arXiv preprints from January 1, 2026 through September 6, 2026 — 23:07:38 EST
For a (molecular) graph $G$ and any real number $α\ne 0$ , the zero-order general Randić index , denote by $^0R_α$, is defined by the following equation: \begin{align*} {^0R_α} (G) =\sum_{v\in G}d_G (v) ^α (α\in \mathbb{R}-\left\{0\right\}) . \end{align*} The deficiency of $G$, denoted by $def(G)$, is equal to the cardinality of...
In this article we establish global a priori estimates on the solution of a generic rough differential equation driven by an $α$-Hölder path covering the full range of regularity $α\in(0,1)$, under the hypothesis of Lipschitz continuity (but neither boundedness nor coercivity) of specific combinations of the coefficients and their...
We show that MI connects subtly and elegantly the two best-known state-of-the-art independence test statistics: the $G^2$ and the Pearson's chi-square statistic $χ^2$. Furthermore, we show that the MI connects directly those statistics by an elegant formula arising from a stochastic Taylor expansion of MI ($δ$-method). MI second order...
We study fibrations of higher categories from the perspective of oriented category theory, a framework which accounts for lax phenomena in higher category theory via systematic enrichment in the Gray tensor product. We give several equivalent characterizations of fibrations of $(\infty,\infty)$-categories and oriented categories, and...
A permutation $π\in S_n$ is (32-1)-avoiding when there do not exist $1 \leq i < i+1 < j \leq n$ such that $π_i > π_{i+1} > π_j$. We determine the maximum inversion number for (32-1)-avoiding permutations and count the number of permutations that achieve this maximum. We then provide a direct construction that enumerates these permutations.
$\texttt{codesign-mcdp}$ is a Python library for formulating and solving $\textit{Monotone Co-Design Problems}$ (MCDPs) in the framework of Censi (2015). A design problem is a relation between two posets, a functionality poset $F$ and a resource poset $R$; given a target functionality, the problem asks for the antichain of minimal...
This paper is devoted to two distinguished families of distributions: special multi-flags and their lower-rank counterparts, Goursat distributions. It is known that these two families are weakly nilpotent but strongly nilpotent only at special points. Local classification of these objects is still open and only recently singularity...
We study an inverse problem for generalized Runcorn's theorem motivated by an apparent paradox of lunar magnetism. In mathematical terms, we characterize square-integrable complex functions $f$ such that $$ \int_{\{ x\in\mathbb{R}^n : a \leq |x| \leq b \}} f(x) \nabla u(x)\cdot\nabla v(x)\,\mathrm{d}\mathrm{V}(x) = 0 $$ for every...
As electric vehicle (EV) adoption grows, quantifying the scheduling burden and economic cost of long-distance travel under the existing charging infrastructure becomes increasingly important for infrastructure planning and policy. This paper presents a scalable, optimization-based framework for scheduling EV charging stops along...
For a finite-dimensional normed space $V$ and a subset $X$ with finite Hausdorff distance from $V$, we prove that the Gromov--Hausdorff distance between $X$ and $V$ is at least the Hausdorff distance between $X$ and $V$, divided by twice the relative Jung constant of $V$. If $V$ furthermore satisfies a certain intersection property,...
We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate. In this setting, we investigate the ``driver'' site frequency spectrum (SFS), or the random measure that records mutant allelic...
Many prediction and decision-making problems in operations research involve path-valued covariates -- data that evolve over time -- for which path signatures have become a canonical feature representation. Their use is justified by a universal approximation theorem, but this is an existence result: it guarantees that a finite-level...
Nonlinear state estimation under structural model uncertainty remains a central challenge in autonomous Guidance, Navigation, and Control (GNC) systems. Classical estimators propagate states using assumed dynamics and incorporate measurements through posterior correction, which under mismatch leads to biased innovations, estimator...
The skew-symmetric discretisation has recently been proposed as a new robust simulation method for weakly approximating stochastic differential equations (SDEs) with non-globally Lipschitz drift. This work develops a pathwise version of the scheme by representing the noise increment as a skew-normal distribution and coupling it with...
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.