arXiv preprints from January 1, 2026 through September 7, 2026 — 09:21:58 EST
We study in this paper the weak propagation of chaos for McKean--Vlasov diffusions with branching, whose induced marginal measures are nonnegative finite measures but not necessary probability measures. The flow of marginal measures satisfies a non-linear Fokker--Planck equation, along which we provide a functional Itô's formula. We...
In the present paper we derive complicated families of orthogonal polynomials in one variable from scratch using the known ones as building blocks. We recall the basics of operational formalism and introduce the notations we use throughout this paper. Then we establish through this formalism the known families of orthogonal...
A tetrahedron is called a path tetrahedron, if it has three mutually orthogonal edges that do not intersect at a single point. A tetrahedron is called a 4-ball tetrahedron, if there exists a sphere tangent to all its edges. We derive two-sided tight bounds for dihedral angle sums of such tetrahedra. In particular, we prove that this...
For any continuous self-map of a compact metric space, we provide sufficient conditions under which the infinite direct product of the map is $ω$-chaotic. We also apply the result to obtain some examples of unusual $ω$-chaotic maps.
Accumulation of amyloid beta proteins is a defining feature of Alzheimer's disease, and is usually accompanied by cerebrovascular pathology. Evidence suggests that amyloid beta and cerebrovascular pathology are mutually reinforcing; in particular, amyloid beta suppresses perfusion by constricting capillaries, and hypoperfusion...
Let $H(\mathbb{D})$ be the space of all analytic functions in the unit disc $\mathbb{D}$. For $g\in H(\mathbb{D})$, the generalized Hilbert operator $\mathcal{H}_{g}$ is defined by $$\mathcal{H}_{g}(f)(z)=\int_{0}^{1}f(t)g'(tz)dt, \ \ z\in \mathbb{D}, f\in H(\mathbb{D}).$$ In this paper, we study the operator $\mathcal{H}_{g}$...
A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the $p$-Laplacian with measurable coefficients. We focus on the sub-critical range $1<p\le \tfrac{2N}{N+2}$...
Using the generalized Temperley trick, we demonstrate the explicit embedding of circular electrical networks into totally non-negative Grassmannians. Building on this result, we show that the effective resistances between boundary nodes of circular electrical networks satisfy the Kalmanson property, and we provide the full...
We prove a polynomial continued fraction identity for the constant $-π/4$, conjectured by the Ramanujan Machine project. The proof proceeds by explicitly solving the underlying second-order linear difference equation. We derive a closed-form expression for the denominator sequence, $q_n = (-1)^n (2n-3)!!\,(n^2+n-1)$, and establish...
Influence diagrams represent decision-making problems with interdependencies between random events, decisions, and consequences. Traditionally, they have been solved using algorithms that determine the expected utility-maximizing decision strategy. In contrast, state-of-the-art solution approaches convert influence diagrams into a...
Two divergent series in Entry 7 of Gauss's diary are extended systematically by introducing additional parameters. Rogers-Fine identities, Ramanujan's continued fractions and Heine's transformation relations of basic hypergeometric series are applied to find equivalent alternative series, which enable us to compute the sums of...
This article gives a formula for associated Stirling numbers of the second kind based on the moment of a sum of independent random variables having a beta distribution. From this formula we deduce, using probabilistic approaches, lower and upper bounds for these numbers.
Several aspects of pluripotential theory are generalized to octonionic plurisubharmonic (OPSH) functions of two variables. We prove the comparison principle for continuous OPSH functions and the quasicontinuity of locally bounded ones. An important tool is a formula of integration by parts for mixed octonionic Monge-Ampère operator....
In the Lagrange-Newton method, where Newton's method is applied to a Lagrangian function that includes equality constraints, all stationary points are saddle points. It is therefore not possible to use a line-search method based on the value of the objective function; instead, the line search can operate on merit functions. In this...
This paper analyzes a two-product make-to-stock queueing system where a single production facility serves two customer classes with independent Poisson arrivals. Customers make strategic join-or-balk decisions without observing current inventory levels. The analysis establishes the existence and uniqueness of Nash equilibria in...
We prove that non-abelian definable, definably simple groups in 1-h-minimal henselian valued fields are essentially already linear algebraic groups. Here, the group is assumed to live in the home sort. We have a similar result in pure algebraically closed valued fields of positive characteristic, under the additional assumption that...
Let $p>3$ be a prime and $b\ge 2$ an integer such that $p$ does not divide $b$. Then $1/p$ has a periodic digit expansion with respect to the basis $b$. The length $q$ of the period is the (multiplicative) order of $b$ mod $p$. In the case $q=p-1$ a formula for the variance of the digits of a period was given previously. This formula...
The ionic Vlasov-Poisson-Boltzmann system is a fundamental model in dilute collisional plasmas. In this work, we study the compressible ionic Euler-Poisson limit of the ionic Vlasov-Poisson-Boltzmann system for the full range of cutoff potentials $-3 < γ\leq 1$. By employing a truncated Hilbert expansion together with a novel weighted...
This paper examines the relationship between shadowing phenomena and the continuity properties of $ω$-limit sets in dynamical systems. We give a necessary and sufficient condition for a shadowable point to be an upper (resp. a lower) semicontinuity point of $ω$-limit sets. Assuming global shadowing, we show that the lower...
Given a smooth projective variety $X$ over an algebraically closed field $k$, we compute the Chow ring of the Hilbert scheme of three points on $X$, $\operatorname{Hilb}^3(X)$, as an algebra with generators and relations over the Chow ring of $X\times\operatorname{Sym}^2(X)$. If in addition the characteristic of $k$ is zero, we extend...
We study optimal transport between probability measures supported on the same finite metric space, where the ground cost is a distance induced by a weighted connected graph. Building on recent work showing that the resulting Kantorovich distance can be expressed as a minimization problem over the set of spanning trees of this...
We study the $L^p$-convergence of Fourier expansions in terms of non-symmetric Heckman-Opdam polynomials of type $A_1$. Using kernel estimates and duality arguments, we prove that the partial sums converge in $ L^p([-π,π],dm_k)$ for $$2-\frac{1}{k+1} < p < 2+\frac{1}{k}.$$
The globalization problem arises when local tensor fields possess a given property (such as being symplectic or Poisson) but cannot be consistently extended to a global object due to incompatibilities on chart overlaps. A notable instance occurs in locally conformal analysis, where local representatives coincide only up to conformal...
We derive a representation formula for the Weyl solution to the Schrödinger operator on the semi-axis for certain classes of potentials. Our approach is based on relations with the initial-boundary value problem for the wave equation with the same potential on the half-line.
Singular perturbations have been used to select solutions of (non-convex) variational problems with a multiplicity of minimizers. The prototype of such an approach is the gradient theory of phase transitions by L. Modica, who specialized some earlier Gamma-convergence results by himself and S. Mortola contained in a seminal paper,...