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arXiv:2412.11086v3 Announce Type: replace Abstract: We study the behavior of perturbations in a compressible one-dimensional inviscid gas with an ambient state consisting of constant pressure and periodically-varying density. We show through asymptotic analysis that long-wavelength perturbations approximately obey a system of dispersive nonlinear wave equations. Computational experiments demonstrate that solutions of the 1D Euler equations..
arXiv:2412.11568v2 Announce Type: replace Abstract: We study the Rellich type theorem (RT) for the Maxwell operator __ D = D____0 on Z3 in a constant anisotropic medium, i.e., the permittivity and permeability of which are constant non-scalar diagonal matrices. We also prove the unique continuation property (UCP) in the exterior of a compact convex set Kint $\subset$ Z3 for the perturbed Maxwell operator __ Dp = D__p __0 on Z3 for which th..
arXiv:2412.11975v4 Announce Type: replace Abstract: The Nielsen-Thomsen sequence plays a pivotal role in refining invariants for C$^*$-algebras beyond the Elliott classification framework. This paper revisits the sequence, introducing the concepts of Nielsen-Thomsen bases, rotation maps and diagonalisable morphisms, to better understand its unnatural splitting. These insights enable novel comparison methods for *-homomorphisms at the level..
arXiv:2501.00710v2 Announce Type: replace Abstract: Given a triangulated category $\mathcal{C}$, we construct a partial compactification, denoted $\mathcal{A}\mathrm{Stab}(\mathcal{C})$, of the quotient of its stability manifold by $\mathbb{C}$. The purpose of $\mathcal{A}\mathrm{Stab}(\mathcal{C})$ is to shed light on the structure of semiorthogonal decompositions of $\mathcal{C}$. A point of $\mathcal{A}\mathrm{Stab}(\mathcal{C})$, calle....
arXiv:2501.07847v2 Announce Type: replace Abstract: We consider nonlinear drift-diffusion equations (both porous medium equations and fast diffusion equations) with measure data. We establish the existence of nonnegative weak solutions satisfying gradient estimates, provided that the drift term belongs to a sub-scaling class relevant to the $L^1$ space. When the drift is divergence-free, this requirement can be relaxed: the drift may belon....
arXiv:2501.10918v2 Announce Type: replace Abstract: In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects every dicut. Edmonds and Giles conjectured that in a weighted digraph, the minimum weight of a dicut is equal to the maximum size of a packing of dijoins. This has been disproved. However, the unweighted version conjectured by Woodall remains open. We prove that the Edmon..
arXiv:2501.12790v4 Announce Type: replace Abstract: Considered in this work is the Yang-Mills field in an extremal Reissner-Nordstr\"om black hole, a physically motivated mathematical model introduced by Bizo\'n and Kahl. The kink is a fundamental, strongly unstable stationary solution in this non-perturbative, variable coefficients model, with a polynomial tail and no explicit form. In this paper, we introduce and extend several virial te..
arXiv:2501.15168v2 Announce Type: replace Abstract: Standard Virtual Element Method (VEM) requires stabilization terms that significantly affect the numerical computation performance. In this work, we propose a stabilization-free VEM for general order \(\mathbf{H}(\operatorname{\mathbf{curl}})\) and \(\mathbf{H}(\operatorname{div})\)-conforming spaces by constructing novel serendipity projectors and corresponding serendipity spaces with mi..
arXiv:2502.00753v4 Announce Type: replace Abstract: Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progres....
arXiv:2502.02108v3 Announce Type: replace Abstract: We study the Schur algebra counterpart of a vast class of quantum wreath products. This is achieved by developing a theory of twisted convolution algebras, inspired by geometric intuition. In parallel, we provide an algebraic Schurification via a Kashiwara-Miwa-Stern-type action on a tensor space. We give a uniform proof of Schur duality, and construct explicit bases of the new Schur alge..
arXiv:2502.03847v2 Announce Type: replace Abstract: A proof of optimal-order error estimates is given for the full discretization of the bulk--surface Cahn--Hilliard system with dynamic boundary conditions in a smooth domain. The numerical method combines a linear bulk--surface finite element discretization in space and linearly implicit backward difference formulae of order one to five in time. The error estimates are obtained by a consis..
arXiv:2503.13157v4 Announce Type: replace Abstract: We introduce and study the notion of ramification ideals in higher ramification theory. After general results on their computation for finite extensions, we discuss their connection with the possibly nontrivial defect of the extensions. We compute them for Artin-Schreier extensions and Kummer extensions of prime degree equal to the residue characteristic, which may or may not have nontriv..
arXiv:2503.15046v2 Announce Type: replace Abstract: We address the enumeration of Eulerian orientations of 4-valent planar maps according to three parameters: the number of vertices, the number of alternating vertices (having in/out/in/out incident edges), and the number of clockwise oriented faces. This is a refinement of the six vertex model studied by Kostov, then Zinn-Justin and Elvey Price, where one only considers the first two param....
arXiv:2504.00854v2 Announce Type: replace Abstract: For curves singularities the dimension of smoothing components in the deformation space is an invariant of the singularity, but in general the deformation space has components of different dimensions. We are interested in the question of what the generic singularities are above these components. To this end we revisit the known examples of non-smoothable singularities and study their def....
arXiv:2504.04181v3 Announce Type: replace Abstract: We study a vectorial $L^\infty$-variational problem of second order, where the supremal functional depends on the vector function $u$ through a linear elliptic operator in divergence form. We prove existence and uniqueness of the minimiser $u_\infty$ under prescribed Dirichlet boundary conditions, together with a characterisation of $u_\infty$ as solution of a specific system of PDEs. Our..
arXiv:2504.10019v3 Announce Type: replace Abstract: This paper extends the article of the Bruns and Conca on SAGBI bases and their computation (J. Symb. Comput. 120 (2024)) in two directions. (i) We describe the extension of the Singular library sagbiNormaliz.sing to the computation of defining ideals of subalgebras of polynomial rings. (ii) We give a complete classification of the algebras of minors for which the generating set is a SAGBI..
arXiv:2504.10633v5 Announce Type: replace Abstract: Currently, there is no general theory for deriving diffusion approximations of queueing systems with high- or infinite-dimensional state descriptors. In this paper, we explore one path for deriving diffusion limit equations of queueing models. The method hinges on a martingale decomposition of dynamics driven by time-changed renewal processes, which are a common feature of many queueing m....
arXiv:2504.11603v4 Announce Type: replace Abstract: The dynamics of weighted translation operators on Lebesgue spaces, Orlicz spaces, and in general on solid Banach function spaces have been studied in numerous papers. Recently, the dynamics of weighted translations on weighted Orlicz spaces have also been studied by Chen and others. The main idea of this paper is to obtain a generalization of these results to the case of general weighted ..
arXiv:2504.14455v3 Announce Type: replace Abstract: We extend the topological results of Lytchak-Nagano and Lytchak-Nagano-Stadler for CAT(0) spaces to the setting of Busemann spaces of nonpositive curvature, i.e., BNPC spaces. We give a characterization of locally BNPC topological manifolds in terms of their links and show that the singular set of a locally BNPC homology manifold is discrete. We also prove that any (globally) BNPC topolog..
arXiv:2504.19273v2 Announce Type: replace Abstract: In this paper, we first study the sharp weak estimate for the $p$-adic $n$-dimensional fractional Hardy operator from $L^p$ to $L^{q,\infty}$. Secondly, we study the sharp bounds for the $m$-linear $n$-dimensional $p$-adic integral operator with a kernel on $p$-adic weighted spaces $H_{\alpha}^{\infty}( \mathbb{Q} _{p}^{n} )$. As an application, the sharp bounds for $p$-adic Hardy and Hil..
arXiv:2504.20549v3 Announce Type: replace Abstract: The Pappas--Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to line bundles on spherical Schubert varieties in affine Grassmannians and to line bundles on unions of Schubert varieties in affine flag varieties. Algebraically, the claim can be reformulated as an equality..
arXiv:2505.07128v2 Announce Type: replace Abstract: In this first work in a series, we prove the local-in-time well-posedness of the IBVP for the vacuum Einstein equations with Dirichlet boundary data on a finite timelike boundary, provided the Brown- York stress tensor of the boundary is a Lorentz metric of the same signature (up to an overall sign) as the induced Lorentz metric on the boundary. This is a convexity-type assumption which i..
arXiv:2505.07352v2 Announce Type: replace Abstract: We establish a Brownian extension to Selberg's central limit theorem for the Riemann zeta function. This implies various limiting distributions for $\zeta$, including an analogue of the reflection principle for the maximum of the Brownian motion: as $T$ diverges, for any $u>0$ we have \[ \frac{1}{T}\cdot {\rm meas}\Big\{0\leq t\leq T:\max_{\sigma\geq \tfrac{1}{2}}\log|\zeta(\sigma+i t)|\g..
arXiv:2505.10809v3 Announce Type: replace Abstract: We prove tilting equivalence for the finite almost derived algebraic cobordism spectrum $\mathrm{dMGL}^{a,fin}$ of perfectoid algebras. More precisely, for any perfectoid algebra, the tilting functor induces a weak equivalence on the corresponding finite almost derived algebraic cobordism spectra. This invariant is a finite-syntomic, derived, and non-$\msthbb{A}^1$-local version of algebr..
arXiv:2505.17707v2 Announce Type: replace Abstract: In this paper, we first obtain the operator norms of the $n$-dimensional Hardy-Littlewood-P\'{o}lya operator $\mathcal{H}$ from weighted Lebesgue spaces $L^p( \mathbb{R} ^n,| x |^{\beta} ) $ to weighted weak Lebesgue spaces $L^{q,\infty}(\mathbb{R} ^n,|x|^{\gamma})$. Next, we obtain the weak bounds for the $m$-linear $n$-dimensional integral operator.
arXiv:2505.23293v3 Announce Type: replace Abstract: Mediangle graphs are a common generalization of median graphs (1-sekeleta of CAT(0) cube complexes) and Coxeter graphs (Cayley graphs of Coxeter systems). Answering a question motivated from geometric group theory, we show that these graphs can be endowed with the structure of a contractible cell complex. We further show that the cells of this complex are products of simplices and simplic..
arXiv:2505.23562v2 Announce Type: replace Abstract: In 1996, Youngs proved that any quadrangulation of the real projective plane is not 3-chromatic. This result has been extended in various directions over the years, including to other non-orientable closed surfaces, higher-dimensional analogues of quadrangulations and circular colorings. In this paper, we provide a generalization which yields some of these extensions of Youngs' theorem.
arXiv:2506.01217v2 Announce Type: replace Abstract: We define two stochastic analogs of a geometric flow on even-dimensional manifolds called $Q$-curvature flow, and use the theory of Dirichlet forms to construct weak solutions to both. The first of these flows, which we call the normalized $Q$ flow (NQF), preserves the intrinsic volume normalization from the deterministic setting. The second, which we call the Liouville $Q$ flow (LQF), ha..
arXiv:2506.02340v3 Announce Type: replace Abstract: In this paper, we obtain an explicit formula for the heat kernel on the Cayley graph of the modular group $PSL_2(Z)$, given by the presentation $\langle a,b\mid a^2=1, b^3=1\rangle$. Our approach extends a method of Chung--Yau by observing that the Cayley graph strongly and regularly covers a weighted infinite line. We solve the spectral problem on this line to obtain an integral expressi..
arXiv:2506.05747v3 Announce Type: replace Abstract: This paper proposes asymmetric perturbation, where only one player's payoff function is perturbed, for solving bilinear saddle-point optimization problems, commonly arising in minimax problems, game theory, and constrained optimization. Symmetric perturbation is known to require decreasing its strength to ensure convergence to a solution, i.e., an equilibrium in the original game, resulti....
arXiv:2506.09267v3 Announce Type: replace Abstract: The problem of estimating the slope parameter in regression between two spatial processes under confounding by an unmeasured spatial process has received widespread attention in the recent statistical literature. Yet, a fundamental question remains unresolved: when is this slope consistently estimable under spatial confounding, with existing insights being largely empirical or estimator-s....
arXiv:2506.11118v2 Announce Type: replace Abstract: The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we give a proof for the effective Banach Category Theorem. Further, we provide a game-theoretic proof of an effective theorem in dynamical systems, namely the c..
arXiv:2506.15340v2 Announce Type: replace Abstract: This work presents a mathematical model for the optimal control of thin-film flows over a flexible substrate influenced by an external force. The objective is to find the optimal distributed force acting on the topography that minimises the differences between actual and desired thin-film profiles. A nonlinear lubrication equation governing the fluid dynamics and appropriate functional se....
arXiv:2506.17895v2 Announce Type: replace Abstract: We study the joint distribution of two randomly weighted sums. Inspired by the practical applications, we assume that the main random variables follow the non-standard bivariate regular variation, symbolically BRV , to put emphasis to the value of inhomogeneity of the risk distribution tails, while the random weights are weakly dependent with main random variables. Under some moment condi....
arXiv:2506.21025v3 Announce Type: replace Abstract: This work develops novel energy-stable parametric finite element methods (ES-PFEM) for the Willmore flow and curvature-dependent geometric gradient flows of surfaces in three dimensions. The key to achieving the energy stability lies in the use of two novel geometric identities: (i) a reformulated variational form of the normal velocity field, and (ii) incorporation of the temporal evolut....
arXiv:2507.01623v4 Announce Type: replace Abstract: In this paper, we present a qualitative and bifurcation analysis of the three-parameter FitzHugh-Nagumo system and its compactified formulation. The study is structured according to three parameter-dependent regimes, for which the associated phase portraits are characterized. In one of these regimes, the system exhibits a double-zero bifurcation with $Z_2$ symmetry, corresponding to a co....
arXiv:2507.02552v4 Announce Type: replace Abstract: This paper investigates the detection and estimation of a single change in high-dimensional linear models. We derive minimax lower bounds for the detection boundary and the estimation rate, which uncover a phase transition governed by the sparsity of the covariance-weighted differential parameter. This form of "inherent sparsity" captures a delicate interplay between the covariance struct....
arXiv:2507.05008v2 Announce Type: replace Abstract: We construct a new quantization $K_t(\mathcal{O}^{sh}_{\mathbb{Z}})$ of the Grothendieck ring of the category $\mathcal{O}^{sh}_{\mathbb{Z}}$ of representations of shifted quantum affine algebras (of simply-laced type). We establish that our quantization is compatible with the quantum Grothendieck ring $K_t(\mathcal{O}^{\mathfrak{b},+}_{\mathbb{Z}})$ for the quantum Borel affine algebra, ....
arXiv:2507.08608v2 Announce Type: replace Abstract: We investigate equidecomposability in the ring of polygons with sides restricted to given directions and using only translations. Extending classical results of Dehn and Hadwiger, we prove that equidecomposability in these rings is equivalent to the equality of some translation invariant simple valuations. We also consider the algebraic structure of direction sets. We show that under mild..
arXiv:2507.10144v2 Announce Type: replace Abstract: The Lanczos method is a fast and memory-efficient algorithm for solving large-scale symmetric eigenvalue problems. However, its rapid convergence can deteriorate significantly when computing clustered eigenvalues due to a lack of cluster robustness. A promising strategy to enhance cluster robustness -- without substantially compromising convergence speed or memory efficiency -- is to use ....
arXiv:2507.10816v2 Announce Type: replace Abstract: We pursue an old conjecture of John Roe about the algebraic K-theory of the algebra of finite propagation, locally trace-class operators, namely that transgressing the algebraic coarse character map on this algebra to a Higson dominated corona coincides with the usual Chern character on the corona.
arXiv:2507.10937v3 Announce Type: replace Abstract: A \emph{matching} from a finite subset \( A \) of an abelian group \( G \) to another subset \( B \) is a bijection \( f : A \to B \) such that \( af(a) \notin A \) for all \( a \in A \). The study of matchings began in the 1990s and was motivated by a conjecture of E.~K.~Wakeford on canonical forms for homogeneous polynomials. The theory was later extended to the linear setting of vector..
arXiv:2507.11365v2 Announce Type: replace Abstract: We classify representations of the mapping class group of a surface of genus $g$ (with at most one puncture or boundary component) up to dimension $3g-3$. Any such representation is the direct sum of a representation in dimension $2g$ or $2g+1$ (given as the action on the (co)homology of the surface or its unit tangent bundle) with a trivial representation. As a corollary, any linear syst..
arXiv:2507.13000v2 Announce Type: replace Abstract: This paper develops a geometric framework for the stability analysis of differential inclusions governed by maximally monotone operators. A key structural decomposition expresses the operator as the sum of a convexified limit mapping and a normal cone. However, the resulting dynamics are often difficult to analyze directly due to the absence of Lipschitz selections and boundedness. To ove....
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