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"We construct a partial compactification of the moduli space, Mk, of SU(2) magnetic monopoles on R3, wherein monopoles of charge k decompose into widely separated 'monopole clusters' of lower charge going off to infinity at comparable rates. The hyperKahler metric on Mk has a complete asymptotic expansion up to the boundary, the leading term of which generalizes the asymptotic metric discovered by Bielawski, Gibbons and Manton when each lower charge is 1"--

Vector bundles. --- Global differential geometry. --- Global analysis (Mathematics) --- Quantum field theory.

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This book lays the algebraic foundations of a Galois theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers.

Group theory --- Functional analysis --- Difference equations --- Galois theory --- Mathematical Theory --- Operations Research --- Mathematics --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Equations aux differences --- Galois [Theorie de ] --- Galois [Theorie van ] --- Vergelijkingen met differenties --- Mathematical analysis. --- Analysis (Mathematics). --- Algebra. --- Analysis. --- Mathematical analysis --- 517.1 Mathematical analysis --- Difference equations. --- Galois theory. --- Equations, Theory of --- Number theory --- Calculus of differences --- Differences, Calculus of --- Equations, Difference

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Finite groups. --- Finite groups --- Groups, Finite --- Group theory --- Modules (Algebra) --- Funcions algebraiques --- Grups finits

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Among all computer-generated mathematical images, Julia sets of rational maps occupy one of the most prominent positions. Their beauty and complexity can be fascinating. They also hold a deep mathematical content. Computational hardness of Julia sets is the main subject of this book. By definition, a computable set in the plane can be visualized on a computer screen with an arbitrarily high magnification. There are countless programs to draw Julia sets. Yet, as the authors have discovered, it is possible to constructively produce examples of quadratic polynomials, whose Julia sets are not computable. This result is striking - it says that while a dynamical system can be described numerically with an arbitrary precision, the picture of the dynamics cannot be visualized. The book summarizes the present knowledge about the computational properties of Julia sets in a self-contained way. It is accessible to experts and students with interest in theoretical computer science or dynamical systems.

Algebra --- Complex analysis --- Computer science --- Computer. Automation --- algebra --- toegepaste informatica --- complexe analyse (wiskunde) --- informatica --- wiskunde --- algoritmen