arXiv preprints from January 1, 2026 through September 18, 2026 — 18:35:00 EST
In a statistical factor model, principal components (or eigenvectors) of a sample covariance matrix serve as estimates of {\it principal directions}, the true drivers of co-movement of a collection of observed variables. We write the often substantial error in these estimates as a sum of two interpretable terms, which we show have...
Bernheim, Braghieri, Martínez-Marquina, and Zuckerman (2021, American Economic Review) develop a theory of chosen preferences and conjecture convergence to pure worldviews at low mindset flexibility. We show that the (original) conjecture fails: with arbitrarily many worldviews, a common-value action sustains mixed worldviews in every...
We prove a sharp concentration inequality for the spectral norm of sparse random tensors with independent Bernoulli entries. Let $T$ be an order-$k$ tensor of dimension $n\times\cdots\times n$ with independent Bernoulli$(p)$ entries, where $k$ is fixed. For any $c,r>0$, we show that $\|T-\mathbb E T\|\le C_{k,r,c}\sqrt{np}$ with...
The interaction of moose (Alces alces) and wolf (Canis lupus) populations in ecosystems such as Isle Royale National Park is a canonical benchmark for ecological modeling of prey-predator dynamics. Mathematical modeling is a useful tool for modeling these interactions. A central challenge in this domain is the inverse problem,...
We introduce a unified framework for supervised and unsupervised learning based on reproducing kernel Hilbert spaces (RKHSs). We introduce the notion of a nested RKHS, namely an increasing sequence of RKHSs whose union is dense in an ambient $L^2$-space. This construction makes it possible to exploit the computational structure of...
Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\geq 1$. For each prescribed volume fraction $s\in(0,1)\setminus\{\frac12\}$, we prove that the isoperimetric region of volume $s \operatorname{Vol}_g(M)$ is unique for a generic set of smooth Riemannian metrics $g$. At half volume, a generic metric has exactly two...
We prove new small-ball inequalities for boolean matrix-series. The leading example is $\mathbb E_s[{\text{det}(I-S^2)^β\,\mathbf 1_{\{\|S\|<1\}}}]\ge e^{-O(βτ)}$, which holds for boolean matrix-series $S=\sum_i s_iA_i$ formed using symmetric matrices $A_1,\dots,A_n$ and uniformly random signs $s\in\{\pm1\}^n$. Specifically, this...
For $r> 2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field of...
We study a singular Dirichlet problem driven by a regularized $p(x)$-Laplacian under a prescribed modular constraint. Using growth and differentiability estimates for the regularized energy, together with coercivity and modular compactness, we construct nonnegative constrained minimizers for a family of nonsingular approximating...
We investigate the sensitivity of solutions to multi-dimensional scalar balance laws with respect to perturbations of the initial data analytically and numerically. Reliable first-order sensitivity information is essential in gradient-based optimization and inverse problems constrained by hyperbolic balance laws. In hyperbolic...
We prove local regularity for axisymmetric suitable weak solutions of the 3D Navier-Stokes equations, which are smooth before the terminal time $t=0$, and satisfy Type~II pointwise bounds at a vanishing length scale $\ell(t)$. We say $\ell(t)$ is an Euler length if it is non-increasing, satisfies a doubling condition, and if...
In this paper we study a weighted generalization of the Poisson-Voronoi tessellation called the Poisson-Laguerre tessellation, where the nuclei of the generating Poisson process additionally carry independent non-negative random weights. For each cell we can define its inradius as the radius of the largest ball centered at the nucleus...
Location estimation exhibits markedly different finite-sample behavior across noise distributions: regular families typically yield root-\(n\) rates, whereas compactly supported laws may admit faster, boundary-driven rates. We question whether a single estimator, without knowledge of the density's shape, can adapt to the instance-wise...
We introduce a new method that brings the combinatorics of hyperplane arrangements into the study of representation zeta functions of compact Lie groups. For the Witten zeta function $ζ_Φ(s)$ associated with a root system $Φ$, our method yields elegant formulas for $ζ_Φ(0)$ and $ζ_Φ'(0)$ in terms of the exponents of various parabolic...
We prove existence and uniqueness of a strong initial trace for every bounded kinetic solution of a stochastic scalar conservation law, although no initial value is prescribed and no nondegeneracy condition is imposed on the flux. The core of the proof is pathwise. After fixing a realization, the rescaled stochastic terms and kinetic...
Principal component analysis (PCA) is a fundamental tool for learning low-dimensional structure from high-dimensional data. When data are collected from multiple sources, the underlying task distributions may exhibit unknown degrees of similarity, with some tasks potentially arising from arbitrary distributions. We propose new...
Motivated by questions and results of Erdős, Sárközy and Sós (1989), and Khalfalah and Szemerédi (2006), we introduce an ergodic approach to additive problems concerning solutions of equations of the form \[ x+y=\lfloorα(n)\rfloor,\quad x,y\in A,n\in\mathbb{N} \] where $α(t)$ is a sufficiently regular function, such as a polynomial or...
We prove that for every acyclic matrix pattern $P$ it holds that $\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$.
We study the Anderson model on hyperbolic lattices. The graph is the Cartesian product of the Euclidean lattice with the regular hyperbolic square lattice having five squares at each vertex. We prove the existence of absolutely continuous spectrum and the absence of singular spectrum on a deterministic set of positive measure at weak...
We develop a new approach to study Bernoulli percolation in dimensions $d>6$. The key idea is to approximate open paths at probability $p'>p$ by a Markov chain of pointed $p$-open clusters. This allows us to transfer sharp information on the two point function from $p$ to $p'$. This inductively gives good asymptotic estimates on the...
We show that unless P = NP, there cannot be a polynomial-time (or even pseudo-polynomial-time) algorithm for output feedback stabilization of a linear dynamical system with a linear controller. This settles one of the best-known open problems in control theory. The result holds in both continuous and discrete time. We also present a...
The classical Whittle formula provides a closed-form expression for the asymptotic Fisher information matrix (FIM) rate of multidimensional, real-valued, zero-mean, purely nondeterministic stationary Gaussian processes (GPs), expressed in terms of their parameterized spectra within a maximum-likelihood framework. However, the Gaussian...
A polyhedron is a planar, $3$-connected graph. We iteratively construct all polyhedra (save for pyramids) from a unique starting graph, namely the square pyramid, via two graph transformations. This builds upon a previous construction, that starts from the full class of pyramids, and applies the same transformations. In a related...
We prove linear stability of Kasner spacetimes in the direction of the Big Bang, up to an explicit finite dimensional space of non-decaying self-similar solutions. The fastest growing solution is the linearization of Taub's explicit Bianchi II solution, describing a Kasner transition. All other non-decaying solutions are inhomogeneous...
We consider generalizations of the classical Fourier uniqueness theorem. First, we construct a family of uniformly discrete sets $Λ\subset \mathbb{R}$, of uniform density one, such that the exponential system $\{e^{2πiλx} : λ\in Λ\}$ is complete in $L^p(S)$ for every $1 \leq p < \infty$ and every measurable set $S \subset \mathbb{R}$...