arXiv preprints from January 1, 2026 through September 6, 2026 — 22:31:32 EST
Let $G$ be a finite group and let $V$ be a finite-dimensional $G$-module over a field $k$. We construct explicit counterexamples in characteristic $2$ to several questions and conjectures of Wehlau concerning Noether numbers. For the $2$-group $G=D_8$, we exhibit a submodule $U\subseteq V$, with $\dim_kU=5$ and $\dim_kV=6$, such that...
We study doubly reflected backward stochastic differential equations with jumps and two completely separated right-continuous with left limits barriers in a filtration generated by an inhomogeneous Levy process. We establish existence and uniqueness results under a stochastic Lipschitz condition on the driver by means of a...
For a (molecular) graph $G$ and any real number $α\ne 0$ , the zero-order general Randić index , denote by $^0R_α$, is defined by the following equation: \begin{align*} {^0R_α} (G) =\sum_{v\in G}d_G (v) ^α (α\in \mathbb{R}-\left\{0\right\}) . \end{align*} The deficiency of $G$, denoted by $def(G)$, is equal to the cardinality of...
The ultrametric skeleton theorem associates with every compact metric probability space $(X,d,μ)$ a compact subset $S\subseteq X$ of ultrametric distortion $O(1/\varepsilon)$ and a probability measure $ν$ supported on $S$ such that $ν(B_d(x,r))\leqμ(B_d(x,C_\varepsilon r))^{1-\varepsilon}$. We prove a stronger version with a...
Nonlinear state estimation under structural model uncertainty remains a fundamental challenge in autonomous Guidance, Navigation, and Control (GNC) systems. Conventional Bayesian filtering separates state propagation from measurement correction, allowing model mismatch to accumulate during propagation, resulting in...
Bönisch, Duhr, and Maggio introduced three meromorphic modular forms \(C_4,C_{6a},C_{6b}\) on \(Γ_0(2)\), arising from a hypergeometric K3 family, and conjectured that they are magnetic of depths \(1,2,2\). Writing \[ C_4=\sum_{n\ge1}c_4(n)q^n,\qquad C_{6a}=\sum_{n\ge1}c_{6a}(n)q^n,\qquad C_{6b}=\sum_{n\ge1}c_{6b}(n)q^n, \] we prove...
Minimum-variance portfolios are well known to be highly sensitive to covariance estimation error. In this paper, we show that by imposing a block diagonal correlation structure, we can derive closed-form expressions for long-only minimum-variance portfolios that make this fragility explicit. These analytical solutions reveal that...
Let $f$ be the density of the Chernoff random variable $\mathrm{argmax}_{t\in\mathbb{R}}\{W(t)-t^2\}$, where $W$ is a two-sided Brownian motion. This note proves the conjecture of Balabdaoui and Wellner (2014) that $f$ is strongly log-concave. The proof was generated in its entirety by GPT-5.6 Sol.
We study nonlinear parabolic equations with delayed diffusion terms governed by finite signed measure kernels. The atom of the kernel at the origin is absorbed into the present-time operator, while the remaining part is treated as a residual delay kernel. Under structural assumptions on the effective present-time operators and a...
We investigate the integrality property of an arithmetic function defined on the set of odd numbers $n\geq 3$. It is constructed via Bernoulli numbers. As a result, we show that this function unifies three distinct number classes -- primes, Carmichael numbers, and Giuga numbers -- into a single integrality criterion. The article is...
Motivated by the open problem of classifying Hopf $2$-cocycles for affine algebraic reductive groups $G$ over $\mathbb{C}$, in particular by \cite[Question 7.2]{EG1} which concerns minimal Hopf $2$-cocycles, we classify (minimal) Hopf $2$-cocycles for affine algebraic reductive groups $G$ whose connected component of the identity is a...
A Schubert variety $X_0$ on a rational homogenous space $X=G/P$ is said to be homologically rigid, if any subvariety $Z$ on $X$ representing the same homology class with $X_0$ must satisfy $Z=g\cdot X_0$ for some $g\in{\rm Aut_0}(X)$. We say $X_0$ is Schur rigid, if furthermore any subvariety $Z$ on $X$ whose homology class is a...
We construct a smooth projective variety in positive characteristic whose $\mathbb{Q}_{\ell}$-coefficient etale cohomology ring is not the scalar extension of any graded $\mathbb{Q}$-algebra, providing an example of a new type of obstruction to characteristic zero liftability.
Let $G$ be a finite group, let $V$ be a finite-dimensional $G$-module over a field $k$, and let $U$ be a $G$-submodule of $V$. Wehlau conjectured that the corresponding Noether numbers satisfy $β\bigl(k[U]^G\bigr)\leq β\bigl(k[V]^G\bigr)$. We disprove this conjecture in characteristic $2$. For the dihedral group $D_8$, we construct an...
This paper investigates the geometry of foliations on smooth algebraic varieties over an algebraically closed field of arbitrary characteristic $p \ge 0$. We address several specific features of foliations in positive characteristic, aiming to highlight both similarities and differences with the characteristic zero case. First, we...
Building on our previous general computational solution to Bellman's Lost-in-a-Forest Problem, we present a new approach and analytical formulas for the previously well-known escape path for the infinite unit-strip forest and unit broadworm by Zalgaller. Earlier studies addressed these problems exclusively through geometric methods....
In this paper, we study and characterise the natural embedding of the Ree-Tits octagons. Keywords: Ree-Tits Octagon, Metasymplectic space, generalised octagon, generalised polygon
The compressible fluid-solid interaction (FSI) with a thermomechanical phase transition is formulated at large strains within the Eulerian frame. For the deviatoric part, the Jeffreys (also called anti-Zener) rheology with an additional viscosity is adopted. The core philosophy governing the mechanical solid-liquid transition is that...
We study doubly reflected backward stochastic differential equations with jumps and two completely separated right-continuous with left limits barriers in a filtration generated by an inhomogeneous Levy process. We establish existence and uniqueness results under a stochastic Lipschitz condition on the driver by means of a...
We study polynomial holomorphic $1$-forms in $\mathbb{C}^2$ that are homologically trivial along the fibers of meromorphic pencils of the form $ φ= \frac{f^p}{g^q}, $ where $f,g$ are holomorphic functions (possibly polynomials) in general position and $(p,q)=1$. We first establish a homological characterization of relative exactness:...
The theorem of Mendel and Naor (2013) associates to every compact metric probability space a large ultrametric subset carrying a probability measure controlled from above on balls by the original measure. Building on Bartal's Ramsey decompositions (2021) and the author's previous proofs of the ultrametric skeleton theorem for doubling...
We construct a vector space whose defining characteristics are rooted in univariate regular variation of random variables. Specifically, the base vector space $\mathbb{V}_b$ consists of random variables whose limiting tail probabilities, when scaled by regularly varying functions of the form $b(s)=s^αL(s)$, are finite. Defining a...
Tarski's geometry of solids reconstructs point-like objects from concentric families of spherical regions rather than taking points as primitive entities. We formalize this reconstruction in Coq within a nominal mereological framework inspired by Lesniewski. The main question is how a regional, point-free geometry can support a...
Two families of polytopes underlie the combinatorics of associative operations. Associahedra and multiplihedra respectively capture the information in the operation itself and in the morphisms that respect that operation. The first applications of these polytopes, from Stasheff, were for modeling homotopy associative spaces and their...
Barry introduced a central transform of integer sequences and proposed four conjectures concerning the Hankel transforms of central transform of four rational families. We prove these four conjectures. The proofs are unified within a common algebraic framework: we interpret the Hankel determinants as Gram determinants and use a basis...