arXiv preprints from January 1, 2026 through September 19, 2026 — 13:16:17 EST
We develop a purely algebraic framework for index-type theorems based on the Rees construction for filtered differential graded algebras (FDGAs). Alongside the classical Rees module we introduce a smooth variant $C^ω_{\mathcal{R}}A$, adapted to analytic arguments, and we study traces and their pointwise and coefficient-wise extensions...
This work is concerned with the flexible part of Onsager theorem for the two-dimensional stochastic incompressible Euler equations. We develop a stochastic and intermittent variant of the Newton--Nash iteration scheme, which incorporates new stochastic pressure, Reynolds stress and intermittency perturbations to formalize the...
We prove a finite-entropy criterion for the entropic conditional central limit theorem. Let $(ξ_i,η_i)_{i\geq 1}$ be independent copies of a pair $(ξ,η)$, and set $W_n=n^{-1/2}\sum_{i=1}^n ξ_i$ and $\boldsymbolη_n=(η_1,\ldots,η_n)$. Under the assumptions that $\mathbb{E}\operatorname{Var}(ξ\midη)<\infty$ and that the conditional law...
We study non-negative solutions of the indefinite sublinear Dirichlet problem $$ \left \{\ \begin{array}{ll} -Δu=λu+m(x)|u|^{α-1}u&\quad\hbox{ in $Ω$},\\ u=0&\quad \hbox{ on $\partial Ω$}, \end{array} \right . $$ where $Ω\subset \mathbb{R}^{N}$ is a smooth bounded domain, $0<α<1$, $m\in L^{\infty }(Ω)$ is a sign-changing or...
We study constructive Kempe factorizations for rational curves on $\operatorname{SO}_4(\mathbb{R})$. Motivated by motion-polynomial factorization and rational matrix curves on real classical groups, we prove that every rational curve of degree $2d$ with $d\ge1$ first factors into $d$ quadratic rational curves and then into a product...
We provide a new instance of singular relative Langlands duality, underlying a Rankin-Selberg integral on $\mathrm{GSO}_6$ due to Bump-Friedberg-Ginzburg. We conclude that this integral represents an essentially self-dual object in the relative Langlands program, and we demonstrate that the Langlands dual automorphic integral computes...
In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodniĭ is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this...
Given two quadratic functions \( f(x) = x^T Ax + 2a^T x + a_0 \) and \( g(x) = x^T Bx + 2b^T x + b_0 ,\) each associated with either the strict inequality ($<0$); non-strict inequality ($\leq 0$); or the equality ($=0$), it is a fundamental question to ask whether or not the joint system has a solution. For homogeneous quadratic...
In this paper, we prove that any two dimensional complete translating soliton in ${\mathbb R}^4$ with finite total curvature is a plane. It is in fact a consequence of our main result.
Integrals of spectral densities are frequently used to summarize spectral characteristics of linear processes. This work studies self-normalization for estimators of such integrals based on sequential periodograms and establishes weak convergence of the corresponding processes. For linear functionals of the spectral density,...
We study higher regularity and its relation to nonresonant behavior for time-periodic solutions of boundary value problems for one-dimensional linear and nonlinear nonautonomous first-order integro-differential strictly hyperbolic systems. The boundary conditions include integral operators and various types of boundary reflections. We...
We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $δ$-shell interaction of strength $τ\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $Σ\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a...
This paper is devoted to developing a framework for stochastic growth models with environmental risk, in which rare but catastrophic shocks interact with capital accumulation and pollution. Building on the Poisson point process formulation studied in arXiv:2511.13568, we extend the model to disasters driven by a marked Hawkes process,...
In resin transfer moulding, complete saturation of the fibre preform is necessary before the resin front reaches the outlet vent(s), to prevent dry-spot formation. In practice, the flow front rarely advances uniformly due to race-tracking effects. We propose a combined estimation and control strategy to address this issue. We use...
We study nonzero-frequency phase-locked motions in asymmetrically coupled Kuramoto networks. Such motions are relative equilibria with fixed phase differences and a nonzero common angular velocity, and we call them dancing equilibria. Their existence requires all coupling sums to have the same nonzero value. We show that neither...
Safety-critical optimization applications, such as real-time power system operation, maintain feasibility at every intermediate step, not merely at convergence. Existing approaches either violate constraints mid-solve (interior-point methods) or enforce feasibility through per-instant quadratic programming subproblems with cubic...
In this paper, we use completed root views to study measure sofic entropy of Poisson actions of locally compact sofic groups. We prove that, for every noncompact locally compact second countable sofic group and every locally compact sofic approximation, each positive-intensity Poisson action has completely positive measure sofic...
This document reports over 15 counterexamples found using GPT Pro over the course of 12 months, for problems in combinatorics, number theory, convexity, analysis, and other areas. The counterexamples concern both (reasonably) well-known problems as well as lesser known instances. The counterexamples are accompanied by the repo...
Let $p$ be an odd prime and let $K/\mathbb{Q}_p$ be a finite unramified extension. From a finite presentation by roots of unity of order prime to $p$, we construct a localized Iwasawa pseudomeasure on $\mathbb{Z}_p^\times$. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on...
Let $J_r^{SU(n)}(K;q)$ denote the reduced $SU(n)$ quantum invariant of a zero-framed knot $K$, colored by the $r$th symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed $n\ge2$ we prove the Chen--Liu--Zhu cyclotomic expansion conjecture: there are unique coefficients...
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a right Auslander-Reiten-Serre (ARS for short) duality $(τ, η)$ in the sense of Iyama, Nakaoka and Palu. We show $(τ, η)$ induces right ARS dualities on the relative theories of $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Under relative structures, we show...
Let $A, T \in M_n(\mathbb{C})$ and let $Δ_A(T) = AT - TA$ denote the inner derivation induced by $A$. We determine when $T$ and $Δ_A(T)$ are nilpotent under the second-order relation $$ Δ_A^2(T) + α\, Δ_A(T) + β\, T = 0, \qquad α, β\in \mathbb{R}, $$ according to the location of the roots of $z^2 + αz + β$. If the roots have...
Fix an integer $p\geq 2$ and $0<s<1$. We construct an ergodic $\times p$-invariant measure $μ$ on $\mathbb T^2$ of dimension $s$, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass $O(w^s)$ to every planar tube of width $w$. The...
The main focuses of the article are limit theorems of Wasserstein projection in the convex order which are useful for inference tasks. The main results rely on the establishment of dual attainment, stability and several useful observations. The first is a clean criterion to invoke the Wasserstein projection dualities on the classical...
We prove a purely classical, coding-theoretic characterization of Hausdorff and packing dimensions in \(\mathbb R^n\). A \emph{ball code} assigns names to closed balls of \(\mathbb R^n\): it is a partial map from a prefix-free set of finite binary strings to balls. For every nonempty \(E\subseteq\mathbb R^n\), the Hausdorff dimension...