This textbook provides a readable account of the examples and fundamental results of groups from a theoretical and geometrical point of view. Topics on important examples of groups (like cyclic groups, permutation groups, group of arithmetical functions, matrix groups and linear groups), Lagrangefs theorem, normal subgroups, factor groups, derived subgroup, homomorphism, isomorphism and automorphism of groups have been discussed in depth. Covering all major topics, this book is targeted to undergraduate students of mathematics with no prerequisite knowledge of the discussed topics. Each section ends with a set of worked-out problems and supplementary exercises to challenge the knowledge and ability of the reader.
Related Subjects:
Mathematics and Computing, Mathematics, Algebra, Group Theory and Generalizations, Number Theory, Linear Algebra
Subject: Mathematics and Statistics
Copyright Year: 2021
ISBN: 978-981-16-6364-2
Format: Book: Generic (Hard cover)
Product Category: Textbooks
This book provides the knowledge of the newly-established supertrigonometric and superhyperbolic functions with the special functions such as Mittag-Leffler, Wiman, Prabhakar, Miller-Ross, Rabotnov, Lorenzo-Hartley, Sonine, Wright and Kohlrausch-Williams-Watts functions, Gauss hypergeometric series and Clausen hypergeometric series. The special functions can be considered to represent a great many of the real-world phenomena in mathematical physics, engineering and other applied sciences. The audience benefits of new and original information and references in the areas of the special functions applied to model the complex problems with the power-law behaviors.
The results are important and interesting for scientists and engineers to represent the complex phenomena arising in applied sciences therefore graduate students and researchers in mathematics, physics and engineering might find this book appealing.
Related Subjects:
Mathematics and Computing, Mathematics, Analysis, Special Functions, Integral Transforms and Operational Calculus
Subject: Mathematics and Statistics
Copyright Year: 2021
ISBN: 978-981-33-6333-5
Format: Book: Generic (Hard cover)
Product Category: Monographs
The focus of this book is on providing students with insights into geometry that can help them understand deep learning from a unified perspective. Rather than describing deep learning as an implementation technique, as is usually the case in many existing deep learning books, here, deep learning is explained as an ultimate form of signal processing techniques that can be imagined.
To support this claim, an overview of classical kernel machine learning approaches is presented, and their advantages and limitations are explained. Following a detailed explanation of the basic building blocks of deep neural networks from a biological and algorithmic point of view, the latest tools such as attention, normalization, Transformer, BERT, GPT-3, and others are described. Here, too, the focus is on the fact that in these heuristic approaches, there is an important, beautiful geometric structure behind the intuition that enables a systematic understanding. A unified geometric analysis to understand the working mechanism of deep learning from high-dimensional geometry is offered. Then, different forms of generative models like GAN, VAE, normalizing flows, optimal transport, and so on are described from a unified geometric perspective, showing that they actually come from statistical distance-minimization problems.
Because this book contains up-to-date information from both a practical and theoretical point of view, it can be used as an advanced deep learning textbook in universities or as a reference source for researchers interested in acquiring the latest deep learning algorithms and their underlying principles. In addition, the book has been prepared for a codeshare course for both engineering and mathematics students, thus much of the content is interdisciplinary and will appeal to students from both disciplines.
Related Subjects:
Mathematics and Computing, Mathematics, Analysis, Functional Analysis, Geometry, Differential Geometry, Computer Science, Artificial Intelligence, Applications of Mathematics, Mathematical Models of Cognitive Processes and Neural Networks, Mathematical and Computational Biology
Subject: Mathematics and Statistics
Copyright Year: 2021
ISBN: 978-981-16-6045-0
Format: Book: Generic (Hard cover)
Product Category:Textbooks
Rich information-theoretic structure in out-of-equilibrium thermodynamics exists in both the classical and quantum regimes, leading to the fruitful interplay among statistical physics, quantum information theory, and mathematical theories such as matrix analysis and asymptotic probability theory. The main purpose of this book is to clarify how information theory works behind thermodynamics and to shed modern light on it.
The book focuses on both purely information-theoretic concepts and their physical implications. From the mathematical point of view, rigorous proofs of fundamental properties of entropies, divergences, and majorization are presented in a self-contained manner. From the physics perspective, modern formulations of thermodynamics are discussed, with a focus on stochastic thermodynamics and resource theory of thermodynamics. In particular, resource theory is a recently developed field as a branch of quantum information theory to quantify guseful resourcesh and has an intrinsic connection to various fundamental ideas of mathematics and information theory. This book serves as a concise introduction to important ingredients of the information-theoretic formulation of thermodynamics.
Related Subjects:
Physical Sciences, Physics and Astronomy, Theoretical, Mathematical and Computational Physics, Mathematical Physics, Materials Science, Condensed Matter, Spintronics, Quantum Physics, Complex Systems
Subject: Physics and Astronomy
Copyright Year: 2022
ISBN: 978-981-16-6643-8
Format: Book: Generic (Soft cover)
Product Category: Monographs