arXiv preprints from January 1, 2026 through September 7, 2026 — 13:40:07 EST
We prove a bound on the heat trace of the Neumann Laplacian on a convex domain that captures the first two terms in its small-time expansion, but is valid for all times and depends on the underlying domain only through very simple geometric characteristics. This is proved via a precise and uniform expansion of the on-diagonal heat...
Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $δ$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge number $|E(F)| > \frac{d-1}{d}(|V(G)| - 1)$, such that if $F$ is not a spanning tree, then $F$ has a...
Over the split-octonion algebra defined over an arbitrary field, we solve all polynomial equations whose coefficients are scalar except for the constant term. As an application, we determine the square and cubic roots of an octonion.
This paper studies AdamW-style Shampoo, an effective variant of the classical Shampoo that won the external tuning track of the AlgoPerf neural network training competition. Our analysis unifies one-sided and two-sided preconditioning. When the exponents of the two preconditioners sum to $1/2$, we establish the convergence rate...
We construct a unified framework of geometrodynamics based on the Finsler geometry to reveal the relationship between spacetime and dynamics.The Lagrangian of electron in electromagnetic field as the Finsler function gives the Finslerian metric, which modifies spacetime metric in the Finsler-Randers space. The geodesic equation gives...
This paper studies irregularity-type invariants of special C-pairs, or "geometric orbifolds" in the sense of Campana. Under mild assumptions on the singularities, we show that the augmented irregularity of a C-pair (X,D) is bounded by its dimension. This generalizes earlier results of Campana, and strengthens known results even in the...
Let $A,B\in\mathbb{H}_n$ and set $H=A+B$. For each integer $k\ge 1$ define $$ Q_k:=\sum_{p=0}^k \binom{k}{p} A^pB^{k-p}, R_k:=\Re\,Q_k=\frac{Q_k+Q_k^*}{2}. $$ Then $H^k=\left.\frac{d^k}{dt^k}e^{Ht}\right|_{t=0}$ and $Q_k=\left.\frac{d^k}{dt^k}(e^{At}e^{Bt})\right|_{t=0}$. We prove that, for $k=3,4,$ $$ λ(H^k)\prec_w...
The paper is devoted to the study of conditions for the Hausdorff-Besicovitch faithfulness of the family of cylinders generated by Cantor series expansions. We show that there exist subgeometric Cantor series expansions for which the corresponding families of cylinders are not faithful for the Hausdorff-Besicovitch dimension on the...
Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of social choice theory, wherein intuitive and seemingly desirable constraints on decision-making necessarily lead to contradictory...
Probabilistic Manifold Decomposition (PMD)\cite{doi:10.1137/25M1738863}, developed in our earlier work, provides a nonlinear model reduction by embedding high-dimensional dynamics onto low-dimensional probabilistic manifolds. The PMD has demonstrated strong performance for time-dependent systems. However, its formulation is for...
We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary invariant Riemannian metric on the space, every geodesic is an orbit of a 1-parameter subgroup of...
TriCG is a short-recurrence iterative method recently introduced by Montoison and Orban [SIAM J. Sci. Comput., 43 (2021), pp. A2502--A2525] for solving symmetric quasi-definite (SQD) linear systems. TriCG takes advantage of the inherent block structure of SQD linear systems and performs substantially better than SYMMLQ. However,...
We report partial progress on the weak coupling limit behavior of observables for the periodic quantum Lorentz gas. Our results indicate that for certain observables, the limit behavior is trivial and can be described via a transport equation, while for other observables, the existence of the limit hinges on the regularity properties...
In control theory, a system which has output depending only on the present and past values of the input is said to be causal (or nonanticipative). Respectively, a system is acausal (or non-causal) if its output depends on future inputs as well. Overall majority of literature in stochastic control theory discusses causal systems. Only...
We establish sufficient conditions for positive (semi-)definiteness, with or without radial unboundedness, for nonquadratic Lyapunov function constructed as sign-indefinite quadratic forms involving the state and the deadzone of a suitable input. We then use these conditions to build weak nonquadratic Lyapunov functions establishing...
This paper provides an extensive study of the $\mathscr{I}$-Miller null ideals $M_\mathscr{I}$, $σ$-ideals on the Baire space parametrized by ideals $\mathscr{I}$ on countable sets. These $σ$-ideals are associated to the idealized versions of Miller forcing in the same way that the meager ideal is associated to Cohen forcing. We...
In this paper, we investigate the log-concavity of the kernel for the parabolic Ornstein-Uhlenbeck operator in a bounded, convex domain. Consequently, we get the preservation of the log-concavity of the initial datum by the related flow. As an application, we give another proof of a Brunn-Minkowski type inequality for the first...
We provide a writeup of a resolution of Erdős Problem #728; this is the first Erdős problem (a problem proposed by Paul Erdős which has been collected in the Erdős Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by OpenAI and Aristotle by Harmonic,...
We study the critical points of the solution of second elliptic equations in divergence and diagonal form with a bounded and positive definite coefficient, under the assumption that the statement of the Hopf lemma holds (sign assumptions on its normal derivatives) along the boundary. The proof combines the argument principle...
We study the Calogero--Moser derivative nonlinear Schrödinger equation (CM-DNLS), a mass-critical and completely integrable dispersive model. Recent works established finite-time blow-up constructions and soliton resolution, describing the asymptotic behaviors of blow-up solutions. In this paper, we go beyond soliton resolution and...
A cograph is a graph that contains no induced path $P_4$ on four vertices or equivalently a graph that can be constructed from vertices by sum and product operations. We study the bipartite Turán problem restricted to cographs: for fixed integers $s \leq t$, what is the maximum number of edges in an $n$-vertex cograph that does not...
In this study, we develop and analyze a deterministic two-strain host-vector model for dengue transmission that incorporates key immuno-epidemiological mechanisms, including temporary cross-immunity, antibody-dependent enhancement (ADE), disease-induced mortality during secondary infections, and explicit vector co-infection. The human...
In this work, we propose a novel layerwise adaptive construction method for neural network architectures. Our approach is based on a goal--oriented dual-weighted residual technique for the optimal control of neural differential equations. This leads to an ordinary differential equation constrained optimization problem with controls...
We prove the uniform solvability of a stationary problem associated to internal waves equation with small viscosity in a two dimensional aquarium with real-analytic boundary, under a Morse--Smale dynamical assumption. This is achieved by using complex deformations of the aquarium, on which the inviscid stationary internal wave...
In this paper we prove a coherent version of geometric Satake equivalence proposed in Cautis-Williams' work arXiv:2306.03023 for type A. In their work, they studied an abelian version of the classical limit Satake category, namely, the Koszul perverse heart of the categorified Coulomb branch for adjoint representations. In this paper...