In this paper, we study the Markoff-Hurwitz equations x0 + ...+x 2 n = ax0 · · ·xn. The variety V defined by this equation admits a group of automorphisms A ∼= Z/2∗· · ·∗Z/2 (an n+1 fold free product). For a solution P on this variety, we consider the number NP (t) of points Q in the A-orbit of P with logarithmic height h(Q) less than t. We show that if a is rational, and P is a non-trivial rat...

The main result proved in the paper is the computation of the explicit equations defining the Hurwitz schemes of coverings with punctures as subschemes of the Sato infinite Grassmannian. As an application, we characterize the existence of certain linear series on a smooth curve in terms of soliton equations.

In this paper, we mainly study the robust stability of linear continuous systems with parameter uncertainties, a more general kind of uncertainties for system matrices is considered, i.e., entries of system matrices are rational functions of uncertain parameters which are varying in intervals. we present a method which can check the robust Hurwitz stability of such uncertain systems in finite s...

In this paper, using the approach of Hurwitz and the necessary conditions given in 4,6], we construct a third order linear diierential equation whose diierential Galois group is the primitive subgroup F SL3 36 S L(3; C) of order 108.

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 .

By means of the Hadamard product, the present paper introduces new classes, Σ t, * a (α, β, ρ) and Σ t a (α, β, ρ) of Hurwitz-Lerch-Zeta function in the punctured disk U * = {z : 0 < |z| < 1}. In addition, the study investigates a number of inclusion relationships, properties and derives some interesting properties depending on some integral properties.

We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a certain hypergeometric differential equation. This generalizes the result of Deligne and Rapoport on the reduction of the modular curve X (p).

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M , the Hurwitz constant of M . It measures how well all g...

The Stieltjes constants γk(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about its only pole at s = 1. We present the relation of γk(1) to the ηj coefficients that appear in the Laurent expansion of the logarithmic derivative of the Riemann zeta function about its pole at s = 1. We obtain novel integral representations of the Stieltjes constants and new d...