Download or read book Group Pattern Matrices written by Roger Chalkley and published by . This book was released on 2021-02-17 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: A group-pattern for a group G of order n is specified by the n x n interior of any multiplication table for G in which the identity element e of G occupies each of the n principal diagonal positions. Thus, G and a list of the elements in G provide a pattern for the components of corresponding group-pattern matrices. As an example, there is at least one list of the elements in a cyclic group of order n for which the corresponding multiplication table specifies the pattern for n x n circulant matrices. This monograph shows that a serious treatment of its subject immediately yields an efficient Mathematica program to obtain all of the automorphisms for finite groups. This viewpoint is quite different from the way that Georg Frobenius introduced "group-matrices" for his research of 1896-1897. That distinction is maintained with the more descriptive phrase "group-pattern matrices" for the perspective here. Chapters may be read in various orders. As an illustration, to supplement an interest in circulant matrices, a reader may prefer to begin directly with Appendices A, B, C, D followed by Chapters 5, 6, 1, and 2. Or, for those whose primary interest may be automorphisms for finite groups, the recommended order would be Chapters 1, 2, 3, Appendix F, and Chapters 18, 19. Chapter 1 provides numerous group-patterns based on multiplication tables where the operations are easy to interpret and apply. Chapter 2 includes an algorithm to decide whether a given n x n matrix having n distinct components in its first row is a group-pattern matrix. An efficient Mathematica program implements that algorithm. When the decision is yes, the program specifies a suitable group and group-pattern. Chapter 3 presents on page 34 the computer-algebra program to obtain all of the automorphisms for finite groups. Chapter 4 is devoted to general algebraic properties of group-pattern matrices that have components in a field. Henceforth, for simplicity of explanation, let all components of matrices be complex numbers. Chapter 5 shows that, at one time, there was considerable interest in various determinants having interesting factorizations. Now, when those determinants are recognized as determinants of group-pattern matrices for abelian groups, their factorizations can be obtained as applications of a general result mentioned next. Chapters 6 through 9 are about group-pattern matrices for which the group is abelian. In particular, each such matrix is diagonalizable and its determinant is therefore equal to the product of the corresponding diagonal components. Moreover, when a group-pattern is specified for an abelian group G of order n, then: (i) there is a nonsingular n x n matrix that diagonalizes each n x n matrix having that pattern; (ii) the set of matrices having that pattern forms a ring; and (iii) that ring is isomorphic to the ring of n x n diagonal matrices. Chapters 10 and 11 show how research of Richard Dedekind in 1886 influenced Georg Frobenius to construct in 1897 a remarkable block-diagonalization for a group-pattern matrix where the group is the nonabelian one of order 6. That example motivated deep research about matrix representations of finite groups. Chapters 12 through 16 show how matrix representations for groups enable a Frobenius block-diagonalization to be obtained for any group-pattern matrix. Each such block-diagonalization is a diagonalization if and only if the group is abelian. There are 23 chapters and 8 appendices. Throughout, various computer-algebra programs are employed. For each one, there is a corresponding Mathematica notebook available that can be downloaded according to directions in Appendix H on page 235.
Download or read book Group Matrices Group Determinants and Representation Theory written by Kenneth W. Johnson and published by Springer Nature. This book was released on 2019-11-08 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book sets out an account of the tools which Frobenius used to discover representation theory for nonabelian groups and describes its modern applications. It provides a new viewpoint from which one can examine various aspects of representation theory and areas of application, such as probability theory and harmonic analysis. For example, the focal objects of this book, group matrices, can be thought of as a generalization of the circulant matrices which are behind many important algorithms in information science. The book is designed to appeal to several audiences, primarily mathematicians working either in group representation theory or in areas of mathematics where representation theory is involved. Parts of it may be used to introduce undergraduates to representation theory by studying the appealing pattern structure of group matrices. It is also intended to attract readers who are curious about ideas close to the heart of group representation theory, which do not usually appear in modern accounts, but which offer new perspectives.
Download or read book The Tripartite Matrix in the Developing Theory and Expanding Practice of Group Analysis written by Earl Hopper and published by Taylor & Francis. This book was released on 2023-08-18 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Tripartite Matrix in the Developing Theory and Expanding Practice of Group Analysis explores the social unconscious in persons, groups and societies in terms of the "un-acknowledged" restraints and constraints of our social and cultural groupings. In this context, Earl Hopper and an international team of contributors elucidate the theory and concept of the tripartite matrix as a tool for the deeper understanding of the human condition and for clinical work in various settings. They consider topics ranging from envy to intersectionality, and from addiction to the inability to mourn. The Tripartite Matrix in the Developing Theory and Expanding Practice of Group Analysis will be of great interest to group analysts, psychoanalytical group therapists, psychoanalysts and psycho-dramatists, as well as to social scientists more generally. Its extensive bibliography will be of particular value to students.
Download or read book Pattern Theory written by Ulf Grenander and published by Oxford University Press. This book was released on 2007 with total page 633 pages. Available in PDF, EPUB and Kindle. Book excerpt: 'Pattern Theory' provides a comprehensice & accessible overview of the modern challenges in signal, data & pattern analysis in speech recognition, computational linguistics, image analysis & computer vision. Aimed at graduate students the text includes numerous exercises & an extensive bibliography.
Download or read book Matrix Groups for Undergraduates written by Kristopher Tapp and published by American Mathematical Soc.. This book was released on 2016-04-07 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: Matrix groups touch an enormous spectrum of the mathematical arena. This textbook brings them into the undergraduate curriculum. It makes an excellent one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups. Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigorous proofs. The story begins and ends with the rotations of a globe. In between, the author combines rigor and intuition to describe the basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, maximal tori, homogeneous spaces, and roots. This second edition includes two new chapters that allow for an easier transition to the general theory of Lie groups.
Download or read book The Random Matrix Theory of the Classical Compact Groups written by Elizabeth S. Meckes and published by Cambridge University Press. This book was released on 2019-08 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides a comprehensive introduction to the theory of random orthogonal, unitary, and symplectic matrices.
Download or read book The Theory of Group Characters and Matrix Representations of Groups written by Dudley Ernest Littlewood and published by American Mathematical Soc.. This book was released on 2005 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: Originally written in 1940, this book remains a classical source on representations and characters of finite and compact groups. The book starts with necessary information about matrices, algebras, and groups. Then the author proceeds to representations of finite groups. Of particular interest in this part of the book are several chapters devoted to representations and characters of symmetric groups and the closely related theory of symmetric polynomials. The concluding chapters present the representation theory of classical compact Lie groups, including a detailed description of representations of the unitary and orthogonal groups. The book, which can be read with minimal prerequisites (an undergraduate algebra course), allows the reader to get a good understanding of beautiful classical results about group representations.
Download or read book The Unitary Group for the Evaluation of Electronic Energy Matrix Elements written by Jürgen Hinze and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the last thirty years, with the development of high speed electronic computers, methods have evolved, which permit an accurate and quantitative, ab initio determina tion of the electronic wavefunctions of atoms and molecules. Thus a detailed elucida tion of the electronic energy and structure of molecules has become possible using quantum mechanics directly. Ho\~ever, it is necessary, if such calculations are to yield accurate and reliable results, to include electron correlation explicitely, which requires in general . configuration mixing procedures with an extremely large 5 number of configurations, of the order of 10 configurations. With eigenvalue problems of this size, the limits of even the largest and fastest computers are reached rapidly, and their solution has become possible only, because direct methods have been deve~ loped which permit the determination of eigenvalues and eigenvectors for such large matrices iteratively without constructing the energy matrix explicitely. These direct methods had been limited to the description of closed shell systems, i. e. systems with a single dominant closed shell reference determinant. This limitation arose, because with an open shell reference or with several reference determinants, no procedures were known, which allowed a rapid calculation of the energy matrix elements between configurations with general and widely different spin couplings, which would be necessary. Recently such methods have been developed, based on early work of Gelfand, Biedenharn and Moshinski using a unitary group representation of different spin coupled states; Paldus achieved an extremely compact description.
Download or read book An Introduction to Matrices Sets and Groups for Science Students written by G. Stephenson and published by Courier Dover Publications. This book was released on 2016-01-14 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This outstanding text offers undergraduate students of physics, chemistry, and engineering a concise, readable introduction to matrices, sets, and groups. Concentrating mainly on matrix theory, the book is virtually self-contained, requiring a minimum of mathematical knowledge and providing all the background necessary to develop a thorough comprehension of the subject. Beginning with a chapter on sets, mappings, and transformations, the treatment advances to considerations of matrix algebra, inverse and related matrices, and systems of linear algebraic equations. Additional topics include eigenvalues and eigenvectors, diagonalisation and functions of matrices, and group theory. Each chapter contains a selection of worked examples and many problems with answers, enabling readers to test their understanding and ability to apply concepts.
Download or read book Direct Methods for Sparse Matrices written by I. S. Duff and published by Oxford University Press. This book was released on 2017-03-09 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of sparse matrices has its root in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis. Efficient use of sparsity is a key to solving large problems in many fields. This second edition is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all examples in the first edition were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer architectures are now much more complex, requiring new ways of adapting algorithms to parallel environments with memory hierarchies. Because the area is such an important one to all of computational science and engineering, a huge amount of research has been done in the last 30 years, some of it by the authors themselves. This new research is integrated into the text with a clear explanation of the underlying mathematics and algorithms. New research that is described includes new techniques for scaling and error control, new orderings, new combinatorial techniques for partitioning both symmetric and unsymmetric problems, and a detailed description of the multifrontal approach to solving systems that was pioneered by the research of the authors and colleagues. This includes a discussion of techniques for exploiting parallel architectures and new work for indefinite and unsymmetric systems.
Download or read book Hadamard Matrices and Their Applications written by K. J. Horadam and published by Princeton University Press. This book was released on 2012-01-06 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: In Hadamard Matrices and Their Applications, K. J. Horadam provides the first unified account of cocyclic Hadamard matrices and their applications in signal and data processing. This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago. The book translates physical applications into terms a pure mathematician will appreciate, and theoretical structures into ones an applied mathematician, computer scientist, or communications engineer can adapt and use. The first half of the book explains the state of our knowledge of Hadamard matrices and two important generalizations: matrices with group entries and multidimensional Hadamard arrays. It focuses on their applications in engineering and computer science, as signal transforms, spreading sequences, error-correcting codes, and cryptographic primitives. The book's second half presents the new results in cocyclic Hadamard matrices and their applications. Full expression of this theory has been realized only recently, in the Five-fold Constellation. This identifies cocyclic generalized Hadamard matrices with particular "stars" in four other areas of mathematics and engineering: group cohomology, incidence structures, combinatorics, and signal correlation. Pointing the way to possible new developments in a field ripe for further research, this book formulates and discusses ninety open questions.
Download or read book Handbook of Linear Algebra Second Edition written by Leslie Hogben and published by CRC Press. This book was released on 2013-11-26 with total page 1906 pages. Available in PDF, EPUB and Kindle. Book excerpt: With a substantial amount of new material, the Handbook of Linear Algebra, Second Edition provides comprehensive coverage of linear algebra concepts, applications, and computational software packages in an easy-to-use format. It guides you from the very elementary aspects of the subject to the frontiers of current research. Along with revisions and updates throughout, the second edition of this bestseller includes 20 new chapters. New to the Second Edition Separate chapters on Schur complements, additional types of canonical forms, tensors, matrix polynomials, matrix equations, special types of matrices, generalized inverses, matrices over finite fields, invariant subspaces, representations of quivers, and spectral sets New chapters on combinatorial matrix theory topics, such as tournaments, the minimum rank problem, and spectral graph theory, as well as numerical linear algebra topics, including algorithms for structured matrix computations, stability of structured matrix computations, and nonlinear eigenvalue problems More chapters on applications of linear algebra, including epidemiology and quantum error correction New chapter on using the free and open source software system Sage for linear algebra Additional sections in the chapters on sign pattern matrices and applications to geometry Conjectures and open problems in most chapters on advanced topics Highly praised as a valuable resource for anyone who uses linear algebra, the first edition covered virtually all aspects of linear algebra and its applications. This edition continues to encompass the fundamentals of linear algebra, combinatorial and numerical linear algebra, and applications of linear algebra to various disciplines while also covering up-to-date software packages for linear algebra computations.
Download or read book An Introduction to Groups and their Matrices for Science Students written by Robert Kolenkow and published by Cambridge University Press. This book was released on 2022-06-02 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text for advanced undergraduates develops group theory from triangle symmetry to particle physics, supported by examples from research.
Download or read book Linear Models in Matrix Form written by Jonathon D. Brown and published by Springer. This book was released on 2015-01-21 with total page 548 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook is an approachable introduction to statistical analysis using matrix algebra. Prior knowledge of matrix algebra is not necessary. Advanced topics are easy to follow through analyses that were performed on an open-source spreadsheet using a few built-in functions. These topics include ordinary linear regression, as well as maximum likelihood estimation, matrix decompositions, nonparametric smoothers and penalized cubic splines. Each data set (1) contains a limited number of observations to encourage readers to do the calculations themselves, and (2) tells a coherent story based on statistical significance and confidence intervals. In this way, students will learn how the numbers were generated and how they can be used to make cogent arguments about everyday matters. This textbook is designed for use in upper level undergraduate courses or first year graduate courses. The first chapter introduces students to linear equations, then covers matrix algebra, focusing on three essential operations: sum of squares, the determinant, and the inverse. These operations are explained in everyday language, and their calculations are demonstrated using concrete examples. The remaining chapters build on these operations, progressing from simple linear regression to mediational models with bootstrapped standard errors.
Download or read book Matrix Theory and Applications written by Charles R. Johnson and published by American Mathematical Soc.. This book was released on 1990 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the lecture notes prepared for the AMS Short Course on Matrix Theory and Applications, held in Phoenix in January, 1989. Matrix theory continues to enjoy a renaissance that has accelerated in the past decade, in part because of stimulation from a variety of applications and considerable interplay with other parts of mathematics. In addition, the great increase in the number and vitality of specialists in the field has dispelled the popular misconception that the subject has been fully researched.
Download or read book Extracellular Matrix Remodeling written by Nicoletta Gagliano and published by MDPI. This book was released on 2019-11-11 with total page 428 pages. Available in PDF, EPUB and Kindle. Book excerpt: The extracellular matrix (ECM) is where cells live. It is composed of collagen and elastic fibers, glycosaminoglycans and proteoglycans, and several glycoproteins. In most tissues, fibril-forming collagen type I is the major constituent of ECM.The function of the ECM goes beyond providing mechanical support to cells and tissues. As cells are embedded into the ECM and interact with its components through their surface receptors, cell–ECM interaction plays a key role in influencing different cell activities, such as cell proliferation and migration. The ECM also sequesters and releases growth factors affecting important cellular pathways. Overall, the ECM strongly influences and affects cell behavior and tissue homeostasis. ECM is a highly dynamic structural network that continuously undergoes controlled remodeling mediated by matrix-degrading enzymes, the matrix metalloproteinases (MMPs), under normal conditions. Quantitative and qualitative deregulation of ECM remodeling and, especially, of collagen turnover, is responsible for the alteration of ECM composition and structure, associated with the development and progression of several pathological conditions, such as organ fibrosis (determined by the abnormal accumulation of ECM components) and tumor invasion. The understanding of the diverse biological roles and properties of the ECM components is pivotal to the development of new therapeutic tools for disease treatment.
Download or read book Research Methodology written by Douglas K. Detterman and published by Praeger. This book was released on 1985-01-01 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt:
