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Matrices, Moments and Quadrature with Applications
This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part.
Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization.
This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.
eBook
Log-Gases and Random Matrices (LMS-34)
Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years.Log-Gases and Random Matricesgives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials.
Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, makingLog-Gases and Random Matricesan indispensable reference work, as well as a learning resource for all students and researchers in the field.
eBook
Topics in Quaternion Linear Algebra
Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations.
Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.
eBook
Positive Definite Matrices
This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.
eBook
Effect of High Intensity Interval Training on Matrix Metalloproteinases in Women with Breast Cancer Receiving Anthracycline-Based Chemotherapy
Anthracycline chemotherapy is commonly used to treat breast cancer yet may increase the level of matrix metalloproteinases (MMP) -2 and -9, which increase the risk of atherosclerosis. While exercise has been shown to reduce the level of MMP in patients with diabetes, high intensity interval training (HIIT) has not been utilized to improve level of MMP in women with breast cancer receiving anthracycline chemotherapy. Thirty women were randomized to either 8-week HIIT or control (CON) group. The CON group was offered the HIIT intervention after 8 weeks. MMP-1, -2 -7, -9, tissue inhibitor of MMP (TIMP) -1, and-2 were measured at baseline and post-intervention. Repeated measures ANCOVA and paired t-test were performed to assess changes in MMP and TIMP. Post-intervention, no significant between-group differences were observed for MMP and TIMP. However, within-group decrease in MMP-9 was observed in the HIIT group [104.3(51.9) to 65.2(69.1); P = 0.01]. MMP-9 in the CON group was not significantly changed [115.5(47.2) to 90.4(67.9);]. MMP-2 significantly increased in both the HIIT group [76.6(11.2) to 83.2(13.1); P = 0.007) and the CON group [69.0(8.9) to 77.6(11.1) P = 0.003). It is unclear whether an 8-week HIIT intervention influences MMP-9 in breast cancer patients undergoing anthracycline chemotherapy. Additional investigations are required to understand the exercise-induced changes in MMP-2 and -9 in women undergoing anthracycline chemotherapy.
Journal Article