arXiv preprints from January 1, 2026 through September 7, 2026 — 23:34:49 EST
In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are elementary of order $4$. Letting $\mathrm{G}$ equal the Galois group of the second Hilbert narrow...
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method (RFM) and extreme learning machines (ELM), transform network training into a linear least-squares...
The first purpose of this work is to provide a friendly introduction to the theory of nonautonomous linear systems of ordinary differential equations, the property of exponential dichotomy and its corresponding spectral theory. The second purpose of this work is disseminate the linearization results carried out by the authors in a...
A ball polyhedron is a finite intersection of congruent balls in $\mathbb{R}^3$. These shapes arise in various contexts in discrete and convex geometry. We focus on Reuleaux polyhedra, the subclass of ball polyhedra whose centers and vertices coincide. Building on Bogosel's recent work on the volume of Meissner polyhedra, we derive a...
Studying Nambu solutions of the rainbow-ladder gap equation in QCD at zero temperature and chemical potential, we prove that the mass function emerges continuously from zero as the interaction strength is increased past the critical point for all positive, asymptotically perturbative kernels almost everywhere continuous in $L^1$ using...
The transition of transit fleets to alternative powertrains offers a potential pathway to reducing the cost of mobility. However, the limited range and long charging durations of battery electric buses (BEBs) introduce significant operational complexities, necessitating innovative scheduling and charging strategies. This study...
In this paper we study directional second-order tangent sets of real and complex analytic sets. For an analytic set $X\subseteq \mathbb K^n$ and a nonzero tangent direction $u\in T_0X$, we compare the geometric directional second-order tangent set $T^2_{0,u}X$, defined through second-order expansions of analytic curves in $X$, with...
Singular stochastic partial differential equations informally refer to the partial differential equations with rough random force that leads to the products in the nonlinear terms becoming ill-defined. Besides the theories of regularity structures and paracontrolled distributions, the technique of convex integration has emerged as a...
We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf K$-forms of $\mathrm{SL}_2^{\mathsf n}$ where $\mathbf K$ is a number field. This gives an effective version of equidistribution results of Ratner and Shah with a polynomial rate. The...
We present a unified operator-theoretic framework for stochastic calculus based on the factorization (Id - E)F = δ_X Π_X D_X F, valid for F_T^X-measurable F in L^2(Ω) when the driving process X has the representation property. For a square-integrable process X with stochastic integral δ_X, we define the operator-covariant derivative...
There exist conformally invariant, higher-derivative, variational analogs of the Yang-Mills condition for connections on vector bundles over a conformal manifold of even dimension greater than or equal to six. We give a compact formula for these analogs and prove that they are a strict weakening of the Yang-Mills condition with...
We develop an operator-theoretic formulation of stochastic calculus for fractional Brownian motion with Hurst parameter H in (0, 1/2). The approach is based on adjointness between stochastic integration and differentiation in the Cameron-Martin space of the driving process. For Gaussian Volterra processes, we establish a canonical...
Let Y be a random variable satisfying specific moment conditions. This paper introduces and investigates probabilistic heterogeneous Stirling numbers of the second kind and probabilistic heterogeneous Bell polynomials. These structures unify several classical and probabilistic families, including those of Stirling, Lah, Bell and...
Let $G$ be an infinite, connected, locally finite planar graph and consider i.i.d.\ Bernoulli$(p)$ site percolation. Write $p_c^{\mathrm{site}}(G)$ and $p_u^{\mathrm{site}}(G)$ for the critical and uniqueness thresholds. Using a well--separated Freudenthal embedding $G\hookrightarrow\mathbb S^2$, we introduce a cycle--separation...
Let $C$ be a smooth projective curve over an algebraically closed field ${\mathbb{F}}$ equipped with the action of a finite group $G$. When $p =\textrm{char}(\mathbb{F})$ divides the order of $G$, the long-standing problem of computing the induced representation of $G$ on the space $H^0(C,Ω^{\otimes m}_C)$ of globally holomorphic...
In this paper, we construct directed strongly regular graphs and divisible design graphs with new parameters merging some basic relations of so-called Tatra associations schemes. We also study the above association schemes, their fusions and isomorphisms.
For Bernoulli site percolation on an infinite, connected, locally finite graph $G=(V,E)$, we obtain quantitative upper bounds on the supercritical disconnection probability \[ \mathbb{P}_p(S\nleftrightarrow\infty) \] for arbitrary finite or infinite sets $S\subset V$ and all $p>p^{\mathrm{site}}_c(G)$. The key quantity is a...
The choice of the tuning parameter in the Lasso is central to its statistical performance in high-dimensional linear regression. In this work, we study tuning regimes under which the Lasso exhibits suboptimal prediction performance, in the sense that a simple refinement improves upon it both on high-probability events and in mean...
In this paper, we propose a novel reformulation of the smooth nonconvex-strongly-concave (NC-SC) minimax problems that casts the problem as a joint minimization. We show that our reformulation preserves not only first-order stationarity, but also global and local optimality, second-order stationarity, and the Kurdyka-Łojasiewicz (KL)...
In this paper we obtain several results related to the $p$-adic interpolation of the classical Cogdell lift, mapping special cycles on Picard modular surfaces to elliptic modular forms. The results have a three-fold nature: in the first part of the paper, we $p$-adically interpolate the adjoint Kudla lift, exploiting the previously...
We establish sharp rates for propagation of chaos in Rényi divergences for interacting diffusion systems at stationarity. Building upon the entropic hierarchy established in Lacker (2023), we show that under strong isoperimetry and weak interaction conditions, one can achieve $\mathsf R_q(μ^1 \,\lVert\, π) = \widetilde O(\frac{d...
For $d\geq 4$, Kalai (1987) characterized all simplicial $(d-1)$-spheres with $g_2=0$, and for $k\geq 2$ and $d\geq 2k$, Murai and Nevo (2013) characterized all simplicial $(d-1)$-spheres with $g_k=0$. In addition, for $d\geq 4$, Nevo and Novinsky (2011) characterized all simplicial $(d-1)$-spheres with $g_2=1$. Motivated by these...
In this paper, we develop a general theory for the transport equation within the framework of Triebel-Lizorkin spaces. We first derive commutator estimates in these spaces, dispensing with the conventional divergence-free condition, via the Bony paraproduct decomposition and vector-valued maximal function inequalities. Building on...
We study the subgradient method for factorized robust signal recovery problems, including robust PCA, robust phase retrieval, and robust matrix sensing. The resulting objectives are nonsmooth and nonconvex, and can have unbounded sublevel sets, so standard analyses based on descent and coercivity do not apply. For locally Lipschitz...
We present a fast, unconditionally energy-stable numerical scheme for simulating vesicle deformation under osmotic pressure using a phase-field approach. The model couples an Allen-Cahn equation for the biomembrane interface with a variable-mobility Cahn-Hilliard equation governing mass exchange across the membrane. Classical...