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...

New family of flat potential (Darboux-Egoroff) metrics on the Hurwitz spaces and corresponding Frobenius structures are found. We consider a Hurwitz space as a real manifold. Therefore the number of coordinates is twice as big as the number of coordinates used in the construction of Frobenius manifolds on Hurwitz spaces found by B.Dubrovin more than 10 years ago. The branch points of a ramified...

This note presents an elementary proof of the familiar Routh-Hurwitz test. The proof is basically one continuity argument, it does not rely on Sturm chains, Cauchy index and the principle of the argument and it is fully self-contained. In the same style an extended Routh-Hurwitz test is derived, which finds the inertia of polynomials.

We define maps which induce mediant convergents of Rosen continued fractions and discuss arithmetic and metric properties of mediant convergents. In particular, we show equality of the ergodic theoretic Lenstra constant with the arithmetic Legendre constant for each of these maps. This value is sufficiently small that the mediant Rosen convergents directly determine the Hurwitz constant of Diop...

The first published proof of key analytic properties of zeta-functions associated with quadratic forms is due to Paul Epstein [2] in 1903 (submitted January 1902). The corresponding name “Epstein zeta-functions” was introduced by Edward Charles Titchmarsh in his article [15] from 1934; soon after, this terminology became common through influential articles of Max Deuring and Carl Ludwig Siegel....

We explore the structure of compact Riemann surfaces by studying their isometry groups. First we give two constructions due to Accola [1] showing that for all g ≥ 2, there are Riemann surfaces of genus g that admit isometry groups of at least some minimal size. Then we prove a theorem of Hurwitz giving an upper bound on the size of any isometry group acting on any Riemann surface of genus g ≥ 2...

This was discovered by Euler in the 18th century, forgotten, and then rediscovered in the 19th century by Hamilton in his work on quaternions. Shortly after Hamilton’s rediscovery of (1.2) Cayley discovered a similar 8-square identity. In all of these sum-of-squares identities, the terms being squared on the right side are all bilinear expressions in the x’s and y’s: each such expression, like ...

Kiister, G., On the Hurwitz product of formal power series and automata, Theoretical Computer Science 83 (1991) 261-273. The Hurwitz (shuffle) product defined on formal power series is generalized to matrices and therefore to automata. The resulting constructions are then used to study commutative power series and abstract families of power series. In particular, the families of power series re...

In this paper we prove a version of Mordell’s inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that 16dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz structu...