The present manuscript constitutes the lecture notes for the “Probability, Statistics and Stochastic Processes” module of the PhD program on Complexity Science, held at ISCTE-IUL during April and May 2012. The aim of the notes is to provide some auxiliary material for the students to follow this 10 hour length module, devoted to the study of the Probability theory and Stochastic Processes, as w...

We prove that every module over a commutative homogeneous Koszul algebra has regularity bounded by its regularity over a polynomial ring of which the Koszul algebra is a homomorphic image. From this we derive a result conjectured by George Kempf to the effect that a suffkiently high truncation of any module over a homogeneous Koszul algebra has a linear free resolution. ‘E’ 1992 Academic Press....

Serre’s reduction aims at reducing the number of unknowns and equations of a linear functional system (e.g., system of ordinary or partial differential equations, system of differential time-delay equations, system of difference equations). Finding an equivalent representation of a linear functional system containing fewer equations and fewer unknowns generally simplifies the study of its struc...

in this paper we introduce a generalization of m-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . we will use the structure of the radical of a module in  and get some suitable results about this class of modules. also the relation between injective hull in  and this kind of modules will be investigated in this article.   for a module  we show...

This paper describes the system SYNHESYS, which is a shell for developing expert systems composed of both a connectionist module and a symbolic module. The shell allows for several types of interactions between the two modules, and for several degrees of coupling. SYNHESYS aims at facilitating the process of knowledge acquisition and at improving the performance of the symbolic module. We prese...