An $$ \mathcal{L}- -algorithm is proposed for constructing an infinite sequence of integer solutions to systems linear inequalities in d + 1 variables. The are obtained using recurrence relations order 1. approximation rate estimated by the Diophantine exponent \theta =\frac{m}{n}-\varrho, where ≤ n number inequalities, m = − free variables, and deviation ϱ > 0 can be made arbitrarily small due...

We introduce a renormalization group framework for the study of quasiperiodic skew flows on Lie groups of real or complex n×n matrices, for arbitrary diophantine frequency vectors in R and dimensions d,n. In cases where the group component of the vector field is small, it is shown that there exists an analytic manifold of reducible skew systems, for each diophantine frequency vector. More gener...

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear sub-spaces generated by the best Diophantine approximations. Originally most of these results have been established by the author in [14, 15, 16, 17, 18]. Here we collect all of them togeth...

Normaliz is a program for the computation of Hilbert bases of rational cones and the normalizations of affine monoids. It may also be used for solving diophantine linear systems. In this paper we present the algorithms implemented in the program.

A lot of Scientific and Engineering problems require the solution of large systems of linear equations of the form b Ax in an effective manner. LU-Decomposition offers good choices for solving this problem. Our approach is to find the lower bound of processing elements needed for this purpose. Here is used the so called “Omega calculus”, as a computational method for solving problems via their ...

In this paper we investigate the algebraic properties of multivariate discrete splines. It turns out that multivariate discrete splines are closely related to linear diophantine equations. In particular, we use a solvability condition for a system of linear diophantine equations to obtain a necessary and sufficient condition for the integer translates of a discrete box spline to be linearly ind...