Hodge integrals are a class of intersection numbers on moduli spaces of curves involving the tautological classes λi, which are the Chern classes of the Hodge bundle E. In recent years Hodge integrals have shown a great amount of interconnections with Gromov-Witten theory and enumerative geometry. The classical Hurwitz numbers, counting the numbers of ramified Covers of a curve with an assigned...

Liu, Liu, and Liu, in the paper “A novel three-dimensional autonomous chaos system, Chaos, Solitons and Fractals. 39 (2009) 1950-1958”, introduce a novel three-dimensional autonomous chaotic system. In this paper, the fractional-order case is considered. The lowest order for the system to remain chaotic is found via numerical simulation. Stability analysis of the fractional-order system is stud...

In the qualitative study of systems ordinary differential equations, we can analyze dynamics solutions by observing whether small variations or modifications in initial conditions produce changes future, this intuitive idea was formalized and studied Lyapunov reflects concept stability. For both linear nonlinear stability using criteria to obtain Hurwitz type polynomials. paper, a prey-predator...

In this chapter, we will see a proof of the analytic continuation of the Riemann zeta function ζ(s) and the Dirichlet L function L(s, χ) via the Hurwitz zeta function. This then gives rise to a functional equation for ζ(s) and a direct computation for the value of this function at negative integer points. 1. The Hurwitz zeta function We have already seen the definition of the Riemann zeta funct...

We formulate Goldbach type questions for Gaussian, Hurwitz, Octavian and Eisenstein primes.

Many problems start with two (compact Riemann surface) covers f : X → Pz and g : Y → Pz of the Riemann sphere, Pz, uniformized by a variable z. Some data problems have f and g defined over a number field K, and ask: What geometric relation between f and g hold if they map the values X(OK/p) and Y (OK/p) similarly for (almost) all residue classes of OK/p. Variants on Davenport’s problem interpre...

In 1826 N. Abel found a generalization of the binomial formula. In 1902 Abel’s theorem was further generalized by A. Hurwitz. In this paper we describe constructions that provide infinitely many identities each being a generalization of a Hurwitz’s identity. Moreover, we give combinatorial interpretations of all these identities as the forest volumes of certain directed graphs. Published by Els...

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group Sn. We also fin...

The classical Hurwitz numbers which count coverings of a complex curve have an analog when the curve is endowed with a theta characteristic. These “spin Hurwitz numbers”, recently studied by Eskin, Okounkov and Pandharipande, are interesting in their own right. By the authors’ previous work, they are also related to the Gromov-Witten invariants of Kähler surfaces. We prove a recursive formula f...