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The book is devoted to the theory of pairs of compact convex sets and in particular to the problem of finding different types of minimal representants of a pair of nonempty compact convex subsets of a locally convex vector space in the sense of the Rådström-Hörmander Theory. Minimal pairs of compact convex sets arise naturally in different fields of mathematics, as for instance in non-smooth analysis, set-valued analysis and in the field of combinatorial convexity. In the first three chapters of the book the basic facts about convexity, mixed volumes and the Rådström-Hörmander lattice are presented. Then, a comprehensive theory on inclusion-minimal representants of pairs of compact convex sets is given. Special attention is given to the two-dimensional case, where the minimal pairs are uniquely determined up to translations. This fact is not true in higher dimensional spaces and leads to a beautiful theory on the mutual interactions between minimality under constraints, separation and decomposition of convex sets, convexificators and invariants of minimal pairs. This theory throws light upon both sides of the collection of all compact convex subsets of a locally vector space, namely the geometric and the algebraic one. From the algebraic point of view the collection of all nonempty compact convex subsets of a topological vector space is an ordered semi group with cancellation property under the inclusion of sets and the Minkowski-addition. From this approach pairs of nonempty compact convex sets correspond to fractions of elements from the semi group and minimal pairs to relatively prime fractions.

This book constitutes selected best papers from the 10th International Conference on Artificial Evolution, EA 2011, held in Angers, France, in October 2011. Initially, 33 full papers and 10 post papers were carefully reviewed and selected from 64 submissions. This book presents the 19 best papers selected from these contributions. The papers are organized in topical sections on ant colony optimization; multi-objective optimization; analysis; implementation and robotics; combinatorial optimization; learning and parameter tuning; new nature inspired models; probabilistic algorithms; theory and evolutionary search; and applications.

The standard dictionary of the English language micrographically printed in one volume

Thomas Nelson's NKJV Compact End-of-Verse Reference Bible presents the complete text of the popular New King James Version in a small, portable size with easy-to-read print and cross references that appear at the end of the verses.