setting.
- [428] arXiv:2402.06623 (replaced) [pdf, html, other]
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Title: Taut smoothings of arcs and curvesComments: Second version: 32 pages, 22 figures. Exposition has been improved, parts of the text have been reorganised, and proofs have been expanded / details clarified. Theorem 4.8 is newSubjects: Geometric Topology (math.GT); Combinatorics (math.CO); Differential Geometry (math.DG)
We study the geometric and combinatorial effect of smoothing an intersection point in a collection of arcs or curves on a surface. We prove that all taut arcs with fixed endpoints and all taut 1-manifolds with at least two non-disjoint components on an orientable surface with negative Euler characteristic admit a taut smoothing, and also that all taut arcs with free endpoints admit a smoothing that is either taut or becomes taut after removing at most one intersection. We deduce that for every Riemannian metric on a surface, the shortest properly immersed arcs with at least $k$ self-intersections have exactly $k$ self-intersections when the endpoints of the arc are fixed, and at most $k+1$ self-intersections otherwise, and that the arc length spectrum is "coarsely ordered" by self-intersection number. Along the way, we obtain partial analogous results in the case of curves.
- [429] arXiv:2402.07355 (replaced) [pdf, html, other]
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Title: Sampling from the Mean-Field Stationary DistributionSubjects: Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML)
We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term. Our main insight is to decouple the two key aspects of this problem: (1) approximation of the mean-field SDE via a finite-particle system, via uniform-in-time propagation of chaos, and (2) sampling from the finite-particle stationary distribution, via standard log-concave samplers. Our approach is conceptually simpler and its flexibility allows for incorporating the state-of-the-art for both algorithms and theory. This leads to improved guarantees in numerous settings, including better guarantees for optimizing certain two-layer neural networks in the mean-field regime. A key technical contribution is to establish a new uniform-in-$N$ log-Sobolev inequality for the stationary distribution of the mean-field Langevin dynamics.
- [430] arXiv:2402.08281 (replaced) [pdf, other]
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Title: On two modular geometric realizations of an affine Hecke algebraComments: v1: 107 pages; v2: 110 pages, minor revisionSubjects: Representation Theory (math.RT)
In this paper we construct equivalences of monoidal categories relating three geometric or representation-theoretic categorical incarnations of the affine Hecke algebra of a connected reductive algebraic group $G$ over a field of positive characteristic: a category of Harish-Chandra bimodules for the Lie algebra of $G$; the derived category of equivariant coherent sheaves on (a completed version of) the Steinberg variety of the Frobenius twist $G^{(1)}$ of $G$; a derived category of constructible sheaves on the affine flag variety of reductive group which is Langlands dual to $G^{(1)}$. These constructions build on the localization theory developed by the first author with Mirković and Rumynin and previous work of ours (partly joint with L. Rider), and provide an analogue for positive-characteristic coefficients of a construction of the first author. As an application, we prove a conjecture by Finkelberg-Mirković giving a geometric realization of the principal block of algebraic representations of $G$.
- [431] arXiv:2402.08560 (replaced) [pdf, html, other]
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Title: Failure of almost uniformly convergence for noncommutative martingalesComments: 7 pages, final version incorporating referees' comments, to appear in Probability Theory and Related FieldsSubjects: Operator Algebras (math.OA); Functional Analysis (math.FA)
In this paper, we provide a counterexample to show that in sharp contrast to the classical case, the almost uniform convergence may not happen for truly noncommutative $L_p$-martingales when $1\leq p<2$. The same happens to ergodic averages. The proof consists of some sharp estimates of the distributional function of a sequence of matrices and some non standard transference techniques, which might admit further applications.
- [432] arXiv:2402.09160 (replaced) [pdf, html, other]
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Title: At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized LaplacianComments: Added new results in Section 3 (Theorem 3.12 - Proposition 3.16)Subjects: Combinatorics (math.CO); Spectral Theory (math.SP)
For a graph with largest normalized Laplacian eigenvalue $\lambda_N$ and (vertex) coloring number $\chi$, it is known that $\lambda_N\geq \chi/(\chi-1)$. Here we prove properties of graphs for which this bound is sharp, and we study the multiplicity of $\chi/(\chi-1)$. We then describe a family of graphs with largest eigenvalue $\chi/(\chi-1)$. We also study the spectrum of the $1$-sum of two graphs (also known as graph joining or coalescing), with a focus on the maximal eigenvalue. Finally, we give upper bounds on $\lambda_N$ in terms of $\chi$.
- [433] arXiv:2403.07101 (replaced) [pdf, html, other]
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Title: Advanced-Step Real-time Iterations with Four Levels -- New Error Bounds and Fast Implementation in acadosComments: 6 pages, 2 figures, accepted for L-CSSSubjects: Optimization and Control (math.OC)
The Real-Time Iteration (RTI) is an online nonlinear model predictive control algorithm that performs a single Sequential Quadratic Programming (SQP) per sampling time. The algorithm is split into a preparation and a feedback phase, where the latter one performs as little computations as possible solving a single prepared quadratic program. To further improve the accuracy of this method, the Advanced-Step RTI (AS-RTI) performs additional Multi-Level Iterations (MLI) in the preparation phase, such as inexact or zero-order SQP iterations on a problem with a predicted state estimate. This paper extends and streamlines the existing local convergence analysis of AS-RTI, such as analyzing MLI of level A and B for the first time, and significantly simplifying the proofs for levels C and D. Moreover, this paper provides an efficient open-source implementation in acados, making it widely accessible to practitioners.
- [434] arXiv:2403.07170 (replaced) [pdf, html, other]
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Title: Cyclical Long Memory: Decoupling, Modulation, and ModelingComments: 35 pages, 7 figuresSubjects: Statistics Theory (math.ST); Probability (math.PR)
A new model for general cyclical long memory is introduced, by means of random modulation of certain bivariate long memory time series. This construction essentially decouples the two key features of cyclical long memory: quasi-periodicity and long-term persistence. It further allows for a general cyclical phase in cyclical long memory time series. Several choices for suitable bivariate long memory series are discussed, including a parametric fractionally integrated vector ARMA model. The parametric models introduced in this work have explicit autocovariance functions that can be used readily in simulation, estimation, and other tasks.
- [435] arXiv:2403.07482 (replaced) [pdf, html, other]
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Title: Linking Invariants for Valuations and Orderings on FieldsSubjects: Number Theory (math.NT)
The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of $\mathbb{Q}$ with unitriangular Galois groups, and contain the Legendre and Redei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the q-th roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient, and relate it to Massey product elements in Galois cohomology.
- [436] arXiv:2403.09120 (replaced) [pdf, html, other]
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Title: On the Miyaoka-Yau inequality for manifolds with nef anti-canonical line bundleComments: In the second version, we added the assumption about the numerical dimension to Theorem 1.3 (existence for cscK metrics). Some surface examples are appended. Comments on the equality condition for the Miyaoka-Yau inequality is updatedSubjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG)
Based on the recent work of K.~Zhang, we discuss the Miyaoka-Yau type inequality for projective manifolds with nef anti-canonical line bundle, assuming the lower bound of the delta-invariant introduced by Fujita and Odaka.
- [437] arXiv:2403.09618 (replaced) [pdf, html, other]
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Title: Dynamically accelerating the power iteration with momentumComments: 29 pages, 6 figures, 4 tablesSubjects: Numerical Analysis (math.NA)
In this paper, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method can be applied to real diagonalizable matrices, is provably convergent with acceleration in the symmetric case, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power-like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations.
- [438] arXiv:2403.10115 (replaced) [pdf, html, other]
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Title: Fast Generation of Feasible Trajectories in Direct Optimal ControlSubjects: Optimization and Control (math.OC)
This paper examines the question of finding feasible points to discrete-time optimal control problems. The optimization problem of finding a feasible trajectory is transcribed to an unconstrained optimal control problem. An efficient algorithm, called FP-DDP, is proposed that solves the resulting problem using Differential Dynamic Programming preserving feasibility with respect to the system dynamics in every iteration. Notably, FP-DDP admits global and rapid local convergence properties induced by a combination of a Levenberg-Marquardt method and an Armijo-type line search. The efficiency of FP-DDP is demonstrated against established methods such as Direct Multiple Shooting, Direct Single Shooting, and state-of-the-art solvers.
- [439] arXiv:2403.11888 (replaced) [pdf, html, other]
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Title: Paintability of $r$-chromatic graphsComments: 13 pages, a terminology error was correctedSubjects: Combinatorics (math.CO)
The paintability of a graph is a coloring parameter defined in terms of an online list coloring game. In this paper we ask, what is the paintability of a graph $G$ of maximum degree $\Delta$ and chromatic number $r$? By considering the Alon-Tarsi number of $G$, we prove that the paintability of $G$ is at most $\left(1 - \frac{1}{4r+1} \right ) \Delta + 2$. We also consider the DP-paintability of $G$, which is defined in terms of an online DP-coloring game. By considering the strict type $3$ degeneracy parameter recently introduced by Zhou, Zhu, and Zhu, we show that when $r$ is fixed and $\Delta$ is sufficiently large, the DP-paintability of $G$ is at most $\Delta - \Omega( \sqrt{\Delta \log \Delta})$.
- [440] arXiv:2403.15338 (replaced) [pdf, other]
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Title: Topoi with enough pointsComments: 30 pagesSubjects: Category Theory (math.CT); Algebraic Geometry (math.AG); Logic (math.LO)
We extend Deligne's original argument showing that locally coherent topoi have enough points, clarified using collage diagrams. We show that our refinement of Deligne's technique can be adapted to recover every existing result of this kind, including the most recent results about $\kappa$-coherent $\kappa$-topoi. Our presentation allows us to relax the cardinality assumptions typically imposed on the sites involved. We show that a larger class of locally finitely presentable toposes have enough points and that a closed subtopos of a topos with enough points has enough points.
- [441] arXiv:2403.18496 (replaced) [pdf, html, other]
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Title: Nijenhuis deformations of Poisson algebras and $F$-manifold algebrasComments: 28 pages; Twisted Rota-Baxter operators on Poisson algebras are considered and some examples are added in this new versionSubjects: Rings and Algebras (math.RA); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
The notion of pre-Poisson algebras was introduced by Aguiar in his study of zinbiel algebras and pre-Lie algebras. In this paper, we first introduce NS-Poisson algebras as a generalization of both Poisson algebras and pre-Poisson algebras. An NS-Poisson algebra has an associated sub-adjacent Poisson algebra. We show that a Nijenhuis operator and a twisted Rota-Baxter operator on a Poisson algebra deforms the structure into an NS-Poisson algebra. The semi-classical limit of an NS-algebra deformation and a suitable filtration of an NS-algebra produce NS-Poisson algebras. On the other hand, F-manifold algebras were introduced by Dotsenko as the underlying algebraic structure of F-manifolds. We also introduce NS-F-manifold algebras as a simultaneous generalization of NS-Poisson algebras, F-manifold algebras and pre-F-manifold algebras. In the end, we show that Nijenhuis deformations of F-manifold algebras and the semi-classical limits of NS-pre-Lie algebra deformations have NS-F-manifold algebra structures.
- [442] arXiv:2403.18809 (replaced) [pdf, html, other]
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Title: $L^\infty$-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decompositionComments: 25 pages, 3 figures, 5 tablesSubjects: Dynamical Systems (math.DS); Numerical Analysis (math.NA)
Extended dynamic mode decomposition (EDMD) is a well-established method to generate a data-driven approximation of the Koopman operator for analysis and prediction of nonlinear dynamical systems. Recently, kernel EDMD (kEDMD) has gained popularity due to its ability to resolve the challenging task of choosing a suitable dictionary by using the kernel's canonical features and, thus, data-informed observables. In this paper, we provide the first pointwise bounds on the approximation error of kEDMD. The main idea consists of two steps. First, we show that the reproducing kernel Hilbert spaces of Wendland functions are invariant under the Koopman operator. Second, exploiting that the learning problem given by regression in the native norm can be recast as an interpolation problem, we prove our novel error bounds by using interpolation estimates. Finally, we validate our findings with numerical experiments.
- [443] arXiv:2403.19017 (replaced) [pdf, html, other]
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Title: Pole Placement and Feedback Stabilization for Discrete Linear Ensemble SystemsSubjects: Optimization and Control (math.OC); Dynamical Systems (math.DS)
We consider discrete ensembles of linear, scalar control systems with single-inputs. Assuming that all the individual systems are unstable, we investigate whether there exist linear feedback control laws that can asymptotically stabilize the ensemble system. We provide necessary/sufficient conditions for feasibility of pole placement in the left half plane and for feedback stabilizability of the ensemble systems.
- [444] arXiv:2404.07953 (replaced) [pdf, other]
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Title: Floer Homology with DG Coefficients. Applications to cotangent bundlesComments: 125 pages, 3 figures. v2: we have partially rewritten the introduction and we have added some referencesSubjects: Symplectic Geometry (math.SG); Algebraic Topology (math.AT); Differential Geometry (math.DG)
We define Hamiltonian Floer homology with differential graded (DG) local coefficients for symplectically aspherical manifolds. The differential of the underlying complex involves chain representatives of the fundamental classes of the moduli spaces of Floer trajectories of arbitrary dimension. This setup allows in particular to define and compute Floer homology with coefficients in chains on fibers of fibrations over the free loop space of the underlying symplectic manifold. We develop the DG Floer toolset, including continuation maps and homotopies, and we also define and study symplectic homology groups with DG local coefficients. We define spectral invariants and establish general criteria for almost existence of contractible periodic orbits on regular energy levels of Hamiltonian systems inside Liouville domains.
In the case of cotangent bundles, we prove a Viterbo isomorphism theorem with DG local coefficients. This serves as a stepping stone for applications to the almost existence of contractible closed characteristics on closed smooth hypersurfaces. In this context, our methods allow to access for the first time the dichotomy between closed manifolds that are aspherical and those that are not. - [445] arXiv:2404.09121 (replaced) [pdf, html, other]
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Title: Stability conditions on crepant resolutions of quotients of product varietiesComments: 19 pages, minor updatesSubjects: Algebraic Geometry (math.AG)
We construct stability conditions on crepant resolutions of certain quotients of product varieties, giving as a special case the first examples of stability conditions on strict Calabi-Yau varieties of arbitrary dimension. Along the way, we prove the crepant resolutions are derived equivalent to the corresponding quotient stacks, verifying an instance of a conjecture of Bondal and Orlov.
- [446] arXiv:2404.10203 (replaced) [pdf, html, other]
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Title: Finitely and non-finitely related wordsComments: 25 pages, 2 figuresSubjects: Group Theory (math.GR); Rings and Algebras (math.RA)
An algebra is finitely related (or has finite degree) if its term functions are determined by some finite set of finitary relations. Nilpotent monoids built from words, via Rees quotients of free monoids, have been used to exhibit many interesting properties with respect to the finite basis problem. We show that much of this intriguing behaviour extends to the world of finite relatedness by using interlocking patterns called chain, crown, and maelstrom words. In particular, we show that there are large classes of non-finitely related nilpotent monoids that can be used to construct examples of: ascending chains of varieties alternating between finitely and non-finitely related; non-finitely related semigroups whose direct product are finitely related; the addition of an identity element to a non-finitely related semigroup to produce a finitely related semigroup.
- [447] arXiv:2404.11039 (replaced) [pdf, html, other]
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Title: Nilpotent symplectic alternating algebras IComments: 25 pagesJournal-ref: Journal of Algebra, Volume 423, 2015, Pages 615-635, ISSN 0021-8693Subjects: Rings and Algebras (math.RA)
We develop a structure theory for nilpotent symplectic alternating algebras.
- [448] arXiv:2404.11696 (replaced) [pdf, html, other]
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Title: Graviton topologyComments: 29 pages, 1 figureSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)
Over the past three decades, it has been shown that discrete and continuous media can support topologically nontrivial waves. Recently, it was shown that the same is true of the vacuum, namely, right (R) and left (L) circularly polarized photons are topologically nontrivial. Here, we study the topology of another class of massless particles, namely gravitons. We show that the collection of all gravitons forms a topologically trivial vector bundle over the lightcone, allowing us to construct a globally smooth basis for gravitons. The graviton bundle also has a natural geometric splitting into two topologically nontrivial subbundles, consisting of the R and L gravitons. The R and L gravitons are unitary irreducible bundle representations of the Poincaré group, and are thus elementary particles; their topology is characterized by the Chern numbers $\mp 4$. This nontrivial topology obstructs the splitting of graviton angular momentum into spin and orbital angular momentum.
- [449] arXiv:2404.11908 (replaced) [pdf, html, other]
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Title: Asymptotics of Spectral Functions of Lower Energy Forms on Weakly 1-Complete ManifoldsComments: 12 pagesSubjects: Complex Variables (math.CV); Differential Geometry (math.DG)
In this paper, we show that the optimal fundamental estimate holds true on a weakly $1$-complete manifold with mild conditions, then we establish the weak Morse inequalities for lower energy forms on the manifold. We also study the case for $q$-convex manifolds.
- [450] arXiv:2404.14384 (replaced) [pdf, other]
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Title: A New Optimization Model for Multiple-Control Toffoli Quantum Circuit DesignSubjects: Optimization and Control (math.OC); Emerging Technologies (cs.ET); Quantum Physics (quant-ph)
As quantum technology is advancing, the efficient design of quantum circuits has become an important area of research. This paper provides an introduction to the MCT quantum circuit design problem for reversible Boolean functions without assuming a prior background in quantum computing. While this is a well-studied problem, optimization models that minimize the true objective have only been explored recently. This paper introduces a new optimization model and symmetry-breaking constraints that improve solving time by up to two orders of magnitude compared to earlier work when a Constraint Programming solver is used. Experiments with up to seven qubits and using up to 15 quantum gates result in several new best-known circuits, obtained by any method, for well-known benchmarks. Finally, an extensive comparison with other approaches shows that optimization models may require more time but can provide superior circuits with optimality guarantees.
- [451] arXiv:2404.15868 (replaced) [pdf, html, other]
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Title: On a general notion of a polynomial identity and codimensionsComments: 26 pages; minor improvementsSubjects: Rings and Algebras (math.RA); Category Theory (math.CT); Quantum Algebra (math.QA); Representation Theory (math.RT)
Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter-Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.
- [452] arXiv:2404.16707 (replaced) [pdf, html, other]
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Title: A self-improving property of Riesz potentials in BMOSubjects: Functional Analysis (math.FA)
In this paper we prove that for non-negative measurable functions $f$, \begin{align*} I_\alpha f \in BMO(\mathbb{R}^n) \text{ if and only if } I_\alpha f \in BMO^\beta(\mathbb{R}^n) \text{ for } \beta \in (n-\alpha,n]. \end{align*} Here $I_\alpha$ denotes the Riesz potential of order $\alpha$ and $BMO^\beta$ represents the space of functions of bounded $\beta$-dimensional mean oscillation.
- [453] arXiv:2404.18614 (replaced) [pdf, html, other]
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Title: A Few Projective Classes of (Non-Hausdorff) Topological SpacesComments: Typos corrected. 21 pages, 2 figuresSubjects: General Topology (math.GN)
A class of topological spaces is projective (resp., $\omega$-projective) if and only if projective systems of spaces (resp., with a countable cofinal subset of indices) in the class are still in the class. A certain number of classes of Hausdorff spaces are known to be, or not to be, ($\omega$-) projective. We examine classes of spaces that are not necessarily Hausdorff. Sober and compact sober spaces form projective classes, but most classes of locally compact spaces are not even $\omega$-projective. Guided by the fact that the stably compact spaces are exactly the locally compact, strongly sober spaces, and that the strongly sober spaces are exactly the sober, coherent, compact, weakly Hausdorff (in the sense of Keimel and Lawson) spaces, we examine which classes defined by combinations of those properties are projective. Notably, we find that coherent sober spaces, compact coherent sober spaces, as well as (locally) strongly sober spaces, form projective classes.
- [454] arXiv:2404.19703 (replaced) [pdf, html, other]
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Title: $C$-embedding, Lindel\"ofness, \v{C}ech-completenessComments: Version 2: some corrections after referee reportSubjects: General Topology (math.GN)
We show that in the class of Lindelöf Čech-complete spaces the property of being $C$-embedded is quite well-behaved. It admits a useful characterization that can be used to show that products and perfect preimages of $C$-embedded spaces are again $C$-embedded. We also show that both properties, Lindelöf and Čech-complete, are needed in the product result.
- [455] arXiv:2405.03423 (replaced) [pdf, html, other]
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Title: Generalized Baer and Generalized Quasi-Baer Rings of Skew Generalized Power SeriesSubjects: Rings and Algebras (math.RA)
Let $R$ be a ring with identity, $(S,\leq)$ an ordered monoid, $\omega:S \to End(R)$ a monoid homomorphism, and $A= R\left[\left[S,\omega \right]\right]$ the ring of skew generalized power series. The concepts of generalized Baer and generalized quasi-Baer rings are generalization of Baer and quasi-Baer rings, respectively. A ring $R$ is called generalized right Baer (generalized right quasi-Baer) if for any non-empty subset $S$ (right ideal $I$) of $R$, the right annihilator of $S^n \space{0.1cm}(I^n)$ is generated by an idempotent for some positive integer $n$. Left cases may be defined analogously. A ring $R$ is called generalized Baer (generalized quasi-Baer) if it is both generalized right and left Baer (generalized right and left quasi-Baer) ring. In this paper, we examine the behavior of a skew generalized power series ring over a generalized right Baer (generalized right quasi-Baer) ring and prove that, under specific conditions, the ring $A$ is generalized right Baer (generalized right quasi-Baer) if and only if $R$ is a generalized right Baer (generalized right quasi-Baer) ring.
- [456] arXiv:2405.05782 (replaced) [pdf, html, other]
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Title: Minimax problems for ensembles of affine-control systemsComments: 21 pages, 1 Figure. Correction of typos and new section with a numerical exampleSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
In this paper, we consider ensembles of affine-control systems in $\mathbb{R}^d$, and we study simultaneous optimal control problems related to the worst-case minimization. After proving that such problems admit solutions, denoting with $(\Theta^N)_N$ a sequence of compact sets that parametrize the ensembles of systems, we first show that the corresponding minimax optimal control problems are $\Gamma$-convergent whenever $(\Theta^N)_N$ has a limit with respect to the Hausdorff distance. Besides its independent interest, the previous result is crucial role for establishing the Pontryagin Maximum Principle (PMP) when the ensemble is parametrized by a set $\Theta$ consisting of infinitely many points. Namely, we first approximate $\Theta$ by finite and increasing-in-size sets $(\Theta^N)_N$ for which the PMP is known, and then we derive the PMP for the $\Gamma$-limiting problem. The same strategy can be pursued in applications, where we can reduce infinite ensembles to finite ones to compute the minimizers numerically. We bring as a numerical example the Schrödinger equation for a qubit with uncertain resonance frequency.
- [457] arXiv:2405.05873 (replaced) [pdf, html, other]
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Title: Duality for Cohen--Macaulay Complexes through Combinatorial SheavesComments: 61 pages. Update: the CM Duality Theorem is now stated and proved for possibly non-commmutative rings. Added a proof of CM duality with coefficients in arbitrary (co)sheaves (Section 4.3)Subjects: Algebraic Topology (math.AT); Group Theory (math.GR)
We prove a duality theorem for Cohen--Macaulay simplicial complexes. This is a generalisation of Poincaré Duality, framed in the language of combinatorial sheaves. Our treatment is self-contained and accessible for readers with a working knowledge of simplicial complexes and (co)homology. The main motivation is a link with Bieri-Eckmann duality for discrete groups, which is explored in a companion paper.
- [458] arXiv:2405.06723 (replaced) [pdf, html, other]
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Title: Positive formula for the product of conjugacy classes on the unitary groupComments: New version with corrected typos, improved presentation and simplified proofs in Section 6. 48 pages, 29 figures with colorsSubjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Combinatorics (math.CO); Probability (math.PR); Symplectic Geometry (math.SG)
The convolution product of two conjugacy classes of the unitary group $U_n$ is described by a probability distribution on the space of central measures. Relating this convolution to the quantum cohomology of Grassmannians and using recent results describing the structure constants of the latter, we give a manifestly positive formula for the density of the probability distribution for the product of generic conjugacy classes. In the same flavor as the hive model of Knutson and Tao, this formula is given in terms of a subtraction-free sum of volumes of explicit polytopes. As a consequence, this expression also provides a positive and explicit formula for the volume of $SU_n$-valued flat connections on the three-holed two dimensional sphere, which was first given by Witten in terms of an infinite sum of characters.
- [459] arXiv:2405.08592 (replaced) [pdf, html, other]
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Title: Horocycle flows on abelian covers of surfaces of negative curvatureComments: 42 pagesSubjects: Dynamical Systems (math.DS)
We consider the unit speed parametrization of the horocycle flow on infinite Abelian covers of compact surfaces of negative curvature. We prove an asymptotic result for the ergodic integrals of sufficiently regular functions. In the case of constant curvature, where the unit speed and the uniformly contracting parametrizations of horocycles coincide, we recover a result by Ledrappier and Sarig. Our method, which does not use symbolic dynamics, is based on a general Fourier decomposition for Abelian covers and on the study of spectral theory of weighted (and twisted) transfer operators for the geodesic flow acting on appropriate anisotropic Banach spaces. Finally, as a byproduct result, we obtain a power deviation estimate for the horocycle ergodic averages on compact surfaces, without requiring any pinching condition as in previous results.
- [460] arXiv:2405.09726 (replaced) [pdf, other]
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Title: Boundary layer expansions of the steady MHD equations in a bounded domainSubjects: Analysis of PDEs (math.AP)
In this paper, we investigate the validity of boundary layer expansions for the MHD system in a rectangle. We describe the solution up to any order when the tangential magnetic field is much smaller or much larger than the tangential velocity field, thereby extending a previous work of S.J. Ding, Z.L. Lin and F. Xie.
- [461] arXiv:2405.11214 (replaced) [pdf, html, other]
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Title: Maximizing the index of signed complete graphs with spanning trees on $k$ pendant verticesSubjects: Combinatorics (math.CO)
A signed graph $\Sigma=(G,\sigma)$ consists of an underlying graph $G=(V,E)$ with a sign function $\sigma:E\rightarrow\{-1,1\}$. Let $A(\Sigma)$ be the adjacency matrix of $\Sigma$ and $\lambda_1(\Sigma)$ denote the largest eigenvalue (index) of $\Sigma$.Define $(K_n,H^-)$ as a signed complete graph whose negative edges induce a subgraph $H$. In this paper, we focus on the following problem: which spanning tree $T$ with a given number of pendant vertices makes the $\lambda_1(A(\Sigma))$ of the unbalanced $(K_n,T^-)$ as large as possible? To answer the problem, we characterize the extremal signed graph with maximum $\lambda_1(A(\Sigma))$ among graphs of type $(K_n,T^-)$.
- [462] arXiv:2405.15091 (replaced) [pdf, other]
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Title: Nilpotent Symplectic Alternating AlgebrasComments: 125 pages, PhD thesisSubjects: Rings and Algebras (math.RA)
We develop a structure theory for nilpotent symplectic alternating algebras. We then give a classification of all nilpotent symplectic alternating algebras of dimension up to 10 over any field. The study reveals a new subclasses of powerful groups that we call powerfully nilpotent groups and powerfully soluble groups.
- [463] arXiv:2405.16354 (replaced) [pdf, html, other]
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Title: Universal lower bounds for Dirichlet eigenvaluesSubjects: Spectral Theory (math.SP)
Let $\Omega \subset \mathbb{R}^d$ be a bounded domain and let $\lambda_1, \lambda_2, \dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for $\lambda_n$ that are independent of the domain $\Omega$. A well--known such inequality follows from the Berezin--Li--Yau approach. The purpose of this paper is to point out a certain degree of flexibility in the Li--Yau approach. We use it to prove a new type of two-point inequality which are strictly stronger than what is implied by Berezin-Li-Yau itself. For example, when $d=2$, one has $ 2 \lambda_n + \lambda_{2n} \geq 10 \pi n/|\Omega|.$
- [464] arXiv:2406.00931 (replaced) [pdf, html, other]
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Title: Cohomological splitting over rationally connected basesComments: 36 pages, comments welcome! v2: updates on exposition and correction of typosSubjects: Symplectic Geometry (math.SG); Algebraic Geometry (math.AG)
We prove a cohomological splitting result for Hamiltonian fibrations over enumeratively rationally connected symplectic manifolds As a key application, we prove that the cohomology of a smooth, projective family over a smooth (stably) rational projective variety splits additively over any field. The main ingredients in our arguments include the theory of Fukaya-Ono-Parker (FOP) perturbations developed by the first and third author, which allows one to define integer-valued Gromov-Witten type invariants, and variants of Abouzaid-McLean-Smith's global Kuranishi charts tailored to concrete geometric problems.
- [465] arXiv:2406.02007 (replaced) [pdf, html, other]
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Title: On Ramsey degrees, compactness and approximabilitySubjects: Logic (math.LO); Combinatorics (math.CO)
One of the consequences of the Compactness Principle in structural Ramsey theory is that the small Ramsey degrees cannot exceed the corresponding big Ramsey degrees, thereby justifying the choice of adjectives. However, it is unclear what happens in the realm of dual Ramsey degrees due to the lack of the compactness argument that applies to that setting. In this paper we present a framework within which both "direct" and dual Ramsey statements can be stated and reasoned about in a uniform fashion. We introduce the notion of approximability which yields a general compactness argument powerful enough to prove statements about both "direct" and dual Ramsey phenomena. We conclude the paper with an application of the new strategies by generalizing Voigt's $\star$-version of the Infinite Ramsey Theorem to a large class of relational structures and deriving a Ramsey statement for "loose colorings" of enumerated Fra\"ıssé limits.
- [466] arXiv:2406.05099 (replaced) [pdf, other]
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Title: Canonicalizing zeta generators: genus zero and genus oneDaniele Dorigoni, Mehregan Doroudiani, Joshua Drewitt, Martijn Hidding, Axel Kleinschmidt, Oliver Schlotterer, Leila Schneps, Bram VerbeekComments: 92 pages. Submission includes ancillary data files. v2: Typos correctedSubjects: Quantum Algebra (math.QA); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Number Theory (math.NT)
Zeta generators are derivations associated with odd Riemann zeta values that act freely on the Lie algebra of the fundamental group of Riemann surfaces with marked points. The genus-zero incarnation of zeta generators are Ihara derivations of certain Lie polynomials in two generators that can be obtained from the Drinfeld associator. We characterize a canonical choice of these polynomials, together with their non-Lie counterparts at even degrees $w\geq 2$, through the action of the dual space of formal and motivic multizeta values. Based on these canonical polynomials, we propose a canonical isomorphism that maps motivic multizeta values into the $f$-alphabet. The canonical Lie polynomials from the genus-zero setup determine canonical zeta generators in genus one that act on the two generators of Enriquez' elliptic associators. Up to a single contribution at fixed degree, the zeta generators in genus one are systematically expanded in terms of Tsunogai's geometric derivations dual to holomorphic Eisenstein series, leading to a wealth of explicit high-order computations. Earlier ambiguities in defining the non-geometric part of genus-one zeta generators are resolved by imposing a new representation-theoretic condition. The tight interplay between zeta generators in genus zero and genus one unravelled in this work connects the construction of single-valued multiple polylogarithms on the sphere with iterated-Eisenstein-integral representations of modular graph forms.
- [467] arXiv:2406.05553 (replaced) [pdf, html, other]
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Title: Universality in Random Persistent Homology and Scale-Invariant FunctionalsSubjects: Probability (math.PR); Algebraic Topology (math.AT)
In this paper, we prove a universality result for the limiting distribution of persistence diagrams arising from geometric filtrations over random point processes. Specifically, we consider the distribution of the ratio of persistence values (death/birth), and show that for fixed dimension, homological degree and filtration type (Cech or Vietoris-Rips), the limiting distribution is independent of the underlying point process distribution, i.e., universal. In proving this result, we present a novel general framework for universality in scale-invariant functionals on point processes. Finally, we also provide a number of new results related to Morse theory in random geometric complexes, which may be of an independent interest.
- [468] arXiv:2406.05979 (replaced) [pdf, html, other]
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Title: Robustly non-convex hypersurfaces in contact manifoldsComments: 38 pages + references. 3 figures. v2 has small corrections and improvements to exposition. Comments welcome!Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS)
We construct the first examples of hypersurfaces in any contact manifold of dimension 5 and larger that cannot be $C^2$-approximated by convex hypersurfaces. This contrasts sharply with the foundational result of Giroux in dimension $3$ and the work of Honda-Huang in the $C^0$ case. The main technical step is the construction of a Bonatti-Diaz type blender in the contact setting.
- [469] arXiv:2406.06741 (replaced) [pdf, html, other]
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Title: On non-isomorphic universal sofic groupsComments: 14 pagesSubjects: Group Theory (math.GR); Logic (math.LO)
We show that there are $2^{\aleph_0}$ non-isomorphic universal sofic groups. This proves a conjecture of Simon Thomas.
- [470] arXiv:2406.07452 (replaced) [pdf, html, other]
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Title: Function spaces on Corson-like compactaSubjects: General Topology (math.GN)
For an index set $\Gamma$ and a cardinal number $\kappa$ the $\Sigma_{\kappa}$-product of real lines $\Sigma_{\kappa}(\mathbb{R}^{\Gamma})$ consist of all elements of $\mathbb{R}^{\Gamma}$ with $<\kappa$ nonzero coordinates. A compact space is $\kappa$-Corson if it can be embedded into $\Sigma_{\kappa}(\mathbb{R}^{\Gamma})$ for some $\Gamma$. We also consider a class of compact spaces wider than the class of $\omega$-Corson compact spaces, investigated by Nakhmanson and Yakovlev as well as Marciszewski, Plebanek and Zakrzewski called $NY$ compact spaces. For a Tychonoff space $X$, let $C_{p}(X)$ be the space of real continuous functions on the space $X$, endowed with the pointwise convergence topology. We present here a characterisation of $\kappa$-Corson compact spaces $K$ for regular, uncountable cardinal numbers $\kappa$ in terms of function spaces $C_{p}(K)$, extending a theorem of Bell and Marciszewski and a theorem of Pol. We also prove that classes of $NY$ compact spaces and $\omega$-Corson compact spaces $K$ are preserved by linear homeomorphisms of function spaces $C_{p}(K)$.
- [471] arXiv:2406.08621 (replaced) [pdf, html, other]
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Title: On the extension problems for three 33-stem homotopy groupsComments: arXiv admin note: text overlap with arXiv:2406.02713Subjects: Algebraic Topology (math.AT)
This paper tackles the extension problems for three far-unsatble homotopy groups $\pi_{39}(S^{6})$, $\pi_{40}(S^{7})$, and $\pi_{41}(S^{8})$ localized at 2, the puzzles having remained unsolved for forty-five years. By a Toda bracket indexed by 1 included in $\pi_{39}(S^{6}_{(2)})$, which makes better use of the deuspension property of homotopy classes, we address the problems. As a corollary, through Thomeier's 8-step backward theorem of the metastable homotopy theory, together with the results of Oda, Mukai and Miyauchi, we show a table of the 33-stem homotopy groups $\pi_{33+n}(S^{n}_{(2)})$, ($2\leq n\leq 9$, $n\geq27$).
- [472] arXiv:2406.09360 (replaced) [pdf, html, other]
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Title: On Arratia's coupling and the Dirichlet law for the factors of a random integerComments: 36 pages; minor correctionsSubjects: Number Theory (math.NT); Probability (math.PR)
Let $x \ge 2$, let $N_x$ be an integer chosen uniformly at random from the set $\mathbb Z \cap [1, x]$, and let $(V_1, V_2, \ldots)$ be a Poisson--Dirichlet process of parameter $1$. We prove that there exists a coupling of these two random objects such that $$ \mathbb E \, \sum_{i \ge 1} |\log P_i- V_i\log x| \ll 1, $$ where the implied constant is absolute and $N_x = P_1P_2 \cdots$ is the unique factorization of $N_x$ into primes or ones with the $P_i$'s being non-increasing. This establishes a conjecture of Arratia (2002) arXiv:1305.0941 who proved that the left-hand side in the above estimate can be made $\ll \log\!\log x$. We also use this coupling to give a probabilistic proof of the Dirichlet law for the average distribution of the integer factorization into $k$ parts proved in 2023 by Leung arXiv:2206.14728 and we improve on its error term.
- [473] arXiv:2406.12781 (replaced) [pdf, html, other]
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Title: Asymptotic behaviour of determinants through the expansion of the Moyal star productComments: 39 pages; v2: minor correctionsSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)
We work out a generalization of the Szegö limit theorems on the determinant of large matrices. We focus on matrices with nonzero leading principal minors and elements that decay to zero exponentially fast with the distance from the main diagonal, but we relax the constraint of the Toeplitz structure. We obtain an expression for the asymptotic behaviour of the determinant written in terms of the factors of a left and right Wiener-Hopf type factorization of an appropriately defined symbol. For matrices with elements varying slowly along the diagonals (e.g., in locally Toeplitz sequences), we propose to apply the analogue of the semiclassical expansion of the Moyal star product in phase-space quantum mechanics. This is a systematic method that provides approximations up to any order in the typical scale of the inhomogeneity and allows us to obtain explicit asymptotic formulas.
- [474] arXiv:2406.13398 (replaced) [pdf, other]
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Title: Non-additive derived functors via chain resolutionsComments: 44 pages; changed title, minor changes throughout the textSubjects: Category Theory (math.CT)
Let $F\colon \mathcal{C} \to \mathcal{E}$ be a functor from a category $\mathcal{C}$ to a homological (Borceux-Bourn) or semi-abelian (Janelidze-Márki-Tholen) category $\mathcal{E}$. We investigate conditions under which the homology of an object $X$ in $\mathcal{C}$ with coefficients in the functor $F$, defined via projective resolutions in $\mathcal{C}$, remains independent of the chosen resolution. Consequently, the left derived functors of $F$ can be constructed analogously to the classical abelian case.
Our approach extends the concept of chain homotopy to a non-additive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of Bourn-Janelidze, originally introduced in the context of subtractive categories. This method is applicable when $\mathcal{C}$ is a pointed regular category with finite coproducts and enough projectives, provided these projectives are closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor $F$ meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts - conditions that amount to additivity when $\mathcal{C}$ and $\mathcal{E}$ are abelian categories.
Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples. - [475] arXiv:2406.13447 (replaced) [pdf, other]
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Title: High-probability minimax lower boundsComments: 38 pages, 3 figuresSubjects: Statistics Theory (math.ST); Information Theory (cs.IT); Machine Learning (cs.LG); Machine Learning (stat.ML)
The minimax risk is often considered as a gold standard against which we can compare specific statistical procedures. Nevertheless, as has been observed recently in robust and heavy-tailed estimation problems, the inherent reduction of the (random) loss to its expectation may entail a significant loss of information regarding its tail behaviour. In an attempt to avoid such a loss, we introduce the notion of a minimax quantile, and seek to articulate its dependence on the quantile level. To this end, we develop high-probability variants of the classical Le Cam and Fano methods, as well as a technique to convert local minimax risk lower bounds to lower bounds on minimax quantiles. To illustrate the power of our framework, we deploy our techniques on several examples, recovering recent results in robust mean estimation and stochastic convex optimisation, as well as obtaining several new results in covariance matrix estimation, sparse linear regression, nonparametric density estimation and isotonic regression. Our overall goal is to argue that minimax quantiles can provide a finer-grained understanding of the difficulty of statistical problems, and that, in wide generality, lower bounds on these quantities can be obtained via user-friendly tools.
- [476] arXiv:2406.14247 (replaced) [pdf, html, other]
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Title: Formal groups over non-commutative ringsSubjects: Algebraic Topology (math.AT)
We develop an extension of the usual theory of formal group laws where the base ring is not required to be commutative and where the formal variables need neither be central nor have to commute with each other.
We show that this is the natural kind of formal group law for the needs of algebraic topology in the sense that a (possibly non-commutative) complex oriented ring spectrum is canonically equipped with just such a formal group law. The universal formal group law is carried by the Baker-Richter spectrum M{\xi} which plays a role analogous to MU in this non-commutative context.
As suggested by previous work of Morava the Hopf algebra B of "formal diffeomorphisms of the non-commutative line" of Brouder, Frabetti and Krattenthaler is central to the theory developed here. In particular, we verify Morava's conjecture that there is a representation of the Drinfeld quantum-double D(B) through cohomology operations in M{\xi}. - [477] arXiv:2406.14824 (replaced) [pdf, html, other]
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Title: On the minimal period of integer tilingsComments: 7 pages. Added a remark in Section 2 and corrected some typosSubjects: Number Theory (math.NT); Combinatorics (math.CO)
If a finite set $A$ tiles the integers by translations, it also admits a tiling whose period $M$ has the same prime factors as $|A|$. We prove that the minimal period of such a tiling is bounded by $\exp(c(\log D)^2/\log\log D)$, where $D$ is the diameter of $A$. In the converse direction, given $\epsilon>0$, we construct tilings whose minimal period has the same prime factors as $|A|$ and is bounded from below by $D^{3/2-\epsilon}$. We also discuss the relationship between minimal tiling period estimates and the Coven-Meyerowitz conjecture.
- [478] arXiv:2406.15114 (replaced) [pdf, html, other]
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Title: Approximate Controllability of Linear Fractional Impulsive Evolution Equations in Hilbert SpacesSubjects: Optimization and Control (math.OC); Dynamical Systems (math.DS)
In this paper, we investigate the approximate controllability of linear fractional impulsive evolution equations in Hilbert spaces. We provide a representation of solutions utilizing impulsive operators. Necessary and sufficient conditions for the approximate controllability of linear fractional impulsive evolution equations are established in terms of the impulsive resolvent operator. An example is presented to demonstrate the application of the obtained theoretical results.
- [479] arXiv:2406.15122 (replaced) [pdf, html, other]
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Title: Convolutional dynamical sampling and some new resultsSubjects: Information Theory (cs.IT)
In this work, we explore the dynamical sampling problem on $\ell^2(\mathbb{Z})$ driven by a convolution operator defined by a convolution kernel. This problem is inspired by the need to recover a bandlimited heat diffusion field from space-time samples and its discrete analogue. In this book chapter, we review recent results in the finite-dimensional case and extend these findings to the infinite-dimensional case, focusing on the study of the density of space-time sampling sets.
- [480] arXiv:2406.15164 (replaced) [pdf, html, other]
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Title: Some Cases of the Erd\H{o}s-Lov\'asz Tihany Conjecture for Claw-free GraphsComments: 18 pages, 3 figuresSubjects: Combinatorics (math.CO)
The Erdős-Lovász Tihany Conjecture states that any $G$ with chromatic number $\chi(G) = s + t - 1 > \omega(G)$, with $s,t \geq 2$ can be split into two vertex-disjoint subgraphs of chromatic number $s, t$ respectively. We prove this conjecture for pairs $(s, t)$ if $t \leq s + 2$, whenever $G$ has a $K_s$, and for pairs $(s, t)$ if $t \leq 4 s - 3$, whenever $G$ contains a $K_s$ and is claw-free. We also prove the Erdős Lovász Tihany Conjecture for the pair $(3, 10)$ for claw-free graphs.
- [481] arXiv:2406.15697 (replaced) [pdf, html, other]
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Title: On the topology and geometry of certain $13$-manifoldsSubjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
This paper gives the classifications of certain manifolds $\mathcal{M}$ of dimension $13$ up to diffeomorphism, homeomorphism, and homotopy equivalence, whose cohomology rings are isomorphic to $H^\ast(\mathrm{CP}^3\times S^7;\mathbb{Z})$. Moreover, we prove that either $\mathcal{M}$ or $\mathcal{M}\#\Sigma^{13}$ admits a metric of non-negative sectional curvature where $\Sigma^{13}$ is a certain exotic sphere of dimension 13.
- [482] arXiv:2406.15907 (replaced) [pdf, html, other]
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Title: Rates of Convergence of the Magnetization in the Tensor Curie-Weiss Potts ModelComments: 30 pagesSubjects: Probability (math.PR); Statistics Theory (math.ST)
In this paper, we derive distributional convergence rates for the magnetization vector and the maximum pseudolikelihood estimator of the inverse temperature parameter in the tensor Curie-Weiss Potts model. Limit theorems for the magnetization vector have been derived recently in Bhowal and Mukherjee (2023), where several phase transition phenomena in terms of the scaling of the (centered) magnetization and its asymptotic distribution were established, depending upon the position of the true parameters in the parameter space. In the current work, we establish Berry-Esseen type results for the magnetization vector, specifying its rate of convergence at these different phases. At most points in the parameter space, this rate is $N^{-1/2}$ ($N$ being the size of the Curie-Weiss network), while at some "special" points, the rate is either $N^{-1/4}$ or $N^{-1/6}$, depending upon the behavior of the fourth derivative of a certain "negative free energy function" at these special points. These results are then used to derive Berry-Esseen type bounds for the maximum pseudolikelihood estimator of the inverse temperature parameter whenever it lies above a certain criticality threshold.
- [483] arXiv:2406.18307 (replaced) [pdf, html, other]
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Title: Five-Lee-weight linear codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$Subjects: Information Theory (cs.IT)
In this study, linear codes having their Lee-weight distributions over the semi-local ring $\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^{2}=1$ are constructed using the defining set and Gauss sums for an odd prime $q $. Moreover, we derive complete Hamming-weight enumerators for the images of the constructed linear codes under the Gray map. We finally show an application to secret sharing schemes.
- [484] arXiv:2406.18397 (replaced) [pdf, html, other]
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Title: Second Maximum of a Gaussian Random Field and Exact (t-)Spacing testComments: 5 figures, 22 pages main document, 2 pages supplementsSubjects: Statistics Theory (math.ST); Machine Learning (cs.LG); Differential Geometry (math.DG); Probability (math.PR); Machine Learning (stat.ML)
In this article, we introduce the novel concept of the second maximum of a Gaussian random field on a Riemannian submanifold. This second maximum serves as a powerful tool for characterizing the distribution of the maximum. By utilizing an ad-hoc Kac Rice formula, we derive the explicit form of the maximum's distribution, conditioned on the second maximum and some regressed component of the Riemannian Hessian. This approach results in an exact test, based on the evaluation of spacing between these maxima, which we refer to as the spacing test.
We investigate the applicability of this test in detecting sparse alternatives within Gaussian symmetric tensors, continuous sparse deconvolution, and two-layered neural networks with smooth rectifiers. Our theoretical results are supported by numerical experiments, which illustrate the calibration and power of the proposed tests. More generally, this test can be applied to any Gaussian random field on a Riemannian manifold, and we provide a general framework for the application of the spacing test in continuous sparse kernel regression.
Furthermore, when the variance-covariance function of the Gaussian random field is known up to a scaling factor, we derive an exact Studentized version of our test, coined the $t$-spacing test. This test is perfectly calibrated under the null hypothesis and has high power for detecting sparse alternatives. - [485] arXiv:2406.20013 (replaced) [pdf, html, other]
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Title: Heights on 'Hybrid orbits' in Shimura varietiesSubjects: Number Theory (math.NT)
We prove the 'hybrid conjecture' which is a common generalisation of the Andreé-Oort conjecture and the André-Pink-Zannier conjecture, in the case of Shimura varieties of abelian type.
- [486] arXiv:2407.00288 (replaced) [pdf, html, other]
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Title: A Rank-Two Case of Local-Global Compatibility for $l = p$Comments: 9 pagesSubjects: Number Theory (math.NT)
We prove the classical $l = p$ local-global compatibility conjecture for certain regular algebraic cuspidal automorphic representations of weight 0 for GL$_2$ over CM fields. Using an automorphy lifting theorem, we show that if the automorphic side comes from a twist of Steinberg at $v | l$, then the Galois side has nontrivial monodromy at $v$. Based on this observation, we will give a definition of the Fontaine-Mazur $\mathcal{L}$-invariants attached to certain automorphic representations.
- [487] arXiv:2407.00793 (replaced) [pdf, html, other]
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Title: From clonal interference to Poissonian interacting trajectoriesComments: 38 pages, 9 figures; changes in version 2: Acknowledgements were added and a few minor typos removedSubjects: Probability (math.PR)
We consider a population whose size $N$ is fixed over the generations, and in which random beneficial mutations arrive at a rate of order $1/\log N$ per generation. In this so-called Gerrish-Lenski regime, typically a finite number of contending mutations is present together with one resident type. These mutations compete for fixation, a phenomenon addressed as clonal interference. We study a system of Poissonian interacting trajectories (PIT) which arise as a large population scaling limit of the logarithmic sizes of the contending clonal subpopulations. We prove that this system exhibits an a.s.\ positive asymptotic rate of fitness increase (speed of adaptation), which turns out to be finite if and only if fitness increments have a finite expectation. We relate this speed to heuristic predictions from the literature. Furthermore, we derive a functional central limit theorem for the fitness of the resident population in the PIT. A main result of this work is that the Poissonian interacting trajectories arise as a large-population limit of a continuous time Moran model with strong selection.
- [488] arXiv:2407.00822 (replaced) [pdf, html, other]
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Title: ROM inversion of monostatic data lifted to full MIMOSubjects: Numerical Analysis (math.NA)
The Lippmann--Schwinger--Lanczos (LSL) algorithm has recently been shown to provide an efficient tool for imaging and direct inversion of synthetic aperture radar data in multi-scattering environments [17], where the data set is limited to the monostatic, a.k.a. single input/single output (SISO) measurements. The approach is based on constructing data-driven estimates of internal fields via a reduced-order model (ROM) framework and then plugging them into the Lippmann-Schwinger integral equation. However, the approximations of the internal solutions may have more error due to missing the off diagonal elements of the multiple input/multiple output (MIMO) matrix valued transfer function. This, in turn, may result in multiple echoes in the image. Here we present a ROM-based data completion algorithm to mitigate this problem. First, we apply the LSL algorithm to the SISO data as in [17] to obtain approximate reconstructions as well as the estimate of internal field. Next, we use these estimates to calculate a forward Lippmann-Schwinger integral to populate the missing off-diagonal data (the lifting step). Finally, to update the reconstructions, we solve the Lippmann-Schwinger equation using the original SISO data, where the internal fields are constructed from the lifted MIMO data. The steps of obtaining the approximate reconstructions and internal fields and populating the missing MIMO data entries can be repeated for complex models to improve the images even further. Efficiency of the proposed approach is demonstrated on 2D and 2.5D numerical examples, where we see reconstructions are improved substantially.
- [489] arXiv:2407.01174 (replaced) [pdf, html, other]
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Title: A problem of Erd\H{o}s about rich distancesComments: 6 pages, 4 figuresSubjects: Combinatorics (math.CO)
An old question posed by Erdős asked whether there exists a set of $n$ points such that $c \cdot n$ distances occur more than $n$ times. We provide an affirmative answer to this question, showing that there exists a set of $n$ points such that $\lfloor \frac{n}{4}\rfloor$ distances occur more than $n$ times. We also present a generalized version, finding a set of $n$ points where $c_m \cdot n$ distances occurring more than $n+m$ times.
- [490] arXiv:2407.01203 (replaced) [pdf, other]
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Title: An elementary proof of a criterion for subfunctors of Ext to be closedComments: 17 pages. v2: font size reduced, minor typos fixedSubjects: Category Theory (math.CT); Representation Theory (math.RT)
Let $\mathcal{A}$ be an abelian category and let $F$ be a subbifunctor of the additive bifunctor $\text{Ext}_{\mathcal{A}}^{1}(-,-)\colon \mathcal{A}^{\text{op}}\times \mathcal{A}\to \mathsf{Ab}$. Buan proved in [4] that $F$ is closed if, and only if, $F$ has the $3\times 3$-lemma property, a certain diagrammatic property satisfied by the class of $F$-exact sequences. The proof of this result relies on the theory of exact categories and on the Freyd--Mitchell embedding theorem, a very well-known overpowered result. In this paper we provide a proof of Buan's result only by means of elementary methods in abelian categories. To achieve this we survey the required theory of subfunctors leading us to a self-contained exposition of this topic.
- [491] arXiv:2407.01803 (replaced) [pdf, html, other]
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Title: Error analysis for a viscoelastic phase separation modelComments: 31 pages; 1 figureSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
We consider systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element approximation in space and a variational time discretization strategy. The proposed method preserves the energy-dissipation structure of the underlying system exactly and allows to establish a fully discrete nonlinear stability estimate in natural norms based on the concept of relative energy. These estimates are used to derive order optimal error estimates for the method under minimal smoothness assumptions on the problem data, despite the presence of various strong nonlinearities in the equations. The theoretical results and main properties of the method are illustrated by numerical simulations which also demonstrate the capability to reproduce the relevant physical effects observed in experiments.
- [492] arXiv:2407.02364 (replaced) [pdf, other]
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Title: On the stochastic selection of integral curves of a rough vector fieldSubjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA); Probability (math.PR)
We prove that for bounded, divergence-free vector fields b in L^1_{loc}((0,1];BV(\T^d;\R^d)), there exists a unique incompressible measure on integral curves of b. We recall the vector field constructed by Depauw in [Depauw, C. R. Math. Acad. Sci. Paris, 2003], which lies in the above class, and prove that for this vector field, the unique incompressible measure on integral curves exhibits stochasticity.
- [493] arXiv:2407.02377 (replaced) [pdf, html, other]
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Title: The weak form of the SDOF and MDOF equation of motion, part II: A numerical method for the SDOF problemSubjects: Numerical Analysis (math.NA)
A new, more efficient, numerical method for the SDOF problem is presented. Its construction is based on the weak form of the equation of motion, as obtained in part I of the paper, using piece-wise polynomial functions as interpolation functions. The approximation rate can be arbitrarily high, proportional to the degree of the interpolation functions, tempered only by numerical instability. Moreover, the mechanical energy of the system is conserved. Consequently, all significant drawbacks of existing algorithms, such as the limitations imposed by the Dahlqvist Barrier theorem and the need for introduction of numerical damping, have been overcome.
- [494] arXiv:2407.02847 (replaced) [pdf, other]
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Title: Existence of solutions for a semilinear parabolic system with singular initial dataSubjects: Analysis of PDEs (math.AP)
Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ (u(\cdot,0),v(\cdot,0))=(\mu,\nu) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with $pq>1$, and $(\mu,\nu)$ is a pair of nonnegative Radon measures or locally integrable nonnegative functions in ${\mathbb R}^N$. In this paper we establish sharp sufficient conditions on the initial data for the existence of solutions to problem~(P) using uniformly local Morrey spaces and uniformly local weak Zygmund type spaces.
- [495] arXiv:2407.03164 (replaced) [pdf, html, other]
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Title: Matrices with Hyperbolical Krein Space Numerical RangeSubjects: Functional Analysis (math.FA)
This paper is devoted to matrices with hyperbolical Krein space numerical range. This shape characterizes the 2-by-2 case and persists for certain classes of matrices, independently of their size. Ne\-cessary and sufficient conditions for low dimensional tridiagonal matrices to have this shape are obtained only involving the matrix entries
- [496] arXiv:2407.03250 (replaced) [pdf, html, other]
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Title: When big data actually are low-rank, or entrywise approximation of certain function-generated matricesComments: Fixed the definition of the function $f_1$ in Section 7.2 and figure captionsSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG)
The article concerns low-rank approximation of matrices generated by sampling a smooth function of two $m$-dimensional variables. We refute an argument made in the literature that, for a specific class of analytic functions, such matrices admit accurate entrywise approximation of rank that is independent of $m$. We provide a theoretical explanation of the numerical results presented in support of this argument, describing three narrower classes of functions for which $n \times n$ function-generated matrices can be approximated within an entrywise error of order $\varepsilon$ with rank $\mathcal{O}(\log(n) \varepsilon^{-2} \mathrm{polylog}(\varepsilon^{-1}))$ that is independent of the dimension $m$: (i) functions of the inner product of the two variables, (ii) functions of the squared Euclidean distance between the variables, and (iii) shift-invariant positive-definite kernels. We extend our argument to low-rank tensor-train approximation of tensors generated with functions of the multi-linear product of their $m$-dimensional variables. We discuss our results in the context of low-rank approximation of attention in transformer neural networks.
- [497] arXiv:1907.05940 (replaced) [pdf, html, other]
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Title: Finding irrelevant vertices in linear time on bounded-genus graphsComments: This version is a far generalisation of the techniques and results of earlier versions of this arxiv submissionSubjects: Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
The irrelevant vertex technique provides a powerful tool for the design of parameterized algorithms for a wide variety of problems on graphs. A common characteristic of these problems, permitting the application of this technique on surface-embedded graphs, is the fact that every graph of large enough treewidth contains a vertex that is irrelevant, in the sense that its removal yields an equivalent instance of the problem. The straightforward application of this technique yields algorithms with running time that is quadratic in the size of the input graph. This running time is due to the fact that it takes linear time to detect one irrelevant vertex and the total number of irrelevant vertices to be detected is linear as well. Using advanced techniques, sub-quadratic algorithms have been designed for particular problems, even in general graphs. However, designing a general framework for linear-time algorithms has been open, even for the bounded-genus case. In this paper we introduce a general framework that enables finding in linear time an entire set of irrelevant vertices whose removal yields a bounded-treewidth graph, provided that the input graph has bounded genus. Our technique consists in decomposing any surface-embeddable graph into a tree-structured collection of bounded-treewidth subgraphs where detecting globally irrelevant vertices can be done locally and independently. Our method is applicable to a wide variety of known graph containment or graph modification problems where the irrelevant vertex technique applies. Examples include the (Induced) Minor Folio problem, the (Induced) Disjoint Paths problem, and the $\mathcal{F}$-Minor-Deletion problem.
- [498] arXiv:2102.11076 (replaced) [pdf, html, other]
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Title: Kernel Ridge Riesz Representers: Generalization, Mis-specification, and the Counterfactual Effective DimensionSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Econometrics (econ.EM); Statistics Theory (math.ST)
Kernel balancing weights provide confidence intervals for average treatment effects, based on the idea of balancing covariates for the treated group and untreated group in feature space, often with ridge regularization. Previous works on the classical kernel ridge balancing weights have certain limitations: (i) not articulating generalization error for the balancing weights, (ii) typically requiring correct specification of features, and (iii) justifying Gaussian approximation for only average effects.
I interpret kernel balancing weights as kernel ridge Riesz representers (KRRR) and address these limitations via a new characterization of the counterfactual effective dimension. KRRR is an exact generalization of kernel ridge regression and kernel ridge balancing weights. I prove strong properties similar to kernel ridge regression: population $L_2$ rates controlling generalization error, and a standalone closed form solution that can interpolate. The framework relaxes the stringent assumption that the underlying regression model is correctly specified by the features. It extends Gaussian approximation beyond average effects to heterogeneous effects, justifying confidence sets for causal functions. I use KRRR to quantify uncertainty for heterogeneous treatment effects, by age, of 401(k) eligibility on assets. - [499] arXiv:2105.03067 (replaced) [pdf, html, other]
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Title: The $s$-value: evaluating stability with respect to distributional shiftsComments: Camera ready version of 37th Conference on Neural Information Processing Systems (NeurIPS 2023)Subjects: Methodology (stat.ME); Statistics Theory (math.ST); Machine Learning (stat.ML)
Common statistical measures of uncertainty such as $p$-values and confidence intervals quantify the uncertainty due to sampling, that is, the uncertainty due to not observing the full population. However, sampling is not the only source of uncertainty. In practice, distributions change between locations and across time. This makes it difficult to gather knowledge that transfers across data sets. We propose a measure of instability that quantifies the distributional instability of a statistical parameter with respect to Kullback-Leibler divergence, that is, the sensitivity of the parameter under general distributional perturbations within a Kullback-Leibler divergence ball. In addition, we quantify the instability of parameters with respect to directional or variable-specific shifts. Measuring instability with respect to directional shifts can be used to detect the type of shifts a parameter is sensitive to. We discuss how such knowledge can inform data collection for improved estimation of statistical parameters under shifted distributions. We evaluate the performance of the proposed measure on real data and show that it can elucidate the distributional instability of a parameter with respect to certain shifts and can be used to improve estimation accuracy under shifted distributions.
- [500] arXiv:2108.06713 (replaced) [pdf, html, other]
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Title: Channel State Acquisition in Uplink NOMA for Cellular-Connected UAV: Exploitation of Doppler and Modulation DiversitiesComments: 17 pages, 8 figures, 1 Table, submitted to IEEE Open Journal of the Communications SocietySubjects: Signal Processing (eess.SP); Statistics Theory (math.ST)
Integration of unmanned aerial vehicles (UAVs) for surveillance or monitoring applications into fifth generation (5G) New Radio (NR) cellular networks is an intriguing problem that has recently tackled a lot of interest in both academia and industry. For an efficient spectrum usage, we consider a recently-proposed sky-ground nonorthogonal multiple access (NOMA) scheme, where a cellular-connected UAV acting as aerial user (AU) and a static terrestrial user (TU) are paired to simultaneously transmit their uplink signals to a base station (BS) in the same time-frequency resource blocks. In such a case, due to the highly dynamic nature of the UAV, the signal transmitted by the AU experiences both time dispersion due to multipath propagation effects and frequency dispersion caused by Doppler shifts. On the other hand, for a static ground network, frequency dispersion of the signal transmitted by the TU is negligible and only multipath effects have to be taken into account. To decode the superposed signals at the BS through successive interference cancellation, accurate estimates of both the AU and TU channels are needed. In this paper, we propose channel estimation procedures that suitably exploit the different circular/noncircular modulation formats (modulation diversity) and the different almost-cyclostationarity features (Doppler diversity) of the AU and TU by means of widely-linear time-varying processing. Our estimation approach is semi-blind since Doppler shifts and time delays of the AU are estimated based on the received data only, whereas the remaining relevant parameters of the AU and TU channels are acquired relying also on the available training symbols, which are transmitted by the AU and TU in a nonorthogonal manner.
- [501] arXiv:2109.14507 (replaced) [pdf, html, other]
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Title: A Deformation Quantization for Non-Flat Spacetimes and Applications to QFTComments: Major revisionJournal-ref: Journal of Physics A: Mathematical and Theoretical, 2024Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We provide a deformation quantization, in the sense of Rieffel, for \textit{all} globally hyperbolic spacetimes with a Poisson structure. The Poisson structures have to satisfy Fedosov type requirements in order for the deformed product to be associative. We apply the novel deformation to quantum field theories and their respective states and we prove that the deformed state (i.e.\ a state in non-commutative spacetime) has a singularity structure resembling Minkowski, i.e.\ is \textit{Hadamard}, if the undeformed state is Hadamard. This proves that the Hadamard condition, and hence the quantum field theoretical implementation of the equivalence principle is a general concept that holds in spacetimes with quantum features (i.e. a non-commutative spacetime).
- [502] arXiv:2206.03441 (replaced) [pdf, html, other]
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Title: Robust Sparse Mean Estimation via Sum of SquaresComments: Fixed minor oversight in runtime calculationSubjects: Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
We study the problem of high-dimensional sparse mean estimation in the presence of an $\epsilon$-fraction of adversarial outliers. Prior work obtained sample and computationally efficient algorithms for this task for identity-covariance subgaussian distributions. In this work, we develop the first efficient algorithms for robust sparse mean estimation without a priori knowledge of the covariance. For distributions on $\mathbb R^d$ with "certifiably bounded" $t$-th moments and sufficiently light tails, our algorithm achieves error of $O(\epsilon^{1-1/t})$ with sample complexity $m = (k\log(d))^{O(t)}/\epsilon^{2-2/t}$. For the special case of the Gaussian distribution, our algorithm achieves near-optimal error of $\tilde O(\epsilon)$ with sample complexity $m = O(k^4 \mathrm{polylog}(d))/\epsilon^2$. Our algorithms follow the Sum-of-Squares based, proofs to algorithms approach. We complement our upper bounds with Statistical Query and low-degree polynomial testing lower bounds, providing evidence that the sample-time-error tradeoffs achieved by our algorithms are qualitatively the best possible.
- [503] arXiv:2206.05245 (replaced) [pdf, html, other]
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Title: List-Decodable Sparse Mean Estimation via Difference-of-Pairs FilteringComments: Added fact about taking roots in SoS proofs (Fact 2.9)Subjects: Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
We study the problem of list-decodable sparse mean estimation. Specifically, for a parameter $\alpha \in (0, 1/2)$, we are given $m$ points in $\mathbb{R}^n$, $\lfloor \alpha m \rfloor$ of which are i.i.d. samples from a distribution $D$ with unknown $k$-sparse mean $\mu$. No assumptions are made on the remaining points, which form the majority of the dataset. The goal is to return a small list of candidates containing a vector $\widehat \mu$ such that $\| \widehat \mu - \mu \|_2$ is small. Prior work had studied the problem of list-decodable mean estimation in the dense setting. In this work, we develop a novel, conceptually simpler technique for list-decodable mean estimation. As the main application of our approach, we provide the first sample and computationally efficient algorithm for list-decodable sparse mean estimation. In particular, for distributions with "certifiably bounded" $t$-th moments in $k$-sparse directions and sufficiently light tails, our algorithm achieves error of $(1/\alpha)^{O(1/t)}$ with sample complexity $m = (k\log(n))^{O(t)}/\alpha$ and running time $\mathrm{poly}(mn^t)$. For the special case of Gaussian inliers, our algorithm achieves the optimal error guarantee of $\Theta (\sqrt{\log(1/\alpha)})$ with quasi-polynomial sample and computational complexity. We complement our upper bounds with nearly-matching statistical query and low-degree polynomial testing lower bounds.
- [504] arXiv:2206.12511 (replaced) [pdf, html, other]
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Title: Cost-efficiency in Incomplete MarketsComments: 31 pages. Examples and Counterexamples have been relegated to a separate document, upon journal editor's requestSubjects: Portfolio Management (q-fin.PM); Probability (math.PR); Mathematical Finance (q-fin.MF)
This paper studies the topic of cost-efficiency in incomplete markets. A payoff is called cost-efficient if it achieves a given probability distribution at some given investment horizon with a minimum initial budget. Extensive literature exists for the case of a complete financial market. We show how the problem can be extended to incomplete markets and how the main results from the theory of complete markets still hold in adapted form. In particular, we find that in incomplete markets, the optimal portfolio choice for non-decreasing preferences that are diversification-loving (a notion introduced in this paper) must be "perfectly" cost-efficient. This notion of perfect cost-efficiency is shown to be equivalent to the fact that the payoff can be rationalized, i.e., it is the solution to an expected utility problem.
- [505] arXiv:2207.06325 (replaced) [pdf, html, other]
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Title: Non-Myopic Multifidelity Bayesian OptimizationJournal-ref: Knowledge-Based Systems 299 (2024): 111959Subjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
Bayesian optimization is a popular framework for the optimization of black box functions. Multifidelity methods allows to accelerate Bayesian optimization by exploiting low-fidelity representations of expensive objective functions. Popular multifidelity Bayesian strategies rely on sampling policies that account for the immediate reward obtained evaluating the objective function at a specific input, precluding greater informative gains that might be obtained looking ahead more steps. This paper proposes a non-myopic multifidelity Bayesian framework to grasp the long-term reward from future steps of the optimization. Our computational strategy comes with a two-step lookahead multifidelity acquisition function that maximizes the cumulative reward obtained measuring the improvement in the solution over two steps ahead. We demonstrate that the proposed algorithm outperforms a standard multifidelity Bayesian framework on popular benchmark optimization problems.
- [506] arXiv:2301.07289 (replaced) [pdf, html, other]
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Title: Impact of symmetry inheritance on conformally flat spacetimeSubjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
The goal of this research paper is to investigate curvature inheritance symmetry in conformally flat spacetime. Curvature inheritance symmetry in conformally flat spacetime is shown to be a conformal motion. We have proven that a conformally flat spacetime reduces to Einstein spacetime if admits curvature inheritance symmetry. A few results on conformally flat spacetimes that obey Einstein's field equation with or without a cosmological constant, if admits the curvature inheritance symmetry. The energy-momentum tensor is to be covariantly constant in a 4-dimensional relativistic perfect fluid spacetime which is also conformally flat spacetime, admits curvature inheritance, and obeys Einstein's field equations in the presence of a cosmological constant. Moreover, it is also obtained that such spacetimes with perfect fluid satisfy the the vacuum-like equation of state consecutively it is dark matter. Finally, in the third part of the article, the case compatible with all Theorems from Theorem \ref{Th2.1} to Theorem \ref{Th2.5n} is shown. On the other hand, it has also been emphasized that it is an example of de Sitter spacetime. It has been demonstrated that this spacetime also has a conformal killing vector.
- [507] arXiv:2302.13351 (replaced) [pdf, html, other]
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Title: Optimal local identifying and local locating-dominating codesComments: 28 pages, 9 figuresSubjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO)
We introduce two new classes of covering codes in graphs for every positive integer $r$. These new codes are called local $r$-identifying and local $r$-locating-dominating codes and they are derived from $r$-identifying and $r$-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small $n$ optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid, and the king grid. We prove that seven out of eight of our constructions have optimal densities.
- [508] arXiv:2303.10256 (replaced) [pdf, html, other]
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Title: PINNSim: A Simulator for Power System Dynamics based on Physics-Informed Neural NetworksComments: presented at the 23rd Power Systems Computation Conference (PSCC 2024) and published in Electric Power Systems ResearchJournal-ref: Electric Power Systems Research, vol. 235, p. 110796, Oct. 2024Subjects: Systems and Control (eess.SY); Machine Learning (cs.LG); Numerical Analysis (math.NA)
The dynamic behaviour of a power system can be described by a system of differential-algebraic equations. Time-domain simulations are used to simulate the evolution of these dynamics. They often require the use of small time step sizes and therefore become computationally expensive. To accelerate these simulations, we propose a simulator - PINNSim - that allows to take significantly larger time steps. It is based on Physics-Informed Neural Networks (PINNs) for the solution of the dynamics of single components in the power system. To resolve their interaction we employ a scalable root-finding algorithm. We demonstrate PINNSim on a 9-bus system and show the increased time step size compared to a trapezoidal integration rule. We discuss key characteristics of PINNSim and important steps for developing PINNSim into a fully fledged simulator. As such, it could offer the opportunity for significantly increasing time step sizes and thereby accelerating time-domain simulations.
- [509] arXiv:2304.00841 (replaced) [pdf, html, other]
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Title: Hypergraph AnimalsComments: 16 pages, 5 figures plus several in-line figuresSubjects: Molecular Networks (q-bio.MN); Disordered Systems and Neural Networks (cond-mat.dis-nn); Combinatorics (math.CO)
Here we introduce simple structures for the analysis of complex hypergraphs, hypergraph animals. These structures are designed to describe the local node neighbourhoods of nodes in hypergraphs. We establish their relationships to lattice animals and network motifs, and we develop their combinatorial properties for sparse and uncorrelated hypergraphs. We make use of the tight link of hypergraph animals to partition numbers, which opens up a vast mathematical framework for the analysis of hypergraph animals. We then study their abundances in random hypergraphs. Two transferable insights result from this analysis: (i) it establishes the importance of high-cardinality edges in ensembles of random hypergraphs that are inspired by the classical Erdös-Renyí random graphs; and (ii) there is a close connection between degree and hyperedge cardinality in random hypergraphs that shapes animal abundances and spectra profoundly. Both findings imply that hypergraph animals can have the potential to affect information flow and processing in complex systems. Our analysis of also suggests that we need to spend more effort on investigating and developing suitable conditional ensembles of random hypergraphs that can capture real-world structures and their complex dependency structures.
- [510] arXiv:2306.05186 (replaced) [pdf, html, other]
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Title: Inference through innovation processes tested in the authorship attribution taskSubjects: Methodology (stat.ME); Information Theory (cs.IT); Applied Physics (physics.app-ph); Data Analysis, Statistics and Probability (physics.data-an)
Urn models for innovation capture fundamental empirical laws shared by several real-world processes. The so-called urn model with triggering includes, as particular cases, the urn representation of the two-parameter Poisson-Dirichlet process and the Dirichlet process, seminal in Bayesian non-parametric inference. In this work, we leverage this connection to introduce a general approach for quantifying closeness between symbolic sequences and test it within the framework of the authorship attribution problem. The method demonstrates high accuracy when compared to other related methods in different scenarios, featuring a substantial gain in computational efficiency and theoretical transparency. Beyond the practical convenience, this work demonstrates how the recently established connection between urn models and non-parametric Bayesian inference can pave the way for designing more efficient inference methods. In particular, the hybrid approach that we propose allows us to relax the exchangeability hypothesis, which can be particularly relevant for systems exhibiting complex correlation patterns and non-stationary dynamics.
- [511] arXiv:2306.11609 (replaced) [pdf, html, other]
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Title: Error-induced extinction in a multi-type critical birth-death processComments: 34 pages, 7 figuresSubjects: Populations and Evolution (q-bio.PE); Probability (math.PR)
Extreme mutation rates in microbes and cancer cells can result in error-induced extinction (EEX), where every descendant cell eventually acquires a lethal mutation. In this work, we investigate critical birth-death processes with $n$ distinct types as a birth-death model of EEX in a growing population. Each type-$i$ cell divides independently $(i)\to(i)+(i)$ or mutates $(i)\to(i+1)$ at the same rate. The total number of cells grows exponentially as a Yule process until a cell of type-$n$ appears, which cell type can only die at rate one. This makes the whole process critical and hence after the exponentially growing phase eventually all cells die with probability one. We present large-time asymptotic results for the general $n$-type critical birth-death process. We find that the mass function of the number of cells of type-$k$ has algebraic and stationary tail $(\text{size})^{-1-\chi_k}$, with $\chi_k=2^{1-k}$, for $k=2,\dots,n$, in sharp contrast to the exponential tail of the first type. The same exponents describe the tail of the asymptotic survival probability $(\text{time})^{-\chi_n}$. We present applications of the results for studying extinction due to intolerable mutation rates in biological populations.
- [512] arXiv:2306.17720 (replaced) [pdf, html, other]
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Title: Conformal duality of the nonlinear Schr\"odinger equation: Theory and applications to parameter estimationComments: 19 pages, 5 figuresSubjects: Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI); Nuclear Theory (nucl-th); Quantum Physics (quant-ph)
The nonlinear Schrödinger equation (NLSE) is a rich and versatile model, which in one spatial dimension has stationary solutions similar to those of the linear Schrödinger equation as well as more exotic solutions such as solitary waves and quantum droplets. Here we present the unified theory of the NLSE, showing that all stationary solutions of the local one-dimensional cubic-quintic NLSE can be classified according to a single number called the cross-ratio. Any two solutions with the same cross-ratio can be converted into one another using a conformal transformation, and the same also holds true for traveling wave solutions. Further, we introduce an optimization afterburner that relies on this conformal symmetry to substantially improve NLSE parameter estimation from noisy empirical data. The new method therefore should have far reaching practical applications for nonlinear physical systems.
- [513] arXiv:2306.17815 (replaced) [pdf, html, other]
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Title: Bayesian Optimization with Formal Safety Guarantees via Online Conformal PredictionComments: 15 pages, 10 figures, this work has been published in IEEE Journal of Selected Topics in Signal ProcessingSubjects: Machine Learning (cs.LG); Information Theory (cs.IT); Signal Processing (eess.SP)
Black-box zero-th order optimization is a central primitive for applications in fields as diverse as finance, physics, and engineering. In a common formulation of this problem, a designer sequentially attempts candidate solutions, receiving noisy feedback on the value of each attempt from the system. In this paper, we study scenarios in which feedback is also provided on the safety of the attempted solution, and the optimizer is constrained to limit the number of unsafe solutions that are tried throughout the optimization process. Focusing on methods based on Bayesian optimization (BO), prior art has introduced an optimization scheme -- referred to as SAFEOPT -- that is guaranteed not to select any unsafe solution with a controllable probability over feedback noise as long as strict assumptions on the safety constraint function are met. In this paper, a novel BO-based approach is introduced that satisfies safety requirements irrespective of properties of the constraint function. This strong theoretical guarantee is obtained at the cost of allowing for an arbitrary, controllable but non-zero, rate of violation of the safety constraint. The proposed method, referred to as SAFE-BOCP, builds on online conformal prediction (CP) and is specialized to the cases in which feedback on the safety constraint is either noiseless or noisy. Experimental results on synthetic and real-world data validate the advantages and flexibility of the proposed SAFE-BOCP.
- [514] arXiv:2308.09571 (replaced) [pdf, other]
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Title: Physics-Informed Boundary Integral Networks (PIBI-Nets): A Data-Driven Approach for Solving Partial Differential EquationsJournal-ref: Journal of Computational Science, Elsevier, 2024Subjects: Machine Learning (cs.LG); Dynamical Systems (math.DS); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
Partial differential equations (PDEs) are widely used to describe relevant phenomena in dynamical systems. In real-world applications, we commonly need to combine formal PDE models with (potentially noisy) observations. This is especially relevant in settings where we lack information about boundary or initial conditions, or where we need to identify unknown model parameters. In recent years, Physics-Informed Neural Networks (PINNs) have become a popular tool for this kind of problems. In high-dimensional settings, however, PINNs often suffer from computational problems because they usually require dense collocation points over the entire computational domain. To address this problem, we present Physics-Informed Boundary Integral Networks (PIBI-Nets) as a data-driven approach for solving PDEs in one dimension less than the original problem space. PIBI-Nets only require points at the computational domain boundary, while still achieving highly accurate results. Moreover, PIBI-Nets clearly outperform PINNs in several practical settings. Exploiting elementary properties of fundamental solutions of linear differential operators, we present a principled and simple way to handle point sources in inverse problems. We demonstrate the excellent performance of PIBI- Nets for the Laplace and Poisson equations, both on artificial datasets and within a real-world application concerning the reconstruction of groundwater flows.
- [515] arXiv:2308.13451 (replaced) [pdf, html, other]
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Title: Gotta match 'em all: Solution diversification in graph matching matched filtersComments: 27 pages, 12 figures, 3 tablesSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Combinatorics (math.CO); Applications (stat.AP); Methodology (stat.ME)
We present a novel approach for finding multiple noisily embedded template graphs in a very large background graph. Our method builds upon the graph-matching-matched-filter technique proposed in Sussman et al., with the discovery of multiple diverse matchings being achieved by iteratively penalizing a suitable node-pair similarity matrix in the matched filter algorithm. In addition, we propose algorithmic speed-ups that greatly enhance the scalability of our matched-filter approach. We present theoretical justification of our methodology in the setting of correlated Erdos-Renyi graphs, showing its ability to sequentially discover multiple templates under mild model conditions. We additionally demonstrate our method's utility via extensive experiments both using simulated models and real-world dataset, include human brain connectomes and a large transactional knowledge base.
- [516] arXiv:2308.15048 (replaced) [pdf, html, other]
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Title: Optimal ratcheting of dividend payout under Brownian motion surplusComments: To appear in SICONSubjects: Mathematical Finance (q-fin.MF); Optimization and Control (math.OC); General Finance (q-fin.GN)
This paper is concerned with a long standing optimal dividend payout problem subject to the so-called ratcheting constraint, that is, the dividend payout rate shall be non-decreasing over time and is thus self-path-dependent. The surplus process is modeled by a drifted Brownian motion process and the aim is to find the optimal dividend ratcheting strategy to maximize the expectation of the total discounted dividend payouts until the ruin time. Due to the self-path-dependent control constraint, the standard control theory cannot be directly applied to tackle the problem. The related Hamilton-Jacobi-Bellman (HJB) equation is a new type of variational inequality. In the literature, it is only shown to have a viscosity solution, which is not strong enough to guarantee the existence of an optimal dividend ratcheting strategy. This paper proposes a novel partial differential equation method to study the HJB equation. We not only prove the the existence and uniqueness of the solution in some stronger functional space, but also prove the strict monotonicity, boundedness, and $C^\infty$-smoothness of the dividend ratcheting free boundary. Based on these results, we eventually derive an optimal dividend ratcheting strategy, and thus solve the open problem completely. Economically speaking, we find that if the surplus volatility is above an explicit threshold, then one should pay dividends at the maximum rate, regardless the surplus level. Otherwise, by contrast, the optimal dividend ratcheting strategy relays on the surplus level and one should only ratchet up the dividend payout rate when the surplus level touches the dividend ratcheting free boundary. Moreover, our numerical results suggest that one should invest into those companies with stable dividend payout strategies since their income rates should be higher and volatility rates smaller.
- [517] arXiv:2309.14416 (replaced) [pdf, html, other]
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Title: Homotopy, Symmetry, and Non-Hermitian Band TopologyComments: 44 pages, 12 figures, published versionJournal-ref: Rep. Prog. Phys. 87 078002 (2024)Subjects: Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Mathematical Physics (math-ph); Optics (physics.optics); Quantum Physics (quant-ph)
Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time ($\mathcal{PT}$) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases in $\mathcal{PT}$-symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous $\mathcal{PT}$ symmetry breaking is captured by Chern-Euler and Chern-Stiefel-Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.
- [518] arXiv:2310.17582 (replaced) [pdf, html, other]
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Title: Convergence of flow-based generative models via proximal gradient descent in Wasserstein spaceSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC); Statistics Theory (math.ST)
Flow-based generative models enjoy certain advantages in computing the data generation and the likelihood, and have recently shown competitive empirical performance. Compared to the accumulating theoretical studies on related score-based diffusion models, analysis of flow-based models, which are deterministic in both forward (data-to-noise) and reverse (noise-to-data) directions, remain sparse. In this paper, we provide a theoretical guarantee of generating data distribution by a progressive flow model, the so-called JKO flow model, which implements the Jordan-Kinderleherer-Otto (JKO) scheme in a normalizing flow network. Leveraging the exponential convergence of the proximal gradient descent (GD) in Wasserstein space, we prove the Kullback-Leibler (KL) guarantee of data generation by a JKO flow model to be $O(\varepsilon^2)$ when using $N \lesssim \log (1/\varepsilon)$ many JKO steps ($N$ Residual Blocks in the flow) where $\varepsilon $ is the error in the per-step first-order condition. The assumption on data density is merely a finite second moment, and the theory extends to data distributions without density and when there are inversion errors in the reverse process where we obtain KL-$W_2$ mixed error guarantees. The non-asymptotic convergence rate of the JKO-type $W_2$-proximal GD is proved for a general class of convex objective functionals that includes the KL divergence as a special case, which can be of independent interest. The analysis framework can extend to other first-order Wasserstein optimization schemes applied to flow-based generative models.
- [519] arXiv:2311.01375 (replaced) [pdf, html, other]
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Title: Monotone Generative Modeling via a Gromov-Monge EmbeddingComments: 21 pages excluding referencesSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
Generative adversarial networks (GANs) are popular for generative tasks; however, they often require careful architecture selection, extensive empirical tuning, and are prone to mode collapse. To overcome these challenges, we propose a novel model that identifies the low-dimensional structure of the underlying data distribution, maps it into a low-dimensional latent space while preserving the underlying geometry, and then optimally transports a reference measure to the embedded distribution. We prove three key properties of our method: 1) The encoder preserves the geometry of the underlying data; 2) The generator is $c$-cyclically monotone, where $c$ is an intrinsic embedding cost employed by the encoder; and 3) The discriminator's modulus of continuity improves with the geometric preservation of the data. Numerical experiments demonstrate the effectiveness of our approach in generating high-quality images and exhibiting robustness to both mode collapse and training instability.
- [520] arXiv:2312.05831 (replaced) [pdf, html, other]
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Title: Physics-Aware Multifidelity Bayesian Optimization: a Generalized FormulationJournal-ref: Computers & Structures 296 (2024): 107302Subjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
The adoption of high-fidelity models for many-query optimization problems is majorly limited by the significant computational cost required for their evaluation at every query. Multifidelity Bayesian methods (MFBO) allow to include costly high-fidelity responses for a sub-selection of queries only, and use fast lower-fidelity models to accelerate the optimization process. State-of-the-art methods rely on a purely data-driven search and do not include explicit information about the physical context. This paper acknowledges that prior knowledge about the physical domains of engineering problems can be leveraged to accelerate these data-driven searches, and proposes a generalized formulation for MFBO to embed a form of domain awareness during the optimization procedure. In particular, we formalize a bias as a multifidelity acquisition function that captures the physical structure of the domain. This permits to partially alleviate the data-driven search from learning the domain properties on-the-fly, and sensitively enhances the management of multiple sources of information. The method allows to efficiently include high-fidelity simulations to guide the optimization search while containing the overall computational expense. Our physics-aware multifidelity Bayesian optimization is presented and illustrated for two classes of optimization problems frequently met in science and engineering, namely design optimization and health monitoring problems.
- [521] arXiv:2401.07187 (replaced) [pdf, html, other]
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Title: A Survey on Statistical Theory of Deep Learning: Approximation, Training Dynamics, and Generative ModelsComments: 33 pages, no figures,Invited for review in Annual Review of Statistics and Its Application (In review)Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
In this article, we review the literature on statistical theories of neural networks from three perspectives. In the first part, results on excess risks for neural networks are reviewed in the nonparametric framework of regression or classification. These results rely on explicit constructions of neural networks, leading to fast convergence rates of excess risks, in that tools from the approximation theory are adopted. Through these constructions, the width and depth of the networks can be expressed in terms of sample size, data dimension, and function smoothness. Nonetheless, their underlying analysis only applies to the global minimizer in the highly non-convex landscape of deep neural networks. This motivates us to review the training dynamics of neural networks in the second part. Specifically, we review papers that attempt to answer ``how the neural network trained via gradient-based methods finds the solution that can generalize well on unseen data.'' In particular, two well-known paradigms are reviewed: the Neural Tangent Kernel (NTK) paradigm, and Mean-Field (MF) paradigm. In the last part, we review the most recent theoretical advancements in generative models including Generative Adversarial Networks (GANs), diffusion models, and in-context learning (ICL) in the Large Language Models (LLMs). The former two models are known to be the main pillars of the modern generative AI era, while ICL is a strong capability of LLMs in learning from a few examples in the context. Finally, we conclude the paper by suggesting several promising directions for deep learning theory.
- [522] arXiv:2401.10553 (replaced) [pdf, other]
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Title: Single-set cubical categories and their formalisation with a proof assistant (extended version)Subjects: Logic in Computer Science (cs.LO); Category Theory (math.CT)
We introduce a single-set axiomatisation of cubical $\omega$-categories, including connections and inverses. We justify these axioms by establishing a series of equivalences between the category of single-set cubical $\omega$-categories, and their variants with connections and inverses, and the corresponding cubical $\omega$-categories. We also report on the formalisation of cubical $\omega$-categories with the Isabelle/HOL proof assistant, which has been instrumental in developing the single-set axiomatisation.
- [523] arXiv:2402.08052 (replaced) [pdf, html, other]
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Title: Jackiw-Teitelboim Gravity, Random Disks of Constant Curvature, Self-Overlapping Curves and Liouville $\text{CFT}_{1}$Comments: 14 pages, 1 figure; v2: typos corrected, refs updatedSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We propose a microscopic definition of finite cut-off JT quantum gravity on the disk, both in the discretized and in the continuum points of view. The discretized formulation involves a new model of so-called self-overlapping random polygons. The measure is not uniform, implying that the degrees of freedom are not in one-to-one correspondence with the shape of the boundary. The continuum formulation is based on a boundary $\text{CFT}_{1}$ from which we predict some critical exponents of the self-overlapping polygon model. The coupling to an arbitrary bulk matter CFT is also discussed.
- [524] arXiv:2402.08264 (replaced) [pdf, html, other]
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Title: On Iiro Honkala's contributions to identifying codesSubjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO)
A set $C$ of vertices in a graph $G=(V,E)$ is an identifying code if it is dominating and any two vertices of $V$ are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala's contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes.
- [525] arXiv:2402.17977 (replaced) [pdf, html, other]
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Title: Null Infinity as a Weakly Isolated HorizonComments: 21 pages. At referee's suggestion version 1 of this paper was split into two papers. Consequently the current version v2 contains only the first three sections of v1. The remaining sections appear in a paper entitled "Null Infinity and Horizons: New Approach to Fluxes and Charges', arXiv:2407.03254. Both papers are at press at Phy. Rev. DSubjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
Null infinity arises as a boundary of the Penrose conformal completion of an asymptotically flat physical space-time. We first note that null infinity is a weakly isolated horizon (WIH), and then show that its familiar properties can be derived from the general WIH framework. This seems quite surprising because physics associated with black hole (and cosmological) WIHs is very different from that extracted at null infinity. We show that these differences can be directly traced back to the fact that null infinity is a WIH in the conformal completion rather than the physical space-time. In particular, the BMS group at null infinity stems from the symmetry group of WIHs. In a companion paper, we obtain fluxes and charges associated with symmetries associated with both null infinity and black hole (and cosmological) horizons using a new Hamiltonian framework. The fact that is there is a single mathematical framework underlying these different situations paves the way to explore the relation between horizon dynamics in the strong field region and waveforms at infinity. It should also be useful in the analysis of black hole evaporation in quantum gravity.
- [526] arXiv:2402.19389 (replaced) [pdf, html, other]
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Title: Fault-tolerance of the [[8,1,3]] non-CSS codeSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT)
We present a fault-tolerant [[8, 1, 3]] non-CSS quantum error correcting code and study its logical error rates. We choose the unitary encoding procedure for stabilizer codes given by Gottesman and modify it to suit the setting of a class of non- CSS codes. Considering two types of noise models for this study, namely the depolarising noise and anisotropic noise, to depict the logical error rates obtained in decoding, we adopt the procedure of the bare ancilla method presented by Brown et al. to reorder the measurement sequence in the syndrome extraction step and upgrade it to obtain higher pseudo-thresholds and lower leading order terms of logical error rates.
- [527] arXiv:2403.02284 (replaced) [pdf, other]
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Title: Graphical Quadratic AlgebraSubjects: Logic in Computer Science (cs.LO); Category Theory (math.CT); Optimization and Control (math.OC)
We introduce Graphical Quadratic Algebra (GQA), a string diagrammatic calculus extending the language of Graphical Affine Algebra with a new generator characterised by invariance under rotation matrices. We show that GQA is a sound and complete axiomatisation for three different models: quadratic relations, which are a compositional formalism for least-squares problems, Gaussian stochastic processes, and Gaussian stochastic processes extended with non-determinisms. The equational theory of GQA sheds light on the connections between these perspectives, giving an algebraic interpretation to the interplay of stochastic behaviour, relational behaviour, non-determinism, and conditioning. As applications, we discuss various case studies, including linear regression, probabilistic programming, and electrical circuits with realistic (noisy) components.
- [528] arXiv:2403.02686 (replaced) [pdf, html, other]
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Title: Extending echo state property for quantum reservoir computingComments: 16 pages, 14 figuresSubjects: Quantum Physics (quant-ph); Dynamical Systems (math.DS); Machine Learning (stat.ML)
The echo state property (ESP) represents a fundamental concept in the reservoir computing (RC) framework that ensures output-only training of reservoir networks by being agnostic to the initial states and far past inputs. However, the traditional definition of ESP does not describe possible non-stationary systems in which statistical properties evolve. To address this issue, we introduce two new categories of ESP: $\textit{non-stationary ESP}$, designed for potentially non-stationary systems, and $\textit{subspace/subset ESP}$, designed for systems whose subsystems have ESP. Following the definitions, we numerically demonstrate the correspondence between non-stationary ESP in the quantum reservoir computer (QRC) framework with typical Hamiltonian dynamics and input encoding methods using non-linear autoregressive moving-average (NARMA) tasks. We also confirm the correspondence by computing linear/non-linear memory capacities that quantify input-dependent components within reservoir states. Our study presents a new understanding of the practical design of QRC and other possibly non-stationary RC systems in which non-stationary systems and subsystems are exploited.
- [529] arXiv:2403.03352 (replaced) [pdf, html, other]
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Title: Characterization of admissible quasisymmetriesComments: 18 pagesSubjects: Plasma Physics (physics.plasm-ph); Mathematical Physics (math-ph)
We solve "half" the problem of finding three-dimensional quasisymmetric magnetic fields that do not necessarily satisfy force balance. This involves determining which hidden symmetries are admissible as quasisymmetries, and then showing explicitly how to construct quasisymmetric magnetic fields given an admissible symmetry. The admissibility conditions take the form of a system of overdetermined nonlinear partial differential equations involving second derivatives of the symmetry's infinitesimal generator.
- [530] arXiv:2403.16456 (replaced) [pdf, other]
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Title: Flux Quantization on 11-dimensional SuperspaceComments: 62 pages; v3: added section 2.1.7 and various minor improvements; v2: fixed lemma 2.48 and added lemma 3.6Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Algebraic Topology (math.AT); Differential Geometry (math.DG)
Flux quantization of the C-field in 11d supergravity is arguably necessary for the (UV-)completion of the theory, in that it determines the torsion charges carried by small numbers of M-branes. However, hypotheses about C-field flux-quantization ("models of the C-field") have previously been discussed only in the bosonic sector of 11d supergravity and ignoring the supergravity equations of motion. Here we highlight a duality-symmetric formulation of on-shell 11d supergravity on superspace, observe that this naturally lends itself to completion of the theory by flux quantization, and indeed that 11d super-spacetimes are put on-shell by carrying quantizable duality-symmetric super-C-field flux; the proof of which we present in detail.
- [531] arXiv:2404.00666 (replaced) [pdf, other]
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Title: Accelerated Parameter-Free Stochastic OptimizationSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
We propose a method that achieves near-optimal rates for smooth stochastic convex optimization and requires essentially no prior knowledge of problem parameters. This improves on prior work which requires knowing at least the initial distance to optimality d0. Our method, U-DoG, combines UniXGrad (Kavis et al., 2019) and DoG (Ivgi et al., 2023) with novel iterate stabilization techniques. It requires only loose bounds on d0 and the noise magnitude, provides high probability guarantees under sub-Gaussian noise, and is also near-optimal in the non-smooth case. Our experiments show consistent, strong performance on convex problems and mixed results on neural network training.
- [532] arXiv:2404.06929 (replaced) [pdf, html, other]
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Title: Exact solution of a two-parameter extended Bariev modelSubjects: Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)
An exactly solvable strongly correlated electron model with two independent parameters is constructed in the frame of the quantum inverse scattering method, which can be seen as a generalization of the Bariev model. Through the Bethe ansatz method, a set of Bethe ansatz equations is derived. In the thermodynamic limit, to study the ground state of the model, we obtain the integral equations for the density of Bethe roots. Numerical validation are done to confirm the accuracy of our analytic results.
- [533] arXiv:2404.08826 (replaced) [pdf, other]
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Title: Strongly Tail-Optimal Scheduling in the Light-Tailed M/G/1Comments: 33 pages, 8 figures. SIGMETRICS 2024Subjects: Performance (cs.PF); Probability (math.PR)
We study the problem of scheduling jobs in a queueing system, specifically an M/G/1 with light-tailed job sizes, to asymptotically optimize the response time tail. This means scheduling to make $\mathbf{P}[T > t]$, the chance a job's response time exceeds $t$, decay as quickly as possible in the $t \to \infty$ limit. For some time, the best known policy was First-Come First-Served (FCFS), which has an asymptotically exponential tail: $\mathbf{P}[T > t] \sim C e^{-\gamma t}$. FCFS achieves the optimal *decay rate* $\gamma$, but its *tail constant* $C$ is suboptimal. Only recently have policies that improve upon FCFS's tail constant been discovered. But it is unknown what the optimal tail constant is, let alone what policy might achieve it.
In this paper, we derive a closed-form expression for the optimal tail constant $C$, and we introduce *$\gamma$-Boost*, a new policy that achieves this optimal tail constant. Roughly speaking, $\gamma$-Boost operates similarly to FCFS, but it pretends that small jobs arrive earlier than their true arrival times. This significantly reduces the response time of small jobs without unduly delaying large jobs, improving upon FCFS's tail constant by up to 50% with only moderate job size variability, with even larger improvements for higher variability. While these results are for systems with full job size information, we also introduce and analyze a version of $\gamma$-Boost that works in settings with partial job size information, showing it too achieves significant gains over FCFS. Finally, we show via simulation that $\gamma$-Boost has excellent practical performance. - [534] arXiv:2404.14648 (replaced) [pdf, html, other]
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Title: Pseudorandom Permutations from Random Reversible CircuitsComments: v2: added references and comparison to subsequent work, removed claim in previous Section 7.3 with error in proofSubjects: Computational Complexity (cs.CC); Cryptography and Security (cs.CR); Probability (math.PR)
We study pseudorandomness properties of permutations on $\{0,1\}^n$ computed by random circuits made from reversible $3$-bit gates (permutations on $\{0,1\}^3$). Our main result is that a random circuit of depth $n \cdot \tilde{O}(k^2)$, with each layer consisting of $\approx n/3$ random gates in a fixed nearest-neighbor architecture, yields almost $k$-wise independent permutations. The main technical component is showing that the Markov chain on $k$-tuples of $n$-bit strings induced by a single random $3$-bit nearest-neighbor gate has spectral gap at least $1/n \cdot \tilde{O}(k)$. This improves on the original work of Gowers [Gowers96], who showed a gap of $1/\mathrm{poly}(n,k)$ for one random gate (with non-neighboring inputs); and, on subsequent work [HMMR05,BH08] improving the gap to $\Omega(1/n^2k)$ in the same setting.
From the perspective of cryptography, our result can be seen as a particularly simple/practical block cipher construction that gives provable statistical security against attackers with access to $k$~input-output pairs within few rounds. We also show that the Luby--Rackoff construction of pseudorandom permutations from pseudorandom functions can be implemented with reversible circuits. From this, we make progress on the complexity of the Minimum Reversible Circuit Size Problem (MRCSP), showing that block ciphers of fixed polynomial size are computationally secure against arbitrary polynomial-time adversaries, assuming the existence of one-way functions (OWFs). - [535] arXiv:2405.09277 (replaced) [pdf, other]
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Title: Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representationComments: v1: 41 pages; v2: 46 pages, several typos are corrected, a more detailed discussion of non-invertible symmetry is provided; v3: typos are corrected, more discussion on Hopf symmetries addedSubjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and delve into a detailed discussion of its non-invertible symmetries. We also show that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model. We also introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra.
- [536] arXiv:2406.00208 (replaced) [pdf, html, other]
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Title: Progresses on some open problems related to infinitely many symmetriesSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Pattern Formation and Solitons (nlin.PS); Classical Physics (physics.class-ph)
The quest to reveal the physical essence of the infinitely many symmetries and conservation laws that are intrinsic to integrable systems has historically posed a significant challenge at the confluence of physics and mathematics. This scholarly investigation delves into five open problems related to these boundless symmetries within integrable systems by scrutinizing their multi-wave solutions, employing a fresh analytical methodology. For a specified integrable system, there exist various categories of $n$-wave solutions. Each sub-wave comprising the $n$-wave solution may possess free parameters, including center, width, and periodic parameters. It is evident that these solutions are translation invariant with respect to all these free parameters. We postulate that the entirety of the recognized infinite symmetries merely constitute linear combinations of these finite wave parameter translation symmetries. The conjecture intimates that the currently known infinitely many symmetries are not exhaustive, and an indeterminate number of symmetries remain to be discovered. This conjecture further indicates that by imposing an infinite array of symmetry constraints, it becomes feasible to derive exact multi-wave solutions. By considering the renowned KdV equation and the Burgers equation as simple examples, the conjecture is substantiated for the $n$-soliton solutions. It is unequivocal that any linear combination of the wave parameter translation symmetries retains its status as a symmetry associated with the particular solution. This observation suggests that by introducing a ren-variable and a ren-symmetric derivative which serve as generalizations of the Grassmann variable and the super derivative, it may be feasible to unify classical integrable systems, supersymmetric integrable systems, and ren-symmetric integrable systems within a cohesive hierarchical framework.
- [537] arXiv:2406.15860 (replaced) [pdf, other]
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Title: Elliptic Deformation of the Gaiotto-Rap\v{c}\'{a}k Corner VOA and the Associated Partially Symmetric PolynomialsComments: 44 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Combinatorics (math.CO); Quantum Algebra (math.QA); Representation Theory (math.RT)
We construct the elliptic Miura transformation and use it to obtain the expression of the currents of elliptic corner VOA. We subsequently prove a novel combinatorial formula that is essential for deriving the quadratic relations of the currents. In addition, we give a conjecture that relates the correlation function of the currents of elliptic corner VOA to a certain family of partially symmetric polynomials. The elliptic Macdonald polynomials, constructed recently by Awata-Kanno-Mironov-Morozov-Zenkevich, and Fukuda-Ohkubo-Shiraishi, can be obtained as a particular case of this family.
- [538] arXiv:2407.02864 (replaced) [pdf, html, other]
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Title: A quantum approach for optimal controlComments: 39 pagesSubjects: Quantum Physics (quant-ph); Optimization and Control (math.OC)
In this work, we propose a novel variational quantum approach for solving a class of nonlinear optimal control problems. Our approach integrates Dirac's canonical quantization of dynamical systems with the solution of the ground state of the resulting non-Hermitian Hamiltonian via a variational quantum eigensolver (VQE). We introduce a new perspective on the Dirac bracket formulation for generalized Hamiltonian dynamics in the presence of constraints, providing a clear motivation and illustrative examples. Additionally, we explore the structural properties of Dirac brackets within the context of multidimensional constrained optimization problems.
Our approach for solving a class of nonlinear optimal control problems employs a VQE-based approach to determine the eigenstate and corresponding eigenvalue associated with the ground state energy of a non-Hermitian Hamiltonian. Assuming access to an ideal VQE, our formulation demonstrates excellent results, as evidenced by selected computational examples. Furthermore, our method performs well when combined with a VQE-based approach for non-Hermitian Hamiltonian systems. Our VQE-based formulation effectively addresses challenges associated with a wide range of optimal control problems, particularly in high-dimensional scenarios. Compared to standard classical approaches, our quantum-based method shows significant promise and offers a compelling alternative for tackling complex, high-dimensional optimization challenges.