Chamber behavior of double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers CP¹→CP¹ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.     

Sums Involving the Hurwitz Zeta Function

We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers.     

Hodge Integrals and Hurwitz Numbers via Virtual Localization

We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P¹ with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. Our proof uses virtual localization on the moduli space of stable maps. We describe how the proof could be simplified by the proper algebro-geometric definition of a 'relative space'. Such a space has recently been defined by J. Li.     

Polynomial Hurwitz Numbers and Intersections on \overline M _{0,k}

We express Hurwitz numbers of polynomials of arbitrary topological type in terms of intersection numbers on the moduli space of curves of genus zero with marked points.     

Some Relations for Double Hurwitz Numbers

We obtain new relations for Hurwitz numbers of functions with one polynomial and one arbitrary critical value. (All other critical values are supposed to be simple.) This is a straightforward generalization of our earlier results on Hurwitz numbers of functions with one polynomial critical value.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 91–94, 2005Original Russian Text Copyright © by S. V. ShadrinSupported in part by grants RFBR-01-01-00660, RFBR-02-01-22004, and NSh-1972.2003.1.     

Wreath Hurwitz numbers, colored cut-and-join equations, and 2-Toda hierarchy

Let G be arbitrary finite group,define H G· (t;p +,p) to be the generating function of G-wreath double Hurwitz numbers.We prove that H G· (t;p +,p) satisfies a differential equation called the colored cutand-join equation.Furthermore,H G·(t;p +,p) is a product of several copies of tau functions of the 2-Toda hierarchy,in independent variables.These generalize the corresponding results for ordinary Hurwitz numbers.     

Explicit formulae for one-part double Hurwitz numbers with completed 3-cycles

We prove two explicit formulae for one-part double Hurwitz numbers with completed 3-cycles. We define “combinatorial Hodge integrals” from these numbers in the spirit of the celebrated ELSV formula. The obtained results imply some explicit formulae and properties of the combinatorial Hodge integrals.     

Enumerative Geometry for Plane Cubic Curves in Characteristic 2

Consider the plane cubic curves over an algebraically closed field of characteristic 2. By blowing up the parameter space P⁹ twice we obtain a variety B of complete cubics. We then compute the characteristic numbers for various families of cubics by intersecting cycles on B.     

Derivatives of the Hurwitz Zeta function for rational arguments

The functional equation for the Hurwitz Zeta function ζ(s,a) is used to obtain formulas for derivatives of ζ(s,a) at negative odd s and rational a. For several of these rational arguments, closed-form expressions are given in terms of simpler transcendental functions, like the logarithm, the polygamma function, and the Riemann Zeta function.     

New inequalities for the Hurwitz zeta function

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function.     

-analogue triangular numbers and distance geometry

The so-called ``-identities' play a major role in classical combinatorics. Most of them can be viewed as arising somehow in the context of hypergeometric series. Here we present a ``sum of squares' identity involving -analogues of the triangular numbers that, by contrast, arises in the context of distance geometry.

     

The chromatic numbers of double coverings of a graph

If we fix a spanning subgraph H of a graph G, we can define a chromatic number of H with respect to G and we show that it coincides with the chromatic number of a double covering of G with co-support H. We also find a few estimations for the chromatic numbers of H with respect to G.     

Rough convergence of double sequences of fuzzy numbers

In this paper, we define the concepts of rough convergence and rough Cauchy sequence of double sequences of fuzzy numbers. Then, we investigate some relations between rough limit set and extreme limit points of such sequences.     

Asymptotic positivity of Hurwitz product traces: Two proofs

Consider the polynomial tr(A+tB)^m in t for positive hermitian matrices A and B with m∈N. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative coefficients only. We prove that they are at least asymptotically positive, for the nontrivial case of AB≠0. More precisely, we show—once complex-analytically, once combinatorially—that the k-th coefficient is positive for all integer m?m₀, where m₀ depends on A, B and k.     

A new approximate functional equation for Hurwitz zeta function for rational parameter

For Hurwitz zeta function ζ(s, (a/k)) witha = 1,2,3,…,k, we obtain a new simple approximate functional equation (uniform ink andt) in critical strip. Our method should prove to be an alternative approach to Atkinson’s method in dealing with , whereL(s, x) is Dirichlet L-series moduloq and s = σ +it.     

On Miki's identity for Bernoulli numbers

We give a short proof of Miki's identity for Bernoulli numbers,

     

Some double sequence spaces of interval numbers defined by Orlicz function

In this paper we introduce some interval valued double sequence spaces defined by Orlicz function and study different properties of these spaces like inclusion relations, solidity, etc. We establish some inclusion relations among them. Also we introduce the concept of double statistical convergence for interval number sequences and give an inclusion relation between interval valued double sequence spaces.     

Shortened recurrence relations for Bernoulli numbers

Starting with two little-known results of Saalschütz, we derive a number of general recurrence relations for Bernoulli numbers. These relations involve an arbitrarily small number of terms and have Stirling numbers of both kinds as coefficients. As special cases we obtain explicit formulas for Bernoulli numbers, as well as several known identities.