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The Riemann boundary problem on Riemann surfaces (mathematics and its applications—Soviet series)
37-l
Book Reviews
Yu.L. Rodin: “The Riemann Boundary Problem on Riemann Surfaces” (Mathematics and Its Applications-Soviet Series). D. Reidel Publishing Co., 1988, 199 pages, Dfl. 13O.OO/US !$64.00/&39.50, ISBN 90-277-2653-l. Let L be a closed contour separating the complex plane into domains T+ and T- and let G(t) be a matrix (or function) defined on L. The Riemann boundary problem on the plane is to determine analytic matrices (functions) F ‘(2) in T * such that Ff( t) = G( t)F-( t), t E L. This problem occurs quite often in mathematics and in physical problems such as contact problems in elasticity, dispersion relations in quantum mechanics and soliton theory. This book is intended both for the initiate and the specialist. Chapter 1 contains the more classical aspects of Riemann surfaces. Chapter 2 is devoted to relations between the problem and complex line bundles. Chapter 3 is devoted to the matrix problem and its connections with complex vector bundles. Open surfaces of infinite genus are studied in Chapter 4. Generalized analytic functions of Bers-Vekua are addressed in Chapter 5. Chapter 6 is devoted to physical applications. The book concludes with Appendices, Notation, References, and an Index. (WFA)
R. Seydel: “From Equilibrium to Chaos” (Practical Bifurcation and Stability Amsterdam, 1988, 367 pages, US $55.00, ISBN O-444-01250-8.
Analysis).
Elsevier,
Requiring only a basic knowledge of calculus, this book focuses on computational methods to present an introduction to the nonlinear phenomena of bifurcation theory. Motivating examples and geometrical interpretations are very much in vogue and the spectrum of problems is wide. The first part of the book consists of two chapters that introduce stability and bifurcation. The second part, consisting of Chapters 3-7, concentrates on practical aspects and numerical methods. Chapter 3 shows what computational difficulties can arise and what kind of numerical methods are required to get around these difficulties. Chapter 4 gives an account of the principles of continuation, the procedure by which ‘parameter studies’ for nonlinear problems will be carried out. Chapters 5 and 6 treat basic computational methods for handling bifurcations, both for systems of algebraic equations and for ordinary differential equation boundaryvalue problems. In Chapter 7, branching phenomena of periodic solutions are handled and related numerical methods are outlined. The third and final part, Chapters 8 and 9, focuses on qualitative aspects. Singularity theory and catastrophe theory, which help in the interpretation of numerical results, are introduced in Chapter 8. Chapter 9 is an introduction to chaotic behavior. (WFA)
Book Reviews
Yu.L. Rodin: “The Riemann Boundary Problem on Riemann Surfaces” (Mathematics and Its Applications-Soviet Series). D. Reidel Publishing Co., 1988, 199 pages, Dfl. 13O.OO/US !$64.00/&39.50, ISBN 90-277-2653-l. Let L be a closed contour separating the complex plane into domains T+ and T- and let G(t) be a matrix (or function) defined on L. The Riemann boundary problem on the plane is to determine analytic matrices (functions) F ‘(2) in T * such that Ff( t) = G( t)F-( t), t E L. This problem occurs quite often in mathematics and in physical problems such as contact problems in elasticity, dispersion relations in quantum mechanics and soliton theory. This book is intended both for the initiate and the specialist. Chapter 1 contains the more classical aspects of Riemann surfaces. Chapter 2 is devoted to relations between the problem and complex line bundles. Chapter 3 is devoted to the matrix problem and its connections with complex vector bundles. Open surfaces of infinite genus are studied in Chapter 4. Generalized analytic functions of Bers-Vekua are addressed in Chapter 5. Chapter 6 is devoted to physical applications. The book concludes with Appendices, Notation, References, and an Index. (WFA)
R. Seydel: “From Equilibrium to Chaos” (Practical Bifurcation and Stability Amsterdam, 1988, 367 pages, US $55.00, ISBN O-444-01250-8.
Analysis).
Elsevier,
Requiring only a basic knowledge of calculus, this book focuses on computational methods to present an introduction to the nonlinear phenomena of bifurcation theory. Motivating examples and geometrical interpretations are very much in vogue and the spectrum of problems is wide. The first part of the book consists of two chapters that introduce stability and bifurcation. The second part, consisting of Chapters 3-7, concentrates on practical aspects and numerical methods. Chapter 3 shows what computational difficulties can arise and what kind of numerical methods are required to get around these difficulties. Chapter 4 gives an account of the principles of continuation, the procedure by which ‘parameter studies’ for nonlinear problems will be carried out. Chapters 5 and 6 treat basic computational methods for handling bifurcations, both for systems of algebraic equations and for ordinary differential equation boundaryvalue problems. In Chapter 7, branching phenomena of periodic solutions are handled and related numerical methods are outlined. The third and final part, Chapters 8 and 9, focuses on qualitative aspects. Singularity theory and catastrophe theory, which help in the interpretation of numerical results, are introduced in Chapter 8. Chapter 9 is an introduction to chaotic behavior. (WFA)