In this volume we will present some applications of special functions in computer science. This largely consists of adaptations of articles that have appeared in the literature . Here they are presented in a format made accessible for the non-expert by providing some context. The material on group representations and Young tableaux is introductory in nature. However, the algebraic approach of Chapter 2 is original to the authors and has not appeared previously . Similarly, the material and approach based on Appell states, so formulated, is presented here for the first time . As in all volumes of this series, this one is suitable for self-study by researchers . It is as well appropriate as a text for a course or advanced seminar . The solutions are tackled with the help of various analytical techniques, such as g- erating functions, and probabilistic methods/insights appear regularly . An interesting feature is that, as has been the case in classical applications to physics, special functions arise- here in complexity analysis. And, as in physics, their appearance indicates an underlying Lie structure. Our primary audience is applied mathematicians and theoretical computer scientists . We are quite sure that pure mathematicians will find this volume interesting and useful as well .

Interconnections between graph theory and algebraic structure theory have always led to innovative solutions to problems in both areas and new research topics in Mathematics and other scientific fields. This reprint aims to emphasize the new theoretical aspects and practical applications of these two research areas. It contains 18 articles selected for publication in the Special Issue "Algebraic Structures and Graph Theory," printed by the MDPI journal Mathematics.

In this book the authors present an alternative set theory dealing with a more relaxed notion of infiniteness, called finitely supported mathematics (FSM). It has strong connections to the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel (ZF) set theory with atoms and to the theory of (generalized) nominal sets. More exactly, FSM is ZF mathematics rephrased in terms of finitely supported structures, where the set of atoms is infinite (not necessarily countable as for nominal sets). In FSM, 'sets' are replaced either by `invariant sets' (sets endowed with some group actions satisfying a finite support requirement) or by `finitely supported sets' (finitely supported elements in the powerset of an invariant set). It is a theory of `invariant algebraic structures' in which infinite algebraic structures are characterized by using their finite supports.

The geometrical theory of nonlinear differential equations originates from classical works by S. Lie and A. Bäcklund. It obtained a new impulse in the sixties when the complete integrability of the Korteweg-de Vries equation was found and it became clear that some basic and quite general geometrical and algebraic structures govern this property of integrability.

This book offers a self-contained guide to advanced algorithms and their applications in various fields of science. Gathering contributions by authoritative researchers in the field of mathematics, statistics and computer science, it aims at offering a comprehensive and up-to-date view of algorithms, including the theory behind them, as well as practical considerations, current limitations and solutions.