In the series De Gruyter Textbook
This book is a comprehensive guide to Linear Algebra and covers all the fundamental topics such as vector spaces, linear independence, basis, linear transformations, matrices, determinants, inner products, eigenvectors, bilinear forms, and canonical forms. It also introduces concepts such as fields, rings, group homomorphism, and binary operations early on, which gives students a solid foundation to understand the rest of the material.
Unlike other books on Linear Algebra that are either too theory-oriented with fewer solved examples or too problem-oriented with less good quality theory, this book strikes a balance between the two. It provides easy-to-follow theorem proofs and a considerable number of worked examples with various levels of difficulty. The fundamentals of the subject are explained in a methodical and straightforward way.
This book is aimed at undergraduate and graduate students of Mathematics and Engineering Mathematics who are studying Linear Algebra. It is also a useful resource for students preparing for exams in higher education competitions such as NET, GATE, lectureships, etc. The book includes some of the most recent and challenging questions from these exams.
This book makes a balance between well-explained theory content and solved examples.
It contains the B.Sc Mathematics syllabus of Linear Algebra of almost all universities.
This book is the most suitable for lectureship competition exams.
This book presents a novel journey of the Gauss hypergeometric function and contains the different versions of the Gaussian hypergeometric function, including its classical version. In particular, the $q$-Gauss or basic Gauss hypergeometric function, Gauss hypergeometric function with matrix arguments, Gauss hypergeometric function with matrix parameters, the matrix-valued Gauss hypergeometric function, the finite field version, the extended Gauss hypergeometric function, the $(p, q)$- Gauss hypergeometric function, the incomplete Gauss hypergeometric function and the discrete analogue of Gauss hypergeometric function.
All these forms of the Gauss hypergeometric function and their properties are presented in such a way that the reader can understand the working algorithm and apply the same for other special functions. This book is useful for UG and PG students, researchers and faculty members working in the field of special functions and related areas.
The unique book containin various forms of Gauss hypergeometric function.
Combination of various forms of special functions.
Represents the analysis of distinct forms of special functions
In the series De Gruyter STEM
Currently, nonstandard analysis is barely considered in university teaching. The author argues that nonstandard analysis is valuable not only for teaching, but also for understanding standard analysis and mathematics itself. An axiomatic approach wich pays attention to different language levels (for example, in the distinction between sums of ones and the natural numbers of the theory) leads naturally to a nonstandard theory. For motivation historical ideas of Leibniz can be taken up. The book contains an elaborated concept that follows this approach and is suitable, for example, as a basis for a lecture-supplementary course.
The monograph part presents all major approaches to nonstandard analysis and discusses logical, model-theoretic, and set-theoretic investigations to reveal possible mathematical reasons that may lead to reservations about nonstandard analysis. Also various foundational positions as well as ontological, epistemological, and application-related issues are addressed. It turns out that the one-sided preference for standard analysis is justified neither from a didactic, mathematical nor philosophical point of view.
Thus, the book is especially valuable for students and instructors of analysis who are also interested in the foundations of their subject.
Explains the most important approaches to nonstandard analysis.
Provides interesting insights into the foundations of mathematics.
Demonstrates how introductory courses can be meaningfully complemented by nonstandard analysis.
Volume 11 in the series Advances in Analysis and Geometry
This book is intended to provide a fast, interdisciplinary introduction to the basic results of p-adic analysis and its connections with mathematical physics and applications. The book revolves around three topics: (1) p-adic heat equations and ultradiffusion; (2) fundamental solutions and local zeta functions, Riesz kernels, and quadratic forms; (3) Sobolev-type spaces and pseudo-differential evolution equations. These topics are deeply connected with very relevant current research areas. The book includes numerous examples, exercises, and snapshots of several mathematical theories. This book arose from the need to quickly introduce mathematical audience the basic concepts and techniques to do research in p-adic analysis and its connections with mathematical physics and other areas. The book is addressed to a general mathematical audience, which includes computer scientists, theoretical physicists, and people interested in mathematical analysis, PDEs, etc.
p-Adic analysis has received much attention in the last thirty-five years due to its connections with mathematical physics, biology, computer science, etc.
The proposed book introduces the p-adic analysis for an interdisciplinary audience
In the series De Gruyter Textbook
Analysis and Probability on graphs is an introduction to random graphs, Markov chains on digraphs, entropy of Markov Chains, and discrete Lyapunov exponents and Hausdorff dimension, requiring only minimal background in probability, mathematical analysis, and graphs. This textbook includes constructive discussions about the motivation of basic concepts, and many worked-out problems in each chapter, making it ideal for classroom use or self-study.
Covers topics that are known to the field but have never been collected in a single book.
Includes worked-out problems and unsolved problems.
Uses abstract notions and ideas to present probability theory, random graphs, Markov chains on digraphs.
Author / Editor information
Shmuel Friedland, University of Illinois; Mohsen Aliabadi, University of California, USA.
A textbook for advanced math or computer science majors taking probability, graph theory, combinatorics, or matrix theory courses.
Combinatorics and Graph Theory
Mathematics
Probability and Statistics
This self-contained book explains how to count graph configurations to obtain topological invariants for 3-manifolds and links in these 3-manifolds, and it investigates the properties of the obtained invariants. The simplest of these invariants is the linking number of two disjoint knots in the ambient space described in the beginning of the book as the degree of a Gauss map.
Mysterious knot invariants called gquantum invariantsh were introduced in the mid-1980s, starting with the Jones polynomial. Witten explained how to obtain many of them from the perturbative expansion of the Chern?Simons theory. His physicist viewpoint led Kontsevich to a configuration-counting definition of topological invariants for the closed 3-manifolds where knots bound oriented compact surfaces. The book's first part shows in what sense an invariant previously defined by Casson for these manifolds counts embeddings of the theta graph. The second and third parts describe a configuration-counting invariant Z generalizing the above invariants. The fourth part shows the universality of Z with respect to some theories of finite-type invariants. The most sophisticated presented generalization of Z applies to small pieces of links in 3-manifolds called tangles. Its functorial properties and its behavior under cabling are used to describe the properties of Z.
The book is written for graduate students and more advanced researchers interested in low-dimensional topology and knot theory.