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arXiv:2604.25106v3 Announce Type: replace Abstract: We develop a spectral theory for continuous- and discrete-time Blahut--Arimoto (BA) dynamics, centered on the relaxation kernel $ \G = \E_p[K^*_X \otimes K^*_X] $. Five main results are established. (i) Along the continuous-time BA flow, the free energy satisfies the exact $ \chi^2 $-dissipation identity $ \dot F_\beta = -\D(q) $, where $ \D(q)=\chi^2(\T q \| q) $ is the Pearson $ \chi^2 ....
arXiv:2604.25228v2 Announce Type: replace Abstract: For any Tychonoff space $X$ let $D(X)$ be either the set $C(X)$ of all continuous functions on $X$ or the set $C^*(X)$ of all bounded continuous functions on $X$. When $D(X)$ is endowed with the point convergence topology, we write $D_p(X)$. Zakrzewski \cite[Theorem 3.12]{kz} proved that if $X$ and $Y$ are $\sigma$-compact spaces and there is a continuous linear map $T:C_p(X)\to C_p(Y)$ s....
arXiv:2604.25382v2 Announce Type: replace Abstract: We study the class of selfless C*-probability spaces introduced by Robert. It is known that a selfless tracial algebra has strict comparison and a unique trace. We prove that for separable tracial C*-algebras, selflessness is equivalent to approximate selflessness, a finitary condition: for every finite set $F$, every $N \geq 1$ and $\varepsilon > 0$ there exists a unitary $u$ with $|\tau..
arXiv:2604.25709v2 Announce Type: replace Abstract: We give a criterion for certain generic nondegenerate surfaces in a fake weighted projective $3$-space to have Picard number $>1$. These algebraic surfaces are of general type. We do this by considering degenerations (along an edge), keeping track of the geometric genus, and using vanishing cohomology classes to construct a rational Picard class on the surface not proportional to the cano..
arXiv:2604.26249v2 Announce Type: replace Abstract: Submersions with definite folds are submersions on manifolds with boundary whose restrictions to the boundary are definite fold maps. In this paper, we study differential-topological properties of manifolds with boundary admitting such maps into Euclidean spaces. When the target is $\mathbb{R}$, we obtain restrictions on the diffeomorphism types of the source manifolds by using results fo..
arXiv:2605.00550v2 Announce Type: replace Abstract: In this article, we investigate the asymptotic profile of solutions to the Cauchy problem for a nonlinear beam equation with two variable coefficients in the subcritical nonlinear case. In contrast to our previous result [6], in which the asymptotic profile is governed by the linear heat kernel and the nonlinear effect is asymptotically negligible, the asymptotic profile in the present se..
arXiv:2605.03097v2 Announce Type: replace Abstract: We prove a full support theorem for the relative good moduli space of the universal compactified Jacobian $\bar{\pi}\colon \overline{J}_{g,n}^{d,\phi}\to \overline{\mathcal{M}}_{g,n}$, showing that every direct summand appearing in the BBDG decomposition of $\mathrm{R}\bar{\pi}_*\mathrm{IC}(\overline{J}_{g,n}^{d,\phi})$ has full support on the base $\overline{\mathcal{M}}_{g,n}$. Moreover..
arXiv:2605.03284v2 Announce Type: replace Abstract: A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we study the set $\Delta(G)$ of conjugacy classes of nontrivi....
arXiv:2605.03734v2 Announce Type: replace Abstract: We study the existence of local and global strong solutions for the stochastic tamed Navier--Stokes equations on the whole space $\mathbb{R}^3$, driven by multiplicative Wiener noise and compensated L\'evy jump noise. For $p > 3$, we first prove the existence of a pathwise unique maximal local $L^p$-strong solution for divergence-free, $\mathcal{F}_0$-measurable initial data in $L^p(\Omeg..
arXiv:2605.03781v4 Announce Type: replace Abstract: Using standard-normal critical-value calibration (SNC) to construct a kernel-smoother-based confidence interval faces a fundamental challenge: the normalization makes a small estimation bias become a non-negligible inferential bias. This paper takes a different route by replacing the SNC control with empirical Bernstein tail control. The resulting confidence intervals control stochastic v....
arXiv:2605.03940v4 Announce Type: replace Abstract: We describe a dynamical system in which a symbolic field is coupled to a geometric field via a bipartite Hilbert-Schmidt kernel. The system is fully described by a retarded functional differential equation (RFDE) on the history space, subject to Lipschitz and small gain conditions. We show that the RFDE is well-posed under constant input and that it admits a compact global attractor. The ..
arXiv:2605.05442v2 Announce Type: replace Abstract: On metric measure spaces with sub-Gaussian heat kernel behavior in small time, we obtain a sufficient condition to solve Wick renormalized stochastic quantization equations with polynomial interaction. Given the power of the nonlinearity, the local solution condition depends on the Hausdorff dimension $d_h$, the walk dimension $d_w$, and the maximal spatial H\"older regularity of the heat....
arXiv:2605.08091v2 Announce Type: replace Abstract: The analysis of a total least square problem (TLS) can be reduced to that of an associated core problem, which typically has lower dimension and improved solubility properties. Nevertheless, even a core problem may remain reducible, admitting further decomposition into irreducible component subproblems with simpler structure and better analytical properties. However, no systematic and inv....
arXiv:2605.09683v3 Announce Type: replace Abstract: For words in the variables $X$ and $Y$ satisfying the commutation relation of the $q$-deformed generalized Ore algebra, $XY-qYX= \mu I + \nu Y$, we show that the corresponding normal ordering coefficients can be given an interpretation in terms of mixed placements of rooks and files. In particular, the associated $q$-deformed Ore-Stirling and Ore-Lah numbers are treated in detail. We show....
arXiv:2605.10282v2 Announce Type: replace Abstract: This paper addresses the problem of universal learning under model misspecification with log-loss. In this setting, the learner operates with a hypothesis class of models denoted by $\Theta$, while the true data-generating process belongs to a broader class $\Phi \supset \Theta$, and may lie outside the assumed hypothesis space. Classical approaches have characterized the minimax regret a....
arXiv:2605.10677v2 Announce Type: replace Abstract: Generalized Euler numbers have previously been studied mainly from a combinatorial viewpoint. The purpose of this paper is to explore them from $p$-adic and analytic perspectives. To this end, we introduce congruential Euler numbers, a new family extending generalized Euler numbers. We establish several $p$-adic congruences for these numbers, including an answer to a conjecture related to..
arXiv:2605.11833v2 Announce Type: replace Abstract: Each self-similar dendrite K with a finite self-similar boundary defines a finite acyclic edge-labeled bipartite graph G, called the sprout of K. The paper shows that the sprout G determines the combinatorial properties of the dendrite K and its topological structure.
arXiv:2605.11988v3 Announce Type: replace Abstract: This paper proposes a conjectural framework for Alperin's Main Problem of Block Theory from 1976. The character sets considered here are defined by nonvanishing at given elements, not only by degree conditions. From this point of view, McKay's conjecture is usually recovered as a first degree-level consequence. The guiding idea is that the right local objects governing character values ar..
arXiv:2605.12880v3 Announce Type: replace Abstract: Ribbon decomposition matrices give determinantal formulas for skew Schur functions that include as special cases the classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz formulas. We prove that certain elements of Lusztig's dual canonical basis, called Temperley-Lieb immanants, are Schur-positive when evaluated on ribbon decomposition matrices. We conjecture that this positivity holds f..
arXiv:2605.13562v5 Announce Type: replace Abstract: Let $\Sigma_a\subset B^3(r(a))\subset\mathbb{H}^3$ ($a>1/2$) be the critical hyperbolic catenoid of the Mori family, a free boundary minimal surface in the geodesic ball. The Medvedev conjecture [15] states ind$(\Sigma_a)=4$ for all $a>1/2$. We study its strong form: ind$(\Sigma_a)=4$ and nul$(\Sigma_a)=2$. The nullity condition nul$(\Sigma_a)=2$ combines the mode-$|k|=1$ result $\text{nu....
arXiv:2605.13582v2 Announce Type: replace Abstract: We give a trajectory-based proof of transfer-of-regularity estimates \`a la Bouchut-H\"ormander for kinetic equations at the weak scale of local diffusion. The proof is based on kinetic mollification along critical kinetic trajectories, whose endpoint map has precisely the kinetic scaling. Instead of using Fourier multipliers or the Kolmogorov fundamental solution, we work directly in phy..
arXiv:2605.14661v2 Announce Type: replace Abstract: Fluid antenna is a new reconfigurable antenna technology that can dynamically adjust the positions or ports of radiating elements and therefore provides a new degree of freedom for wireless communications. However, the associated port selection is a challenging large-scale combinatorial optimization problem and difficult to solve. Existing manually designed heuristic algorithms are not on....
arXiv:2605.16121v2 Announce Type: replace Abstract: We investigate involutive, non-combinatorial solutions of the braid equation, viewing them as special deformations of the permutation map. Utilizing these solutions, we identify the associated quantum algebra and introduce it as the $\mathfrak{gl}_{k,m}$ Yangian. This newly derived Yangian is distinct from the known Yangian of the general linear Lie superalgebra; crucially, as a Hopf alge....
arXiv:2605.16564v2 Announce Type: replace Abstract: Discontinuous coefficient fields arise in many computational physics problems and are often represented as cellwise constant data tied to a given spatial discretization. Such representations are inherently mesh-dependent, requiring interpolation or projection whenever they are transferred to a different discretization. In this work, we develop \emph{Parallel and Adaptive Mesh-Free Approxi....
arXiv:2605.17122v4 Announce Type: replace Abstract: We generalize the fundamental bounds of Delsarte thesis (1973) on codes of given degree and designs of given strength in the new setting of Bannai et al. (2025). We assume the scheme is weakly metric in the sense of (Sol\'e, 1989). We give upper bounds on the size of codes of given degree, and also on the size of codes with a given number of pairwise distances. Codes meeting these bounds ....
arXiv:2605.17218v2 Announce Type: replace Abstract: In this paper, we prove that there exists an absolute constant $g_0$ such that, for every integer $k\ge 3$, every graph $G$ with $\delta(G)\ge k$ and $g(G)\ge g_0$ contains an induced subdivision of $K_{k+1}$. This fully resolves a problem raised by K\"{u}hn and Osthus (originally attributed to Shi), and improves a recent result of Gir\~{a}o and Hunter. Our proof uses some ideas from Gir\..
arXiv:2605.17674v3 Announce Type: replace Abstract: The first purpose of this paper is to give the fnite transcendence of Frobenius traces for elliptic curves over $\mathbb{Q}$ without the assumption of complex multiplication (CM). This result generalizes the previous work by Luca and Zudilin, who obtained similar transcendence results specifically for the CM case. The second purpose is to give the finite transcendence of Frobenius traces ..
arXiv:2605.18426v2 Announce Type: replace Abstract: Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely irreducible, and de Rham of regular parallel weights. As an application, we prove some new cases of the Langlands-Clozel-Fontaine-Mazur conjecture of $\mathrm{GL}..
arXiv:2605.18528v2 Announce Type: replace Abstract: A growing lesson from neural network optimization is that optimizer design should respect how the model is parametrized. Scale-invariant methods become important because their normalized layerwise updates can not only support hyperparameter transfer across model sizes but exploit input-output matrix norm geometry. At the same time, stochastic gradient noises in deep learning are often far....
arXiv:2605.18694v2 Announce Type: replace Abstract: Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process. To manage this realistic and challenging setting, new mechanisms, such as gradient clipping and gradient normalization, have been introduced to ensure the convergence of first-order algorithms. However, adaptive gradient methods, a famous class of modern optimizers th..
arXiv:2605.18924v2 Announce Type: replace Abstract: We prove, and mechanize in Rocq, an abstract obstruction theorem for closure predicates $C : \mathsf{Form} \to \mathsf{Prop}$ over the closed implication-falsity fragment $A,B ::= \bot \mid A \to B$. Evaluation completeness, $\mathsf{Eval}(C)$, says that every formula-valued behavior of codes is represented up to closure equivalence, where $A \simeq_C B$ abbreviates $C(A \to B) \land C(B ....
arXiv:2605.19097v2 Announce Type: replace Abstract: In this paper, we discuss some topological properties of the graph-directed iterated function system (GDIFS) of injective contractions. Further, we establish the existence of a Lipschitz bijection between two different bi-Lipschitz graph-directed attractors. In general, two different bi-Lipschitz graph-directed attractors defined on the same graph need not be bi-Lipschitz equivalent. Howe..
arXiv:2605.19709v2 Announce Type: replace Abstract: A long-standing assumption in the literature on switched linear systems is that static, homogeneous of degree one feedbacks form the most general class of controllers necessary and sufficient for stabilization. In this paper, we provide a rigorous justification. More specifically, we prove by construction that if a switched linear system admits a stabilizing full-information controller, w..
arXiv:2605.19896v2 Announce Type: replace Abstract: Monitoring the integrity of elastic structures using ultrasonic waves requires the efficient identification of material parameters from measured surface displacements. The displacement field is governed by Cauchy's equation of motion, i.e., an elastic wave equation. Consequently, defect localization leads to a high-dimensional spatial parameter identification problem for a hyperbolic syst....
arXiv:2605.19914v2 Announce Type: replace Abstract: This paper develops a time-inconsistent and path-dependent singular control framework incorporating a running minimum process. We derive a verification theorem that characterizes equilibria under substantially weaker regularity conditions than those imposed in the existing literature, and we obtain a stronger notion of equilibrium by enlarging the class of feasible perturbations. We first....
arXiv:2605.20039v2 Announce Type: replace Abstract: We give a complete constructive description of all finite dimensional nilpotent Lie algebras of smooth vector fields in three variables, including intransitive algebras. The description is organized by the rank and dimension of the center, which serves as the key invariant. Since every nonabelian solvable algebra lives in the normalizer of a nilpotent algebra, our normal forms provide the..
arXiv:2605.20629v3 Announce Type: replace Abstract: In recent work (Forum Math. Sigma, 2024), we established a correspondence between MAT-labeled graphs arising from hyperplane arrangements and regular vines from probability theory. In this paper, we extend this connection to Arrow's single-peaked domains in social choice theory. We show that MAT-labeled complete graphs, regular vines, and maximal Arrow's single-peaked domains arise from t....
arXiv:2605.20860v2 Announce Type: replace Abstract: Let $K$ be a totally real number field of odd degree in which $2$ is inert. Let $l \geq 5$ be a prime with $l \nmid [K:\mathbb{Q}]$ and $\gcd(\frac{l-1}{2}, [K:\mathbb{Q}])=1$. We prove that if $2$ is inert in $K$, $l$ is non-Wieferich, i.e., $2^{l-1} \not\equiv 1 \pmod{l^2}$, and $l$ is totally ramified in $K$, then the asymptotic Fermat's Last Theorem holds over each $n$-th layer $K_{n,....
arXiv:2605.20945v3 Announce Type: replace Abstract: A group is self-simulable if all its computable actions admit SFT covers, which means roughly that they can be implemented with finitely many tiling constraints. We prove that a graph product of infinite finitely-generated groups is self-simulable if and only if its defining graph has no disconnecting clique consisting of amenable groups. In particular, a right-angled Artin group (a.k.a.\..
arXiv:2605.21450v3 Announce Type: replace Abstract: Holland-Simpson thin-wire finite-difference time-domain (FDTD) simulations of obliquely oriented closed-loop antennas exhibit persistent low-frequency parasitic currents because the current-deposition operator fails to conserve charge. This deposition operator, together with an interpolation operator that samples the tangential electric field along the wire, can be realized as regularizat....
arXiv:2605.21769v3 Announce Type: replace Abstract: We investigate the finite-time blow-up of solutions to a Tricomi-type equation with scale-invariant potential and power nonlinearities in the oscillatory regime. For smooth, compactly supported, nonnegative initial data, we prove nonexistence of global-in-time solutions when the power nonlinearity lies below the positive root of an explicit Strauss-type polynomial naturally associated wit....
arXiv:2605.22652v3 Announce Type: replace Abstract: We study inequalities between integer-valued knot invariants arising from classical knot theory, four-dimensional topology, knot homologies, and knot polynomials. We present a directed graph consisting of 46 inequalities between 33 knot invariants. Using these inequalities together with parity constraints, we construct and propagate a database NewDB, for knots up to 13 crossings, extendin..
arXiv:2605.22745v2 Announce Type: replace Abstract: We develop a first and second fundamental theorem for $n$--tuples of bosonic and fermionic matrices, by developing graded analogues of the classical case.
arXiv:2605.24148v2 Announce Type: replace Abstract: We address a new case of the Harris-Viehmann conjecture, which establishes a parabolic induction formula on the cohomology groups associated to non-basic local Shimura data. It follows that all supercuspidal representations on a Shimura variety are concentrated along the basic locus, making the conjecture relevant to the Langlands program. Historically, many cases of the Harris-Viehmann c..
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