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arXiv:2511.01480v2 Announce Type: replace Abstract: In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert u_{x_{i}}\vert}\right]\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\Omega\times I, \] where $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ for $n\geq2$, $I\subset\mathbb{R}$ is a bounded open interva....
arXiv:2511.02641v4 Announce Type: replace Abstract: Tilting bundles translate geometry into non-commutative algebra via derived equivalences. We prove the existence of, and classify, $d$-tilting bundles consisting of line bundles on $d$-dimensional smooth toric Fano stacks of Picard number one or two. Their endomorphism algebras give natural examples of $d$-representation infinite algebras and are closely related to the derived McKay corre....
arXiv:2511.07712v4 Announce Type: replace Abstract: Let $G$ be a simple graph on $n$ vertices and $m$ edges with chromatic number $\chi$, and let $\lambda_n$ denote the least adjacency eigenvalue. Solving a conjecture of Fan, Yu and Wang~[Electron. J. Combin., 2012], we prove that when $3\le \chi\le n-1$, the chromatic number satisfies the following upper bound: $$ \chi \le \left(\frac{n}{2}+1+\lambda_n\right) + \sqrt{\left(\frac{n}{2}+1+\....
arXiv:2511.11151v2 Announce Type: replace Abstract: Fix a bounded $3$-polygon $(\Omega; x_1, x_2, x_3)$ with three marked boundary points $x_1, x_2, x_3\in\partial\Omega$ and suppose $(\Omega^{\delta}; x_1^{\delta}, x_2^{\delta}, x_3^{\delta})$ is an approximation of $(\Omega; x_1, x_2, x_3)$ on $\delta$-scaled hexagonal lattice. We consider uniform spanning tree (UST) in $\Omega^{\delta}$ with wired boundary conditions. Conditional on the....
arXiv:2511.13547v2 Announce Type: replace Abstract: This is the first of a pair of papers where we construct and investigate a closed monoidal structure on the category of generalized algebraic theories (in the sense of Cartmell). In the present text, as a starting point, we define the tensor product, $A \otimes B$, between two generalized algebraic theories $A$ and $B$. This is done syntactically via an algorithm that uses the axioms of ....
arXiv:2511.15212v2 Announce Type: replace Abstract: If a complex $X$ is a subcomplex of a diagrammatically reducible 2-complex $Y$ that has locally indicable fundamental group, then $X$ has locally indicable fundamental group. This is a consequence of the Corson-Trace characterization of diagrammatic reducibility. In this paper we use a Corson-Trace like characterization of diagrammatic reducibility away from a subcomplex to obtain a consi..
arXiv:2511.15360v2 Announce Type: replace Abstract: Direct-search algorithms are derivative-free optimization techniques that operate by polling the variable space along specific directions forming positive spanning sets (PSSs). When the problem variables are constrained to lie on a Riemannian manifold, polling must be performed along tangent directions. Although Riemannian variants of direct search have already been proposed and endowed w....
arXiv:2511.15940v2 Announce Type: replace Abstract: Modeling tumor growth accurately is essential for understanding cancer progression and informing treatment strategies. To estimate the parameters in the tumor growth model described by a nonlinear PDE, we adopt Physics-Informed Neural Networks (PINNs) and DeepONet, which show advantages especially when the observation data is scarce and contains noise. With the help of real-life lab data,..
arXiv:2511.16889v2 Announce Type: replace Abstract: In this paper, we study the quantization of classical observables (i.e., functions) on a K\"ahler manifold $X$ as differential operators acting on holomorphic sections of tensor powers $L^{\otimes k}$ of the pre-quantum line bundle $L$. We prove two global results as follows. (1). For a general smooth function $f \in C^\infty(X)$, we construct higher order generalizations of Kostant-Sour....
arXiv:2511.20173v2 Announce Type: replace Abstract: Generalizing work of Marin [12], we construct in a unified way all the "braids and ties'' algebras available in literature and new ones.
arXiv:2512.02342v3 Announce Type: replace Abstract: The stochastic Polyak step size (SPS) has proven to be a promising choice for stochastic gradient descent (SGD), delivering competitive performance relative to state-of-the-art methods on smooth convex and non-convex optimization problems, including deep neural network training. However, extensions of this approach to non-smooth settings remain in their early stages, often relying on inte....
arXiv:2512.04428v2 Announce Type: replace Abstract: We investigate the lifespan of solutions to the higher-order semilinear parabolic equation $$u_t+(-\Delta)^m u=|u|^p, \quad x \in \mathbb{R}^n, t>0 $$ with initial data. We focus on the precise asymptotic behavior of the lifespan of nontrivial solutions. By combining the test function method and semigroup estimates, we derive both upper and lower bounds for the lifespan of solutions $$T_{..
arXiv:2512.05155v2 Announce Type: replace Abstract: Surface holonomy plays a central role in higher gauge theory, bundle gerbes and the geometric formulation of Wess--Zumino terms in string theory. In this work, we consider the relation between surface holonomy and nonabelian multiplicative integration on surfaces. In this framework, we interpret the local Stokes law as a curvature obstruction law for higher holonomy and investigate its co....
arXiv:2512.06425v2 Announce Type: replace Abstract: We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces $C_0(X)$ of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space....
arXiv:2512.08611v2 Announce Type: replace Abstract: Compact Runge-Kutta (cRK) Flux Reconstruction (FR) methods are a variant of RKFR methods for hyperbolic conservation laws with a compact stencil including only immediate neighboring finite elements. We extend cRKFR methods to handle hyperbolic equations with stiff source terms and non-conservative products. To handle stiff source terms, we use IMplicit EXplicit (IMEX) time integration sch....
arXiv:2512.09167v2 Announce Type: replace Abstract: We show that a bounded planar simply connected domain $\Omega$ is a $W^{1,\,1}$-extension domain if and only if for every pair $x,y$ of points in $\Omega^c$ there exists a curve $\gamma \subset \Omega^c$ connecting $x$ and $y$ with $$ \int_\gamma \frac{1}{\chi_{\mathbb R^2\setminus \partial\Omega}(z)}\,ds(z) \le C|x-y|.$$ Consequently, a planar Jordan domain $\Omega$ is a $W^{1,\,1}$-exte..
arXiv:2512.10468v2 Announce Type: replace Abstract: We consider the problem of reconstruction of an $n\times n$ matrix with coefficients depending rationally on $x\in \mathbb P^1$ from the data of: (a) its characteristic polynomial and (b) a line bundle of degree $g+n-1$, with $g$ the geometric genus of the spectral curve, represented by a choice of $g+n+1$ points forming a (non-positive) divisor of the given degree. We thus provide a rec..
arXiv:2512.11598v3 Announce Type: replace Abstract: We present a numerical method for simulating rarefied gases that interact with moving boundaries and rigid bodies. The gas is described by the BGK equation in Lagrangian form and solved using an Arbitrary Lagrangian-Eulerian method, in which grid points move with the local mean velocity of the gas. The main advantage of the moving grid is that the algorithm can deal well with cases where ....
arXiv:2512.12195v5 Announce Type: replace Abstract: Let $\mathcal{G}_k$ denote the gauge group of the principal $G_2$--bundle over $S^4$ classified by $k\in \pi_4(BG_2)\cong \mathbb Z$. Motivated by the $p$--local homotopy classification of these gauge groups, due to Kishimoto--Theriault--Tsutaya and Kameko, we study the low-degree mod~$2$ cohomology of the classifying spaces $B\mathcal{G}_k$ as unstable modules over the Steenrod algebra. ....
arXiv:2512.14912v2 Announce Type: replace Abstract: We develop mathematical programming approximations to tackle the stochastic knapsack problem. In this problem, the decision maker considers items for which either weights or values, or both, are random. The aim is to select a subset of these items to be included into their knapsack. We study both static and dynamic variants of this problem: in the static setting, the decision about which ....
arXiv:2512.18891v3 Announce Type: replace Abstract: We prove that every categorical model of dependent type theory with dependent sums and products, intensional identity types and univalent universes presents via its $\infty$-localisation an elementary $\infty$-topos, that is, a finitely complete, locally cartesian closed $\infty$-category with enough univalent universal morphisms. We also show that elementary $\infty$-toposes have small s..
arXiv:2512.20134v3 Announce Type: replace Abstract: In this paper, we study the cylindricity of $\Bbbk$-forms of singular del Pezzo surfaces obtained by blowing up weighted projective planes $\mathbb{P}(1,1,m)$ over an arbitrary field $\Bbbk$ of characteristic zero. As an application, we obtain vertical cylinders on higher-dimensional fibrations whose generic fibers are such $\Bbbk$-forms.
arXiv:2512.20807v2 Announce Type: replace Abstract: We develop a comparison, base-change, and descent framework for the algebraic $K$-theory of non-commutative $n$-ary $\Gamma$-semirings. Working in the Quillen-exact (and Waldhausen) setting of bi-finite, slot-sensitive $\Gamma$-modules and perfect complexes, we construct functorial maps on $K$-theory induced by extension and restriction of scalars under explicit $\Gamma$-flatness hypothes....
arXiv:2512.24262v2 Announce Type: replace Abstract: We introduce the complete lifted control system associated with a control system on a smooth manifold by replacing each vector field with its complete lift to the tangent bundle. We prove that complete lifted control systems are never controllable on the whole tangent bundle, due to the invariance of the zero section. Motivated by this obstruction and by the invariance of complete lifts u..
arXiv:2601.00672v2 Announce Type: replace Abstract: In this paper, we study the finite element operator network (FEONet), an operator-learning method for parametric problems, originally introduced in J. Y. Lee, S. Ko, and Y. Hong, Finite Element Operator Network for Solving Elliptic-Type Parametric PDEs, SIAM J. Sci. Comput., 47(2), C501-C528, 2025. FEONet realizes the parameter-to-solution map on a finite element space and admits a traini....
arXiv:2601.02323v2 Announce Type: replace Abstract: We study the crossing matrix of a braid and introduce a polynomial invariant for braid systems that is invariant under Hurwitz equivalence. As an application to the study of surface braids and surface links, we also define an invariant that can be used as an indicator of the necessity of Euler fusion or fission between braid systems.
arXiv:2601.06895v3 Announce Type: replace Abstract: This paper derives closed-form expressions for four-parameter Euler sums containing generalized harmonic numbers with rational arguments, expressed in terms of the polylogarithm and Lerch transcendent, and reducible (under admissible parameter choices) to Riemann zeta and Hurwitz zeta values. It further obtains closed forms for related polylogarithmic integrals.
arXiv:2601.09553v3 Announce Type: replace Abstract: We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving $\mathbb{Z}^d$-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for $d\ge 2$ due to Candela and Szegedy) via a measurable selection procedure. We also establish a non-ergodic vertical nilcharacter version of our main result. The non-ergodic version of the Host-Kra-..
arXiv:2601.12818v2 Announce Type: replace Abstract: The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {\em polynomial} weakly metric associati..
arXiv:2601.15576v2 Announce Type: replace Abstract: In this note, we discuss a number of open problems in K-stability theory.
arXiv:2601.18376v3 Announce Type: replace Abstract: We present a nesting-free normal form for the formalism of nested conditions and constraints in the context of finite lattices of subgraphs.
arXiv:2601.19050v3 Announce Type: replace Abstract: We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map ..
Persuasive Privacy - arxiv.org - 1 month ago - eng
arXiv:2601.22945v2 Announce Type: replace Abstract: We propose a novel framework for measuring privacy from a Bayesian game-theoretic perspective. This framework enables the creation of new, purpose-driven privacy definitions that are rigorously justified, while also allowing for the assessment of existing privacy guarantees through game theory. We show that pure and probabilistic differential privacy are special cases of our framework, an..
arXiv:2602.01001v2 Announce Type: replace Abstract: In this article, we are interested in finding rational points on certain superelliptic curves.
arXiv:2602.01460v3 Announce Type: replace Abstract: Policy-gradient methods are widely used in reinforcement learning, yet training often becomes unstable or slows down as learning progresses. We study this phenomenon through the noise-to-signal ratio (NSR) of a policy-gradient estimator, defined as the estimator variance (noise) normalized by the squared norm of the true gradient (signal). Our main result is that, for (i) finite-horizon l....
arXiv:2602.02250v2 Announce Type: replace Abstract: Kullback-Leibler (KL) divergence regularization is widely used in reinforcement learning, but it becomes infinite under support mismatch and can degenerate in low-noise regimes. Using a unified information-geometric framework, we introduce KL analogs by replacing the Fisher-Rao geometry in the dynamical formulation of the KL with transport-based geometries, and derive closed-form expressi....
arXiv:2602.02938v2 Announce Type: replace Abstract: From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve on the boundary of a smooth convex body can be approximated by a sequence of billiard trajectories inside of it. We establish the other direction by proving that, for Riemannian billiard tables (under mild assumptions), there ..
arXiv:2602.03330v2 Announce Type: replace Abstract: We identify a structural extremal principle governing residual \(L^2\)-norms over operator-ordered covariance envelopes. In contrast to the centered setting, where such quantities reduce to trace expressions involving covariance operators, the non-centered framework generates mixed terms that cannot be recovered from covariance ordering alone. We show that the worst-case squared residual..
arXiv:2602.05682v2 Announce Type: replace Abstract: The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. It is known that, for positive pretzel knots, the $\Delta$-unknotting number coincides with the second coefficient of their Conway polynomial. In this paper, we compute the $\Delta$-unknotting number for positive pretzel knots. As a consequence of t..
arXiv:2602.09815v4 Announce Type: replace Abstract: The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces $\mathrm{Ran}_{\leqslant n}(X)$ of the Ran space given by all subsets of $X$ of size at most $n$, and their first homotopy groups. These groups are known to be trivial for $n\geqslant 3$ when $X$ is a path connected ..
arXiv:2602.11051v3 Announce Type: replace Abstract: Consider a discrete-time simple random walk $(X_t)_{t\ge 0}$ on an infinite, connected, locally finite simple graph $G$, and let \[ R_t := |\{X_0,\ldots,X_t\}| \] denote its range. The main result of this revised note is that positive vertex isoperimetry already forces linear expected range, with no bounded-degree assumption: if \[ \iota_V(G) := \inf_{0<|S|<\infty} \frac{|\partial_V S|}{|....
arXiv:2602.13032v2 Announce Type: replace Abstract: We consider (random) strategic interactions in a large population consisting of a variety of players. A rational player chooses actions that maximise certain utility functions, while a behavioural player chooses actions based on preferences such as avoid-the-crowd or follow-the-majority. We specifically study a turn-by-turn dynamic process in which players choose their actions sequentiall....
arXiv:2602.14848v3 Announce Type: replace Abstract: We consider an inverse problem governed by the initial-boundary value problem for the thermo-piezoelectric Kelvin-Voigt dynamical system \[ \left\{ \begin{aligned} \rho(z,t) u_{tt} &= \frac{d}{dz} \big( \Gamma(\Theta) u_{zt} +p_1 u_z +p_2(z,t)\phi_z^0 +p_2(z,t)\chi_z -\beta \Theta \big), \\[1ex] 0 &=-\frac{d}{dz} \big( p_2(z,t)u_z -p_3(z,t)\phi_z^0 -p_3(z,t)\chi_z\big), \\ b(z,t)\Theta_t ....
arXiv:2602.16572v3 Announce Type: replace Abstract: Over an arbitrary commutative ring $R$, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space $\mathbf{Q}(X)$ of quantum cellular automata (QCA) on a given metric space $X$. In most cases of interest, $\pi_0 \mathbf{Q}(X)$ classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences $\..
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