arXiv preprints from January 1, 2026 through September 5, 2026 — 04:19:01 EST
Fushchych and Shtelen defined approximate symmetry by expanding the solution of a perturbed differential equation in the small parameter, replacing the original equation by a triangular system for the expansion coefficients. We extend this construction to arbitrary order. A coefficient-map formulation yields the order-$p$...
We establish finite-energy Cauchy stability for vacuum and matter spacetimes with $T^2$ symmetry on $T^3$, under general constitutive equations satisfying hyperbolicity and mild asymptotic conditions at vacuum and large mass-energy density. First, using the area of the symmetry orbits as a time function, we introduce the JKL...
We prove a Michael-Simon inequality for $2$-varifolds in $\mathbb{R}^N$, in arbitrary codimension, for anisotropies sufficiently close to the area functional. Building on the ideas of Almgren and De Philippis-Pigati, we introduce a projection method for anisotropic stress measures yielding geometric inequalities that apply in...
We prove a Michael-Simon inequality for varifolds of every dimension and codimension with respect to anisotropic energies. Combined with Allard's anisotropic regularity theorem, this result yields the regularity of hypersurfaces with bounded anisotropic mean curvature in every dimension. Using the results of De Philippis and Pigati,...
We prove that deciding whether an ideal in a polynomial ring contains a trinomial is impossible on a Turing machine. More precisely, from an integer polynomial $P$ we compute generators of an ideal $I_P$ in a polynomial ring over $\mathbb{Q}$ such that $I_P$ contains a trinomial if and only if $P$ has an integral zero. By the MRDP...
We prove interior Hölder regularity for solutions of the stationary kinetic Fokker-Planck equation in non-divergence form in space dimension one. The estimate is independent of any small-oscillation condition on the diffusion coefficient.
Let $R$ be a ring that is essentially of finite type over a field and $I\subseteq R$ be an ideal. In this article it is shown that the minimal number of generators of the symbolic powers $I^{(n)}$ is bounded above by a polynomial in $n$. Furthermore, if $R$ is assumed to be a domain and $\mathcal I=\{I_n\}_{n\ge 0}$ is an...
We complete the classification of non-Kähler steady pluriclosed soliton on compact complex surfaces, as initiated by Streets. The underlying complex structure must be a Hopf surface, with any finite primary cover being of class 1. Moreover, the soliton metric is unique up to biholomorphism.
For an integer $t\geq-1$, let $θ_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and let $R_t=\mathbb{Z}[θ_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(θ_t)}$. We prove that if the additive order index $q=[\mathcal{O}_t:R_t]$ is prime, then $[\mathcal{O}_t^\times:R_t^\times]=q$ exactly for $(t,q)=(3,3)$ and $(5,7)$, and...
Hyperanalytic functions, in the sense established by the mathematician Avron Douglis, are Douglis algebra-valued functions defined via a hypercomplex structure rather than the standard Cauchy-Riemann equations characteristic of traditional complex analysis. The classes of polyhyperanalytic and meta-hyperanalytic functions represent...
In a remarkable work published in Analysis and PDE 8 (2015) 1541-1563, M. Fazly, J. Wei, and X. Xu establish the following pointwise estimate \[-Δu \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac 2{p+1- \frac 8{n(n-4)}}} |x|^{\frac σ2} u^\frac{p+1}2 \quad \text{in } \mathbf R^n\] for any bounded, positive, $C^4$-solution $u$ to...
We solve the Nielsen realisation problem for high-dimensional aspherical manifolds in a new class of special cases. Mainly, we focus on actions of cyclic groups of prime order. A novelty is that it gives topological solutions to the problem with certain high-dimensional fixed point sets, whereas previous solutions of this type were...
The problem of global minimization of nonconvex quadratic functions subject to box constraints is studied. Applying Gershgorin theorem we reduce the main matrix A to its diagonal form which is used to obtain difference of convex representation of the quadratic function. Then ε-subdifferentials of DC components are studied and used to...
For every $1\le d<n$, we prove joint equidistribution, as the height tends to infinity, of $d$-dimensional $K$-subspaces in a rigid adelic space over a number field $K$, together with their archimedean Grassmannian images and the $K$-linear isometry classes of the normalized subspace and quotient. We identify the leading constant,...
Let $ζ(G)$ denote the minimum number of parts in a partition of $V(G)$ in which every part induces either a clique or an independent set. Erdős and Gimbel asked whether, for $G_n\sim G(n,1/2)$, the difference $χ(G_n)-ζ(G_n)$ tends to infinity with high probability. We resolve this problem along the full sequence $n\to\infty$ and prove...
We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-Δ)$ of two-dimensional Schrödinger operators $H=-Δ+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave...
The independence complex and the perfect matching complex of a graph are simplicial complexes encoding, respectively, its independent sets and perfect matchings. Determining the homotopy types and other topological properties of these complexes is, in general, a difficult problem, and explicit descriptions are known only for...
Let $\mathcal{A}$ be the normalized class of analytic functions $f$ in the unit disc $\mathbb{D} := \{z \in \mathbb{C} : \vert{}z\vert{} < 1\}$. For $β> 1$, let $\mathcal{N}(β)$ denote the subclass of $\mathcal{A}$ satisfying $\text{Re}\{1 + z f''(z)/f'(z)\} < β$ for $z \in \mathbb{D}$. The main purpose of this paper is to establish...
The concept of an $E$-frame, recently introduced in frame theory, is obtained by applying an infinite invertible complex matrix to a sequence of elements of a Hilbert space $\mathcal{H}$. Here, $E$ is considered as a matrix mapping on the sequence space $\bigoplus_{n=1}^{\infty}\mathcal{H}$. A natural question is to determine the...
Vector-valued multiscale representations are essential when signals or fields take values in $\mathbb{R}^m$ and component interactions carry meaningful information. Most multiwavelet and super-wavelet constructions are formulated in scalar Hilbert-space settings and typically produce channelwise scalar coefficients followed by...
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...