Constructible Convex Sets

Generalisations of Gödel’s universe of constructible sets

Computing the Canonical Representation of Constructible Sets

In the basic paper defining the Gröbner Cover [16] for discussing parametric polynomial systems of equations, we 8 introduced algorithms that have been improved since then. We used our own algorithm buildtree for computing 9 the initial Comprehensive Gröbner System (CGS), needed for the Gröbner Cover, now substituted in the Singular [7] 10 library “grobcov.lib” by themore efficientKapur–Sun–Wan...

Computing with Constructible Sets in Maple

Constructible sets are the geometrical objects naturally attached to triangular decompositions, as polynomial ideals are the algebraic concept underlying the computation of Gröbner bases. This relation becomes even more complex and essential in the case of polynomial systems with infinitely many solutions. In this paper, we introduce ConstructibleSetTools a new module of the RegularChains libra...

Convex Sets and Convex Combinations

Convexity is one of the most important concepts in a study of analysis. Especially, it has been applied around the optimization problem widely. Our purpose is to define the concept of convexity of a set on Mizar, and to develop the generalities of convex analysis. The construction of this article is as follows: Convexity of the set is defined in the section 1. The section 2 gives the definition...

Convex Sets and Convex Functions

In this section, we introduce one of the most important ideas in economic modelling, in the theory of optimization and, indeed in much of modern analysis and computatyional mathematics: that of a convex set. Almost every situation we will meet will depend on this geometric idea. As an independent idea, the notion of convexity appeared at the end of the 19 century, particularly in the works of M...