In this paper we construct sets of type (d1, d2) in the projective Hjelmslev plane. For computational purposes we restrict ourself to planes over Zps with p a prime and s > 1, but the method is described over general Galois rings. The existence of sets of type (d1, d2) is equivalent to the existence of a solution of a Diophantine system of linear equations. To construct these sets we prescribe ...

in this paper we study necessary and sufficient conditions for some types of linear combinations of wave packet frames to be a frame for l^2(r^d). further, we illustrate our results with some examples and applications.

We discuss some easy statements dealing with linear inhomogeneous Diophantine approximation. Surprisingly, we did not find of them in the literature. In particular, prove a precise version Kronecker approximation theorem and related result on coprime

In this paper we show that any system of equations over a free nilpotent group of class c is either unitary or nullary. In fact, such a system either has a most general solution (akin to the most general solution of a system of linear diophantine equations), or every solution has a proper generalization. In principle we provide an algorithm for determining whether or not a most general solution...

We demonstrate that the answer to the question posed in the title is “yes” and “no”: “no” if the set of permissible operations is restricted to {+,−,×,/,mod,<}; “yes” if we are also allowed a gcd-oracle as a permissible operation. It has been shown (see [Sto76, MST91]) that no strongly polynomial algorithm exists for the problem of finding the greatest common divisor (gcd) of two arbitrary inte...

Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.

Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.

Geometry and Diophantine equations have been ever-present in mathematics. Diophantus of Alexandria was born in the 3rd century (as far as we know), but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a linear Diophantine equation can be viewed as a special case of a system of wo...