Related papers: G-functions, motives, and unlikely intersections -…
A well-known conjecture of Gross and Zagier states that the values of the higher automorphic Green's function at pairs of points with complex multiplication in the upper half-plane are proportional to the logarithm of an algebraic number.…
Recently, we have witnessed tremendous applications of algebraic intersection theory to branches of mathematics, that previously seemed very distant. In this article we review some of them. Our aim is to provide a unified approach to the…
Using generalized hypergeometric functions to perform symbolic manipulation of equations is of great importance to pure and applied scientists. There are in the literature a great number of identities for the Meijer-G function. On the other…
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…
We present certain new properties about the intersection numbers on moduli spaces of curves $\bar{\sM}_{g,n}$, including a simple explicit formula of $n$-point functions and several new identities of intersection numbers. In particular we…
This article aims to reinforce the broad applicability of the umbral approach to address complex mathematical challenges and contribute to various scientific and engineering endeavors. The umbral methods are used to reformulate the…
We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd…
A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval $(0,1)$ is studied. This type of polynomials have direct applications in…
In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof…
Applying advances in exact computations of supersymmetric gauge theories, we study the structure of correlation functions in two-dimensional N=(2,2) Abelian and non-Abelian gauge theories. We determine universal relations among correlation…
We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different…
A relationship between two old mathematical subjects is observed: the theory of hypergeometric functions and the separability in classical mechanics. Separable potential perturbations of the integrable billiard systems and the Jacobi…
Let $G=(V,E)$ be a simple connected graph and $d_i$ be the degree of its $i$th vertex. In a recent paper [J. Math. Chem. 46 (2009) 1369-1376] the first geometric-arithmetic index of a graph $G$ was defined as $$GA_1=\sum_{ij\in E}\frac{2…
The goal of the paper is two-fold. At first, we attempt to give a survey of some recent applications of symmetric polynomials and divided differences to intersection theory. We discuss: polynomials universally supported on degeneracy loci;…
The g-convexity of functions on manifolds is a generalization of the convexity of functions on Rn. It plays an essential role in both differential geometry and non-convex optimization theory. This paper is concerned with g-convex smooth…
Using the connection between intersection theory on the Deligne-Mumford spaces and the edge scaling of the GUE matrix model (see math.CO/9903176, math.AG/0101147), we express the n-point functions for the intersection numbers as…
In this article, we show a new general linear independence criterion related to values of $G$-functions, including the linear independence of values at algebraic points of contiguous hypergeometric functions, which is not known before. Let…
This work is devoted to the study of the relationships between graph theory and the qualitative analysis of ordinary differential equations, with a special focus on two-dimensional systems. In particular, we reinterpret classical results…
We present the theory of multifunctions applied to graphs. Its interesting feature is that walks are recognized as iterations. We consider the graphs with arbitrary number of vertices which are determined by multifunctions. The mutually…
In the first part of the paper we give a definition of G_q-function and we establish a regularity result, obtained as a combination of a q-analogue of the Andre'-Chudnovsky Theorem [And89, VI] and Katz Theorem [Kat70, \S 13]. In the second…