In this paper, we show how to handle linear Diophantine constraints incre-mentally by using several variations of the algorithm by Contejean and Devie (hereafter called ABCD) for solving linear Diophantine systems 4, 5]. The basic algorithm is based on a certain enumeration of the potential solutions of a system, and termination is ensured by an adequate restriction on the search. This algorith...

The contribution is focused on a control design and simulation of multi input output (MIMO) linear continuous-time systems. Suitable and efficient tools for description and controller derivation are algebraic notions as rings, polynomial matrices, and Diophantine equations. The generalized MIMO PI controller design is studied for stable and unstable systems. A unified approach through matrix Di...

Following the development of weighted asymptotic approximation properties matrices, we introduce analogous uniform (that is, study improvability Dirichlet's Theorem). An added feature is use general norms, rather than supremum norm, to quantify approximation. In terms homogeneous dynamics, an $m \times n$ matrix are governed by a trajectory in $\mathrm{SL}_{m+n}({\mathbb R})/\mathrm{SL}_{m+n}({...

It is known that under linear-fractional unimodular transformations $$ \upalpha \mapsto {\upalpha}^{\prime }=\frac{a\upalpha +b}{c\upalpha +d} , the continuedfraction expansions of real numbers α and α′ coincide up to a finite number initial incomplete quotients. For this reason, rates approximation these by their convergents continued fractions are same. This result generalized l × k matrices....

Introduction Something happened to me recently I would wager has happened to many who read this note. Teaching a new topic, you cannot understand one of the proofs. Your first attempt to fill the gap fails. You look through your books for an answer. Next, you ask colleagues, go to the library, maybe even use the interlibrary loan. All in vain. Then it strikes you that, in fact, you cannot answe...

We consider deducibility constraints, which are equivalent to particular Diophantine systems, arising in the automatic verification of security protocols, in presence of associative and commutative symbols. We show that deciding such Diophantine systems is, in general, undecidable. Then, we consider a simple subclass, which we show decidable. Though the solutions of these problems are not neces...