We present a new method that addresses the various deficiencies of the state-ofthe-art non-linear invariant generation methods for hybrid systems. Present approaches for non-linear invariant generation are limited to linear systems, or they relay on non scalable methods which have high complexity. Moreover, for hybrid systems with discrete transitions, differential rules, and local conditions t...

Mixed-integer polynomial programming (MIPP) problems are one class of mixed-integer nonlinear programming (MINLP) problems where objective function and constraints are restricted to the polynomial functions. Although the MINLP problem is NP-hard, in special cases such as MIPP problems, an efficient algorithm can be extended to solve it. In this research, we propose an algorit...

In this paper, we extend the idea of comprehensive Gröbner bases given by Weispfenning (1992) to border bases for zero dimensional parametric polynomial ideals. For this, we introduce a notion of comprehensive border bases and border system, and prove their existence even in the cases where they do not correspond to any term order. We further present algorithms to compute comprehensive border b...

There is a close relation between comprehensive Gröbner bases and non-parametric Gröbner bases over commutative von Neumann regular rings. By this relation, Gröbner bases over a commutative von Neumann regular ring can be viewed as an alternative to comprehensive Gröbner bases. (Therefore, this Gröbner basis is called an “alternative comprehensive Gröbner basis (ACGB)”.) In the first part of th...

This paper addresses the critical issue of the development of intelligent or assisting tools for the scientist who is working in the initial problem formulation and mathematical model representation stage of research. In particular, examples of that representation in fluid dynamics and instability theory are discussed. The creation of a mathematical model that is ready for application of certai...

In this paper we present algorithms to compute finite state automata which, given any rational language, recognize the languages of normal words and n-chains. We also show how these automata can be used to compute the Hilbert series and Poincaré series for any algebra with a rational set of leading words of its minimal Gröbner basis.

Parametric Gröbner bases have been studied for more than 15 years and are now a further developed subject. Here we propose a general study of parametric standard bases, that is with local orders. We mainly focus on the commutative case but we also treat the case of differential operators rings. We will be concerned by two aspects: a theoretical aspect with existence theorems and a practical asp...

We showed special types of comprehensive Gröbner bases can be defined and calculated as the applications of Gröbner bases in polynomial rings over commutative Von Neumann regular rings in [5] and [6]. We called them discrete comprehensive Gröbner bases, since there is a strict restriction on specialization of parameters, that is parameters can take values only 0 and 1. In this paper, we show th...