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f42973ce99eaf1a86307743b5536ad8b07a979fb00af2a2a4d275fd5318c40c6
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2026-01-21T00:00:00-05:00
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A Quadratically-Constrained Convex Approximation for the AC Optimal Power Flow
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arXiv:2501.05623v2 Announce Type: replace Abstract: We introduce a quadratically-constrained approximation (QCAC) of the AC optimal power flow (AC-OPF) problem. Unlike existing approximations like the DC-OPF, our model does not rely on typical assumptions such as high reactance-to-resistance ratio, near-nominal voltage magnitudes, or small angle differences, and preserves the structural sparsity of the original AC power flow equations, making it suitable for decentralized power systems optimization problems. To achieve this, we reformulate the AC-OPF problem as a quadratically constrained quadratic program. The nonconvex terms are expressed as differences of convex functions, which are then convexified around a base point derived from a warm start of the nodal voltages. If this linearization results in a non-empty constraint set, the convexified constraints form an inner convex approximation. Our experimental results, based on Power Grid Library instances of up to 30,000 buses, demonstrate the effectiveness of the QCAC approximation with respect to other well-documented conic relaxations and a linear approximation. We further showcase its potential advantages over the well-documented second-order conic relaxation of the power flow equations in two proof-of-concept case studies: optimal reactive power dispatch in transmission networks and PV hosting capacity in distribution grids.
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https://arxiv.org/abs/2501.05623
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e0f441914a6435eb716f51d8e0fd58bcc7601f2a445be12cd5f3a8b202aa5287
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2026-01-21T00:00:00-05:00
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Positive solutions for fractional-order boundary value problems with or without dependence of integer-order ones
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arXiv:2501.05810v2 Announce Type: replace Abstract: We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P} \left\{ \begin{array}{l} D_{0+}^\alpha u + h(t)f(u) = 0, \quad 0<1, \\[1ex] u(0)=u(1)=0, \end{array} \right. \end{align} where $D_{0+}^\alpha$ is the Riemann-Liouville fractional derivative of order $\alpha\in(1,2]$. Firstly, by considering the first eigenvalue $\lambda_1(\alpha)$ of the corresponding eigenvalue problem, we establish the existence of positive solutions for both sublinear and superlinear cases involving $\lambda_1(\alpha)$, thereby extending existing results in the literature. In addition, we address the issue of non-existence, which reinforces the sharpness of both hypotheses. Secondly, we demonstrate the uniqueness of positive solutions. For the sublinear case, we impose certain monotonicity conditions on $f$. For the superlinear case, we assume that $h$ satisfies a specific condition to ensure the uniqueness of positive solutions when $\alpha =2.$ Near $\alpha =2,$ we prove uniqueness by leveraging the non-degeneracy of the unique solution, which represents a novel approach to studying fractional-order differential equations. Finally, we apply this methodology to establish the multiple existence of at least three positive solutions for H\'{e}non-type problems, which is also a new contribution.
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https://arxiv.org/abs/2501.05810
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442b03a8e011cef1c0bd53dcdfb975e3c628d053de609d2f14c162724f07b947
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2026-01-21T00:00:00-05:00
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The Born approximation for the fixed energy Calder\'on problem
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arXiv:2501.05889v2 Announce Type: replace Abstract: The Born approximation of a potential in the context of the Calder\'on inverse problem is an object that can be formally defined in terms of spectral data of the Dirichlet-to-Neumann map of the corresponding Schr\"odinger operator. In this article, we prove, in the case of radial potentials in the Euclidean ball and any fixed energy, that the Born approximation is well-defined as a compactly supported radial distribution, and that the Calder\'on problem can be reformulated as recovering a potential from its Born approximation. In addition, we show that the Born approximation depends locally on the potential and captures exactly its singularities, and that the functional that maps the Born approximation to the potential is H\"older continuous. We also prove that the Born approximation converges to the potential in the high-energy limit. Moreover, we give an explicit formula for the Fourier transform of the Born approximation at any fixed energy, and illustrate how it can be used as the basis of an accurate procedure to approximate a potential from its Dirichlet-to-Neumann map.
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https://arxiv.org/abs/2501.05889
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ec60192ea71d7d1ed0c37181f7ea69b036b1387408781b9ec34339e5733364b6
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2026-01-21T00:00:00-05:00
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Eco-evolutionary dynamics of a trait-structured predator-prey model
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arXiv:2501.07379v3 Announce Type: replace Abstract: The coupling between evolutionary and ecological changes (eco-evolutionary dynamics) has been shown to be relevant among diverse species, and is also of interest outside of ecology, i.e. in cancer evolution. These dynamics play an important role in determining survival in response to climate change, motivating the need for mathematical models to capture this often complex interplay. Models incorporating eco-evolutionary dynamics often sacrifice analytical tractability to capture the complexity of real systems, do not explicitly consider the effect of population heterogeneity, or focus on long-term behaviour. In order to capture population heterogeneity, both transient, and long-term dynamics, while retaining tractability, we generalise a moment-based method applicable in the regime of small segregational variance to the case of time-dependent mortality and birth. These results are applied to a predator-prey model, where ecological parameters such as the contact rate between species are trait-structured. The trait-distribution of the prey species is shown to be approximately Gaussian with constant variance centered on the mean trait, which is asymptotically governed by an autonomous ODE. In this way, we make explicit the impact of eco-evolutionary dynamics on the transient behaviour and long-term fate of the prey species.
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https://arxiv.org/abs/2501.07379
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990a1a6547e7449eda7d41ba6714f9fad6cb7d034aa4c151d6b887e6d2bc7042
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2026-01-21T00:00:00-05:00
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Asymptotic-Preserving Neural Networks based on Even-odd Decomposition for Multiscale Gray Radiative Transfer Equations
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arXiv:2501.08166v2 Announce Type: replace Abstract: We present a novel Asymptotic-Preserving Neural Network (APNN) approach utilizing even-odd decomposition to tackle the nonlinear gray radiative transfer equations (GRTEs). Our AP loss demonstrates consistent stability concerning the small Knudsen number, ensuring the neural network solution uniformly converges to the diffusion limit solution. This APNN method alleviates the rigorous conservation requirements while simultaneously incorporating an auxiliary deep neural network, distinguishing it from the APNN method based on micro-macro decomposition for GRTE. Several numerical problems are examined to demonstrate the effectiveness of our proposed APNN technique.
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https://arxiv.org/abs/2501.08166
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efb4389f7f5b0c23a994a707534a96b23d53c6bf1fc6012b6c6a152d6bd1b3d2
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2026-01-21T00:00:00-05:00
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Stable determination of the potential for the Helmholtz equation in the high frequency limit from boundary measurements
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arXiv:2501.10751v2 Announce Type: replace Abstract: We establish a triple logarithmic stability estimate of determining the potential in a Helmholtz equation from a partial Dirichlet-to-Neumann map in the high frequency limit. This estimate is proved under the assumption that the potential is known near the boundary of a domain when the dimension is greater than or equal to $3$. In addition, we show a triple logarithmic stability for an interior impedance problem.
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https://arxiv.org/abs/2501.10751
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bf1784806c8e799b7319ac9c795e21e07ab403ee45800e0ccfea4f76c36bff69
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2026-01-21T00:00:00-05:00
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D2D Coded Caching Schemes for Multiaccess Networks with Combinatorial Access Topology
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arXiv:2501.10756v2 Announce Type: replace Abstract: This paper considers wireless device-to-device (D2D) coded caching in a multiaccess network, where the users communicate with each other and each user can access multiple cache nodes. Access topologies derived from two combinatorial designs known as the $t$-design and $t$-group divisible design ($t$-GDD), referred to as the $t$-design and $t$-GDD topologies respectively, which subsume a few other known topologies, have been studied for the multiaccess coded caching (MACC) network by Cheng \textit{et al.} in \cite{MACC_des}. These access topologies are extended to a multiaccess D2D coded caching (MADCC) network and novel MADCC schemes are proposed. MADCC network has been studied so far only for the cyclic wrap-around topology. Apart from the proposed novel MADCC schemes, MADCC schemes are also derived from the existing MACC schemes in \cite{MACC_des}. To compare the performance of different MADCC schemes, the metrics of load per user and subpacketization level are used while keeping the number of caches and cache memory size same. The proposed MADCC scheme with $t$-design topology performs better in terms of subpacketization level while achieving the same load per user compared to the MADCC scheme derived from the MACC scheme with $t$-design topology in \cite{MACC_des}. The proposed MADCC scheme with $t$-GDD topology performs better in terms of load per user while achieving the same subpacketization level compared to the MADCC scheme derived from the MACC scheme with $t$-GDD topology in \cite{MACC_des} in some cases. Compared to the existing MADCC scheme with cyclic wrap-around topology, the proposed MADCC scheme with $t$-design topology performs better in terms of load per user, and the proposed MADCC scheme with $t$-GDD topology performs better in terms of subpacketization level at the expense of an increase in load per user.
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https://arxiv.org/abs/2501.10756
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26196cd5d4d0e276af2a86d18d6e14d93b0cb4fdf27a9a0c75acf56c7ed3a752
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2026-01-21T00:00:00-05:00
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Surfaces with flat normal connection in 4-dimensional space forms
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arXiv:2501.15780v3 Announce Type: replace Abstract: Let $N$ be a Riemannian, Lorentzian or neutral $4$-dimensional space form with constant sectional curvature $L_0$. In this paper, noticing the linearly dependent condition, we obtain characterizations of space-like surfaces in $N$ with flat normal connection and parallel normal vector fields. In addition, we obtain a generic characterization of space-like surfaces in $N$ with flat normal connection and $K\equiv L_0$ which do not admit any parallel normal vector fields. For time-like surfaces in $N$ with flat normal connection, we obtain analogous results.
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https://arxiv.org/abs/2501.15780
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9597e9ab5744eeb050ddec35b95eba3eaf08342a3a0e1ec57835eba51dd1e9e5
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2026-01-21T00:00:00-05:00
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A Lyapunov analysis of Korpelevich's extragradient method with fast and flexible extensions
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arXiv:2502.00119v2 Announce Type: replace Abstract: We develop a Lyapunov-based analysis of Korpelevich's extragradient method and show that it achieves an $o(1/k)$ last-iterate convergence rate of the constructed Lyapunov function. This Lyapunov function simultaneously upper bounds several standard measures of optimality, which allows our analysis to sharpen existing last-iterate convergence guarantees for these measures. Moreover, the same analysis enables the design of a class of flexible extensions of the extragradient method in which extragradient steps are adaptively blended with user-specified directions via a Lyapunov-guided line-search procedure. These extensions retain global convergence under practical assumptions and can attain superlinear rates when the directions are chosen appropriately. Numerical experiments confirm the simplicity and efficiency of the proposed framework.
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https://arxiv.org/abs/2502.00119
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dcabefac1480c8082b58c4cd004c65ac57c8ed95e22e3ffb83d218c4c3712866
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2026-01-21T00:00:00-05:00
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Extreme-Scale EV Charging Infrastructure Planning for Last-Mile Delivery Using High-Performance Parallel Computing
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arXiv:2502.05152v2 Announce Type: replace Abstract: This paper addresses stochastic charger location and allocation problems under queue congestion for last-mile delivery using electric vehicles (EVs). The objective is to decide where to open charging stations and how many chargers of each type to install, subject to budgetary and waiting-time constraints. We formulate the problem as a mixed-integer non-linear program, where each station-charger pair is modeled as a multiserver queue with stochastic arrivals and service times to capture the notion of waiting in fleet operations. The model is extremely large, with billions of variables and constraints for a typical metropolitan area; even loading the model in solver memory is difficult, let alone solving it. To address this challenge, we develop a Lagrangian-based dual decomposition framework that decomposes the problem by station and leverages parallelization on high-performance computing systems, where the subproblems are solved by using a cutting plane method and their solutions are collected at the master level. We also develop a three-step rounding heuristic to transform the fractional subproblem solutions into feasible integral solutions. Computational experiments on data from the Chicago metropolitan area with hundreds of thousands of households and thousands of candidate stations show that our approach produces high-quality solutions in cases where existing exact methods cannot even load the model in memory. We also analyze various policy scenarios, demonstrating that combining existing depots with newly built stations under multiagency collaboration substantially reduces costs and congestion. These findings offer a scalable and efficient framework for developing sustainable large-scale EV charging networks.
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https://arxiv.org/abs/2502.05152
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fb6244aba545c220d2a9971ac0b6bbeb929126e88c545c87256237b29e499559
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2026-01-21T00:00:00-05:00
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Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers
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arXiv:2502.05993v2 Announce Type: replace Abstract: Fix $n$ a positive integer. Take the $n$-th metallic number $\phi_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $\phi_1$ is the golden number) and let $\Phi_n(q)$ be its $q$-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an algebraic continued fraction which admits an expansion into a Taylor series around $q=0$, with integral coefficients. By using the notion of Hankel continued fraction introduced by the first author in 2016 we determine explicitly the first $n+2$ sequences of shifted Hankel determinants of $\Phi_n$ and show that they satisfy the following properties: 1) They are periodic and consist of $-1,0,1$ only. 2) They satisfy a three-term Gale-Robinson recurrence, i.e. they form discrete integrable dynamical systems. 3) They are all completely determined by the first sequence. This article thus validates a conjecture formulated by V. Ovsienko and the second author in a recent paper and establishes new connections between $q$-deformations of real numbers and sequences of Catalan or Motzkin numbers.
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https://arxiv.org/abs/2502.05993
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e6aaf8bab054d107d49ded92f46656cd2e353feedb956415daf145b1c74edc61
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2026-01-21T00:00:00-05:00
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Homeomorphism groups of basilica, rabbit and airplane Julia sets
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arXiv:2502.07762v2 Announce Type: replace Abstract: The airplane, the basilica and the Douady rabbit (and, more generally, rabbits with more than two ears) are well-known Julia sets of complex quadratic polynomials. In this paper we study the groups of all homeomorphisms of such fractals and of all automorphisms of their laminations. In particular, we identify them with some kaleidoscopic group or universal groups and thus realize them as Polish permutation groups. From these identifications, we deduce algebraic, topological and geometric properties of these groups.
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https://arxiv.org/abs/2502.07762
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999d1168ec64480921377ca8e801932d68162ec34ffa37782a16ab0ba4c57b0b
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2026-01-21T00:00:00-05:00
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Newton-Mandelbrot set and Murase-Mandelbrot set
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arXiv:2502.14872v2 Announce Type: replace Abstract: We obtain four extended Newton's methods and three extended Mandelbrot's recurrence formulas from the Wasan (Japanese mathematics in the Edo period (1603-1868)). Furthermore, two extended Newton's methods relate to one of the extended Mandelbrot's recurrence formulas. We lead four types of extended Mandelbrot recurrence formulas. Next, we show that these become the same extended Mandelbrot set, and connected, closed set. These show the originality of Wasan.
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https://arxiv.org/abs/2502.14872
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d7fd1489872e4ff2a5259fc9c92f4ec69c9f13b2137ebd8cf5da91dad90e279e
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2026-01-21T00:00:00-05:00
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A Jacobian-free Newton-Krylov method for cell-centred finite volume solid mechanics
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arXiv:2502.17217v2 Announce Type: replace Abstract: This study investigates the efficacy of Jacobian-free Newton-Krylov methods in finite-volume solid mechanics. Traditional Newton-based approaches require explicit Jacobian matrix formation and storage, which can be computationally expensive and memory-intensive. In contrast, Jacobian-free Newton-Krylov methods approximate the Jacobian's action using finite differences, combined with Krylov subspace solvers such as the generalised minimal residual method (GMRES), enabling seamless integration into existing segregated finite-volume frameworks without major code refactoring. This work proposes and benchmarks the performance of a compact-stencil Jacobian-free Newton-Krylov method against a conventional segregated approach on a suite of test cases, encompassing varying geometric dimensions, nonlinearities, dynamic responses, and material behaviours. Key metrics, including computational cost, memory efficiency, and robustness, are evaluated, along with the influence of preconditioning strategies and stabilisation scaling. Results show that the proposed Jacobian-free Newton-Krylov method outperforms the segregated approach in all linear and nonlinear elastic cases, achieving order-of-magnitude speedups in many instances; however, divergence is observed in elastoplastic cases, highlighting areas for further development. It is found that preconditioning choice impacts performance: a LU direct solver is fastest in small to moderately-sized cases, while a multigrid method is more effective for larger problems. The findings demonstrate that Jacobian-free Newton-Krylov methods are promising for advancing finite-volume solid mechanics simulations, particularly for existing segregated frameworks where minimal modifications enable their adoption. The described implementations are available in the solids4foam toolbox for OpenFOAM, inviting the community to explore, extend, and compare these procedures.
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https://arxiv.org/abs/2502.17217
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376589d30d9ebc05dfa48254fd389b9eec82ea03a72c8dd8c5211dd9b4a4a2d5
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2026-01-21T00:00:00-05:00
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Multiplier modules of Hilbert C*-modules revisited
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arXiv:2502.17959v3 Announce Type: replace Abstract: The theory of multiplier modules of Hilbert C*-modules is reconsidered to obtain more properties of these special Hilbert C*-modules. The property of a Hilbert C*-module to be a multiplier C*-module is shown to be an invariant with respect to the consideration as a left or right Hilbert C*-module in the sense of a imprimitivity bimodule in strong Morita equivalence theory. The interrelation of the C*-algebras of ''compact'' operators, the Banach algebras of bounded module operators and the Banach spaces of bounded module operators of a Hilbert C*-module to its C*-dual Banach C*-module, are characterized for pairs of Hilbert C*-modules and their respective multiplier modules. The structures on the latter are always isometrically embedded into the respective structures on the former. Examples are given for which continuation of these kinds of bounded module operators from the initial Hilbert C*-module to its multiplier module fails. However, existing continuations turn out to be always unique. Similarly, bounded modular functionals from both kinds of Hilbert C*-modules to their respective C*-algebras of coefficients are compared, and eventually existing continuations are shown to be unique.
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https://arxiv.org/abs/2502.17959
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4bf92377a06906efbdbfe497883855520281b8957e17be44118e4fb0c048f063
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2026-01-21T00:00:00-05:00
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Rigidity of the escaping set of certain H\'enon maps
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arXiv:2502.19358v3 Announce Type: replace Abstract: Let $H$ be a H\'enon map of the form $H(x,y)=(y,p(y)-ax)$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the actions of automorphisms of $\mathbb{C}^2$ if the degree of $H=d\le |a|$. Specifically, every automorphism of $\mathbb{C}^2$ that preserves $U^+$, essentially takes the form $C \circ H^s$ where $s \in \mathbb{Z}$, and $C(x,y)=(\eta x, \eta^d y)$ with $\eta$ some $(d^2-1)$-root of unity. Consequently, we show that the automorphisms of the short $\mathbb{C}^2$'s, obtained as the sub-level sets of the (positive) Green's function corresponding to the H\'enon map $H$ for strictly positive values, are essentially linear maps of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism groups of these short $\mathbb{C}^2$'s are the same, finite, and form a subgroup of $\mathbb{Z}_{d^2-1}$.
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https://arxiv.org/abs/2502.19358
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4f0dd5a882ace49ecd8d242d44887a8a0a20113919e90a2b05530789cfe4a35f
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2026-01-21T00:00:00-05:00
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Zhuk's bridges, centralizers, and similarity
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arXiv:2503.03551v2 Announce Type: replace Abstract: This is the second of three papers motivated by the author's desire to understand and explain "algebraically" one aspect of Dmitriy Zhuk's proof of the CSP Dichotomy Theorem. In this paper we extend Zhuk's "bridge" construction to arbitrary meet-irreducible congruences of finite algebras in locally finite varieties with a Taylor term. We then connect bridges to centrality and similarity. In particular, we prove that Zhuk's bridges and our "similarity bridges" (defined in our first paper) convey the same information in locally finite Taylor varieties.
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https://arxiv.org/abs/2503.03551
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7a4d60870d2e74a042e12e360efb927654d7c26a203ca6b5dab4ebe1e59a70b5
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2026-01-21T00:00:00-05:00
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A positive product formula of integral kernels of $k$-Hankel transforms
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arXiv:2503.03554v4 Announce Type: replace Abstract: The $k$-Hankel transform $F_{k,1}$ (or the $(k,1)$-generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in $(k,a)$-generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of $F_{k,1}$. Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure $\sigma_{x,t}^{k,1}(\xi)$. We will then study the representing measure $\sigma_{x,t}^{k,1}(\xi)$ and analyze the support of the measure, and derive a weak Huygens's principle for the deformed wave equation in $(k,1)$-generalized Fourier analysis.
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https://arxiv.org/abs/2503.03554
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1a115dac645159259be73d476f002c4495ba50e5036db3dc1f907b22a152a46f
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2026-01-21T00:00:00-05:00
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Consecutive Patterns, Kostant's Problem and Type $A_6$
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arXiv:2503.07809v2 Announce Type: replace Abstract: For a permutation $w$ in the symmetric group $\mathfrak{S}_{n}$, let $L(w)$ denote the simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. We first prove that $L(w)$ is Kostant negative whenever $w$ consecutively contains certain patterns. We then provide a complete answer to Kostant's problem in type $A_{6}$ and show that the indecomposability conjecture also holds in type $A_{6}$, that is, applying an indecomposable projective functor to a simple module outputs either an indecomposable module or zero.
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https://arxiv.org/abs/2503.07809
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0792692886f03a9d7a06d156e3917140deb53aa5dd5c92be33a50dcc09e794cf
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2026-01-21T00:00:00-05:00
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Exotic spherical flexible octahedra and counterexamples to the Modified Bellows Conjecture
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arXiv:2503.09582v2 Announce Type: replace Abstract: In 2014 the author showed that in the three-dimensional spherical space, alongside with three classical types of flexible octahedra constructed by Bricard, there exists a new type of flexible octahedra, which was called exotic. In the present paper we give a geometric construction for exotic flexible octahedra, describe their configuration spaces, and calculate their volumes. We show that the volume of an exotic flexible octahedron is nonconstant during the flexion, and moreover the volume remains nonconstant if we replace any set of vertices of the octahedron with their antipodes. So exotic flexible octahedra are counterexamples to the Modified Bellows Conjecture proposed by the author in 2015.
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https://arxiv.org/abs/2503.09582
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ea1fe4b42051e878736e9d8d682d24c20f5433ae4b5af9866d91a26547eca850
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2026-01-21T00:00:00-05:00
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Rigorous analysis of shape transitions in frustrated elastic ribbons
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arXiv:2503.11779v2 Announce Type: replace Abstract: Ribbons are elastic bodies of thickness $t$ and width $w$ with $t\ll w\ll 1$ (after appropriate nondimensionalization). Many ribbons in nature have a non-trivial internal geometry, making them incompatible with Euclidean space. This incompatibility -- expressed mathematically as a failure of the Gauss-Codazzi equations for surfaces -- can trigger shape transitions between narrow and wide ribbons. These transitions depend on the internal geometry: ribbons whose incompatibility arises from failure of the Gauss equation always exhibit a transition, whereas those whose incompatibility arises from failure of the Codazzi equations, may or may not. We give the first rigorous analysis of this phenomenon, mainly for ribbons whose first fundamental form is flat. For Gauss-incompatible ribbons we identify the natural energy scaling of the problem and prove the existence of a shape transition. For Codazzi-incompatible ribbons we give a necessary condition for a transition to occur. Furthermore, our study reveals a fundamental distinction: the transition is "microscopic" for Gauss-incompatible ribbons, persisting as the width tends to $0$, whereas it is "mesoscopic" for Codazzi-incompatible ribbons, observable only at small but finite width. The results are obtained by calculating the $\Gamma$-limits, as $t,w\to 0$, for narrow ribbons ($w^2 \ll t$), and wide ribbons (taking $t$ to zero and then $w$), in the natural energy scalings dictated by the internal geometry.
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https://arxiv.org/abs/2503.11779
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24731ce9bbe985206a583f47f9bd4c5f3ac26ee348db017998830076da6c0ef3
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2026-01-21T00:00:00-05:00
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Spanning trees in directed square cycles
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arXiv:2503.12561v2 Announce Type: replace Abstract: We classify weakly connected spanning closed (WCSC) subgraphs of $\overrightarrow{C_n^2}$, the square of a directed $n$-vertex cycle. Then we show that every spanning tree of $\overrightarrow{C_n^2}$ is contained in a unique nontrivial WCSC subgraph of $\overrightarrow{C_n^2}$. As a result, we obtain a purely combinatorial derivation of the formula for the number of directed spanning trees of $\overrightarrow{C_n^2}$. Moreover, we obtain the formula for the number of directed spanning trees of $\overrightarrow{C_n^2}$, which is a Jacobsthal number.
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https://arxiv.org/abs/2503.12561
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c83ec53f0f5b9089b47ad27129a10596de20b051350412e808992e8997325908
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2026-01-21T00:00:00-05:00
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Characterising 1-rectifiable metric spaces via connected tangent spaces
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arXiv:2503.15159v2 Announce Type: replace Abstract: We prove that in a complete metric space $X$, $1$-rectifiability of a set $E\subset X$ with $\mathcal{H}^1(E)<\infty$ and positive lower density $\mathcal{H}^1$-a.e. is implied by the property that all tangent spaces are connected metric spaces.
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https://arxiv.org/abs/2503.15159
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6725c1b94652cd257a44cbd552345b5179fd4f1d5bf6a3c325d5cc7870cec58b
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2026-01-21T00:00:00-05:00
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Canonical torus action on symplectic singularities
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arXiv:2503.15791v3 Announce Type: replace Abstract: We show that any symplectic singularity lying on a smoothable projective symplectic variety locally admits a good action of an algebraic torus of dimension $r \geq 1$, which is canonical. Under mild assumptions, $r=1$ is also confirmed. In particular, it admits (canonical) good $\mathbb{C}^*$-action. This proves Kaledin's conjecture conditionally but in a substantially stronger form. Our key idea is to use Donaldson-Sun theory on local Kahler metrics in complex differential geometry to connect with the theory of Poisson deformations of symplectic varieties. For general symplectic singularities, we prove the same assertion -- namely, the existence of a canonical (local) torus action -- assuming that the Donaldson-Sun theory extends to such singularities along with suitable singular (hyper)Kahler metrics. Conversely, our results can be also used to study local behaviour of such metrics around the germ. For instance, we show that such singular hyperKahler metric around isolated singularity is close to a metric cone in a polynomial order, and satisfies $r=1$ i.e., has a good canonical (local) $\mathbb{C}^*$-action, as the complexification of the cone metric rescaling. Our theory also fits well to singularities on many hyperKahler reductions.
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https://arxiv.org/abs/2503.15791
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7c9f752126f563420dc184f0f172a6ebba0c9861e0d486678f993142dd894424
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2026-01-21T00:00:00-05:00
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High-dimensional sparse recovery from function samples Decoders, guarantees and instance optimality
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arXiv:2503.16209v2 Announce Type: replace Abstract: We investigate the reconstruction of multivariate functions from samples using sparse recovery techniques. For Square Root Lasso, Orthogonal Matching Pursuit, and Compressive Sampling Matching Pursuit, we demonstrate both theoretically and empirically that they allow us to recover functions from a small number of random samples. In contrast to Basis Pursuit Denoising, the deployed decoders only require a search space $V_J$ spanned by dictionary elements indexed by $J$ and a sparsity parameter $n$ to guarantee an $L_2$-approximation error decaying no worse than a best $n$-term approximation error and the truncation error with respect to the search space $V_J$ and the uniform norm. We show that this happens simultaneously for all admissible functions if the number of samples scales as $n\log^2 n\log |J|$, coming from known bounds for the RIP for matrices built upon bounded orthonormal systems. As a consequence, we obtain bounds for sampling widths in function classes. In addition, we establish lower bounds on the required sample complexity, which show that the log-factor in $\vert J \vert$ is indeed necessary to obtain such {\em instance-optimal} error guarantees. Finally, we conduct several numerical experiments to show that our theoretical bounds are reasonable and compare the discussed decoders in practice.
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https://arxiv.org/abs/2503.16209
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9f5efb090d3223a7802fb465e3dd908872e904be2f331ec2a6dd8eeeeaf4008e
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2026-01-21T00:00:00-05:00
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Two billiard domains whose billiard maps are Lazutkin conjugates are the same
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arXiv:2503.19550v3 Announce Type: replace Abstract: This paper aimed to show that two billiards whose billiard maps share the same expression in Lazutkin coordinates are isometric. However, the results are incorrect as they rely on erroneous computations in Lazutkin's book concerning the expansion of the billiard map in Lazutkin coordinates.
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https://arxiv.org/abs/2503.19550
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356a2db27151b7597e0bee759ae522a5cb5b43fdc0863a141c286200e8550f69
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2026-01-21T00:00:00-05:00
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The prime number theorem over integers of power-free polynomial values
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arXiv:2504.00804v3 Announce Type: replace Abstract: Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge2$ be an integer. The number of integers $n$ such that $f(n)$ is $k$-free is widely studied in the literature. In principle, one expects that $f(n)$ is $k$-free infinitely often, if $f$ has no fixed $k$-th power divisor. In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Inspired by their work, one may expect that this generalization of the PNT also holds over integers of power-free polynomial values. In this note, we establish such variants of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning.
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https://arxiv.org/abs/2504.00804
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3728976ca68c0f3e608f477b483bbfa192254ed756137b9a754640d565691687
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2026-01-21T00:00:00-05:00
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A Tail-Respecting Explicit Numerical Scheme for L\'evy-Driven SDEs With Superlinear Drifts
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arXiv:2504.07255v2 Announce Type: replace Abstract: We present an explicit numerical approximation scheme, denoted by $\{X^n\}$, for the effective simulation of solutions $X$ to a multivariate stochastic differential equation (SDE) with a superlinearly growing $\kappa$-dissipative drift, where $\kappa>1$, driven by a multiplicative heavy-tailed L\'evy process that has a finite $p$-th moment, with $p>0$. We show that the strong $L^q$-convergence $\sup_{t\in[0,T]}\mathbf E \|X^n_t-X_t\|^q=\mathcal O (h_n^{\gamma})$ holds true for any $q\in (0,p+\kappa-1)$, which is exactly the range where the $q$-moment of the solution is known to be finite. Additionally, for any $q\in (0,p)$ we establish strong uniform convergence: $\mathbf E\sup_{t\in[0,T]} \|X^n_t-X_t\|^q=\mathcal{O} ( h_n^{\delta} )$. In both cases we determine the convergence rates $\gamma$ and $\delta$. In the special case of SDEs driven solely by a Brownian motion, our numerical scheme preserves super-exponential moments of the solution. The scheme $\{X^n\}$ is realized as a combination of a well-known Euler method with a Lie-Trotter type splitting technique.
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https://arxiv.org/abs/2504.07255
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b4f3a59efaa6828ee51f5158620e0076a7af82166fecbc809fb9d27fa03a38b7
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2026-01-21T00:00:00-05:00
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Linear complementary dual quasi-cyclic codes of index 2
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arXiv:2504.09126v3 Announce Type: replace Abstract: We provide a polynomial approach to investigate linear complementary dual (LCD) quasi-cyclic codes over finite fields. We establish necessary and sufficient conditions for LCD quasi-cyclic codes of index 2 with respect to the Euclidean, Hermitian, and symplectic inner products. As a consequence of these characterizations, we derive necessary and sufficient conditions for LCD one-generator quasi-cyclic codes. Furthermore, using these characterizations, we construct some new quasi-cyclic LCD codes over small fields.
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https://arxiv.org/abs/2504.09126
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e3a2841feaa0d8527ea3472722625d202a061fd7c6ede8a026b3a8b4fcfc0c1d
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2026-01-21T00:00:00-05:00
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Estimate for the first Dirichlet eigenvalue of $p-$Laplacian on non-compact manifolds
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arXiv:2504.09856v2 Announce Type: replace Abstract: In this paper, we establish a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on bounded domains of a complete, non-compact Riemannian manifold with non-negative Ricci curvature.
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https://arxiv.org/abs/2504.09856
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1a46758c0b062a82bc5cfce0fd00a8066d3ac88196af6c1a6b4e50dbe0ebfaac
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2026-01-21T00:00:00-05:00
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A structure-preserving numerical method for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems
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arXiv:2504.11892v2 Announce Type: replace Abstract: A conforming finite element scheme with mixed explicit-implicit time discretization for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems in a bounded domain with periodic boundary conditions is presented. The system consists of the Navier-Stokes equations, together with a quasi-incompressibility constraint, coupled with the cross-diffusion Maxwell-Stefan equations. The numerical scheme preserves the partial masses and the quasi-incompressibility constraint and dissipates the discrete energy. Numerical experiments in two space dimensions illustrate the convergence of the scheme and the structure-preserving properties.
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https://arxiv.org/abs/2504.11892
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935b6ecaf18baf89c3be51284667d6334b2adba7bc9eebf8a60c7bbae9cd7cb6
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2026-01-21T00:00:00-05:00
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A functional limit theorem for a dynamical system with an observable maximised on a Cantor set
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arXiv:2504.12534v2 Announce Type: replace Abstract: We consider heavy-tailed observables maximised on a dynamically defined Cantor set and prove convergence of the associated point processes as well as functional limit theorems. The Cantor structure, and its connection to the dynamics, causes clustering of large observations: this is captured in the `decorations' on our point processes and functional limits, an application of the theory developed in a paper by the latter three authors.
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https://arxiv.org/abs/2504.12534
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c1c4aef0130c28221c9e2ba4d1f6bc9a926d69b031deca5f6cb6faf15f6e70a1
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2026-01-21T00:00:00-05:00
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Linear Complementary Pairs of Quasi-Cyclic and Quasi-Twisted Codes
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arXiv:2504.15231v2 Announce Type: replace Abstract: In this paper, we provide a polynomial characterization of linear complementary pairs of quasi-cyclic and quasi-twisted codes of index 2. We also give several examples of linear complementary pairs of quasi-cyclic and quasi-twisted codes with optimal security parameters.
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https://arxiv.org/abs/2504.15231
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11f55ec9965a8ec3aef3ff0e2e67878c4d67e1c1d5468dac40dda93e16124d4c
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2026-01-21T00:00:00-05:00
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$k$-Inductive and Interpolation-Inspired Barrier Certificates for Stochastic Dynamical Systems
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arXiv:2504.15412v2 Announce Type: replace Abstract: In this paper, we introduce two new types of barrier certificates that are based on multiple functions rather than a single one. A conventional barrier certificate for a stochastic dynamical system is a nonnegative real-valued function whose expected value does not increase as the system evolves. This requirement guarantees that the barrier certificate forms a nonnegative supermartingale and can be used to derive a lower bound on the probability that the system remains safe. A key advantage of such certificates is that they can be automatically searched for using tools such as optimization programs instantiated with a fixed template. When this search is unsuccessful, the common practice is to modify the template and attempt the synthesis again. Drawing inspiration from logical interpolation, we first propose an alternative framework that uses a collection of functions to jointly serve as a barrier certificate. We refer to this construct as an interpolation-inspired barrier certificate. Nonetheless, we observe that these certificates still require one function in the collection to satisfy a supermartingale condition. Motivated by recent work in the literature, we next combine k-induction with interpolation-inspired certificates to relax this supermartingale constraint. We develop a general and more flexible notion of barrier certificates, which we call k-inductive interpolation-inspired barrier certificates. This formulation encompasses multiple ways of integrating interpolation-inspired barrier certificates with k-induction. We highlight two specific instantiations among these possible combinations. For polynomial systems, we employ sum-of-squares (SOS) programming to synthesize the corresponding set of functions. Finally, through our case studies, we show that the proposed methods enable the use of simpler templates and yield tighter lower bounds on the safety probability.
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https://arxiv.org/abs/2504.15412
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3334c9c77ec6ccba94f77592b2851c8afb60ce6d5bdb070866548feac3f0bbf9
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$\eta$-Einstein Sasakian Lie algebras
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arXiv:2504.16033v2 Announce Type: replace Abstract: We study $\eta$-Einstein Sasakian structures on Lie algebras, that is, Sasakian structures whose associated Ricci tensor satisfies an Einstein-like condition. We divide into the cases in which the Lie algebra's centre is non-trivial (and necessarily one-dimensional) and those where it is zero. In the former case we show that any Sasakian structure on a unimodular Lie algebra is $\eta$-Einstein. As for centreless Sasakian Lie algebras, we devise a complete characterisation under certain dimensional assumptions regarding the action of the Reeb vector. Using this result, together with the theory of normal $j$-algebras and modifications of Hermitian Lie algebras, we construct new examples of $\eta$-Einstein Sasakian Lie algebras and solvmanifolds, and provide effective restrictions for their existence.
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https://arxiv.org/abs/2504.16033
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2026-01-21T00:00:00-05:00
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A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms
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arXiv:2504.16900v2 Announce Type: replace Abstract: In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into $2^{12}$ classes, based on the behaviour of the covariant derivative $\nabla^g\Phi$ of the fundamental $2$-form $\Phi$. This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms $S$ and $h$, which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving $S$ and $h$, which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of $\mathcal{H}$-parallel almost contact manifolds, defined by the condition $(\nabla^g_X\Phi)(Y,Z)=0$ for all horizontal vector fields, $X,Y,Z\in\mathcal{H}$.
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https://arxiv.org/abs/2504.16900
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be363a9765dcf236006f9f2be671f1c2c47ae691bb27369c8320ea8a03b23b3c
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2026-01-21T00:00:00-05:00
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A finite volume Simo-Reissner beam method for moored floating body dynamics
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arXiv:2504.18248v2 Announce Type: replace Abstract: This paper presents a novel finite volume mooring line model based on the geometrically exact Simo-Reissner beam model for analysing the interaction between a floating rigid body and its mooring lines. The coupled numerical model is implemented entirely within a finite volume-based discretisation framework using a popular computational fluid dynamics C++ toolbox, OpenFOAM. Unlike existing methods for modelling mooring lines, which rely on lumped mass models or finite element-based approaches, this work simulates the mooring cables using non-linear beam models implemented in a finite volume framework to account for bending, tensile, and torsional loading. This advancement makes the current work particularly valuable for simulating extreme sea conditions. The coupled model developed in this study has been validated and verified using experimental and numerical data for a floating box moored with four catenary mooring lines under regular wave conditions featuring different wave heights and periods. The results demonstrate strong agreement with both experimental and numerical data, highlighting the model's accuracy in capturing mooring dynamics and floating body motion.
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https://arxiv.org/abs/2504.18248
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b4499232409494100bb8dcc9461f8b22bf17f1e35d506018bc87a37cdda6d986
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2026-01-21T00:00:00-05:00
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Homogeneity in Coxeter groups and split crystallographic groups
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arXiv:2504.18354v2 Announce Type: replace Abstract: We prove that affine Coxeter groups, even hyperbolic Coxeter groups and one-ended hyperbolic Coxeter groups are homogeneous in the sense of model theory. More generally, we prove that many (Gromov) hyperbolic groups generated by torsion elements are homogeneous. In contrast, we construct split crystallographic groups that are not homogeneous, and hyperbolic (in fact, virtually free) Coxeter groups that are not homogeneous (or, to be more precise, not $\mathrm{EAE}$-homogeneous). We also prove that, on the other hand, irreducible split crystallographic groups and torsion-generated hyperbolic groups are almost homogeneous. We also prove that finitely generated abelian-by-finite groups are homogeneous if and only if they are profinitely homogeneous, i.e., any tuple of words from the group is profinitely rigid. We use this to deduce that affine Coxeter groups are profinitely homogeneous, a result of independent interest in the profinite context.
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https://arxiv.org/abs/2504.18354
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daea961507df6526d297f1a1886cb0a93e98c8870757b7b469bc0bd7e31688b8
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2026-01-21T00:00:00-05:00
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On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs
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arXiv:2504.19751v2 Announce Type: replace Abstract: We investigate two recently introduced graph parameters, both of which measure the complexity of the tree decompositions of a given graph. Recall that the treewidth ${\rm tw}(G)$ of a graph $G$ measures the largest number of vertices required in a bag of every tree decomposition of $G$. Similarly, the tree-independence number ${\rm tree\textnormal{-}}\alpha(G)$ and the tree-chromatic number ${\rm tree\textnormal{-}}\chi(G)$ measure the largest independence number, respectively the largest chromatic number, required in a bag of every tree decomposition of $G$. Recently, Dallard, Milani\v{c}, and \v{S}torgel asked (JCTB, 2024) whether for all graphs $G$ it holds that ${\rm tw}(G)+1 \leq {\rm tree\textnormal{-}}\alpha(G) \cdot {\rm tree\textnormal{-}}\chi(G)$. We provide a negative answer for this question in a strong form: for every function $f\colon {\mathbb N} \rightarrow {\mathbb N}$, there exists a graph $G$ such that ${\rm tw}(G) > {\rm tree\textnormal{-}}\alpha(G) \cdot f({\rm tree\textnormal{-}}\chi(G))$. On the other hand, we complement this result with an upper bound, by showing that ${\rm tw}(G)+1 \leq {\rm tree\textnormal{-}}\alpha(G)^2 \cdot {\rm tree\textnormal{-}}\chi(G)$ for every graph $G$.
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https://arxiv.org/abs/2504.19751
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3eed913f9fc5b1b861961a8f41aaf9c702d06b8886fd1d865ebb8ac1911fe930
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Global Activity Scores
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arXiv:2505.00711v4 Announce Type: replace Abstract: We introduce a new global sensitivity measure, the global activity scores. We establish its theoretical connection with Sobol' sensitivity indices and demonstrate its performance through numerical examples. In these examples, we compare global activity scores with Sobol' sensitivity indices, derivative-based sensitivity measures, and activity scores. The results show that in the presence of noise or high variability, global activity scores outperform derivative-based measures and activity scores, while in noiseless settings the three approaches yield similar results.
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https://arxiv.org/abs/2505.00711
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f918db989ed0512eda2d474a816aa756ad851239d34f5a127fd2a22237d6afaa
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2026-01-21T00:00:00-05:00
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p-adic Heisenberg-Robertson-Schrodinger and p-adic Maccone-Pati Uncertainty Principles
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arXiv:2505.02838v2 Announce Type: replace Abstract: Let $\mathcal{X}$ be a p-adic Hilbert space. Let $A:\mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded self-adjoint linear operators. For $x \in \mathcal{D}(A)$ with $\langle x, x \rangle =1$, define $ \Delta_x(A):= \|Ax- \langle Ax, x \rangle x \|.$ Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad \quad \quad \max\{\Delta_x(A), \Delta_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \end{align*} and \begin{align*} (2) \quad \quad \quad \max\{\Delta_x(A), \Delta_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \end{align*} We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.
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https://arxiv.org/abs/2505.02838
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16b9b8ba2105cdf99c23e4c1f89f1b07334291fbbae64f6ba3b03d7be4641e8e
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2026-01-21T00:00:00-05:00
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Rationality patterns
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arXiv:2505.07151v2 Announce Type: replace Abstract: In this paper, we establish general categorical frameworks that extend Loewy's classification scheme for finite-dimensional real irreducible representations of groups and Borel--Tits' criterion for the existence of rational forms of representations of $\bar{F}\otimes_F G$ for a connected reductive algebraic group $G$ over a field $F$ of characteristic zero and its algebraic closure $\bar{F}$. We also discuss applications of these general formalisms to the theory of Harish-Chandra modules, specifically to classify irreducible Harish-Chandra modules over fields $F$ of characteristic zero and to identify smaller fields of definition of irreducible Harish-Chandra modules over $\bar{F}$, particularly in the case of cohomological irreducible essentially unitarizable modules.
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https://arxiv.org/abs/2505.07151
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bfba56d65dc2ac8c0dbf63cb811c8b8ae5f2c689a3b57cff64ec1f2c207e504a
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2026-01-21T00:00:00-05:00
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High-Order Hermite Optimization: Fast and Exact Gradient Computation in Open-Loop Quantum Optimal Control using a Discrete Adjoint Approach
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arXiv:2505.09857v3 Announce Type: replace Abstract: This work introduces the High-Order Hermite Optimization (HOHO) method, an open-loop discrete adjoint method for quantum optimal control. Our method is the first of its kind to efficiently compute exact (discrete) gradients when using continuous, parameterized control pulses while solving the forward equations (e.g. Schrodinger's equation or the Linblad master equation) with an arbitrarily high-order Hermite Runge-Kutta method. The HOHO method is implemented in QuantumGateDesign$.$jl (https://github.com/leespen1/QuantumGateDesign.jl), an open-source software package for the Julia programming language, which we use to perform numerical experiments comparing the method to Juqbox$.$jl (https://github.com/LLNL/Juqbox.jl). For realistic model problems we observe speedups up to 775x.
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https://arxiv.org/abs/2505.09857
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67d04df2a72dcac29a819c353b1e0eb06384c230185d1e4cc83297d8ab8112b3
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2026-01-21T00:00:00-05:00
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Long-Term Average Impulse Control with Mean Field Interactions
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arXiv:2505.11345v2 Announce Type: replace Abstract: This paper analyzes and explicitly solves a class of long-term average impulse control problems with a specific mean-field interaction. The underlying process is a general one-dimensional diffusion with appropriate boundary behavior. The model is motivated by applications such as the optimal long-term management of renewable resources and financial portfolio management. Each individual agent seeks to maximize her long-term average reward, which consists of a running reward and income from discrete impulses, where the unit intervention price depends on the market through a stationary supply rate, the specific mean field variable to be considered. In a competitive market setting, we establish the existence of and explicitly characterize an equilibrium strategy within a large class of policies under mild conditions. Additionally, we formulate and solve the mean field control problem, in which agents cooperate with each other, aiming to realize a common maximal long-term average profit. To illustrate the theoretical results, we examine a stochastic logistic growth model and a population growth model in a stochastic environment with impulse control.
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https://arxiv.org/abs/2505.11345
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65659f1688de8ef53801ca58e77b791890ea915c6c1e12b5440e09d9a3b01891
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2026-01-21T00:00:00-05:00
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Existence of Friedrich-Wintgen Bound States in the Continuum: Cavity with a Thin Waveguide Opening
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arXiv:2505.12297v2 Announce Type: replace Abstract: Bound states in the continuum (BICs) are localized states embedded within a continuum of propagating waves. Perturbations that disrupt BICs typically induce ultra-strong resonances, a phenomenon enabling diverse applications in photonics. This work investigates the existence of BICs in two-dimensional electromagnetic cavities coupled to thin waveguides for H-polarized waves. Our focus is on Friedrich-Wintgen BICs (FW-BICs), which arise from destructive interference between two resonant modes and were identified numerically in rectangular cavities with waveguide openings by Lyapina et al. [J. Fluid Mech., 780 (2015), pp. 370--387]. Here, we rigorously establish the existence of FW-BICs in a broader class of cavity geometries by introducing perturbations to the refractive index under regularity constraints. We show that BICs correspond to intersections of two curves derived implicitly from the governing equations constructed via the mode-matching method. Crucially, we prove that such intersections are guaranteed for sufficiently small waveguide widths, provided that two eigenvalues of the cavity cross and the associated eigenfunctions exhibit non-vanishing coupling to the radiation channel at the cavity-waveguide interface. Furthermore, our approach remains applicable for studying the emergence of FW-BICs under parameter-dependent boundary perturbations to the cavity.
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https://arxiv.org/abs/2505.12297
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2026-01-21T00:00:00-05:00
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Semiparametric Off-Policy Inference for Optimal Policy Values under Possible Non-Uniqueness
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arXiv:2505.13809v4 Announce Type: replace Abstract: Off-policy evaluation (OPE) constructs confidence intervals for the value of a target policy using data generated under a different behavior policy. Most existing inference methods focus on fixed target policies and may fail when the target policy is estimated as optimal, particularly when the optimal policy is non-unique or nearly deterministic. We study inference for the value of optimal policies in Markov decision processes. We characterize the existence of the efficient influence function and show that non-regularity arises under policy non-uniqueness. Motivated by this analysis, we propose a novel \textit{N}onparametric \textit{S}equenti\textit{A}l \textit{V}alue \textit{E}valuation (NSAVE) method, which achieves semiparametric efficiency and retains the double robustness property when the optimal policy is unique, and remains stable in degenerate regimes beyond the scope of existing asymptotic theory. We further develop a smoothing-based approach for valid inference under non-unique optimal policies, and a post-selection procedure with uniform coverage for data-selected optimal policies. Simulation studies support the theoretical results. An application to the OhioT1DM mobile health dataset provides patient-specific confidence intervals for optimal policy values and their improvement over observed treatment policies.
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https://arxiv.org/abs/2505.13809
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fc3630d845bde84468c578fd6e8877e3b1d147e481f8c66b58344564a777a437
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Periodic operators over a component domain and homogenization of some class of quasi-linear elliptic problems in two-component domain with interfacial resistance
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arXiv:2505.14944v3 Announce Type: replace Abstract: This paper addresses the periodic homogenization of quasilinear elliptic PDEs in a two-component domain with an interfacial thermal barrier. It introduces a periodic extension operator that ensures strong convergence of function sequences in the Sobolev space. Moreover, two families of quasilinear elliptic problems in two-component domains with interfacial resistance will be considered here. One family with \(L^2\) data and another family with \(L^1\) data.
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https://arxiv.org/abs/2505.14944
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2026-01-21T00:00:00-05:00
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An adaptive proximal safeguarded augmented Lagrangian method for nonsmooth DC problems with convex constraints
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arXiv:2505.15369v2 Announce Type: replace Abstract: A proximal safeguarded augmented Lagrangian method for minimizing the difference of convex (DC) functions over a nonempty, closed and convex set with additional linear equality as well as convex inequality constraints is presented. Thereby, all functions involved may be nonsmooth. Iterates (of the primal variable) are obtained by solving convex optimization problems as the concave part of the objective function gets approximated by an affine linearization. Under the assumption of a modified Slater constraint qualification, both convergence of the primal and dual variables to a generalized Karush-Kuhn-Tucker (KKT) point is proven, at least on a subsequence. Numerical experiments and comparison with existing solution methods are presented using some classes of constrained and nonsmooth DC problems.
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https://arxiv.org/abs/2505.15369
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77dd3e6c896b95553d4eb65adcf8bd528f781b5720e2c788b01e38c01288c3a1
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2026-01-21T00:00:00-05:00
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On groups with EDT0L word problem
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arXiv:2505.20057v3 Announce Type: replace Abstract: We prove that the word problem for the infinite cyclic group is not EDT0L, and obtain as a corollary that a finitely generated group with EDT0L word problem must be torsion. In addition, we show that the property of having an EDT0L word problem is invariant under change of generating set and passing to finitely generated subgroups. This represents significant progress towards the conjecture that all groups with EDT0L word problem are finite (i.e. precisely the groups with regular word problem).
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https://arxiv.org/abs/2505.20057
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0bda761be76360f41bb244575c420d23185d2ff9558acb470c4aafa78fed1b5b
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2026-01-21T00:00:00-05:00
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Locating Extremal Periodic Orbits for the Planar Circular Restricted Three Body Problem using Polynomial Sum-of-Squares Optimization
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arXiv:2505.23430v2 Announce Type: replace Abstract: With an increasing interest in the design of long and complex space missions, the search for orbits that require the least amount of fuel is of fundamental interest. This paper develops existing computational models for locating Unstable Periodic Orbits (UPOs) in polynomial dynamical systems using Sum-of-Squares (SOS) optimization technique and proposes a numerical framework to converge UPOs for the Planar Circular Restricted Three-Body Problem (PCR3BP) in astrodynamics. This is done by developing the polynomial SOS optimization technique with extension to systems with non-polynomial and Hamiltonian dynamics. First, we demonstrate and exploit the dependency of convergence of tight bounds on an observable of interest with varying scaling factors for large polynomial degrees. SOS optimization is then used to compute nonnegative polynomials, the minimization sublevel sets of which, approximately localise parts of the corresponding UPO. Improvements in current non-linear optimization techniques are suggested to compute a large number of points inside the relevant sublevel sets. Such points provide good initial conditions for UPO computations with existing algorithms. The distinguishing feature of such UPOs is that they optimize the long-time average of an input observable of interest which is a function of state variables. For the PCR3BP this means that such orbits in space can be traversed indefinitely in time without continuous fuel expenditure. As practical applications to space mission designs, we converge UPOs that minimise transmitted power required by satellites for the Earth-Moon system in a communication relay problem by minimizing the infinite-time average of sum of squares of distances of a satellite from Earth and the Moon.
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https://arxiv.org/abs/2505.23430
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e040eb5e54bbc9cf9d405c6ce4e3f2954e69c362af779ee5dc4d345b4c84924a
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2026-01-21T00:00:00-05:00
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Rydberg Atomic Receivers for Multi-Band Communications and Sensing
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arXiv:2505.24168v3 Announce Type: replace Abstract: Harnessing multi-level electron transitions, Rydberg Atomic REceivers (RAREs) can detect wireless signals across a wide range of frequency bands, from Megahertz to Terahertz. This capability enables multi-band wireless communications and sensing (CommunSense). Existing research on multi-band RAREs primarily focuses on experimental demonstrations, lacking a tractable model to mathematically characterize their mechanisms. This issue leaves the multi-band RARE as a black box and poses challenges in its practical applications. To fill in this gap, this paper investigates the underlying mechanism of multiband RAREs and explores their optimal performance. For the first time, an analytical transfer function with a closed-form expression for multi-band RAREs is derived by solving the quantum response of Rydberg atoms. It shows that a multiband RARE simultaneously serves as a multi-band atomic mixer for down-converting multi-band signals and a multi-band atomic amplifier that reflects its sensitivity to each band. Further analysis of the atomic amplifier unveils that the intrinsic gain at each frequency band can be decoupled into a global gain term and a Rabi attention term. The former determines the overall sensitivity of a RARE to all frequency bands of wireless signals. The latter influences the allocation of the overall sensitivity to each frequency band, representing a unique attention mechanism of multi-band RAREs. The optimal design of the global gain is provided to maximize the overall sensitivity of multi-band RAREs. Subsequently, the optimal Rabi attentions are also derived to maximize the practical multi-band CommunSense performance. An experiment platform is built to validate the effectiveness of the derived transfer function, and numerical results confirm the superiority of multi-band RAREs.
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https://arxiv.org/abs/2505.24168
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2edd465323a9cb2bdf71f404fc934fa0b7dbd2936528bc8795f76719917a2d9a
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2026-01-21T00:00:00-05:00
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Convergence of spectra of digraph limits
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arXiv:2506.04426v2 Announce Type: replace Abstract: The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon.
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https://arxiv.org/abs/2506.04426
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7ca1ea716d8127abbc8a3dc5c42e446a2ffa296550fac39b8024130a94cb4fed
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2026-01-21T00:00:00-05:00
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Rejection-Sampled Linear Codes for Lossy Compression and Channel Simulation
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arXiv:2506.09239v2 Announce Type: replace Abstract: We show that linear codes combined with rejection sampling can yield a capacity-achieving scheme for simulating additive exchangeable noise channels. Specifically, our scheme achieves an amount of communication within $\log e + 1$ bits from the excess functional information lower bound. Hence, it can be used in lossy source coding to achieve the rate-distortion function. We discuss practical implementations based on BCH codes and polar codes. For the simulation of binary symmetric channels, the BCH-based construction with a blocklength of $n = 63$ attains a rate comparable to the PolarSim with $n = 4096$, while significantly reducing the latency. The polar-based construction asymptotically achieves the channel capacity with polynomial average complexity. Furthermore, using the idea from greedy rejection sampling, we propose an algorithm to construct capacity-achieving schemes based on any linear codes. Experiments reveal that our construction can outperform conventional covering codes for lossy source coding with Hamming distortion for a certain range of distortion levels, and performs well even when the blocklength is small (e.g., $n = 24$).
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https://arxiv.org/abs/2506.09239
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2a984b5b9934e719c97da3c9fbd6e66d215e3f52d2fd2465e9c155b738d429b5
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2026-01-21T00:00:00-05:00
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Sobolev regularity for the $\overline\partial$-Neumann operator and transverse vector fields
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arXiv:2506.10372v2 Announce Type: replace Abstract: On a bounded smooth pseudoconvex domain in $\mathbb{C}^n$ with $n >2$, inspired by the compactness condition introduced by Yue Zhang, we present the new sufficient condition for the exact regularity of the $\overline\partial$-Neumann operator via the transverse vector fields.
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https://arxiv.org/abs/2506.10372
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90bb5478c3a035e650c69b0a1e90efb86ee8d912862bd6eb12172863006c8881
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2026-01-21T00:00:00-05:00
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Two axes in non-commutative algebras with a Frobenius form
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arXiv:2506.11303v5 Announce Type: replace Abstract: Throughout this paper $A$ is a commutative non-associative algebra over a field $\mathbb{F}$ of characteristic not $2.$ In addition $A$ posses a Frobenius form. We obtain detailed information about the multiplication in $A$ given two axes of type half in $A.$
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https://arxiv.org/abs/2506.11303
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15c87441a7f771825008ff9ff8df04ad016165ac7793f76f57f27989f9934cea
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2026-01-21T00:00:00-05:00
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Serre's question on thin sets in projective space
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arXiv:2506.13471v2 Announce Type: replace Abstract: We answer a question of Serre from the 1980s on rational points of bounded height on projective thin sets, in degree at least $4$. For degrees $2$ and $3$ we improve the known bounds in general. The focus is on thin sets of type II, namely corresponding to the images of ramified dominant quasi-finite covers of projective space, as thin sets of type I are already well understood via dimension growth results by the third author in 2002 (published in 2023) by a global variant of Heath-Brown's $p$-adic determinant method. For type II, we obtain a uniform affine variant of Serre's question which implies the projective case and for which the implicit constant is furthermore polynomial in the degree. We are able to avoid logarithmic factors when the degree is at least $5$ and we prove our results over any global field, of any characteristic. A key ingredient for obtaining the affine variant comes from Binyamini-Cluckers-Novikov (2024) and Binyamini-Cluckers-Kato (2025) where a question of the third author was answered by providing bounds, for rational points on irreducible curves, which are quadratic in the degree. A second key ingredient is an adaptation of Salberger's global determinant method to the case of weighted polynomials. The third key ingredient is the design of our affine variant of Serre's question, for weighted polynomials which are not necessarily weighted homogeneous.
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https://arxiv.org/abs/2506.13471
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a44e3aea56b8cfed8110321265578a673fc0452ece1638d7ded3335d7e7f79ad
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2026-01-21T00:00:00-05:00
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Instantaneous blowup for interacting SDEs with superlinear drift
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arXiv:2506.21164v2 Announce Type: replace Abstract: We consider a system of interacting SDEs on the integer lattice with multiplicative noise and a drift satisfying the finite Osgood's condition. We show instantaneous everywhere blowup for initial profiles decaying slower than $\exp \left( -\sqrt{\big|\log |x|\big|}\right)$. We employ the splitting-up method to compare the interacting system to a one-dimensional SDE which blows up.
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https://arxiv.org/abs/2506.21164
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2544d6c0b9e0df219609edc125a348077a22bb40ffc2485f5009f28f3e50e513
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2026-01-21T00:00:00-05:00
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On a result by Meshulam
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arXiv:2506.22553v2 Announce Type: replace Abstract: In 1996, Meshulam proved that every sequence generated by applying projections onto affine subspaces, drawn from a finite collection in Euclidean space, must be bounded. In this paper, we extend his result not only from affine subspaces to convex polyhedral subsets, but also from Euclidean to general Hilbert space. Various examples are provided to illustrate the sharpness of the results.
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https://arxiv.org/abs/2506.22553
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684aad527a356ac5fb68f47c85ef9a8343eff69eeecec0097c7ff9ed6c1472fb
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2026-01-21T00:00:00-05:00
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Degrees of non-Gorenstein canonical Fano threefolds with Picard number one
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arXiv:2507.04615v3 Announce Type: replace Abstract: We show that the optimal upper bound for the anticanonical degrees of non-Gorenstein $\mathbb{Q}$-factorial canonical Fano threefolds with Picard number one is 200/3.
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https://arxiv.org/abs/2507.04615
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30058bda2695aa8c660d03bcd2b65a3adcdf8d859ac5c7279ef70772c6fd25df
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2026-01-21T00:00:00-05:00
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Weinstein neighbourhood theorems for stratified subspaces
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arXiv:2507.04897v2 Announce Type: replace Abstract: By analogy with Weinstein's neighbourhood theorem, we prove a uniqueness result for symplectic neighbourhoods of a large family of stratified subspaces. This result generalizes existing constructions, e.g., in the search for exotic Lagrangians. Along the way, we prove a strong version of Moser's trick and a (non-symplectic) tubular neighbourhood theorem for these stratified subspaces.
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https://arxiv.org/abs/2507.04897
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9cb00f719e8568669a96d86c065c72194c8b20cea3a40e16bc6b946504a4dd29
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2026-01-21T00:00:00-05:00
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Elementary equivalence and diffeomorphism groups of smooth manifolds
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arXiv:2507.07427v3 Announce Type: replace Abstract: Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and $M$ and $N$ are $C^r$--diffeomorphic. This strengthens a previously known result by Takens and Filipkiewicz, which asserts that for integer regularities, a group isomorphism between diffeomorphism groups of closed manifolds necessarily arises from a diffeomorphism of the underlying manifolds. We prove an analogous result for groups of diffeomorphisms preserving smooth volume forms, in dimension at least two.
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https://arxiv.org/abs/2507.07427
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15545aca57a477c43497548550c85c8ba56e2fcacf95f25152aaa1f0e31a8f38
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2026-01-21T00:00:00-05:00
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$f$-algebra products on AL and AM-spaces
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arXiv:2507.08435v2 Announce Type: replace Abstract: We characterize all $f$-algebra products on AM-spaces by constructing a canonical AM-space $W_X$ associated to each AM-space $X$, such that the $f$-algebra products on $X$ correspond bijectively to the positive cone $(W_X)_+$. This generalizes the classical description of $f$-algebra products on $C(K)$ spaces. We also identify the unique product (when it exists) that embeds $X$ as a closed subalgebra of $C(K)$, and study AM-spaces for which this product exists -- the so-called AM-algebras. Finally, we investigate AM-spaces that admit only the zero product, providing a characterization in the AL-space case and examples showing that no simple characterization exists in general.
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https://arxiv.org/abs/2507.08435
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7d26f0bbd89e61033194dde071748da11ebf1fc850ca67831003d7f1c13c2f55
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2026-01-21T00:00:00-05:00
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Refinement of the theory and convergence of the Sinc convolution -- beyond Stenger's conjecture
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arXiv:2507.12406v3 Announce Type: replace Abstract: The Sinc convolution is an approximate formula for indefinite convolutions proposed by Stenger. The formula was derived based on the Sinc indefinite integration formula combined with the single-exponential transformation. Although its efficiency has been confirmed in various fields, several theoretical issues remain unresolved. The first contribution of this study is to resolve those issues by refining the underlying theory of the Sinc convolution. This contribution includes an essential resolution of Stenger's conjecture. The second contribution of this study is to improve the convergence rate by replacing the single-exponential transformation with the double-exponential transformation. Theoretical analysis and numerical experiments confirm that the modified formula achieves superior convergence compared to Stenger's original formula.
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https://arxiv.org/abs/2507.12406
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2d899885d7dec671bc7503a075f8a007415a01a80e33da9bcb522e2dde5e5c7a
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2026-01-21T00:00:00-05:00
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Degenerations of families of bands and strings for gentle algebras
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arXiv:2507.13945v2 Announce Type: replace Abstract: Let $A$ be a gentle algebra. For every collection of string and band diagrammes, we consider the constructible subset of the variety of representations containing all modules with this underlying diagramme. We study degenerations of such sets. We show that these sets are defined by vectors of integers which we call $h$-vectors and which are related to a restricted version of the hom-order. We provide combinatorial criteria for the existence of a degeneration, involving the removal of an arrow or the resolving of a type of configuration called "reaching".
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https://arxiv.org/abs/2507.13945
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3f44e8892045b7036bfd9cdc78eb2cd8dfd5681ae0487300cafa1dac1322e5f9
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2026-01-21T00:00:00-05:00
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The inference of Fokker-Planck equations via transport maps
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arXiv:2507.15091v3 Announce Type: replace Abstract: We present a framework, which, from the trajectories detailing the spatiotemporal dynamics of a population, simultaneously reconstructs a transport map as well as the Fokker-Planck equation governing the coarse-grained probability distribution. Leveraging the Knothe-Rosenblatt rearrangement, we model the transport map from a fixed reference distribution to the target distribution, and derive the velocity fields of the flows from the trajectory of transport maps. Exploiting the velocity fields, we circumvent spatial gradients to infer the Fokker-Planck equation's potential and diffusivity. The sparsity of trajectories injects uncertainty, which we treat in a Bayesian setting using variational inference. The approach is applied to inferring the Fokker-Planck dynamics in spaces of up to five dimensions, demonstrating both accurate identification of the system and efficiency with respect to data size.
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https://arxiv.org/abs/2507.15091
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f588ab1ddbc72795dca7cfe84d9a3fc2d3f445e1bc2fdb3e3093f2ca8d39d655
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2026-01-21T00:00:00-05:00
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Structured linear factor models for tail dependence
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arXiv:2507.16340v2 Announce Type: replace Abstract: A common object to describe the extremal dependence of a $d$-variate random vector $X$ is the stable tail dependence function $L$. Various parametric models have emerged, with a popular subclass consisting of those stable tail dependence functions that arise for linear and max-linear factor models with heavy tailed factors. The stable tail dependence function is then parameterized by a $d \times K$ matrix $A$, where $K$ is the number of factors and where $A$ can be interpreted as a factor loading matrix. We study estimation of $L$ under an additional assumption on $A$ called the `pure variable assumption'. Both $K \in \{1, \dots, d\}$ and $A \in [0, \infty)^{d \times K}$ are treated as unknown, which constitutes an unconventional parameter space that does not fit into common estimation frameworks. We suggest two algorithms that allow to estimate $K$ and $A$, and provide finite sample guarantees for both algorithms. Remarkably, the guarantees allow for the case where the dimension $d$ is larger than the sample size $n$. The results are illustrated with numerical experiments and two case studies.
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https://arxiv.org/abs/2507.16340
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58a7cf9ead1667ba744878d771fa23d14d4591e952c87ffcbb342e7f0f6e1b59
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2026-01-21T00:00:00-05:00
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Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Pressure Metric
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arXiv:2507.17077v2 Announce Type: replace Abstract: In this paper, we study complex analytic aspects of the moduli space $\Bcal_d^{fm}$ of degree $d\ge2$ fixed-point-marked Blaschke products. We define a complex structure on $\Bcal_d^{fm}$ and prove the simultaneous uniformization theorem for fixed-point-marked quasi-Blaschke products. As an application, we show that the pressure semi-norms on the space of Blaschke products are non-degenerate outside of the super-attracting locus $\mathcal{SA}^{fm}_d$, which is a codimension-1 subspace of $\Bcal^{fm}_d$.
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https://arxiv.org/abs/2507.17077
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66345d5c98a58c29eb06be097988d2ec9e16ccc9b763e22baa3a9c077c08d665
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2026-01-21T00:00:00-05:00
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The norms for symmetric and antisymmetric tensor products of the weighted shift operators
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arXiv:2507.17181v2 Announce Type: replace Abstract: In the present paper, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$, $$\|S_{\alpha}^{l_1}\odot\cdots \odot S_{\alpha}^{l_k}\odot S_{\alpha}^{*l_{k+1}}\odot\cdots \odot S_{\alpha}^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_{\alpha}^{{l_{i}}}\right\|, \text{ for any} \ (l_1,l_2\cdots l_n)\in\mathbb N^n$$ if and only if the weight satisfies the regularity condition, we partially solve \cite[Problem 6 and Problem 7]{GA}. It will be seen that most weighted shift operators on function spaces, including weighted Bergman shift, Hardy shift, Dirichlet shift, etc, satisfy the regularity condition. Moreover, at the end of the paper, we solve \cite[Problem 1 and Problem 2]{GA}.
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https://arxiv.org/abs/2507.17181
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d23d8e6c14ecc9049d0a920909001056b7a22ae08111afe3a11582efb943c68f
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2026-01-21T00:00:00-05:00
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Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$
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arXiv:2507.20710v2 Announce Type: replace Abstract: Let $\Sigma_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in\{0,1\}$. The Johnson kernel $\mathcal{K}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}(\Sigma_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup $\mathrm{IO}_n\subset\mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization of $F_n$. Long standing questions are whether the groups $\mathcal{K}_g^b$ and $[\mathrm{IO}_n,\mathrm{IO}_n]$ or their abelianizations $(\mathcal{K}_g^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_n,\mathrm{IO}_n]^{\mathrm{ab}}$ are finitely generated for $g\ge3$ (respectively, $n\ge3$). During the last 15 years, these questions were answered positively for $g\ge4$ and $n\ge4$, respectively. Nevertheless, the cases of $g=3$ and $n=3$ remained completely unsettled. In this paper, we prove that the abelianizations $(\mathcal{K}_3^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_3,\mathrm{IO}_3]^{\mathrm{ab}}$ are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group.
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https://arxiv.org/abs/2507.20710
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f911429b34d4cc9c64325574c4a67fe8a1738958dcf7784215fb7aaeed516bcf
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2026-01-21T00:00:00-05:00
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On admissibility in post-hoc hypothesis testing
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arXiv:2508.00770v3 Announce Type: replace Abstract: The validity of classical hypothesis testing requires the significance level $\alpha$ be fixed before any statistical analysis takes place. This is a stringent requirement. For instance, it prohibits updating $\alpha$ during (or after) an experiment due to changing concern about the cost of false positives, or to reflect unexpectedly strong evidence against the null. Perhaps most disturbingly, witnessing a p-value $p\ll\alpha$ vs $p= \alpha- \epsilon$ for tiny $\epsilon > 0$ has no (statistical) relevance for any downstream decision-making. Following recent work of Gr\"unwald (2024), we develop a theory of post-hoc hypothesis testing, enabling $\alpha$ to be chosen after seeing and analyzing the data. To study "good" post-hoc tests we introduce $\Gamma$-admissibility, where $\Gamma$ is a set of adversaries which map the data to a significance level. We classify the set of $\Gamma$-admissible rules for various sets $\Gamma$, showing they must be based on e-values, and recover the Neyman-Pearson lemma when $\Gamma$ is the constant map.
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https://arxiv.org/abs/2508.00770
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65584b8015de05ae4b0b61a720e7b9ab56f83bf718451e31c956946d44407796
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2026-01-21T00:00:00-05:00
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Exceptional dual pair correspondences; case of real groups of split rank one
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arXiv:2508.01551v2 Announce Type: replace Abstract: Exceptional real groups have quaternionic forms of split rank 4 that contain dual pairs $G\times G'$, where $G'$ is the split Lie group of the type $G_2$, and $G$ a Lie group of split rank one. In this paper we restrict the minimal representation of the quaternionic group to the dual pair and prove some significant results for the resulting correspondence of representations.
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https://arxiv.org/abs/2508.01551
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db5b4a95f3ab55cadd54930702f2bf359e0fb4932d5268594f5ee9169427089b
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2026-01-21T00:00:00-05:00
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Large AI Models for Wireless Physical Layer
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arXiv:2508.02314v2 Announce Type: replace Abstract: Large artificial intelligence models (LAMs) are transforming wireless physical layer technologies through their robust generalization, multitask processing, and multimodal capabilities. This article reviews recent advancements in applying LAMs to physical layer communications, addressing obstacles of conventional AI-based approaches. LAM-based solutions are classified into two strategies: leveraging pre-trained LAMs and developing native LAMs designed specifically for physical layer tasks. The motivations and key frameworks of these approaches are comprehensively examined through multiple use cases. Both strategies significantly improve performance and adaptability across diverse wireless scenarios. Future research directions, including efficient architectures, interpretability, standardized datasets, and collaboration between large and small models, are proposed to advance LAM-based physical layer solutions for next-generation communication systems.
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https://arxiv.org/abs/2508.02314
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bf4e6cfd56d482bc575a0cd4de2ca086e10b5e1e95fe788d17e5b94f0c724f53
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2026-01-21T00:00:00-05:00
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Coordinate-independent model reductions of chemical reaction networks based on geometric singular perturbation theory
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arXiv:2508.03304v2 Announce Type: replace Abstract: The quasi-steady-state approximation (QSSA) is a standard technique for reducing the complexity of chemical reaction networks (CRNs). The validity of any QSSA-based model is restricted to specific parameter regimes. Selecting the appropriate reduction is not always straightforward. At times, QSSAs are misused outside of their validity regions and, even when a particular QSSA is considered valid in a given parameter regime, other QSSAs may be simultaneously valid, creating ambiguity. Here, we employ a more powerful alternative: a constructive model reduction framework based on coordinate-independent geometric singular perturbation theory (ci-GSPT) and the parametrization method. A key advantage of this approach is its ability to derive reduced models independent of a clear timescale separation in the variables for a specific parameter configuration. We demonstrate our approach on two benchmark systems. For the Michaelis-Menten (MM) reaction, we show that the framework provides a systematic approach by exploring parameter configurations across three orders of magnitude: asymptotically large, small, and `order one'. A consequence of this systematic analysis is a geometric classification that categorizes the resulting model reductions and provides a point of comparison between our approach and common QSSA variants in the literature. For the more complex Kim-Forger model, we show that this approach successfully produces a reduction without the need for a coordinate transformation, showcasing its applicability to larger systems.
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https://arxiv.org/abs/2508.03304
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085b284e3d2a43b8d490d221e962d932c08b18bee09e55e374aa386d5703a786
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2026-01-21T00:00:00-05:00
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A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation
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arXiv:2508.03359v2 Announce Type: replace Abstract: The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of $\limsup$ sets that arise naturally in Diophantine approximation. However, in the setting of dynamical Diophantine approximation, this principle often fails to apply effectively, as the radii of the balls defining the dynamical $\limsup$ sets generally depend on the orbit of the point $x$ itself. In this paper, we develop a dimensional mass transference principle that enables us to recover and extend classical results on shrinking target problems, particularly for the $\beta$-transformation and the Gauss map. Moreover, our result shows that the corresponding $\limsup$ sets have large intersection properties. A potentially interesting feature of our method is that, in many cases, shrinking target problems are closely related to finding an appropriate Gibbs measure, which may reveal new aspects of the link between thermodynamic formalism and dynamical Diophantine approximation.
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https://arxiv.org/abs/2508.03359
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62a8892c87fa9795130b1f247d56b7aae2dc6d74252f05f16db4c2658e6b990b
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2026-01-21T00:00:00-05:00
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Computable Bounds for Strong Approximations with Applications
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arXiv:2508.03833v2 Announce Type: replace Abstract: The Koml\'os$\unicode{x2013}$Major$\unicode{x2013}$Tusn\'ady (KMT) inequality for partial sums is one of the most celebrated results in probability theory. Yet its practical application has been hindered by a lack of practical constants. This paper addresses this limitation for bounded i.i.d. random variables. At the cost of an additional logarithmic factor, we propose a computable version of the KMT inequality that depends only on the variables' range and standard deviation. We also derive an empirical version of the inequality that achieves nominal coverage even when the standard deviation is unknown. We then demonstrate the practicality of our bounds through applications to online change point detection and first hitting time probabilities. As a byproduct of our analysis, we obtain a Cram\'er-type moderate deviation bound for normalized centered partial sums.
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https://arxiv.org/abs/2508.03833
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e9c0004990f7b41f7e02421f9952a47c2ada9cdf1a37ea42ea72ff151d928eea
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2026-01-21T00:00:00-05:00
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Nonparametric Estimation from Correlated Copies of a Drifted Process
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arXiv:2508.05259v2 Announce Type: replace Abstract: This paper presents several situations leading to the observation of multiple correlated copies of a drifted process, and then non-asymptotic risk bounds are established on nonparametric estimators of the drift function $b_0$ and its derivative. For drifted Gaussian processes with a regular enough covariance function, a sharper risk bound is established on the estimator of $b_0'$, and a model selection procedure is provided with theoretical guarantees.
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https://arxiv.org/abs/2508.05259
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4765277b8b8673a3aa173b290a927bdb6aee2f584e8868d04ae3c87295412bd5
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2026-01-21T00:00:00-05:00
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Entropy and polynomial entropy of derived autoequivalences of derived discrete algebras
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arXiv:2508.05794v2 Announce Type: replace Abstract: The aim of this paper is to calculate entropy in the sense of Dimitrov-Haiden-Katzarkov-Kontsevich and polynomial entropy as defined by Fan-Fu-Ouchi of derived autoequivalences of derived discrete algebras over an algebraically closed field.
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https://arxiv.org/abs/2508.05794
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226e0a267def160afbb3ec18dd50d72ca5b5b50747a4288da4ae08bfcb30080a
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2026-01-21T00:00:00-05:00
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Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes
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arXiv:2508.08736v3 Announce Type: replace Abstract: The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability.
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https://arxiv.org/abs/2508.08736
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877ba72b881102228e71d6f331f79dfafb2a03bb3496e9fa397240ad4a554bdf
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2026-01-21T00:00:00-05:00
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Tatuzawa's theorem for Rankin-Selberg $L$-functions
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arXiv:2508.10844v3 Announce Type: replace Abstract: Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. As a corollary, we show that for all $\varepsilon>0$, there exists an effectively computable constant $c>0$ depending only on $(n,n',[F:\mathbb{Q}],\varepsilon)$ such that $L(s,\pi\times\pi')$ has at most one zero (necessarily simple) in the region \[ \mathrm{Re}(s)\geq 1-c/(C(\pi)C(\pi')(|\mathrm{Im}(s)|+1))^{\varepsilon}, \] where $C(\pi)$ and $C(\pi')$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,\pi\times\widetilde{\pi})$ by an idele class character $\chi$ apart from a possible single exceptional zero (necessarily real and simple) that can occur only when $\pi\otimes\chi^2=\pi$. This extends earlier work of Humphries and Thorner.
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https://arxiv.org/abs/2508.10844
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f1b6b0a4b6c207d192a6a3fbe3a3715be87aabe4d0ea52f701d8910a66be6c19
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2026-01-21T00:00:00-05:00
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High-Capacity and Low-PAPR BICM-OFDM Systems Using Non-Equiprobable and Non-Uniform Constellation Shaping With Clipping and Filtering
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arXiv:2508.15639v2 Announce Type: replace Abstract: We address a design of high-capacity and low-peak-to-average power ratio (PAPR) orthogonal frequency-division multiplexing (OFDM) systems based on bit-interleaved coded modulation (BICM) utilizing non-equiprobable and non-uniform (NENU) constellations as well as clipping and filtering (CAF). The proposed constellations are generated using a truncated Gaussian distribution and the merging of constellation points, where the former creates a non-uniform constellation (NUC), and the latter adjusts the number of signal points for further improving the total bit-wise mutual information (BMI). Unlike other exhaustive search-based approaches, the proposed constellations are uniquely determined by only two parameters associated with NUC and cardinality. Due to this property of limited degrees of freedom, the complexity required for the numerical optimization process can be significantly low. We focus on the constellation design based on one dimension, i.e., pulse amplitude modulation (PAM), which facilitates the reduction of demapping complexity for the BICM receiver. The use of CAF at the transmitter can efficiently reduce the PAPR of OFDM signals; however, it introduces clipping noise that may degrade error rate performance, making the application of clipping noise cancellation (CNC) at the receiver essential. Therefore, we optimize the NENU constellations in the presence of CAF and CNC. Simulation results demonstrate that the combination of constellation shaping with CAF and CNC enables BICM-OFDM systems to simultaneously achieve low PAPR and high spectral efficiency over additive white Gaussian noise (AWGN) as well as frequency-selective fading channels. Furthermore, comparative studies confirm that the proposed system significantly outperforms the single-carrier counterpart (i.e., DFT-precoded BICM-OFDM) in terms of PAPR and bit error rate (BER) performance over fading channels.
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https://arxiv.org/abs/2508.15639
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2d751bf1183dcddd8c0a9622ddaa25c3fe29242bc04313e1d3c65311cd598b91
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2026-01-21T00:00:00-05:00
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Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging
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arXiv:2508.16999v2 Announce Type: replace Abstract: This paper proposes a Physics-Informed Kolmogorov-Arnold Network for analyzing elasticity problems in multi-material electronic packaging structures. The method replaces traditional Multi-Layer Perceptrons with Kolmogorov-Arnold Networks within an energy-based Physics-Informed Neural Network framework. By constructing admissible displacement fields satisfying essential boundary conditions and optimizing network parameters through numerical integration, the proposed method effectively handles material property discontinuities. Unlike traditional methods that require domain decomposition and interface constraints for multi-material problems, Kolmogorov-Arnold Networks' trainable B-spline activation functions provide inherent piecewise characteristics. This capability stems from B-splines' local support, which enables effective approximation of discontinuities despite their individual smoothness. Consequently, this approach enables accurate approximation across the entire domain using a single network and simplifying the computational framework. Numerical experiments demonstrate that the proposed method achieves excellent accuracy and robustness in multi-material elasticity problems, validating its practical potential for electronic packaging analysis. Source codes are available at https://github.com/yanpeng-gong/PIKAN-MultiMaterial.
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https://arxiv.org/abs/2508.16999
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923f18dad3264eaeae808ec2754780f1708a815edbefaef4bdd0b0fbc5176cd1
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2026-01-21T00:00:00-05:00
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Local Well-Posedness of the Cahn-Hilliard-Biot System
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arXiv:2508.17893v2 Announce Type: replace Abstract: We show short-time well-posedness of a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot's equations for poroelasticity, including phase-field dependent material properties, with the Cahn-Hilliard equation to model the evolution of the solid, where we further distinguish between the absence and presence of a visco-elastic term of Kelvin-Voigt type. While both problems will be reduced to a fixed-point equation that can be solved using maximal regularity theory along with a contraction argument, the first case relies on a semigroup approach over suitable Hilbert spaces, whereas treating the second case under minimal assumptions with respect to spatial regularity necessitates the application of Banach scales.
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https://arxiv.org/abs/2508.17893
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374528e0529f84bf90925742bf59f85dbb7476020da8cb6e4570faedd3e15e27
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2026-01-21T00:00:00-05:00
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Calogero-Sutherland hyperbolic system and Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function
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arXiv:2508.18864v3 Announce Type: replace Abstract: We prove equivalence of two integral representations for the wave functions of hyperbolic Calogero-Sutherland system. For this we study two families of Baxter operators related to hyperbolic Calogero-Sutherland and rational Ruijsenaars models; the first one as a limit from hyperbolic Ruijsenaars system, while the second one independently. Besides, computing asymptotics of integral representations and also the value at zero point, we identify them with renormalized Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function.
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https://arxiv.org/abs/2508.18864
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2ab5ec1991c07e96f86e89342ec46b6cc1119653b1c72b69a4b43d068a51c8eb
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2026-01-21T00:00:00-05:00
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Asymptotic behavior of the Bergman kernel and associated invariants in weakly pseudoconvex domains
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arXiv:2509.00301v3 Announce Type: replace Abstract: In this paper, we present an explicit description for the boundary behavior of the Bergman kernel function, the Bergman metric, and the associated curvatures along certain sequences converging to an $h$-extendible boundary point.
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https://arxiv.org/abs/2509.00301
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5867cece7e9967a3872786633795fbe0d73ce0ac2746df7e7ee5e5f65e15198e
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2026-01-21T00:00:00-05:00
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Hochschild-Kostant-Rosenberg isomorphism for derived Deligne-Mumford stacks
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arXiv:2509.00501v3 Announce Type: replace Abstract: We prove a Hochschild--Konstant--Rosenberg (HKR) theorem for arbitrary derived Deligne--Mumford (DM) stacks, extending the results of Arinkin-C\u{a}ld\u{a}raru-Hablicsek in the smooth, global quotient case, although with different methods. To formulate our result, we introduce the notion of orbifold inertia stack of a derived DM stack; this supplies a finely tuned derived enhancement of the classical inertia stack, which does not always coincide with the classical truncation of the free loop space. We show that, in characteristic 0, given a derived DM stack, the shifted tangent bundle of its orbifold inertia stack is equivalent to its free loop space. This yields a canonical HKR isomorphism of algebras between the Hochschild homology of a derived DM stack and the cohomology of differential forms on its orbifold inertia stack. Moreover, this isomorphism intertwines the natural circle action and the de Rham differential. Similarly, HKR theorems for derived DM stacks are established for Hochschild cohomology, cyclic homology, negative cyclic homology, and periodic cyclic homology. As applications, we provide a rich supply of computations of Hochschild homology and Hochschild cohomology for interesting derived DM stacks, such as weighted projective lines, root stacks, quotients by algebraic groups, and mapping stacks, among others.
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https://arxiv.org/abs/2509.00501
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97018e32351feb6d6d1ed019fe031643df1ea2908388e169acf3200cfca76948
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2026-01-21T00:00:00-05:00
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Uniformly S-essential submodules and uniformly S-injective uniformly S-envelopes
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arXiv:2509.01638v2 Announce Type: replace Abstract: In this paper, we introduce the notion of uniformly S-essential (u-S-essential) submodules. Let R be a commutative ring and S a multiplicative subset of R. A submodule K of an R-module M is said to be u-S-essential in M if for any submodule L of M, s1(K \cap L) = 0 for some s1 \in S implies s2L = 0 for some s2 \in S. Several properties of this notion are studied. The notions of a u-S-uniform module and a u-S-injective u-S-envelope are also introduced, and we show that these notions are characterized by u-S-essential submodules.
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https://arxiv.org/abs/2509.01638
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2026-01-21T00:00:00-05:00
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Uniformly S-projective relative to a module and its dual
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arXiv:2509.01646v2 Announce Type: replace Abstract: In this article, we introduce the notion of u-S-projective relative to a module. Let S be a multiplicative subset of a ring R and M an R-module. An R-module P is said to be u-S-projective relative to M if for any u-S-epimorphism f : M \to N, the induced map HomR(P, f) : HomR(P,M) \to HomR(P,N) is a u-S-epimorphism. Dually, we also introduce u-S-injective relative to a module. Some properties of these notions are discussed. Several characterizations of u-S-semisimple modules in terms of these notions are given. The notions of u-S-quasi-projective and u-S-quasi-injective modules are also introduced and some of their properties are discussed.
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https://arxiv.org/abs/2509.01646
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On base point freeness for rank one foliations
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arXiv:2509.03109v2 Announce Type: replace Abstract: We prove the base point free theorem for log canonical foliated pairs of rank one on a Q-factorial projective klt threefold. Moreover, we show abundance in the case of numerically trivial log canonical foliated pairs of rank one in any dimension.
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https://arxiv.org/abs/2509.03109
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5052277e606247cd17815b2de2515fe73af955414084a9fd45c9e37806a9e1ed
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2026-01-21T00:00:00-05:00
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Recovery of Sturm-Liouville operators from partial boundary spectral data and applications
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arXiv:2509.04289v2 Announce Type: replace Abstract: We study the inverse Sturm-Liouville problem on a finite interval from partial knowledge of spectral data. Specifically, we show that the potential can be uniquely reconstructed from the knowledge of a fraction of Dirichlet eigenvalues together with the normal derivatives of the corresponding eigenfunctions at both endpoints. We present two novel applications of our spectral result in inverse coefficient determination problems for evolutionary PDEs that include passive wave-based imaging of a medium and active imaging for the time-dependent Schr\"odinger equation with unknown internal sources. Our results yield finite time measurement bounds for such inverse coefficient determination problems. A central innovation is the use of Kahane's interpolation theorem to analyze endpoint time traces of solutions, enabling the recovery without requiring analyticity assumptions or infinite-time data, as in previous approaches. Finally, in the appendix, we present a spectral interpolation theorem for one-dimensional Schr\"odinger operators, which may be of independent interest.
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https://arxiv.org/abs/2509.04289
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f3a8f29772dc61497639684ab8f3e9495b538efb26a3dc7c00d84fd701700ede
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2026-01-21T00:00:00-05:00
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$\mathbb R^{\omega_1}$-Factorizable Spaces and Groups
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arXiv:2509.05105v2 Announce Type: replace Abstract: A topological space $X$ is $\mathbb R^{\omega_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{\omega_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{\omega_1}$-factorizable if and only if $X\times D(\omega_1)$, where $D(\omega_1)$ is a discrete space of cardinality $\omega_1$, is $z$-embedded in the product $\beta X\times \beta D(\omega_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{\omega_1}$-factorizability is hereditary and countably multiplicative, that any $\mathbb R^{\omega_1}$-factorizable space is hereditarily Lindel\"of and hereditarily separable, and that the existence of nonmetrizable $\mathbb R^{\omega_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all $\mathbb R^{\omega_1}$-factorizable spaces are second-countable, while under MA + $\lnot$CH, the countable Fr\'echet--Urysohn fan is $\mathbb R^{\omega_1}$-factorizable.
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https://arxiv.org/abs/2509.05105
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f62c01de4cc0b8b68407da053b6dd340fce9ef6e1b153473febe2e7fc979de1e
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2026-01-21T00:00:00-05:00
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Serrin's overdetermined theorem within Lipschitz domains
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arXiv:2509.05155v3 Announce Type: replace Abstract: Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $\Omega$ satisfies the following Serrin-type overdetermined system $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus \Omega,\quad \Delta u=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*\Omega} - \mathbf{1}_{\Omega}\,dx,$$ in the weak sense if and only if $\Omega$ is a ball. Here $\mathscr H^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. Moreover, a generalization of our method in the anisotropic setting is discussed. Our approach offers an alternative proof to [15] in the case of Lipschitz domains, introducing a novel viewpoint to settle [18, Question 7.1].
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https://arxiv.org/abs/2509.05155
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5cbb5b54da55f06d04a053df013b96068bf872d9c5193834094b61aec0d52147
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2026-01-21T00:00:00-05:00
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Uniformly S-pseudo-injective modules
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arXiv:2509.05843v2 Announce Type: replace Abstract: This paper introduces the notion of uniformly-S-pseudo-injective (u-S-pseudo-injective) modules as a generalization of u-S-injective modules. Let R be a ring and S a multiplicative subset of R. An R-module E is said to be u-S-pseudo-injective if for any submodule K of E, there is s in S such that for any u-S-monomorphism f : K \to E, sf can be extended to an endomorphism g : E \to E. Several properties of this notion are studied. For example, we show that an R-module M is u-S-quasi-injective if and only if M \oplus M is u-S-pseudo-injective. Two classes of rings related to the class of QI-rings are introduced and characterized.
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https://arxiv.org/abs/2509.05843
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b8f223f47c50c13b8fcddfbf50981889a1b8b6a9a49bc5e898aa0feb018c9719
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2026-01-21T00:00:00-05:00
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Fujita Phenomenon for Mixed Local--Nonlocal Diffusion Equations
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arXiv:2509.07405v3 Announce Type: replace Abstract: We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho \mathbf{w}(x), \qquad (x,t)\in \mathbb{R}^N\times (0,\infty), \] where $\mathsf{s}\in (0,1)$, $p>1$, $b\geq 0$, and $\varrho>-1$. The function $\mathsf{h}(t)$ is assumed to belong to the generalized class of regularly varying functions, while $\mathbf{w}$ is a prescribed spatial source. We first revisit the unforced case and establish sharp blow-up and global existence criteria in terms of the critical Fujita exponent, thereby extending earlier results to the wider class of time-dependent coefficients. For the forced problem, we derive nonexistence of global weak solutions under suitable growth conditions on $\mathsf{h}$ and integrability assumptions on $\mathbf{w}$. Furthermore, we provide sufficient smallness conditions on the initial data and the forcing term ensuring global-in-time mild solutions. Our analysis combines semigroup estimates for the mixed operator, test function methods, and asymptotic properties of regularly varying functions. To our knowledge, this is the first study addressing blow-up phenomena for nonlinear diffusion equations with such a class of time-dependent coefficients.
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https://arxiv.org/abs/2509.07405
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3a41c7737b0fc5c944d2115dd63238be0fd601c10824406c93e19e8c8b03a6cf
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Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs
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arXiv:2509.08645v2 Announce Type: replace Abstract: We consider parabolic evolution equations with Lipschitz continuous and strongly monotone spatial operators. By introducing an additional variable, we construct an equivalent system where the operator is a Lipschitz continuous mapping from a Hilbert space $Y \times X$ to its dual, with a Lipschitz continuous inverse. Resulting Galerkin discretizations can be solved with an inexact Uzawa type algorithm. Quasi-optimality of the Galerkin approximations is guaranteed under an inf-sup condition on the selected `test' and `trial' subspaces of $Y$ and $X$. To circumvent the restriction imposed by this inf-sup condition, an a posteriori condition for quasi-optimality is developed that is shown to be satisfied whenever the test space is sufficiently large.
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https://arxiv.org/abs/2509.08645
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2-Distance Coloring of Planar Graphs with Specific Maximum Degree
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arXiv:2509.10861v2 Announce Type: replace Abstract: A k-distance r-coloring of a graph is a coloring of the vertices of the graph such that if the distance between 2 vertices x and y is less or equal to k, then x and y must have distinct colors. A planar graph is a graph that can be drawn with no edge crossing. We will study the 2-distance coloring of planar graphs with maximum degree at least 6.
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https://arxiv.org/abs/2509.10861
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Failure of Lichnerowicz-type result in parabolic geometries of real rank at least $3$
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arXiv:2509.12542v4 Announce Type: replace Abstract: Given a Yamaguchi nonrigid parabolic model geometry $(G,P)$ with $G$ simple of real rank at least $3$, we use techniques developed by Erickson to establish the existence of closed, nonflat, essential, regular, normal Cartan geometries modeled on $(G,P)$. Yamaguchi nonrigidity is a necessary condition for admitting nonflat, regular, normal examples. This rules out Lichnerowicz-type conjectures for these model geometries.
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https://arxiv.org/abs/2509.12542
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Learning the Influence Graph of a Markov Process that Randomly Resets to the Past
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arXiv:2509.16129v2 Announce Type: replace Abstract: Learning the influence graph G of a high-dimensional Markov process is central to many application domains, including social networks, neuroscience, and financial risk analysis. However, in many of these applications, future states of the process are occasionally and unpredictably influenced by a distant past state, thus destroying the Markovianity. To study this practical issue, we propose the past influence model (PIM), which captures the occasional "random resets to past" by modifying the Markovian dynamics in [1], which, in turn, is a non-linear generalization of the dynamics studied in [2], [3]. The recursive greedy algorithm proposed in this paper recovers any bounded degree $G$ when the number of ``jumps back in time" is order-wise smaller than the total number of samples, and the algorithm does not require memory.
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https://arxiv.org/abs/2509.16129
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2026-01-21T00:00:00-05:00
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Perfect Divisibility and Coloring of Some Bull-Free Graphs
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arXiv:2509.18856v3 Announce Type: replace Abstract: A graph $G$ is {\em perfectly divisible} if, for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges, a {\em fork } is a graph obtained from $K_{1,3}$ by subdividing an edge once, and an {\em odd torch} is a graph obtained from an odd hole by adding an edge $xy$ such that $x$ is non-adjacent to any vertex on the odd hole and the set of neighbors of $y$ on the odd hole is a stable set. Chudnovsky and Sivaraman [J. Graph Theory 90 (2019) 54-60] proved that every (odd hole, bull)-free graph and every ($P_5$, bull)-free graph are perfectly divisible. Karthick {\em et al.} [The Electron. J. of Combin. 29 (2022) P3.19.] proved that every (fork, bull)-free graph is perfectly divisible. Chen and Xu [Discrete Appl. Math. 372 (2025) 298-307.] proved that every ($P_7,C_5$, bull)-free graph is perfectly divisible. Let $H\in$\{\{odd~torch\}, $\{P_8,C_5\}\}$. In this paper, we prove that every ($H$, bull)-free graph is perfectly divisible. We also prove that a ($P_6$, bull)-free graph is perfectly divisible if and only if it contains no Mycielski-Gr\"{o}tzsch graph as an induced subgraph. As corollaries, these graphs are $\binom{\omega+1}{2}$-colorable. Notice that every odd torch contains an odd hole, an induced $P_5$, and an induced fork. Therefore, our results generalize their findings. Moreover, we prove that every ($P_6$, bull)-free graph $G$ satisfies $\chi(G)\leq\omega(G)^7$.
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https://arxiv.org/abs/2509.18856
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2026-01-21T00:00:00-05:00
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A nonlocal Aw-Rascle-Zhang system with linear pressure term
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arXiv:2509.20973v2 Announce Type: replace Abstract: In this paper, we study a nonlocal extension of the Aw-Rascle-Zhang traffic model, where the pressure-like term is modeled as a convolution between vehicle density and a kernel function. This formulation captures nonlocal driver interactions and aligns structurally with the Euler-alignment system studied in [23]. Using a sticky particle approximation, we construct entropy solutions to the equation for the cumulative density and prove convergence of approximate solutions to weak solutions of the nonlocal system. The analysis includes well-posedness, stability estimates, and an entropic selection principle.
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https://arxiv.org/abs/2509.20973
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33cde6b8037f33b1ffc0b0dce4256f1c36aa155be0cb4b7cdd0b52549eaa14f3
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2026-01-21T00:00:00-05:00
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Motivic Classes of Isotropic Degeneracy Loci and Symmetric Orbit Closures
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arXiv:2509.24348v2 Announce Type: replace Abstract: We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern--Schwartz--MacPherson classes, K-theory classes, and Cappell--Shaneson L-classes arise as specializations of the motivic Chern and Hirzebruch classes. Our result is the analogue of the result of Anderson--Chen--Tarasca for the degeneracy loci from the ordinary Grassmannians. As applications, we obtain the motivic Chern and Hirzebruch classes of orthogonal and symplectic orbit closures in flag varieties.
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https://arxiv.org/abs/2509.24348
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Academic Papers
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