arXiv preprints from January 1, 2026 through July 20, 2026 — 05:14:13 EST
We study a class of maps having the Collatz function (famously related to the Collatz Conjecture) as an example, under topological and ergodic perspectives, including an approach with thermodynamic formalism. By introducing a key topology and its Borel sigma-algebra we show that recurrence implies periodicity. Moreover, we establish...
We obtain infinitely many boundary operators in the Brownian loop soup in the subcritical phase by analyzing the conformal block expansion of the two-point function that computes the probability of having two marked points on the upper half-plane being separated by Brownian loops. The resulting boundary operators are primary operators...
The output regulation problem for unknown linear systems has been studied using state-based and output-based internal model approaches in the special case with no disturbances. This paper further investigates the output regulation problem for unknown linear systems using a data-driven output-based approach via value iteration. For...
The Akbari-Cameron-Khosrovshahi (ACK) conjecture, which appears to be unresolved, states that for any simple graph $G$ with at least one edge, there exists a nonzero {$\{0,1\}$}-vector in the row space of its adjacency matrix that is not a row of the matrix itself. In this talk, we present a unified framework that includes several...
Previous research on exceptional units has primarily focused on the ring of rational integers or abstract finite rings, often restricted to linear or quadratic constraints. In this paper, we extend the concept of polynomial-type exceptional units to the ring of integers of an arbitrary algebraic number field. We investigate the number...
The Riemann curvature tensor fully encodes local geometry, but its Ricci contraction retains only limited information: only the Ricci tensor and the scalar curvature survive, while the Weyl curvature vanishes identically. We show that contracting instead the double dual of the Riemann tensor unlocks the full curvature structure,...
The main purpose of this article is to show that the special Newton polygon map from the stack of p-adic shtukas to the stack of G-bundles on the Fargues--Fontaine curve is representable in diamonds and sufficiently nice for cohomological considerations (i.e. fdcs). The second purpose is to show that the $\bar{\mathbb{F}}_p$-fibers of...
We give a detailed description of the nearly Kähler $\frac{\mathrm{SL}(3,\mathbb R)}{\mathbb R\times \mathrm{SO}(2)}$, which is one of the pseudo-Riemannian counterparts of the flag manifold $F(\mathbb{C}^3)$. The main result is the classification of totally geodesic almost complex surfaces in this space.
We give a sufficient condition to rule out complete Riemannian metrics with nonnegative scalar curvature on the interiors of handlebodies. In higher dimensions, we give examples of ends of manifolds with positive scalar curvature metrics.
We study the problem of sampling from a target distribution $π(q)\propto e^{-U(q)}$ on $\mathbb{R}^d$, where $U$ can be non-convex, via the Hessian-free high-resolution (HFHR) dynamics, which is a second-order Langevin-type process that has $e^{-U(q)-\frac12|p|^2}$ as its unique invariant distribution, and it reduces to kinetic...
We prove that complete Riemannian manifolds of dimension $n\ge3$ with harmonic curvature and $\frac{n(n+2)}{2(n+1)}$-nonnegative curvature operator of the second kind must be Einstein. In particular, We show that complete Einstein manifolds of dimension $n\ge4$ with $\frac{3n(n-1)^2(n+2)}{2(5n^3+3n^2-30n+16)}$-nonnegative curvature...
Symmetric quiver varieties with potentials are natural generalizations of Nakajima quiver varieties, and their equivariant critical cohomologies provide more flexible settings for geometric representation theory and enumerative geometry. In this paper, we study their geometric properties and show that they behave like universally...
Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $ε>0$ there are finite covers $M_ε\to M$, and $N_ε\to N$, such that the Teichmuller distance (in the suitable moduli space) between $M_ε$ and $N_ε$ is less than $ε$. It is natural to ask how large the...
Recently, Atar and Miyazawa [2] introduced a multi-level GI/G/1 queue with a finite number of levels, where both the arrival and service rates depend on the level corresponding to the current queue length. For this model, they proved that the diffusion limit of its queue length process in heavy traffic is the level-dependent reflected...
Let $λ_K:\bbR^2\rightarrow\{0,1,\ldots\}\cup\{\infty\}$ be the lambda function of a planar comapctum $K$, as defined in MR4488162. It is known that a planar continuum is locally connected if and only if its lambda function vanishes everywhere, or equivalently, $λ_K(K)=\{0\}$. In this article we show that every fractal square $K$...
Spectral estimation plays a fundamental role in frequency-domain identification and related signal processing problems. This paper revisits a 2-D spectral estimation problem formulated in terms of convex optimization. More precisely, we work with the dual optimization problem and show that the full Hessian of the dual function admits...
We define a set $P$ to be a branching $k$-path vertex cover of an undirected forest $F$ if all leaves and isolated vertices (vertices of degree at most $1$) of $F$ belong to $P$ and every path on $k$ vertices (of length $k-1$) contains either a branching vertex (a vertex of degree at least $3$) or a vertex belonging to $P$. We define...
We propose and analyze a linear and partitioned finite element method for fluid-shell interactions under the arbitrary Lagrangian-Eulerian (ALE) framework. We adopt the P1-bubble/P1/P1 elements for the fluid velocity, pressure, and structure velocity, respectively. We show the stability and error estimates of the scheme without...
The sequence reconstruction problem asks for the recovery of a sequence from multiple noisy copies, where each copy may contain up to $r$ errors. In the case of permutations on \(n\) letters under the Hamming metric, this problem is closely related to the parameter $N(n,r)$, the maximum intersection size of two Hamming balls of radius...
We study two types of graphs interpolating between the curve graph and the pants graph from the viewpoint of large-scale geometry. One was introduced by Erlandsson and Fanoni, and the other by Mahan Mj. These graphs were developed independently in different contexts. In this paper, we provide explicit formulae for computing their...
We illustrate how the recent theory of Spectral Submanifolds (SSM) can capture global bifurcations and complex dynamics in mechanical systems even under delay and spatial discretization. Specifically, we build a parameter-dependent SSM-reduced model that predicts global heteroclinic and local bifurcations in a Furuta pendulum under...
We compute the quantum isometry groups of Cuntz-Krieger algebras endowed with the spectral triples coming from the Ahlfors regular structure of the underlying topological Markov chain. This allows us to exhibit a new family of compact quantum groups, mixing features from quantum automorphism groups of graphs and easy quantum groups....
Les perturbations de faible rang de matrices aléatoires ont été au cœur de nombreux travaux ces vingt dernières années. En particulier, les cas non-hermitiens, moins représentés dans la littérature en règle générale, font ici l'objet d'une attention spéciale en raison de leurs applications à la physique et à l'étude des réseaux de...
Motivated by Varadhan's theorem, we introduce Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to their Fréchet counterparts. Given independent and identically distributed samples, we prove uniform laws of large numbers for their empirical versions. Furthermore, we prove central limit...
Community detection seeks to recover mesoscopic structure from network data that may be binary, count-valued, signed, directed, weighted, or multilayer. The stochastic block model (SBM) explains such structure by positing a latent partition of nodes and block-specific edge distributions. In Bayesian SBMs, standard MCMC alternates...