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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....
arXiv:2507.20902v2 Announce Type: replace Abstract: We study sequences of modular representations of the symplectic and special linear groups over finite fields arising from the first homology of congruence subgroups of mapping class groups and automorphism groups of free groups, as well as from the module of coinvariants for the abelianization of the Torelli group. In each case, we determine the composition factors and their multiplicitie..
arXiv:2507.22147v5 Announce Type: replace Abstract: This work focuses on the frequency-domain modeling of a control system with a flexible beam and a rigid body. A simply supported beam is equipped with a spring-loaded control actuator and possesses local damping effect. Using Hamilton's variational principle, the equations of motion are derived in the state space form taking into account interface conditions involving lumped control and l..
arXiv:2508.01973v4 Announce Type: replace Abstract: This article demonstrates how recent developments in the theory of empirical processes allow us to construct a new family of asymptotically distribution-free smooth tests. Their distribution-free property is preserved even when the parameters are estimated, model selection is performed, and the sample size is only moderately large. A computationally efficient alternative to the classical ..
arXiv:2508.08873v2 Announce Type: replace Abstract: We consider a new class of Parareal algorithms, which use ideas from localized reduced basis methods to construct the coarse solver from truncated SVD approximations of the transfer operators mapping initial values for a given time interval to the solution at the end of the interval. By leveraging randomized singular value decompositions, these low-rank approximations are obtained embarra..
arXiv:2508.11073v3 Announce Type: replace Abstract: This paper addresses stochastic optimization of Lipschitz-continuous, nonsmooth and nonconvex objectives over compact convex sets, where only noisy function evaluations are available. While gradient-free methods have been developed for smooth nonconvex problems, extending these techniques to the nonsmooth setting remains challenging. The primary difficulty arises from the absence of a Tay....
arXiv:2508.16047v2 Announce Type: replace Abstract: For site percolation on the triangular lattice, we define two lattice fields that form a logarithmic pair in the sense of conformal field theory. We show that, at the critical point, their two- and three-point correlation functions have well-defined scaling limits, whose structure agrees with that predicted for logarithmic field theories. One of the two fields can be identified with the p..
arXiv:2508.20730v2 Announce Type: replace Abstract: We study the global macroscopic limits associated with kinetic-fluid interaction models for sprays. Motivated by the Vlasov-Navier-Stokes system under the monokinetic ansatz, we consider the pressureless Euler-Navier-Stokes (Euler-NS) system in $\mathbb{R}^{d}$ ($d\geq2$) coupled through the singular drag force $\frac{1}{\tau} \rho (u-v)$, where $\tau$ is the Stokes relaxation time. For i....
arXiv:2508.21515v2 Announce Type: replace Abstract: In this note, we reveal a relation between the weight distribution of a concatenated code ensemble based on the Plotkin construction and those of its component codes. The relation may find applications in the calculation of the ensemble weight distributions for many codes, including Reed-Muller (RM)-like codes.
arXiv:2509.01191v2 Announce Type: replace Abstract: Local log-regular rings are Cohen-Macaulay local domains introduced by Kazuya Kato to expand the theory of toric varieties without a base. In this note, we show that local rings of monoid algebras over regular rings are log-regular. As a corollary, we show that monoid algebras are log-regular rings.
arXiv:2509.02107v2 Announce Type: replace Abstract: We develop high-order numerical schemes to solve random hyperbolic conservation laws using linear programming. The proposed schemes are high-order extensions of the existing first-order scheme introduced in [{\sc S. Chu, M. Herty, M. Luk\'a\v{c}ov\'a-Medvi{\softd}ov\'a, and Y. Zhou}, SIAM J. Sci. Comput., 48 (2026)], where a novel structure-preserving numerical method using a concept of g....
arXiv:2509.04614v2 Announce Type: replace Abstract: We study cluster algebras over $\mathbb{F}_2$. By the Laurent phenomenon there is a map from the set of seeds of the cluster algebra to the corresponding cluster variety. We show that in type $A$, fibers of this map can be described in terms of certain edges of the universal polytope of triangulations of a polygon. Moreover, we show that there is a section of this map giving seeds whose c..
arXiv:2509.06248v2 Announce Type: replace Abstract: Hardy's $Z$-function $Z(t)$ is a real-valued function of the real variable $t$, and whose zeros correspond exactly to the zeros of the Riemann zeta-function on the critical line. In 2012, K. Matsuoka showed that for every non-negative integer $k$, there exists a $T=T(k)>0$ such that $Z^{(k+1)}(t)$ has exactly one zero between consecutive zeros of $Z^{(k)}(t)$ for $t\ge T$ under the Rieman..
arXiv:2509.06527v3 Announce Type: replace Abstract: The aim of this article is to study basic structures and interrelations of $\delta$-rings, perfectoid towers, and lim Cohen--Macaulay sequences over Noetherian rings in positive or mixed characteristic. We also discuss the deformation of perfectoid purity via perfectoid towers. In the latter part of this paper, we discuss some methods for constructing perfectoid towers, dealing with $p$-t..
arXiv:2509.12734v4 Announce Type: replace Abstract: In the Admixture Model, the probability that an individual carries a certain allele at a specific marker depends on the allele frequencies in $K$ ancestral populations and the proportion of the individual's genome originating from these populations. The markers are assumed to be independent. The Linkage Model is a Hidden Markov Model (HMM) that extends the Admixture Model by incorporating....
arXiv:2509.14083v2 Announce Type: replace Abstract: We prove a formula governing the combinatorics of cyclic group actions and extend a lemma of Hasse to account for ramification. Consequently, we obtain the first constructive proofs of two theorems characterizing arithmetic equivalence. We are also able to give new proofs of strong multiplicity one type theorems for Kronecker equivalence and weak Kronecker equivalence.
arXiv:2509.15385v3 Announce Type: replace Abstract: We propose a Variable-Preconditioned Transformed Primal-Dual (VPTPD) method for solving generalized Wasserstein gradient flows based on the structure-preserving JKO scheme. This is a nontrivial extension of the TPD method [Chen et al. (2025) SIAM J. Sci. Comput.] incorporating proximal splitting techniques to address the challenges arising from the nonsmoothness of the objective function.....
arXiv:2509.18025v2 Announce Type: replace Abstract: One can see deep-learning models as compositions of functions within the so-called tame geometry. In this expository note, we give an overview of some topics at the interface of tame geometry (also known as o-minimality), optimization theory, and deep learning theory and practice. To do so, we gradually introduce the concepts and tools used to build convergence guarantees for stochastic g..
arXiv:2509.23076v2 Announce Type: replace Abstract: Let $\Omega$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $\mathcal{X}$ with dual space $\mathcal{X}^*$. This article presents a hybrid algorithm for finding a common element of the set of solutions to a generalized mixed equilibrium problem and the set of common fixed points of a family of a general class of nonlinear nonexpansive ma..
arXiv:2509.24354v2 Announce Type: replace Abstract: An $r$-pattern $P$ is an ordered pair $P=([l],E)$, where $l$ is a positive integer and $E$ is a set of $r$-multisets with elements from $[l]$. An $r$-graph $H$ is said to be $P$-colorable if there is a homomorphism $\phi$: $V(H)\rightarrow [l]$ such that $\{\phi(v_{1}),\ldots,\phi(v_{r})\}\in E$ for every edge $\{v_{1},\ldots,v_{r}\}\in E(H)$. Let $\mathrm{Col}(P)$ denote the family of al....
arXiv:2509.24456v2 Announce Type: replace Abstract: In this self-contained short note, we introduce the new definition of Good Ramanujan Expansion, say G.R.E., for a fixed arithmetic function $F$, building upon a good decay of its coefficients $G$; this, gains $\log-$powers w.r.t. the trivial bound for $G$ and precisely $\log^{1+\eta}$, where the present parameter $\eta>0$ is real. This property alone has important consequences for all the....
arXiv:2510.02501v2 Announce Type: replace Abstract: It was shown by Barron--Shafiee that an analogue of Gromov's non-squeezing theorem holds for affine maps which preserve a power $\omega^k$ of the symplectic form $\omega$ on $\mathbb{R}^{2n}$. In the present paper, we state and prove in two ways an improved version of their result which is closer to the classical affine non-squeezing theorem. One proof closely follows their argument, and ....
arXiv:2510.02776v3 Announce Type: replace Abstract: Let $Q$ be an $s$-vertex $r$-uniform hypergraph, and let $H$ be an $n$-vertex $r$-uniform hypergraph. Denote by $\mathcal{N}(Q,H)$ the number of isomorphic copies of $Q$ in $H$. For a hereditary family $\mathcal{P}$ of $r$-uniform hypergraphs, define $$\pi(Q,\mathcal{P}):=\lim\limits_{n\to \infty}\binom{n}{s}^{-1}\max\{\mathcal{N}(Q,H): H\in \mathcal{P}~~\mbox{and}~~|V(H)|=n\}.$$ For $p\g....
arXiv:2510.07157v5 Announce Type: replace Abstract: Efficient large-scale network allocation requires data-driven pricing mechanisms that internalize stochastic, nonlinear user behavior. We move beyond the classic fully strategic agents to study oblivious users (agents with bounded rationality and imperfect information). Rather than assuming an infinite horizon, our regime acknowledges that real-world flows are too transient to equilibrate....
arXiv:2510.09213v2 Announce Type: replace Abstract: The inverse source problem for the Helmholtz equation poses significant challenges, particularly when sources exhibit complex or discontinuous geometries. Traditional numerical methods suffer from prohibitive computational costs, while machine learning-based approaches such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM) -- though computationally efficient ....
arXiv:2510.10867v2 Announce Type: replace Abstract: Fukaya categories are deep and rich invariants of symplectic manifolds which are notoriously difficult to compute explicitly. In the case of surfaces, however, the situation is simple, combinatorial,and is very well understood (at least by experts). In this expository paper we will give an introduction with many examples to welcome newcomers to the area and hopefully equip them with the t..
arXiv:2510.12035v3 Announce Type: replace Abstract: Webs are planar directed graphs that encode invariant vectors for tensor products of fundamental $U_q(\mathfrak{sl}_n)$-representations. For $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$, web calculus is governed by effective reduction rules and well-understood reduced-web bases, but these features break down in higher rank. In this paper we develop a global combinatorial framework for untagged....
arXiv:2510.12673v2 Announce Type: replace Abstract: In this work, we establish a local smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. In contrast with all previous works, we remove the Ricci curvature condition and completely localize the smoothing. As an application, we prove the compactness of the space of compact manifolds with bounded curvature concentration under Ahl..
arXiv:2510.14122v2 Announce Type: replace Abstract: Many real-world decision-making processes rely on solving mixed-integer nonlinear programs (MINLPs). However, finding high-quality solutions to MINLPs is often computationally demanding, motivating the development of specialized algorithms to improve their tractability. In this work, we propose Mixed-Integer Smoothing Surrogate Optimization with Constraints (MISSOC), a novel optimization ....
arXiv:2510.14479v2 Announce Type: replace Abstract: We consider linear model reduction in both the control and state variables for unconstrained linear-quadratic optimal control problems subject to time-varying parabolic PDEs. The first-order optimality condition for a state-space reduced model naturally leads to a reduced structure of the optimal control. Thus, we consider a control- and state-reduced problem that admits the same minimize..
arXiv:2510.24553v2 Announce Type: replace Abstract: For any element $g$ of compact reductive group $G$ we investigate the asymptotic behavior of its normalized irreducible character in the high-dimension limit, $\frac{\chi_\lambda(g)}{d_\lambda}$. We show that when $G$ is simple the limit vanishes besides identity element. For semisimple groups one gets the same results under the additional assumption that dimensions of irreducible represe..
arXiv:2510.25151v3 Announce Type: replace Abstract: This paper establishes a quantitative stability theory for one-dimensional stochastic differential equations (SDEs) with non-zero drift, driven by a symmetric $\alpha$-stable process for $\alpha\in(1,2)$. Our work generalizes the classical pathwise comparison method, pioneered by Komatsu for uniqueness problems, to address the stability of SDEs featuring both non-zero drift and, crucially....
arXiv:2510.25578v2 Announce Type: replace Abstract: Linear codes with few weights have been a subject of study for many years, as they have applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this article, two distinct classes of $p$-ary linear codes are constructed through the selection of two specific defining sets. Their weight distributions are completely determined for each case b..
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