arXiv preprints from January 1, 2026 through September 9, 2026 — 10:20:06 EST
We address two open questions in streaming PCA via Oja's algorithm: sharp operator-norm convergence for general rank under sub-Gaussian data, and distributional inference for the resulting subspace estimator. Existing convergence analyses, even in the rank-one case, either assume bounded data or leave non-vanishing remainder terms...
Bayesian posteriors can converge under model refinement even when the exact sensitivities used by gradient-based samplers do not. We study this mismatch for discretely scale-invariant rough potentials and its consequences for Metropolized Hamiltonian proposals. For Weierstrass truncations, adjacent classical-force increments grow...
We consider the question of the optimal timing of the sale of an asset with stochastic dynamics. Our analysis is based on the method of the distribution builder introduced by Sharpe, Goldstein and Blythe [SGB00] for the purpose of optimal portfolio selection. Instead of specifying a utility function or risk aversion coefficient, this...
Asset-pricing models typically condition on a fixed information set. This paper endogenises the market's conditioning architecture by allowing portfolios to choose representations whose induced exposures affect prices. Capital allocated across representations determines aggregate positions and the clearing premium, while price...
We introduce a reinforcement learning framework for market making in a limit order book. Our algorithm aims to maximize trading revenue by dynamically submitting market and limit orders of varying sizes across multiple price levels while controlling inventory size. We use multivariate logistic-normal distributions to model order...
In this work we analyze the eigendecomposition of the Hessian matrix of the Robin function $\mathcal{R}(x)$ for the spectral fractional Laplacian in orthogonally invariant domains. We prove that, if $Ω$ a smooth bounded convex domain invariant under the action of an orthogonal transformation $\mathcal{O}$ then, for...
Let $Φ$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $Ξ$-function, set $s(t)=Φ(\sqrt t)$, and define the first Laguerre expression $f(t)=s'(t)^2-s(t)s''(t)$. Csordas and Dimitrov (2000) conjectured that $\log f$ is strictly concave on $(0,\infty)$; Csordas (2015) later restated the assertion as...
We propose a local Legendre frame method for the accurate computation of Caputo fractional derivatives of order \(0<α<1\). On each local subinterval, the function is represented by a restricted Legendre frame obtained from scaled Legendre polynomials on an extended interval. The local coefficients are computed from equispaced samples...
Let $A$ be a commutative simple algebra in a braided finite tensor category $\mathcal{B}$. We identify the largest transparent subalgebra of $A$ as the algebra induced by a central lift of the free-module functor. This identification gives formulas for the Frobenius-Perron dimension and the Müger center of the category of local...
We describe certain finite-time singularities of the Kähler-Ricci flow arising in the analytic minimal model program. Assuming that convergence to an asymptotically conical Kähler-Ricci shrinker is realized by holomorphic maps, we prove that, in a fixed holomorphic gauge, the nearby flow is modeled on the shrinker at the level of...
We study stochastic control of multivariate Hawkes-driven stochastic differential equations with machine learning algorithms in a non-Markovian setting. Due to the path dependence of the memory of the Hawkes intensity, this problem does not fall within classical stochastic control theory outside particular Markovian kernels. We first...
We determine the critical probability for Bernoulli bond percolation on essentially any finite graph. Namely, letting $λ(G)$ denote the spectral radius (maximum eigenvalue) of $G$, we prove that the critical probability is at $1/λ(G)$: above this probability there is typically a component of order $Ω(λ(G))$, whereas below it all...
Let $π: M \to B$ be an elliptic fibration over $B = D^2$ or $B = S^2$ with $n$ nodal fibers over $Δ\subseteq B$. We study the universal liftable braids for $π$: those braids that admit a fiber-preserving lift to $M$ for all choices of coordinates on $(B,Δ)$. When $B = S^2$, we show that nontrivial universal braids do not exist by...
The goal of this paper is to prove an upper bound for the $L^4$ norm of a trigonometric polynomial whose spectrum is a nontrivial strictly monotone polynomial with integer coefficients of degree three and higher, via its $L^2$ norm. Our condition on the coefficients of the trigonometric polynomial in question is that they form a...
We consider the spherical Borel Eisenstein series induced from the trivial representation for a split semisimple linear algebraic group over a number field. We prove that its regularization at the special point corresponding to half the weighted marking of a distinguished coadjoint nilpotent orbit in the Langlands dual Lie algebra is...
The Erdős Matching Conjecture is governed by two competing ways of excluding $s+1$ disjoint edges: one may concentrate all edges on fewer than $k(s+1)$ vertices, or force every edge to meet a fixed $s$-set. We determine a near-optimal range in which the second construction is extremal. For every fixed $k\ge2$, there is $s_0(k)$ such...
We study the 4D Hessian conjecture in Lorentzian signature. For a polynomial potential $φ$ in 4 real variables whose Hessian matrix has inertia index 1 and determinant $-1$, we define a pivot of $φ$ as a direction vector $v$ such that the double derivative $D^2_vφ$ is a constant function. We then prove that the gradient mapping of...
We prove a global-in-time uniform bound for solutions to thermo-diffusive systems of reaction-diffusion equations with $L^\infty$ initial conditions.
We show that any measurable, translation and $\mathrm{SO}(n)$-invariant valuation on polytopes in $\mathbb{R}^n$ is a linear combination of the intrinsic volumes, which extends Hadwiger's classical characterization of rigid motion invariant continuous valuations on convex bodies. This result is based on a novel regularity result for...
Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with an imaginary Killing function $iμ$ if and only if it is an exact $μ$-Kenmotsu manifold, thereby obtaining an extension of a recent result by the first named author that, under the purity...
Let $G$ be an almost simple group whose socle is an alternating, classical, or sporadic group, and let $H$ be a non-parabolic maximal subgroup of $G$. We prove that any maximal subgroup $M$ of $H$ can be generated by at most $7$ elements, and that this bound is sharp when the socle of $G$ is alternating or classical; this improves the...
Let $X_1,X_2, \ldots $ be independent uniform random points in a bounded region $A \subset {\bf R}^d$ having a smooth boundary, $d \geq 1$. Let $B \subset A$ be compact. The _coverage threshold_ of $B$, $R_n$, is the smallest $r$ such that $B$ is covered by the balls of radius $r$ centred on $X_1,\ldots,X_n$. The _maximal spacing_...
This article introduces a numerical optimal control framework for minmax constrained density control for a class of noisy linear parabolic partial differential equations (PDEs), in particular the noisy heat equation. The goal is to transport an initial density to a target density while minimizing a specified cost with respect to...
Distributed target tracking is essential for scalable and robust sensing systems, as it enables multiple radar nodes to cooperatively estimate a target state without relying on a centralized fusion center. In this paper, we present a fully distributed framework for single-target tracking in frequency-modulated continuous-wave (FMCW)...
Deep reinforcement learning (DRL) has recently gained a great attention due to its real-time adaptation and effectiveness in complex optimization problems. This paper investigates the optimal deployment of millimeter-wave (mmWave) base stations (BSs) in a realistic, non-convex campus topology. The optimization problem is NP-hard, due...