We present another proof of a theorem due to Hoffman and Osserman in Euclidean space concerning the determination of a conformal immersion by its Gauss map. Our approach depends on geometric quantities, that is, the hyperbolic Gauss map G and formulae obtained in hyperbolic space. We use the idea that the Euclidean Gauss map and the hyperbolic Gauss map with some compatibility relation determin...

A novel sparse Gauss–Hermite quadrature filter is proposed using a sparse-grid method for multidimensional numerical integration in the Bayesian estimation framework. The conventional Gauss–Hermite quadrature filter is computationally expensive for multidimensional problems, because the number of Gauss–Hermite quadrature points increases exponentially with the dimension. The number of sparse-gr...

Abstract. In this note, we give some estimates of the generalized quadrature formula of Gauss-Jacobi type for phi-preinvex functions.

This theory provides a compact formulation of Gauss-Jordan elimination for matrices represented as functions. Its distinctive feature is succinctness. It is not meant for large computations. 1 Gauss-Jordan elimination algorithm theory Gauss-Jordan-Elim-Fun imports Main begin Matrices are functions: type-synonym ′a matrix = nat ⇒ nat ⇒ ′a In order to restrict to finite matrices, a matrix is usua...

It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turr an, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. 0. Introduction The idea of constructing quadrature rules that are exact for rational functions with prescribed poles, rather than for polynomials, has received...

In this paper, we report that we have solved the shortest vector problem (SVP) over a 128-dimensional lattice, which is currently the highest dimension of the SVP that has ever been solved. The security of lattice-based cryptography is based on the hardness of solving the SVP in lattices. In 2010 Micciancio et al. proposed a Gauss Sieve algorithm for heuristically solving the SVP using list L o...

In the previous article (Found Phys. Lett. 16 325-341), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a finite group.