We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a nonresonant complex rank one local system. Aomoto and Kita determined this GaussManin connection for arrangements in general position. We use their results and an algorithm constructed in this paper to determine this Gauss-Manin connection for all arrangements.

We consider an interval map which is a generalization of the well-known Gauss transformation. In particular, we prove a result concerning the asymptotic behavior of the distribution functions of this map. 1. Introduction. In 1800, Gauss studied the following problem. In modern notation, it reads as follows. Write x ∈ [0, 1) as a regular continued fraction

The paper presents a simple combinatorial method to encode 3-dimensional manifolds, basing on their Heegaard diagrams. The notion of a Gauss diagram of a 3-manifold is introduced. We check the conditions for a Gauss diagram to represent a closed manifold and a manifold with boundary.

We give the best possible upper bound on the number of exceptional values and totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic Bryant surfaces and some partial results on the Osserman problem for algebraic Bryant surfaces. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space.

In this paper we provide a mathematical reconstruction of what might have been Gauss’ own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction presented here is entirely based on an accurate study of Gauss’ own work on terrestrial magnetism. A brief di...

Many problems in applied mathematics require the evaluation of the sum of N Gaussians at M points in space. The work required for direct evaluation grows like NM as N and M increase; this makes it very expensive to carry out such calculations on a large scale. In this paper, an algorithm is presented which evaluates the sum of N Gaussians at M arbitrarily distributed points in C. (N + M) work, ...

Throughout the paper, we denote this equation by E(a, b, c). Since Kummer we know that in general there are 24 hypergeometric series which express solutions of E(a, b, c). The subject of this paper is Gauss hypergeometric functions when some or all of the numbers a, b, c, c − a, c − b, a − b, c − a − b are integers. We refer to these functions and to corresponding hypergeometric equations as de...