The characteristic polynomial and the matchings polynomial of a weighted oriented graph

Let $G^{\sigma }$ be a weighted oriented graph with skew adjacency matrix $S(G^{\sigma })$. Then $G^{\sigma }$ is usually referred as the weighted oriented graph associated to $S(G^{\sigma })$. Denote by $?(G^{\sigma }\text{;}\lambda )$ the characteristic polynomial of the weighted oriented graph $G^{\sigma }$, which is defined as$?(G^{\sigma }\text{;}\lambda )=\mathit{det}(\lambda I_n-S(G^{\sigma }))=\sum _{i=0}^n{a}_i(G^{\sigma }){\lambda }^{n-i}\text{.}$In this paper, we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph. Then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph. In addition, some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given.     

An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equations (NLS) on ${\mathbb{R}}^d$ with random and rough initial data. By working in the framework of $L^p({\mathbb{R}}^d)$ spaces, $p>2$, we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate.     

LeVeque type inequalities and discrepancy estimates for minimal energy configurations on spheres

Let ${\mathbb{S}}^d$ denote the unit sphere in the Euclidean space ${\mathbb{R}}^{d+1}(d\ge 1)$. We develop LeVeque type inequalities for the discrepancy between the rotationally invariant probability measure and the normalized counting measures on ${\mathbb{S}}^d$. We obtain both upper bound and lower bound estimates. We then use these inequalities to estimate the discrepancy of the normalized counting measures associated with minimal energy configurations on ${\mathbb{S}}^d$.     

Cocycle deformations and Galois objects for semisimple Hopf algebras of dimension p3 and pq2

Let p and q be distinct prime numbers. We study the Galois objects and cocycle deformations of the noncommutative, noncocommutative, semisimple Hopf algebras of odd dimension $p^3$ and of dimension $p{q}^2$. We obtain that the $p+1$ non-isomorphic self-dual semisimple Hopf algebras of dimension $p^3$ classified by Masuoka have no non-trivial cocycle deformations, extending his previous results for the 8-dimensional Kac–Paljutkin Hopf algebra. This is done as a consequence of the classification of categorical Morita equivalence classes among semisimple Hopf algebras of odd dimension $p^3$, established by the third-named author in an appendix.     

Cauchy transform and Poisson’s equation

Let $u\in W^{1,p}\cap W_0^{1,p}$, $1?p?\infty$ be a solution of the Poisson equation $\Delta u=h$, $h\in L^p$, in the unit disk. We prove ${6?u6}_{L^p}?a_p{6h6}_{L^p}$ and ${6?u6}_{L^p}?b_p{6h6}_{L^p}$ with sharp constants $a_p$ and $b_p$, for $p=1$, $p=2$, and $p=\infty$. In addition, for $p>2$, with sharp constants $c_p$ and $C_p$, we show ${6?u6}_{L^{\infty }}?c_p{6h6}_{L^p}$ and ${6?u6}_{L^{\infty }}?C_p{6h6}_{L^p}$. We also give an extension to smooth Jordan domains.These problems are equivalent to determining a precise value of the $L^p$ norm of the Cauchy transform of Dirichlet’s problem.     

Pseudo Drazin inverses in associative rings and Banach algebras

Motivated by strongly $\pi$-regular elements and quasipolar elements, we introduce the concept of pseudopolar elements. An element $a\in R$ is called pseudopolar if there exists $p\in R$ such that $p^2=p\in {\mathrm{comm}}^2(a)\text{,}a+p\in U(R)\mathrm{and}{a}^{k}p\in J(R)$ for some positive integer k. This concept can be used exactly to define a pseudo Drazin inverse in associative rings and Banach algebras. We connect pseudopolar rings with strongly $\pi$-regular rings, semiregular rings, uniquely strongly clean rings and uniquely bleached local rings. Some basic properties of pseudo Drazin inverses are obtained in associative rings and Banach algebras.