We establish sufficient conditions for a matrix to be almost totally positive, thus extending a result of Craven and Csordas who proved that the corresponding conditions guarantee that a matrix is strictly totally positive. Then we apply our main result in order to obtain a new criteria for a real algebraic polynomial to be a Hurwitz one. The properties of the corresponding “extremal” Hurwitz p...

Let D2N be the dihedral group of order 2N , Dic4M the dicyclic group of order 4M , SD2m the semidihedral group of order 2 m, and M2m the group of order 2 m with presentation M2m = 〈α, β | α 2m−1 = β2 = 1, βαβ−1 = α2 m−2+1〉. We classify the orbits in Dn 2N , Dic n 4M , SD n 2m , and M n 2m under the Hurwitz action.

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...

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.

We use a variant of a method of Goncharov, Kontsevich, and Zhao [Go2, Z] to meromorphically continue the multiple Hurwitz zeta function ζd(s; θ) = ∑ 0<n1<···<nd (n1 + θ1) −s1 · · · (nd + θd)d , θk ∈ [0, 1), to C, to locate the hyperplanes containing its possible poles, and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of ζd(s; θ).

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...

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.