Download or read book Grassmann Algebra Volume 1 Foundations written by John Browne and published by John M Browne. This book was released on 2012-10-25 with total page 589 pages. Available in PDF, EPUB and Kindle. Book excerpt: Grassmann Algebra Volume 1: Foundations Exploring extended vector algebra with Mathematica Grassmann algebra extends vector algebra by introducing the exterior product to algebraicize the notion of linear dependence. With it, vectors may be extended to higher-grade entities: bivectors, trivectors, … multivectors. The extensive exterior product also has a regressive dual: the regressive product. The pair behaves a little like the Boolean duals of union and intersection. By interpreting one of the elements of the vector space as an origin point, points can be defined, and the exterior product can extend points into higher-grade located entities from which lines, planes and multiplanes can be defined. Theorems of Projective Geometry are simply formulae involving these entities and the dual products. By introducing the (orthogonal) complement operation, the scalar product of vectors may be extended to the interior product of multivectors, which in this more general case may no longer result in a scalar. The notion of the magnitude of vectors is extended to the magnitude of multivectors: for example, the magnitude of the exterior product of two vectors (a bivector) is the area of the parallelogram formed by them. To develop these foundational concepts, we need only consider entities which are the sums of elements of the same grade. This is the focus of this volume. But the entities of Grassmann algebra need not be of the same grade, and the possible product types need not be constricted to just the exterior, regressive and interior products. For example quaternion algebra is simply the Grassmann algebra of scalars and bivectors under a new product operation. Clifford, geometric and higher order hypercomplex algebras, for example the octonions, may be defined similarly. If to these we introduce Clifford's invention of a scalar which squares to zero, we can define entities (for example dual quaternions) with which we can perform elaborate transformations. Exploration of these entities, operations and algebras will be the focus of the volume to follow this. There is something fascinating about the beauty with which the mathematical structures that Hermann Grassmann discovered describe the physical world, and something also fascinating about how these beautiful structures have been largely lost to the mainstreams of mathematics and science. He wrote his seminal Ausdehnungslehre (Die Ausdehnungslehre. Vollständig und in strenger Form) in 1862. But it was not until the latter part of his life that he received any significant recognition for it, most notably by Gibbs and Clifford. In recent times David Hestenes' Geometric Algebra must be given the credit for much of the emerging awareness of Grassmann's innovation. In the hope that the book be accessible to scientists and engineers, students and professionals alike, the text attempts to avoid any terminology which does not make an essential contribution to an understanding of the basic concepts. Some familiarity with basic linear algebra may however be useful. The book is written using Mathematica, a powerful system for doing mathematics on a computer. This enables the theory to be cross-checked with computational explorations. However, a knowledge of Mathematica is not essential for an appreciation of Grassmann's beautiful ideas.
Download or read book Geometric Multivector Analysis written by Andreas Rosén and published by Springer Nature. This book was released on 2019-11-09 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions. The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes’s theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics. The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis.
Download or read book Proceedings of the American Association for the Advancement of Science written by American Association for the Advancement of Science and published by . This book was released on 1892 with total page 604 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Proceedings written by American Association for the Advancement of Science and published by . This book was released on 1892 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Vector Analysis written by James Henry Taylor and published by . This book was released on 1947 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1995 with total page 538 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lists citations with abstracts for aerospace related reports obtained from world wide sources and announces documents that have recently been entered into the NASA Scientific and Technical Information Database.
Download or read book Principles of the Algebra of Physics written by Alexander Macfarlane and published by . This book was released on 1894 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Proceedings of the American Association for the Advancement of Science written by and published by . This book was released on 1892 with total page 602 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Commutative Harmonic Analysis written by David Colella and published by American Mathematical Soc.. This book was released on 1989 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains an array of both expository and research articles which represents the proceedings of a conference on commutative harmonic analysis, held in July 1987 and sponsored by St Lawrence University and GTE Corporation. This book is suitable for those beginning research in commutative harmonic analysis.
Download or read book Statistics on Special Manifolds written by Yasuko Chikuse and published by Springer Science & Business Media. This book was released on 2012-11-12 with total page 425 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering statistical analysis on the two special manifolds, the Stiefel manifold and the Grassmann manifold, this book is designed as a reference for both theoretical and applied statisticians. It will also be used as a textbook for a graduate course in multivariate analysis. It is assumed that the reader is familiar with the usual theory of univariate statistics and a thorough background in mathematics, in particular, knowledge of multivariate calculation techniques.
Download or read book pt 1 Elementary principles in statistical mechanics pt 2 Dynamics Vector analysis and multiple algebra Electromagnetiic theory of light etc written by Josiah Willard Gibbs and published by . This book was released on 1902 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book A History of Vector Analysis written by Michael J. Crowe and published by Courier Corporation. This book was released on 1994-01-01 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt: Prize-winning study traces the rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers to the final acceptance around 1910 of the modern system of vector analysis.
Download or read book Energy Research Abstracts written by and published by . This book was released on 1991 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Fundamentals of Image Data Mining written by Dengsheng Zhang and published by Springer Nature. This book was released on 2021-06-25 with total page 383 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique and useful textbook presents a comprehensive review of the essentials of image data mining, and the latest cutting-edge techniques used in the field. The coverage spans all aspects of image analysis and understanding, offering deep insights into areas of feature extraction, machine learning, and image retrieval. The theoretical coverage is supported by practical mathematical models and algorithms, utilizing data from real-world examples and experiments. Topics and features: Describes essential tools for image mining, covering Fourier transforms, Gabor filters, and contemporary wavelet transforms Develops many new exercises (most with MATLAB code and instructions) Includes review summaries at the end of each chapter Analyses state-of-the-art models, algorithms, and procedures for image mining Integrates new sections on pre-processing, discrete cosine transform, and statistical inference and testing Demonstrates how features like color, texture, and shape can be mined or extracted for image representation Applies powerful classification approaches: Bayesian classification, support vector machines, neural networks, and decision trees Implements imaging techniques for indexing, ranking, and presentation, as well as database visualization This easy-to-follow, award-winning book illuminates how concepts from fundamental and advanced mathematics can be applied to solve a broad range of image data mining problems encountered by students and researchers of computer science. Students of mathematics and other scientific disciplines will also benefit from the applications and solutions described in the text, together with the hands-on exercises that enable the reader to gain first-hand experience of computing.
Download or read book Efficiency and Sustainability in the Energy and Chemical Industries written by Krishnan Sankaranarayanan and published by CRC Press. This book was released on 2004-01-21 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: Translating fundamental principles of irreversible thermodynamics into day-to-day engineering concepts, this reference provides the tools to accurately measure process efficiency and sustainability in the power and chemical industries-helping engineers to recognize why losses occur and how they can be reduced utilizing familiar thermodynamic principles. Compares the present industrial society with an emerging metabolic society in which mass production and consumption are in closer harmony with the natural environment. The first book to utilize classic thermodynamic principles for clear understanding, analysis, and optimization of work flows, environmental resources, and driving forces in the chemical and power industries.
Download or read book Visual Domain Adaptation in the Deep Learning Era written by Gabriela Csurka and published by Morgan & Claypool Publishers. This book was released on 2022-04-05 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: Solving problems with deep neural networks typically relies on massive amounts of labeled training data to achieve high performance/b>. While in many situations huge volumes of unlabeled data can be and often are generated and available, the cost of acquiring data labels remains high. Transfer learning (TL), and in particular domain adaptation (DA), has emerged as an effective solution to overcome the burden of annotation, exploiting the unlabeled data available from the target domain together with labeled data or pre-trained models from similar, yet different source domains. The aim of this book is to provide an overview of such DA/TL methods applied to computer vision, a field whose popularity has increased significantly in the last few years. We set the stage by revisiting the theoretical background and some of the historical shallow methods before discussing and comparing different domain adaptation strategies that exploit deep architectures for visual recognition. We introduce the space of self-training-based methods that draw inspiration from the related fields of deep semi-supervised and self-supervised learning in solving the deep domain adaptation. Going beyond the classic domain adaptation problem, we then explore the rich space of problem settings that arise when applying domain adaptation in practice such as partial or open-set DA, where source and target data categories do not fully overlap, continuous DA where the target data comes as a stream, and so on. We next consider the least restrictive setting of domain generalization (DG), as an extreme case where neither labeled nor unlabeled target data are available during training. Finally, we close by considering the emerging area of learning-to-learn and how it can be applied to further improve existing approaches to cross domain learning problems such as DA and DG.
Download or read book Space Number and Geometry from Helmholtz to Cassirer written by Francesca Biagioli and published by Springer. This book was released on 2016-08-22 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism.
