The connective topological modular forms spectrum, tmf, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of tmf and several tmf-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account of the homotopy groups of spheres through degree 44, with complete proofs, except that the Adams conjecture is used without proof. Also presented are modern stable proofs of classical results which are hard to extract from the literature.
Tools used in this book include a multiplicative spectral sequence generalizing a construction of Davis and Mahowald, and computer software which computes the cohomology of modules over the Steenrod algebra and products therein. Techniques from commutative algebra are used to make the calculation precise and finite. The H∞ ring structure of the sphere and of tmf are used to determine many differentials and relations.
Graduate students and researchers interested in algebraic topology, specifically in stable homotopy theory.
Mathematical Surveys and Monographs Volume: 253
2021; 690 pp; Hardcover
MSC: Primary 18; 55;
Print ISBN: 978-1-4704-5674-0
Product Code: SURV/253
Not yet published
Expected publication date August 14, 2021
This book presents multiprecision algorithms used in number theory and elsewhere, such as extrapolation, numerical integration, numerical summation (including multiple zeta values and the Riemann-Siegel formula), evaluation and speed of convergence of continued fractions, Euler products and Euler sums, inverse Mellin transforms, and complex L-functions.
For each task, many algorithms are presented, such as Gaussian and doubly-exponential integration, Euler-MacLaurin, Abel-Plana, Lagrange, and Monien summation. Each algorithm is given in detail, together with a complete implementation in the free Pari/GP system. These implementations serve both to make even more precise the inner workings of the algorithms, and to gently introduce advanced features of the Pari/GP language.
This book will be appreciated by anyone interested in number theory, specifically in practical implementations, computer experiments and numerical algorithms that can be scaled to produce thousands of digits of accuracy.
Graduate students and researchers interested in high precision numerical computations in number theory.
Mathematical Surveys and Monographs Volume: 254
2021; 429 pp; Softcover
MSC: Primary 11; 65; 30;
Print ISBN: 978-1-4704-6351-9
Product Code: SURV/254
Not yet published
Expected publication date August 14, 2021
This volume contains the proceedings of the 17th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory (AGC2T-17), held from June 10?14, 2019, at the Centre International de Rencontres Mathematiques in Marseille, France. The conference was dedicated to the memory of Gilles Lachaud, one of the founding fathers of the AGC2T series.
Since the first meeting in 1987 the biennial AGC2T meetings have brought together the leading experts on arithmetic and algebraic geometry, and the connections to coding theory, cryptography, and algorithmic complexity. This volume highlights important new developments in the field.
Graduate students and research mathematicians interested in explicit methods in arithmetic and algebraic geometry with applications to coding theory, cryptography and algorithmic complexity.
Contemporary Mathematics Volume: 770
2021; 303 pp; Softcover
MSC: Primary 11; 14; 20; 51; 94;
Print ISBN: 978-1-4704-5426-5
Product Code: CONM/770
Not yet published
Expected publication date August 5, 2021
Part of Encyclopedia of Mathematics and its Applications
Not yet published - available from November 2021
FORMAT: Hardback ISBN: 9781316516553
In pioneering work in the 1950s, S. Karlin and J. McGregor showed that probabilistic aspects of certain Markov processes can be studied by analyzing orthogonal eigenfunctions of associated operators. In the decades since, many authors have extended and deepened this surprising connection between orthogonal polynomials and stochastic processes. This book gives a comprehensive analysis of the spectral representation of the most important one-dimensional Markov processes, namely discrete-time birth-death chains, birth-death processes and diffusion processes. It brings together the main results from the extensive literature on the topic with detailed examples and applications. Also featuring an introduction to the basic theory of orthogonal polynomials and a selection of exercises at the end of each chapter, it is suitable for graduate students with a solid background in stochastic processes as well as researchers in orthogonal polynomials and special functions who want to learn about applications of their work to probability.
The first text to bring together all the main results on the spectral representation of the most important one-dimensional Markov processes
Many detailed examples of the spectral analysis of birth-death models and diffusion processes and the probabilistic consequences
Accessible to graduate students with a background in probability
Copyright Year 2022
ISBN 9780367626549
September 2, 2021 Forthcoming by Chapman and Hall/CRC
280 Pages 21 B/W Illustrations
Linear algebra provides the essential mathematical tools to tackle all the problems in Science. Introduction to Linear Algebra is primarily aimed at students in applied fields (e.g. Computer Science and Engineering), providing them with a concrete, rigorous approach to face and solve various types of problems for the applications of their interest. This book offers a straightforward introduction to linear algebra that requires a minimal mathematical background to read and engage with.
Presented in a brief, informative and engaging style
Suitable for a wide broad range of undergraduates
Contains many worked examples and exercises
1. Introduction to Linear Systems. 1.1. Linear systems: First Examples. 1.2. Matrices. 1.3. Matrices and Linear Systems. 1.4. The Gaussian Algorithm. 1.5. Exercises with Solutions. 1.6. Suggested Exercises. 2. Vector Spaces. 2.1. Introduction: The Set of Real Numbers. 2.2. The Vector Space Rn and the Vector Space of Matrices. 2.3. Vector Spaces. 2.4. SubSpaces. 2.5. Exercises with Solutions. 2.6. Suggested Exercises. 3. Linear Combination and Linear Independence. 3.1. Linear Combinations and Generators. 3.2. Linear Independence. 3.3. Exercises with Solutions. 3.4. Suggested Exercises. 4. Basis and Dimension. 4.1. Basis: Definition and Examples. 4.2. The Concept of Dimension. 4.3. Gaussian Algorithm. 4.4. Exercises with Solutions. 4.5. Suggested Exercises. 4.6. Appendix: The Completion Theorem. 5. Linear Transformations. 5.1. Linear Transformations: Definition. 5.2. Linear Maps and Matrices. 5.3. the Composition of Linear transformations. 5.4. Kernel and Image. 5.5. The Rank Nullity Theorem. 5.6. Isomorphism of Vector Spaces. 5.7. Calculation of Kernel and Image. 5.8. Exercises with Solutions. 5.9. Suggested Exercises. 6. Linear Systems. 6.1. Preimage. 6.2. Linear Systems. 6.3. Exercises with Solutions. 6.4. Suggested Exercises. 7. Determinant and Inverse. 7.1. Definition of Determinant. 7.2. Calculating the Determinant: Cases 2 _ 2 and 3 _ 3. 7.3. Calculating the Determinant with a Recursive Method. 7.4. Inverse of a Matrix. 7.5. Calculation of the Inverse with the Gaussian Algorithm. 7.6. The Linear Maps from Rn to Rn. 7.7. Exercises with Solutions. 7.8. Suggested Exercises. 7.9. Appendix. 8. Change of Basis. 8.1. Linear Transformations and Matrices. 8.2. The Identity Map. 8.3. Change of Basis for Linear Transformations. 8.4. Exercises with Solutions. 8.5. Suggested Exercises. 9. Eigenvalues and Eigenvectors. 9.1. Diagonalizability. 9.2. Eigenvalues and Eigenvectors. 9.3. Exercises with Solutions. 9.4. Suggested Exercises. 10. Scalar Products. 10.1. Bilinear Forms. 10.2. Bilinear Forms and Matrices. 10.3. Basis Change. 10.4. Scalar Products. 10.5. Orthogonal Subspaces. 10.6. Gram-Schmidt Algorithm. 10.7. Exercises with Solutions. 10.8. Suggested Exercises. 11. Spectral Theorem. 11.1. Orthogonal Linear Transformations. 11.2. Orthogonal Matrices. 11.3. Symmetric Linear Transformations. 11.4. The Spectral Theorem. 11.5. Exercises with Solutions. 11.6. Suggested Exercises. 11.7. Appendix: The Complex Case. 12. Applications of Spectral Theorem and Quadratic Forms. 12.1. Diagonalization of Scalar Products. 12.2. Quadratic Forms. 12.3. Quadratic Forms and in the Plane. 12.4. Exercises with Solutions. 12.5. Suggested Exercises. 13. Lines and Planes. 13.1. Points and Vectors in R3. 13.2. Scalar Product and Vector Product. 13.3. Lines in R3. 13.4. Planes in R3. 13.5. Exercises with Solutions. 13.6. Suggested Exercises. 14. Introduction to Modular Arithmetic. 14.1. The Principle of Induction. 14.2. The Division Algorithm and Euclid’s Algorithm. 14.3. Congruence Classes. 14.4. Congruences. 14.5. Exercises with Solutions. 14.6. Suggested Exercises. 14.7. Appendix: Elementary Notions of Set Theory. Appendix A. Complex Numbers. A.1. Complex Numbers. A.2. Polar Representation. Appendix B. Solutions of Some Suggested Exercises. Bibliography. Index
Biography
Rita Fioresi and Marta Morigi are professors at the University of Bologna and teach linear
algebra courses to students of all majors. Their research publications are centered into algebra with some applications (R. Fioresi Lie theory & machine learning, M. Morigi Group theory).