Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point.

of the Dissertation Periodic Structure in Two-Dimensional Riemann Problems for Hamilton-Jacobi Equations by John Dominick Pinezich Doctor of Philosophy in Applied Mathematics and Statistics State University of New York at Stony Brook 1998 This dissertation investigates the structure of two-dimensional Riemann problems for Hamilton-Jacobi equations. We show that it is possible for the Riemann so...

Conformal (respectively, anticonformal) automorphisms of the Riemann sphere are provided by Möbius extended Möbius) transformations. A Kleinian group an group) is a discrete transformations and transformations, necessarily containing ones). function finitely generated with invariant connected component its region discontinuity. structural decomposition groups, in terms Klein-Maskit combinatio...

The Ginzburg–Landau equations were proposed in the superconductivity theory to describe mathematically intermediate state of superconductors which normal conductivity is mixed with superconductivity. It turned out that these have interesting and non-trivial generalizations. First all, they can be extended arbitrary compact Riemann surfaces. Next, generalized dimension 3 as dynamical (or hyperbo...

Mathematisch-naturwissenschaftlichen Fakultät Doctor of Philosophy Twisted Second Moments and Explicit Formulae of the Riemann Zeta-Function by Nicolas Martinez Robles Verschiedene Aspekte, die analytische Zahlentheorie und die Riemann zeta-Funktion verbinden, werden erweitert. Dies beinhaltet: 1. explizite Formeln, die eine Verbindung zwischen der Möbiusfunktion und den nichttrivialen Nullstel...

If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L271. We als...

We describe a new deterministic algorithm for the problem of constructing k th power nonresidues in nite elds GF p where p is prime and k is a prime divisor of p We prove under the assumption of the Extended Riemann Hypothesis ERH that for xed n and p our algorithm runs in polynomial time Unlike previous algorithms for this problem this polynomial time bound holds even if k is very large More g...

Because of their close relation to moduli spaces of Riemann surfaces, the mapping class groups of orientable surfaces have been the attention of much mathematical research for a long time. Less well studied is the mapping class group of nonorientable surfaces. But recently, the study of mapping class groups has also been extended to the non-orientable case. This paper contributes to this progra...

The paper focuses on the deterministic complexity of factoring polynomials over finite fields assuming the extended Riemann hypothesis (ERH). By the works of Berlekamp (1967, 1970) and Zassenhaus (1969), the general problem reduces deterministically in polynomial time to finding a proper factor of any squarefree and completely splitting polynomial over a prime field Fp . Algorithms are designed...

In this paper we find, for any arbitrary finite topological type, a compact Riemann surface M, an open domain M ⊂ M with the fixed topological type, and a conformal complete minimal immersion X : M → R3 which can be extended to a continuous map X : M → R3, such that X|∂M is an embedding and the Hausdorff dimension of X(∂M) is 1. We also prove that complete minimal surfaces are dense in the spac...