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Full Description
This book is a collection of problems with detailed solutions which will prove valuable to students and research workers in mathematics, physics, engineering and other sciences. The topics range in difficulty from elementary to advanced level. Almost all the problems are solved in detail and most of them are self-contained. All relevant definitions are given. Students can learn important principles and strategies required for problem solving. Teachers will find this text useful as a supplement, since important concepts and techniques are developed through the problems. The material has been tested in the author's lectures given around the world.The book is divided into two volumes. Volume I presents the introductory problems, for undergraduate and advanced undergraduate students. In Volume II, the more advanced problems, together with detailed solutions, are collected, to meet the needs of graduate students and researchers. The problems included cover most of the new fields in theoretical and mathematical physics, such as Lax representation, Bäcklund transformation, soliton equations, Lie-algebra-valued differential forms, the Hirota technique, the Painlevé test, the Bethe ansatz, the Yang-Baxter relation, chaos, fractals, complexity, etc.
Contents
Volume I: Complex Numbers and Functions; Sums and Product; Discrete Fourier Transform; Algebraic and Transcendental Equations; Matrix Calculations; Matrices and Groups; Matrices and Eigenvalue Problems; Transformations; L'Hospital's Rule; Lagrange Multiplier Method; Linear Difference Equations; Linear Differential Equations; Integration; Continuous Fourier Transform; Complex Analysis; Special Functions; Inequalities; Functional Analysis; Combinatorics; Convex Sets and Functions; Optimalization; Volume II: Lax Representation in Classical Mechanics; Kronecker and Tensor Product; Nambu Mechanics; Gateaux and Frechet Derivative; Stability and Bifurcations; Nonlinear Ordinary Difference Equations; Nonlinear Ordinary Differential Equations; Groups; Generalized Functions; Linear Partial Differential Equations; Nonlinear Partial Differential Equations; Group Theoretical Reduction; Backlund Transformation; Soliton Equations; Lax Pairs for Partial Differential Equations; Hirota Technique; Painleve Test; Lie Algebra; Differential Forms; Lie Derivative; Metric Tensor Fields; Killing Vector Fields; Inequalities; Ising Model and Heisenberg Model; Number Theory; Combinatorial Problems; Fermi Operators; Bose Operators; Lax Representation and Bethe Ansatz; Gauge Transformation; Chaos, Fractals and Complexity
