On the antipode of Hopf algebras with the dual Chevalley property

In this paper, we study the antipode of a finite-dimensional Hopf algebra H with the dual Chevalley property and obtain an annihilation polynomial for the antipode. This generalizes an old result given by Taft and Wilson in 1974. As consequences, we show that 1) the quasi-exponent of H is the same as the exponent of its coradical, that is, $\mathrm{qexp}(H)=\exp ?(H_0)$; 2) $\mathrm{qexp}(H?\mathbb{k}〈S^2〉)=\mathrm{qexp}(H)$.     

On discrete Brunn-Minkowski and isoperimetric type inequalities

We show that the lattice point enumerator ${\mathrm{G}}_n(?)$ satisfies${\mathrm{G}}_n{(tK+sL+{(?1,?t+s?)}^n)}^{1/n}\ge t{\mathrm{G}}_n{(K)}^{1/n}+s{\mathrm{G}}_n{(L)}^{1/n}$ for any $K,L?{\mathbb{R}}^n$ bounded sets with integer points and all $t,s\ge 0$.We also prove that a certain family of compact sets, extending that of cubes ${[?m,m]}^n$, with $m\in \mathbb{N}$, minimizes the functional ${\mathrm{G}}_n(K+t{[?1,1]}^n)$, for any $t\ge 0$, among those bounded sets $K?{\mathbb{R}}^n$ with given positive lattice point enumerator.Finally, we show that these new discrete inequalities imply the corresponding classical Brunn-Minkowski and isoperimetric inequalities for non-empty compact sets.     

Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

A tight bound for conflict-free coloring in terms of distance to cluster

Given an undirected graph $G=(V,E)$, a conflict-free coloring with respect to open neighborhoods (CFON coloring) is a vertex coloring such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for such a coloring is the CFON chromatic number of G, denoted by ${\chi }_{ON}(G)$.In previous work [WG 2020], we showed the upper bound ${\chi }_{ON}(G)\le \mathsf{dc}(G)+3$, where $\mathsf{dc}(G)$ denotes the distance to cluster parameter of G. In this paper, we obtain the improved upper bound of ${\chi }_{ON}(G)\le \mathsf{dc}(G)+1$. We also exhibit a family of graphs for which ${\chi }_{ON}(G)>\mathsf{dc}(G)$, thereby demonstrating that our upper bound is tight.     

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer n we denote by $\mathrm{\#}E({\mathbb{F}}_{q^n})$ the number of rational points on E (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q<50$ and $n\le 1000$.