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Representations of groups --- Dirac equation --- Differential operators --- Differential operators. --- Dirac equation. --- Representations of groups.

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Representations of groups. --- Unitary groups. --- Lifting theory. --- Automorphic forms. --- Trace formulas. --- Measure theory --- Group theory --- Group representation (Mathematics) --- Groups, Representation theory of --- Formulas, Trace --- Automorphic forms --- Discontinuous groups --- Representations of groups --- Automorphic functions --- Forms (Mathematics)

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This comprehensive two-volume textbook presents the whole area of Partial Differential Equations - of the elliptic, parabolic, and hyperbolic type - in two and several variables. Special emphasis is put on the connection of PDEs and complex variable methods. In this second volume the following topics are treated: Solvability of operator equations in Banach spaces, Linear operators in Hilbert spaces and spectral theory, Schauder's theory of linear elliptic differential equations, Weak solutions of differential equations, Nonlinear partial differential equations and characteristics, Nonlinear elliptic systems with differential-geometric applications. While partial differential equations are solved via integral representations in the preceding volume, functional analytic methods are used in this volume. This textbook can be chosen for a course over several semesters on a medium level. Advanced readers may study each chapter independently from the others.

Differential equations, Partial --- Integral representations --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Representations, Integral --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Partial differential equations --- Differential equations, partial. --- Functional analysis. --- Mathematical physics. --- Partial Differential Equations. --- Functional Analysis. --- Mathematical Methods in Physics. --- Physical mathematics --- Physics --- Mathematics --- Partial differential equations. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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If F is a non-Archimedean local field, local class field theory can be viewed as giving a canonical bijection between the characters of the multiplicative group GL(1,F) of F and the characters of the Weil group of F. If n is a positive integer, the n-dimensional analogue of a character of the multiplicative group of F is an irreducible smooth representation of the general linear group GL(n,F). The local Langlands Conjecture for GL(n) postulates the existence of a canonical bijection between such objects and n-dimensional representations of the Weil group, generalizing class field theory. This conjecture has now been proved for all F and n, but the arguments are long and rely on many deep ideas and techniques. This book gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields. It uses only local methods, with no appeal to harmonic analysis on adele groups.

Representations of groups. --- L-functions. --- Algebraic number theory. --- Functions, L --- -Number theory --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Number theory --- Number theory. --- Topological Groups. --- Group theory. --- Number Theory. --- Topological Groups, Lie Groups. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Groups, Topological --- Continuous groups --- Number study --- Numbers, Theory of --- Topological groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups

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In the field of genetic and evolutionary algorithms (GEAs), a large amount of theory and empirical study has focused on operators and test problems, while problem representation has often been taken as given. This book breaks away from this tradition and provides a comprehensive overview on the influence of problem representations on GEA performance. The book summarizes existing knowledge regarding problem representations and describes how basic properties of representations, such as redundancy, scaling, or locality, influence the performance of GEAs and other heuristic optimization methods. Using the developed theory, representations can be analyzed and designed in a theory-guided matter. The theoretical concepts are used for solving integer optimization problems and network design problems more efficiently. The book is written in an easy-to-read style and is intended for researchers, practitioners, and students who want to learn about representations. This second edition extends the analysis of the basic properties of representations and introduces a new chapter on the analysis of direct representations.

Genetic programming (Computer science) --- Genetic algorithms. --- Evolutionary programming (Computer science) --- Representations of groups. --- Representations of algebras. --- Algebra --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Computer programming --- GAs (Algorithms) --- Genetic searches (Algorithms) --- Algorithms --- Combinatorial optimization --- Evolutionary computation --- Learning classifier systems --- Genetic algorithms --- Engineering mathematics. --- Artificial intelligence. --- Operations research. --- Information technology. --- Mathematical and Computational Engineering. --- Artificial Intelligence. --- Operations Research/Decision Theory. --- IT in Business. --- IT (Information technology) --- Technology --- Telematics --- Information superhighway --- Knowledge management --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- AI (Artificial intelligence) --- Artificial thinking --- Electronic brains --- Intellectronics --- Intelligence, Artificial --- Intelligent machines --- Machine intelligence --- Thinking, Artificial --- Bionics --- Cognitive science --- Digital computer simulation --- Electronic data processing --- Logic machines --- Machine theory --- Self-organizing systems --- Simulation methods --- Fifth generation computers --- Neural computers --- Engineering --- Engineering analysis --- Mathematical analysis --- Mathematics --- Applied mathematics. --- Decision making. --- Business—Data processing. --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management --- Management decisions --- Choice (Psychology) --- Problem solving --- Decision making