arXiv preprints from January 1, 2026 through September 6, 2026 — 20:48:51 EST
Let $G$ be an automatic group with associated regular language $L$. We describe a procedure for constructing an automaton which recognises elements of a given submonoid or rational subset $K$ of $G$. This builds on work of Kharlampovich, Miasnikov and Weil, on the case where $K$ is a subgroup of $G$. Our construction succeeds, after...
Paths $X \colon [0,T] \to \mathbb R^d$ are traditionally stored in finite memory as time series. Recent research has underscored the benefits of instead representing them as collections of iterated integrals $\{\int_{0 < u_1 < \ldots < u_n < T} \mathrm{d} X_{u_1} \otimes \cdots \otimes \mathrm{d} X_{u_n}\}_{n = 0}^N$. These two...
In low-rank numerical methods for Vlasov dynamics, the SVD-type truncation procedure may introduce negative entries into the numerical solution. Such negative values are unphysical because the solution is a probability distribution function. We design optimization-based post-processing algorithms to recover nonnegativity while...
We find sharp upper bounds for the initial higher-order Schippers-Schwarzian derivatives $σ_3(f)(0)$ and $σ_4(f)(0)$ for several subclasses of univalent and starlike functions in the unit disk. The functions in these classes are defined by subordination to domains bounded by an exponential-sine curve, a cardioid, or a petal-shaped...
It was asked in [ACTT23] if every faithful, discrete, ergodic pmp action of a mixed identity-free (MIF) group is essentially free. In this short note we give some positive evidence by showing that for a MIF group, any faithful generalized Bernoulli action and any affine action on a compact abelian group is essentially free. This gives...
The problems of characterizing the graphs $G$ which are generically rigid in ${\mathbb R}^d$, or more generally, determining the rank function of the $d$-dimensional rigidity matroid ${\cal R}_d(G)$ of an arbitrary graph $G$, have been solved when $d\leq 2$ but are major open problems in discrete geometry when $d\geq 3$. In this paper...
This paper has several parts. We begin by developing a theory of `piecewise (strong) f-genericity' in NIP groups, where we call a definable set piecewise (strong) f-generic if some union of finitely many translates of it is (strong) f-generic. We show that, in an NIP group, the definable sets that are not piecewise (strong) f-generic...
Oscillatory inverse problems often exhibit highly nonconvex discrepancy landscapes due to signal misalignment and cycle-skipping phenomena, posing significant challenges for uncertainty quantification. While Gibbs posteriors provide a flexible framework for incorporating problem-specific discrepancy measures, their performance depends...
We propose a stochastic model of sea spray as a shot noise process attached to a homogeneous Poisson process whose intensity measure is the sea spray generation function, that is to say the average number of droplets of a given radius ejected into the atmosphere per unit area per unit time. We assume the impulse response function...
In this paper, we study Ricci solitons on the three-dimensional solvable Lie group $\mathrm{Sol}^{3}_{m,n}$ equipped with a left-invariant Riemannian metric, viewed as a generalization of the classical $\mathrm{Sol}^{3}$ geometry. We investigate harmonic maps, harmonic sections, and geodesic curves, including the geodesic properties...
Featuring Hessian-driven damping, two inertial primal dual dynamical systems are proposed for solving smooth saddle point problems with bilinear coupling. For convex-concave functions, we establish a convergence rate $\mathcal{O}\left( \frac{1}{t^2} \right)$ for the primal dual gap; for strongly convex-strongly concave functions, we...
Martinsson and Steiner recently proved that the fractional chromatic number of any $d$-degenerate triangle-free graph $G$ satisfies $χ_f(G) = O\left(\frac{d}{\log d}\right)$. They further conjectured a sharp leading constant $1 + o(1)$. In this paper, we confirm their upper bound conjecture for graphs having girth at least $5$. Our...
Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially...
We extend the Caffarelli-Hardt-Simon perturbation argument for truncated regular minimal cones to the special Lagrangian setting and prove a bridge principle for regular special Lagrangian cones in the spirit of Nathan Smale. Our bridge principle yields a general existence theorem for conically singular special Lagrangian submanifolds...
Alon and Füredi proved that at least $n$ affine hyperplanes are required to cover $\{0,1\}^n\setminus\{\textbf{0}\}$ while avoiding the origin, and that this bound is sharp. We study how small the largest Boolean intersection among the hyperplanes can be in a cover attaining this minimum. Let $F(n)$ denote the minimum possible value...
We introduce the multispecies version of the harmonic process on a one-dimensional chain in contact with boundary reservoirs. This process is a continuous-time Markov chain where each site can host an unbounded number of colored particles. The symmetric bulk dynamics is put in contact with reservoirs, which inject and remove particles...
Two central challenges in diffusion-based sampling are the theoretical one of understanding their remarkable effectiveness even in high-dimensional settings, and the practical one of designing algorithms with certified performance guarantees. We show that these questions are intimately connected via the \emph{denoising growth...
In this paper, we derive a conditional copula representation for expectations of the form $\mathbb{E}[g(\boldsymbol{X})]$, where $\boldsymbol{X}$ is a random vector with arbitrary marginal distributions and $g$ is a measurable function satisfying suitable integrability conditions. The proposed representation explicitly separates the...
This paper studies the convergence of stochastic gradient descent (SGD) when the implemented updates are subject to a persistent and state-dependent bias, in which the desired update is scaled by response functions component-wise. Our first contribution is to demonstrate that SGD in this setting implicitly optimizes a penalized...
In electromagnetic heating and other applications, we need to compute the volume enclosed by an isosurface of the 3D temperature distribution that is numerically represented on a rectangular grid. This situation arises naturally when the temperature distribution is obtained by solving a partial differential equation numerically using...
We consider the two-species particle system introduced by Stevens (2000) related to the doubly parabolic Keller-Segel equation. It consists of $N$ cells and of a varying number of chemoattractant particles. Cells diffuse in the plane and follow the (mollified) empirical gradient of concentration of chemoattractant. Chemoattractant...
For a finite subgroup $G\subset \operatorname{SL}(3,\mathbb{C})$, the Cautis--Logvinenko conjecture states that for each nontrivial irreducible representation $ρ$ of $G$, the image of the sheaf $\mathcal{O}_0\otimes ρ$ under the derived equivalence of Bridgeland--King--Reid is a pure sheaf on the $G$-Hilbert scheme. We prove this when...
We study the quasilinear Liouville equation \[ -Δ_n u=e^u \] on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, if a solution $u$ satisfies the optimal logarithmic lower bound \[ u(x)\ge -\frac{n^2}{n-1}\log r(x)+o(\log r(x)) \quad \text{as }r(x)\to+\infty, \] then the underlying...
The Strong Szegő Limit Theorem is a theorem about the asymptotics of the determinants of large Toeplitz matrices. It can be reformulated as a probabilistic statement about eigenvalue statistics of random unitary matrices. We prove a multivariate generalization of the theorem in this latter form, replacing a single unitary with a...
For a ladder-system space $X_L$ with carrier $S\subseteq E^{ω_1}_ω$, the finite-label uniformization property $M_{<ω}$ characterizes countable metacompactness, and countable metacompactness is equivalent to the $Δ$-property. Both equivalences are known for stationary carriers. For arbitrary carriers, an active-tail formulation gives a...