arXiv preprints from January 1, 2026 through July 20, 2026 — 00:35:14 EST
A family $\mathcal{F}$ of subsets of $[n] := \{1,2,\ldots, n\}$ is called maximal $k$-wise intersecting if every collection of at most $k$ members of $\mathcal{F}$ has a non-empty intersection, and adding any other set to $\mathcal{F}$ breaks this property. An old question by Erdős and Kleitman from 1974 asks for the minimum size of a...
Given $n,m\in\mathbb{N}$ such that $n\ge2m\ge4$, letting $g$ be a conformally Euclidean metric on $\mathbb{R}^n$, we consider the question of positivity of the lower-order $Q$-curvatures $Q_g^{(2k)}$ for $k\in\left\{1,\dotsc,m-1\right\}$ when $Q_g^{(2m)}$ is assumed to be nonnegative and not identically zero. We assume moreover that...
We consider the problem of finding the root vertex of a random uniform attachment tree, when the union of the unlabeled tree and an Erdős-Rényi random graph $\mathbb{G}(n,p)$ is observed. We prove that, as long as $p=o(\log n /n)$, for any $\varepsilon>0$, one can construct a confidence set of vertices of size $K(\varepsilon)$ that...
In this work, we study small-noise asymptotics of risk-sensitive exit-time control problems governed by stochastic differential equations with path-dependent coefficients. Our main result establishes the convergence of the $\log$-transformed exit-time problem to a deterministic control problem with path-dependent coefficients. For its...
We study the small-regularization limit for $L^p$-regularized optimal transport with $1<p<\infty$ and for entropically regularized optimal transport (EOT). The exact first-order (respectively, second-order) asymptotics are determined explicitly under mild assumptions on the source and target measures. Our work generalizes the existing...
Let $G=\Spin(n)$ be the split spin group over an arbitrary field, with $n\ge7$. Extending a Steenrod-theoretic obstruction of Karpenko, we classify the recursively defined integral Weyl invariants $q_i$ in the Benson--Wood generating set that lie in the Chow characteristic image: the only such invariant is $q_3$ for $\Spin(10)$. We...
We give explicit complex polynomials $P,Q$ in three independent standard real Gaussian variables such that \[ {\mathbb E}(P^m)=0,\qquad {\mathbb E}(QP^m)=m!\neq0 \] for every $m\geq1$. In natural complex linear coordinates, $P$ has five terms and total degree $4$. Hence the Gaussian Moments Conjecture is false in every dimension...
In this paper we prove matrix-weighted inequalities for fractional operators and their commutators. We do so by developing the theory of convex body domination for such operators. Using this approach we prove quantitative estimates for the fractional integral operator (or Riesz potential) and its commutators, and prove matrix-weighted...
For smooth nonconvex stochastic multi-objective problems, stochastic multi-gradient descent (SMG) computes an approximate steepest common descent direction of the objectives from stochastic gradients. With unbiased, variance-bounded stochastic gradients, this note establishes a new convergence rate for SMG in terms of the squared...
We give a proof of Fricke's trace identity using the exceptional spin double cover of an orthogonal group in four variables. We also explain how Fricke's identity is related to the double-angle formula from trigonometry as well as an identity for symplectic matrices.
In this survey, we study how tools from stable homotopy theory have manifested and impacted motivic homotopy theory. In particular, we discuss various motivic Adams spectral sequences, periodicity in the motivic stable homotopy groups of spheres, and synthetic spectra. We conclude with many problems for future investigation.
In this paper we study maximal and oscillation inequalities for multi-parameter discrete Radon averaging operators. We develop a robust variant of the multi-parameter circle method within the framework of Discrete Analogues in Harmonic Analysis. In particular, this gives quantitative estimates for these averages and their underlying...
We adapt the classical topological notions of properly discontinuous group actions to the framework of coarse geometry. By substituting the rigid constraints of local compactness and discreteness with uniform coarse equivalences, we systematically resolve the topological obstructions identified by Kapovich. We establish comprehensive...
Given a degree sequence $d$, the realization graph $\mathcal{G_F}(d)$ is the graph whose vertices are all labeled realizations of $d$, where two realizations are adjacent if they differ by a single $2$-switch. We prove that $\mathcal{G_F}(d)$ admits a Hamilton path for every degree sequence $d$. The problem was initiated by Arikati...
We give a definition of coverings and branched coverings on general (non-locally connected) metric spaces and study their lifting properties. Instead of lifting curves the theory is developed by lifting continua. An alternative homotopy theory is depicted and versions generalizing the classical results are given. We study applications...
Dengue remains a major public health challenge in tropical regions, and recurring outbreaks suggest that current intervention strategies are not yet fully effective. Existing mathematical models typically assume unlimited hospital capacity and continuously applied fogging, neglecting practical constraints that strongly influence...
We use the tropical trigonal construction to describe the minimal toroidal resolution of the Prym period map $\overline{R}_4\dashrightarrow\overline{A}_3$.
We consider a novel robust utility maximisation problem under bounded cumulative transition rate uncertainty within the class of non-Markovian marked point processes on a finite state-space. Utility is maximised over the class of admissible controls, while Nature chooses a worst-case biometric scenario from the class of admissible,...
Let $V$ be an $n$-dimensional Euclidean vector space, ,where $n\ge 4$, and $\ell = \lfloor\frac{n}{2}\rfloor$. We prove the sharp pointwise estimate \[ q_2(E) \ge -\frac{2(\ell -1)}{3\ell} \mathrm{Scal}(E) \mathrm{Id}_{Λ^2V^*} \] for every algebraic curvature tensor $E$ on $V$ with nonnegative sectional curvature. Applying...
We investigate entropy production and nonequilibrium transport in a class of random dynamical systems modeling a Knudsen gas confined to a compartmented container. The system consists of a single particle undergoing random billiard motion, with collisions leading to random reflection or transmission through semi-reflecting,...
Disordered systems such as spin glasses and polymers characteristically exhibit random energy landscapes with many macroscopically separated energetic valleys corresponding to near-ground states. This high complexity renders these systems extremely sensitive to perturbations of external parameters. For instance, the support of...
This paper considers a multi-environment factor model in which high-dimensional covariates are collected from heterogeneous environments, with auxiliary labels available in a subset of these environments. The joint distribution of the covariates may vary across environments, whereas the latent structure is decomposed into invariant...
We investigate the combinatorial structure of the tropical characteristic polynomial of symmetric matrices using the tropical permanents of their principal submatrices. We establish new inequalities for the leading coefficients of the tropical characteristic polynomial, revealing concavity properties of the coefficient sequence and...
We give a complete characterization of the (purely infinite) simplicity of Katsura algebras and the associated Steinberg algebras. This is achieved by characterizing when the singular ideals vanish via the self-similar groupoid model derived from Exel and Pardo. Analogous results are given for the algebras arising from the faithful...
We consider the space of $(n-1)$-colored graphs on a fixed set of $N$ vertices. Each edge position of the complete graph $K_N$ has $n$ possible states: the absence of an edge and $n-1$ colors. This gives a natural identification of the space of such graphs with the tensor power $(\mathbb C^n)^{\otimes m}$, where $m=\binom N2$, and...