Using functional properties of the Hurwitz zeta function and symbolic derivatives of the trigonometric functions, the function ζ(2n + 1, p/q) is expressed in several ways in terms of other mathematical functions and numbers, including in particular the Glaisher numbers. 2000 Mathematics Subject Classification. Primary 11M35, 33B99. Secondary 11B75, 33E20.

We use the following convention: A, B, C, A1, B1, C1, A2, B2, C2 are points of E2 T, l1, m1, n1 are real numbers, and X, Y , Z are subsets of E2 T. Let us consider X, Y . We introduce X is parallel to Y as a synonym of X misses Y . Let us consider X, Y , Z. We say that X, Y , Z are concurrent if and only if: (Def. 1) X is parallel to Y and Y is parallel to Z and Z is parallel to X or there exis...

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these ...

Let f : Σ̃ d:1 −→ Π Σ denote a branched cover between closed, connected, and orientable surfaces, where d is the global degree, Π = {Π1, . . . ,Πn}, n is the number of branching points, and Πi is the partition of d given by the local degrees over the i-th branching point. If l(Π) is the total length of Π, then the Riemmann-Hurwitz formula asserts that χ(Σ̃) − l(Π) = d · (χ(Σ) − n). A candidate br...

Introduction: Let C be a nonhyperelliptic smooth curve of genus π. An involution of C is an automorphism φ : C−→C such that φ = id. It induces a double cover γ : C−→C/φ = X , where X is a smooth curve of genus g. We say that C has an involution of genus g. By Hurwitz formula, we know that π ≥ 2g − 1. It is well know that a general smooth curve of genus π ≥ 3 has not nontrivial automorphisms. In...

We consider n-dimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true ...

Introduction The differential equation H(y) = 0 (see below) due to Hurwitz goes back to the famous work “Ueber die Transformation siebenter Ordnung der elliptischen Functionen” of F. Klein [14]. In this work, Klein made an important contribution to the much discussed question of the solvability of algebraic equations by means of elliptic modular functions. In order to investigate the transforma...

In this article we describe the cell decompositions of the moduli space of Riemann surfaces due to Harer, Mumford and Penner and its relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational convex polytopes. We show how to effectively calculate the number of lattice points and the volumes over all cells. These calc...

In this paper, the neutrix limit is used to extend the definition of the Hurwitz zeta function ζ(α, x) and its partial derivatives to the whole complex plane except for non-positive integers α, in particular, the values of ζ(1, x) is obtained. This definition is equivalent to the Hermite’s integral of ζ(α, x) as α 6= 1, 0,−1, . . .. Moreover, some properties of ζ(1, x) are established and we fi...