We calculate the cycle class of the Hurwitz divisor D2 on Mg for g = 2k given by the degree k+1 covers of P with simple ramification points, two of which lie in the same fibre. We also study some aspects of the geometry of the natural map from the Hurwitz space H2k,k+1 to the moduli space M2k .

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).

Ž . Let A be an n = n matrix and let s A be its spectrum. The stability Ž . Ž .4 modulus of A is s A s max Re l: l g s A , and A is said to be stable Ž . if s A 0. The stability of a matrix is related to the Routh]Hurwitz problem on the number of zeros of a polynomial that have negative real parts. Much research has been devoted to the latter. The first solution w x dates back to Sturm 21, p. 3...

= ∑ n1>...>nr>0 q11 . . . qirnrn1 1 . . . n −sr r , with q a root of unit and i = (i1, . . . , ir) a composition. These sums converge when s1 > 1. To study simultaneously these families of polyzêtas, the colored Hurwitz polyzêtas, for a composition s = (s1, . . . , sr) and a tuple of complex numbers ξ = (ξ1, . . . , ξr) and a tuple of parameters in ]−∞; 1[, t = (t1, . . . , tr), are defined by [6]

We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interpreted as a representation-ringvalued relative Riemann-Hurwitz formula for fam...

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.

Simple proofs of the Hermite–Biehler and Routh–Hurwitz theorems are presented. The total nonnegativity of the Hurwitz matrix of a stable real polynomial follows as an immediate corollary. © 2003 Elsevier Inc. All rights reserved. AMS classification: 93D05; 34D99; 12D10; 26C05; 26C10; 30C15; 15A23; 15A48; 15A57