数学

We study the large-scale behavior of the coincidence set of perturbations of global solutions to the classical obstacle problem in $\mathbb{R}^n\setminus B_1$, with blow-down invariant in the $e_n$ direction. In dimensions $n\geq 3$, we…

M.V. Zaicev and S.K. Segal, as well as S. D\u{a}sc\u{a}lescu, B. Ion, C. N\u{a}st\u{a}sescu, and D. Raios Montes studied certain gradings on matrix rings and algebras - 'elementary' gradings. However, examples of gradings on a matrix ring…

We propose an efficient algorithm for approximating the prime counting function $\pi(x)$ using a structured non-uniform partition derived from generalized triangular numbers. The method yields an incremental estimator whose updates require…

We prove that the special linear Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ over a finite field of characteristic $p$ is generated by two random elements with high probability as $|\mathfrak{sl}_n(\textbf{F}_q)|$ tends to infinity,…

We propose a regularized algorithm, Regularized Newton-SLRA (RN-SLRA), for local manifold--affine intersection problems under weak intersection conditions, motivated in particular by structured low-rank approximation (SLRA). Newton-SLRA is…

In this paper, we establish a completed pre-Lie bialgebra structure on the tensor product of a Leibniz-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Zinbiel algebra. We also obtain such a structure on the tensor product of a…

We study the ring-theoretic structure and representation theory of the super Jordan plane $\mathcal{J}$ over fields of characteristic different from $2$. We prove that $\mathcal{J}$ is prime and classify its prime, primitive, and maximal…

We study Ornstein--Uhlenbeck operators on rooted metric trees equipped with a Gaussian-type measure. Using form methods, we construct Dirichlet and Neumann realisations corresponding, respectively, to killing and reflection at the root. The…

List-coloring, introduced independently by Vizing and by Erd\H{o}s, Rubin, and Taylor in the 1970s, generalizes ordinary vertex coloring by assigning to each vertex its own set of admissible colors. A graph is chromatic-choosable if its…

We prove local-global compatibility results at $p \neq \ell$ for the automorphic group determinants constructed by Scholze, generalising the result of Varma to torsion classes appearing in Betti cohomology. Our argument combines the…

We generalize the notion of twisting endomorphisms, first defined by Castryck-Panny-Vercauteran, to the setting of $\mathcal{O}$-oriented supersingular elliptic curves. We give an algorithm to find supersingular elliptic curves over…

Let I=[a,b] and consider the degree n Lagrange interpolation at the nodes x, where x\in S:={x=(x_0,x_1,...,x_n):a=x_0<x_1<...<x_n=b}. Then the norm of the Lagrange interpolation operator is the maximum of the Lebesgue function L(x,t) on I.…

Let $Y$ be a prequantization bundle over an integral symplectic manifold $(\Sigma,\omega)$. Let $L$ be a closed monotone Lagrangian submanifold that admits a Legendrian lift $\mathcal{L}$ in $Y$. Under the assumption that the minimal Maslov…

This paper is concerned with the existence of non-radial positive classical solutions for the critical H\'enon equation \[ -\Delta u=|x|^\alpha u^{\frac{N+2+2\alpha}{N-2}} \qquad \text{in }\mathbb R^N, \] where \(\alpha>0\) and \(N\ge3\),…

We give an overview of the theory of quasi-$F$-singularities, focusing on their connection with singularities in birational geometry.

In this manuscript, we introduce positivity-preserving correction methods for low-rank approximations of the Vlasov equation. The key idea is to formulate structural properties, including positivity-preservation, as constraints and to seek…

We develop a framework to predict whether a family of Selmer groups has average size that is bounded or unbounded. Applying this framework to certain geometric families of abelian varieties over $\mathbb{Q}$, we give a conjectural…

Scalar inequalities in an order parameter often arise as the $2\times2$ shadow of a stronger Hankel determinant statement. We record a moment-representation criterion: positive exponential and Mellin order representations, together with…

Let $A/\mathbb{Q}$ be a modular abelian variety of analytic rank $0$. If $G$ is a non-trivial finite abelian group such that all prime factors of $\lvert G \rvert$ are sufficiently large in terms of $A$, we show that there are infinitely…

We develop a unified framework for Fujita-type blow-up of solutions to the inhomogeneous semilinear heat equation $$\partial_tu-\Delta u=|u|^p+\mathbf{w}(x), \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0, \cdot)=u_0.$$ The…