arXiv preprints from January 1, 2026 through September 7, 2026 — 08:11:09 EST
We prove lossless Strichartz estimates at the critical exponent $q_c = \frac{2(n+1)}{n-1}$ and the endpoint exponent pair $\left(2,\frac{2(n-1)}{n-3}\right)$ for the Schrödinger equation on rectangular tori of dimension $n-1$ with frequency localized initial data on small time windows with length depending on the frequency parameter $λ\gg 1$.
In this paper, we establish diameter bounds for compact Kähler manifolds equipped with Kähler metrics $ω$, assuming the associated measure lies in a specific Orlicz space and satisfies an integrability condition. Firstly, we prove a priori estimates for solutions of the complex Monge-Ampère equation in Orlicz spaces, encompassing...
We give a combinatorial characterization of the group of quasiconformal homeomorphisms of a closed, oriented surface $S$ of genus at least $2$. In particular, we prove they are exactly the automorphisms of a graph of essential quasicircles on $S$ that respect a canonical coarse ordering induced by quality constants. We also discuss...
For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show a generic multiplicity one theorem in equivariant min-max theory, and show in generic sense that there are infinitely many $G$-invariant minimal hypersurfaces in a fixed $G$-homology class. We also establish an equivariant min-max theory...
The α-stable Lévy process, commonly used to describe Lévy flight, is characterized by discontinuous jumps and is widely used to model anomalous transport phenomena. In this study, we investigate the associated exit problem and propose a method to compute the exit time probability, which quantifies the likelihood that a trajectory...
We study equilibrium selection for invariant measures of stochastic dynamical systems with constant step size, under persistent noise and minimal moment assumptions, in a general quasi-Feller framework. Such dynamics arise in projection-based algorithms, learning in games, and systems with discontinuous decision rules, where classical...
We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $Γ< G= G_1 \times \dots \times G_d$ in higher rank semisimple algebraic groups and a trace-preserving irreducible action $G \curvearrowright (\mathcal{N}, τ)$, we show that every...
We derive a generalized Pohozhaev's identity for radial solutions of $p$-Laplace equations, by using the approach in [5], thus extending the work of H. Brézis and L. Nirenberg [2], where this identity was implicitly used for the Laplace equation.
We prove the so-called second case of the fractional Korn inequality for uniform domains. We obtain this result as an application of a novel fractional Korn-type inequality formulated in terms of truncated seminorms, which turns out to be valid for the broader class of John domains. We also obtain weighted estimates in which the...
Let $A$ be an associative algebra with a superinvolution $*$ over a field of characteristic zero, and let $c_n^*(A)$, $n = 1, 2, \ldots$, denote its sequence of $*$-codimensions. It is well known that this sequence is either polynomially bounded or grows exponentially. In the polynomial case, a central problem in PI-theory is the...
We construct a natural chain map from the Kontsevich graph complex to the rational singular chain complex of $B\mathrm{Diff}_\partial(D^{2k})$ when the dimension $2k$ is sufficiently large, generalizing Goussarov and Habiro's theories of surgery on 3-valent graphs in 3-manifolds. Our construction can be considered as a topological...
The McKay correspondence establishes a bijection between the cohomology of a minimal resolution and the irreducible representations of a finite subgroup $Γ\subset \text{SU}(2)$. While traditional proofs rely on static algebraic isomorphisms, we propose a dynamical framework grounded in gauge theory and Morse-Bott theory. We analyze an...
This paper presents a rigorous numerical framework for computing multiple solutions of semilinear elliptic problems by spatiotemporal high-index saddle dynamics (HiSD), which extends the traditional HiSD to the continuous-in-space setting, explicitly incorporating spatial differential operators. To enforce the Stiefel manifold...
Non-asymptotic central limit theorem (CLT) rates play a central role in modern machine learning and operations research. In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($W_p$) distance, for general $p\ge 1$. We focus on two fundamental dependence structures that commonly arise in practice: locally...
The twisted product of functions on $R^{2N}$ is extended to a $*$-algebra of tempered distributions which contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are...
In this paper we establish an asymptotic formula for the signless Laplacian spectral radius of a $2$-dimensional simplicial complex with given $2$-th Betti number. Furthermore, we characterize the $2$-dimensional simplicial complex that achieves the maximum signless Laplacian spectral radius among all-dimensional simplicial complex...
Pseudo-cones serve as the noncompact counterpart of convex bodies in convex geometry. This paper establishes a necessary and sufficient condition for the existence of solutions to the Orlicz-Gauss image problem for pseudo-cones and further demonstrates its connection to spherical optimal transport. Our approach combines the...
We introduce tropical Kummer quartic surfaces in tropical projective $3$-space as the images of certain principally polarized tropical abelian surfaces under tropical theta functions of second order. We study some of their properties, showing that they are included in the tropicalizations of Kummer quartic surfaces defined over...
We establish the global existence of weak entropy solutions for 1D isentropic gas dynamics with general pressure laws ($γ> 1$). To address vacuum degeneracy, we introduce a novel structural regularization via a "Synchronized Dual Translation" strategy. This approach offers a distinct advantage over the flux-modification method of Lu...
The paper studies a general norm minimization problem on a product of normed vector spaces. We establish dual necessary and sufficient optimality conditions and derive explicit formulas for the corresponding solution sets. These formulas are obtained under the assumption that one optimal solution together with its associated dual...
Flow cones of a directed acyclic graph admit a family of unimodular triangulations given by Danilov, Karzanov, and Koshevoy (DKK) whose normal fans are related to (generalizations) of the associahedron and permutahedron. A correspondence between these triangulations for certain graphs and maximal cones of a $g$-vector fan of a gentle...
The Birkhoff Ergodic Theorem establishes pointwise convergence for integrable observables, but for $f\notin L^1$, no normalization yields almost sure convergence. This paper investigates trimmed ergodic sums, where the largest observations are removed, for observables with polynomial tails $¶(f>t)\asymp t^{-1/α}$ in exponentially...
In this paper, we prove Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in both Euclidean and Minkowski context. A supplementary example illustrates that zero Gaussian convex spacelike hypersurfaces are not necessary hyperplanes without additional conditions. We show that a zero Gaussian...
We give a short, self-contained analytic proof of the existence of self-similar Dirichlet forms on pillow-type carpets, a family of infinitely ramified fractals that includes the Sierpiński carpet.
We systematically compute improved asymptotic rank bounds for tensors. Using numerical implicitization, we implement the geometric framework of Kaski and Michałek across all computationally feasible cases. By detecting the absence of low-degree vanishing polynomials on secant varieties, we obtain new asymptotic rank bounds that...