We establish a series of integral formulae involving the Hurwitz zeta function. Applications are given to integrals of Bernoulli polynomials, log Γ(q) and log sin(q).

We present a novel method for checking the Hurwitz stability of a polytope of matrices. First we prove that the polytope matrix is stable if and only if two homogenous polynomials are positive on a simplex, then through a newly proposed method, i.e., the weighted difference substitution method, the latter can be checked in finite steps. Examples show the efficiency of our method.

Let E be an elliptic curve over a field K of characteristic 6= 2 and let N > 1 be an integer prime to char(K). The purpose of this paper is to study the family of genus 2 covers of E of fixed degree N , i.e. those covers f : C → E for which C/K is a curve of genus 2 and deg(f) = N . Since we can (without loss of generality) restrict our attention those covers that are normalized in the sense of...

The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrati...

In defining a special type of group, such as commutative group or finite group, one may be content to add a suitable postulate, or possibly a number of postulates, describing the special property under consideration, to any set of postulates for a general group. I t is, however, of interest to pursue the matter further, to determine whether a simplified set of postulates can be set up which wil...

f(x1, . . . , xd) ∈ Ud (C) ⇐⇒ f(ıx1, . . . , ıxd) ∈ Hd(C) Consequently, many of the properties of H(C) follow immediately from the properties of U (C), and their proofs will be omitted. See [1–7, 13] for more information about U (C) and H(C). We use the notation in [7], and let RHP denote the right half plane. We do not cover well-known topics in one variable such as Routh-Hurwitz, the Edge the...

Hurwitz groups are the nontrivial finite quotients 2 3 7 of the ( 2 , 3 , 7 ) triangle group (x, y\x — y = (xy) = 1) . This paper gives a brief survey of such groups, their significance, and some of their properties, together with a description of all examples known to the author.

In this work, we investigate the dynamical behavior of a fractional order cholera model. All the feasible equilibria for the system are obtained and the conditions for the existence of interior equilibrium are determined. Local stability analysis of the cholera model is studied by using the fractional Routh-Hurwitz stability conditions. Our results indicate the potential of fractional-order cho...

We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function ζ(s, a) about s = 1. Additionally we obtain series and integral representations of a sum Sγ(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum Sγ(n) + n is an impo...

We provide an explicit formula for the Tornheim double series T (a, 0, c) in terms of an integral involving the Hurwitz zeta function. For integer values of the parameters, a = m, c = n, we show that in the most interesting case of even weight N := m + n the Tornheim sum T (m, 0, n) can be expressed in terms of zeta values and the family of integrals