Modular Forms, a Computational Approach (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 79)

This marvellous and highly original book fills a significant gap in the extensive literature on classical modular forms. This is not just yet another introductory text to this theory, though it could certainly be used as such in conjunction with more traditional treatments. Its novelty lies in its computational emphasis throughout: Stein not only defines what modular forms are, but shows in illuminating detail how one can compute everything about them in practice. This is illustrated throughout the book with examples from his own (entirely free) software package SAGE, which really bring the subject to life while not detracting in any way from its theoretical beauty. The author is the leading expert in computations with modular forms, and what he says on this subject is all tried and tested and based on his extensive experience. As well as being an invaluable companion to those learning the theory in a more traditional way, this book will be a great help to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations. There is also a useful Appendix by Gunnells on extensions to more general modular forms, which has enough in it to inspire many PhD theses for years to come. While the book's main readership will be graduate students in number theory, it will also be accessible to advanced undergraduates and useful to both specialists and non-specialists in number theory. —John E. Cremona, University of Nottingham William Stein is an associate professor of mathematics at the University of Washington at Seattle. He earned a PhD in mathematics from UC Berkeley and has held positions at Harvard University and UC San Diego. His current research interests lie in modular forms, elliptic curves, and computational mathematics. Machine Generated Contents Note: 1.1. Preview -- 1.2. The Abstract Definition Of A Vector Space -- 1.3. Some Definitions -- 1.4. Mappings -- 1.5. Triangular Matrices -- 1.6. Block Triangular Matrices -- 1.7. Schur Complements -- 1.8. Other Matrix Products -- 2.1. Some Preliminary Observations -- 2.2. Examples -- 2.3. Upper Echelon Matrices -- 2.4. The Conservation Of Dimension -- 2.5. Quotient Spaces -- 2.6. Conservation Of Dimension For Matrices -- 2.7. From U To A -- 2.8. Square Matrices -- 3.1. Gaussian Elimination Redux -- 3.2. Properties Of Ba And Ac -- 3.3. Extracting A Basis -- 3.4. Computing The Coefficients In A Basis -- 3.5. The Gauss-seidel Method -- 3.6. Block Gaussian Elimination -- 3.7. {0, 1, Infinity} -- 3.8. Review -- 4.1. Change Of Basis And Similarity -- 4.2. Invariant Subspaces -- 4.3. Existence Of Eigenvalues -- 4.4. Eigenvalues For Matrices -- 4.5. Direct Sums -- 4.6. Diagonalizable Matrices -- 4.7. An Algorithm For Diagonalizing Matrices 4.8. Computing Eigenvalues At This Point -- 4.9. Not All Matrices Are Diagonalizable -- 4.10. The Jordan Decomposition Theorem -- 4.11. An Instructive Example -- 4.12. The Binomial Formula -- 4.13. More Direct Sum Decompositions -- 4.14. Verification Of Theorem 4.13 -- 4.15. Bibliographical Notes -- 5.1. Functionals -- 5.2. Determinants -- 5.3. Useful Rules For Calculating Determinants -- 5.4. Eigenvalues -- 5.5. Exploiting Block Structure -- 5.6. The Binet-cauchy Formula -- 5.7. Minors -- 5.8. Uses Of Determinants -- 5.9. Companion Matrices -- 5.10. Circulants And Vandermonde Matrices -- 6.1. Overview -- 6.2. Structure Of The Nullspaces Nbj -- 6.3. Chains And Cells -- 6.4. Computing J -- 6.5. An Algorithm For Computing U -- 6.6. A Simple Example -- 6.7. A More Elaborate Example -- 6.8. Jordan Decompositions For Real Matrices -- 6.9. Projection Matrices -- 6.10. Companion And Generalized Vandermonde Matrices -- 7.1. Four Inequalities -- 7.2. Normed Linear Spaces -- 7.3. Equivalence Of Norms -- 7.4. Norms Of Linear Transformations 7.5. Operator Norms For Matrices -- 7.6. Mixing Tops And Bottoms -- 7.7. Evaluating Some Operator Norms -- 7.8. Inequalities For Multiplicative Norms -- 7.9. Small Perturbations -- 7.10. Bounded Linear Functionals -- 7.11. Extensions Of Bounded Linear Functionals -- 7.12. Banach Spaces -- 7.13. Bibliographical Notes -- 8.1. Inner Product Spaces -- 8.2. A Characterization Of Inner Product Spaces -- 8.3. Orthogonality -- 8.4. Gram Matrices -- 8.5. Projections And Direct Sum Decompositions -- 8.6. Orthogonal Projections -- 8.7. Orthogonal Expansions -- 8.8. The Gram-schmidt Method -- 8.9. Toeplitz And Hankel Matrices -- 8.10. Adjoints -- 8.11. The Riesz Representation Theorem -- 8.12. Normal, Selfadjoint And Unitary Transformations -- 8.13. Auxiliary Formulas -- 8.14. Gaussian Quadrature -- 8.15. Bibliographical Notes -- 9.1. Hermitian Matrices Are Diagonalizable -- 9.2. Commuting Hermitian Matrices -- 9.3. Real Hermitian Matrices -- 9.4. Projections And Direct Sums In Fn -- 9.5. Projections And Rank 9.6. Normal Matrices -- 9.7. Qr Factorization -- 9.8. Schur's Theorem -- 9.9. Areas, Volumes And Determinants -- 9.10. Boundary Value Problems -- 9.11. Bibliographical Notes -- 10.1. Singular Value Decompositions -- 10.2. Complex Symmetric Matrices -- 10.3. Approximate Solutions Of Linear Equations -- 10.4. Fitting A Line In R2 -- 10.5. Fitting A Line In Rp -- 10.6. Projection By Iteration -- 10.7. The Courant-fischer Theorem -- 10.8. Inequalities For Singular Values -- 10.9. Von Neumann's Inequality For Contractive Matrices -- 10.10. Bibliographical Notes -- 11.1. Pseudoinverses -- 11.2. The Moore-penrose Inverse -- 11.3. Best Approximation In Terms Of Moore-penrose Inverses -- 11.4. Drazin Inverses -- 11.5. Bibliographical Notes -- 12.1. A Detour On Triangular Factorization -- 12.2. Definite And Semidefinite Matrices -- 12.3. Characterizations Of Positive Definite Matrices -- 12.4. An Application Of Factorization -- 12.5. Positive Definite Toeplitz Matrices -- 12.6. Detour On Block Toeplitz Matrices 12.7. A Maximum Entropy Matrix Completion Problem -- 12.8. A Class Of A 0 For Which (12.52) Holds -- 12.9. Schur Complements For Semidefinite Matrices -- 12.10. Square Roots -- 12.11. Polar Forms -- 12.12. Matrix Inequalities -- 12.13. A Minimal Norm Completion Problem -- 12.14. A Description Of All Solutions To The Minimal Norm Completion Problem -- 12.15. Bibliographical Notes -- 13.1. Systems Of Difference Equations -- 13.2. Nonhomogeneous Systems Of Difference Equations -- 13.3. The Exponential Eta -- 13.4. Systems Of Differential Equations -- 13.5. Uniqueness -- 13.6. Isometric And Isospectral Flows -- 13.7. Second-order Differential Systems -- 13.8. Stability -- 13.9. Nonhomogeneous Differential Systems -- 13.10. Strategy For Equations -- 13.11. Second-order Difference Equations -- 13.12. Higher Order Difference Equations -- 13.13. Second-order Differential Equations -- 13.14. Higher Order Differential Equations -- 13.15. Wronskians -- 13.16. Variation Of Parameters -- 14.1. Mean Value Theorems 14.2. Taylor's Formula With Remainder -- 14.3. Application Of Taylor's Formula With Remainder -- 14.4. Mean Value Theorem For Functions Of Several Variables -- 14.5. Mean Value Theorems For Vector-valued Functions Of Several Variables -- 14.6. A Contractive Fixed Point Theorem -- 14.7. Newton's Method -- 14.8. A Refined Contractive Fixed Point Theorem -- 14.9. Spectral Radius -- 14.10. The Brouwer Fixed Point Theorem -- 14.11. Bibliographical Notes -- 15.1. Preliminary Discussion -- 15.2. The Implicit Function Theorem -- 15.3. A Generalization Of The Implicit Function Theorem -- 15.4. Continuous Dependence Of Solutions -- 15.5. The Inverse Function Theorem -- 15.6. Roots Of Polynomials -- 15.7. An Instructive Example -- 15.8. A More Sophisticated Approach -- 15.9. Dynamical Systems -- 15.10. Lyapunov Functions -- 15.11. Bibliographical Notes -- 16.1. Classical Extremal Problems -- 16.2. Convex Functions -- 16.3. Extremal Problems With Constraints -- 16.4. Examples -- 16.5. Krylov Subspaces Harry Dym. Includes Bibliographical References (pages 575-578) And Indexes. Linear algebra permeates mathematics, perhaps more so than any other single subject. It plays an essential role in pure and applied mathematics, statistics, computer science, and many aspects of physics and engineering. This book conveys in a user-friendly way the basic and advanced techniques of linear algebra from the point of view of a working analyst. The techniques are illustrated by a wide sample of applications and examples that are chosen to highlight the tools of the trade. In short, this is material that many of us wish we had been taught as graduate students. Roughly the first third of the book covers the basic material of a first course in linear algebra. The remaining chapters are devoted to applications drawn from vector calculus, numerical analysis, control theory, complex analysis, convexity and functional analysis. In particular, fixed point theorems, extremal problems, matrix equations, zero location and eigenvalue location problems, and matrices with nonnegative entries are discussed. Appendices on useful facts from analysis and supplementary information from complex function theory are also provided for the convenience of the reader. In this new edition, most of the chapters in the first edition have been revised, some extensively. The revisions include changes in a number of proofs, either to simplify the argument, to make the logic clearer or, on occasion, to sharpen the result. New introductory sections on linear programming, extreme points for polyhedra and a Nevanlinna-Pick interpolation problem have been added, as have some very short introductory sections on the mathematics behind Google, Drazin inverses, band inverses and applications of SVD together with a number of new exercises. Covers classical modular forms. This book is suitable to those who wish to use modular forms in applications, such as in the explicit solution of Diophantine equations.