Format: Hardback, 676 pages, height x width: 254x178 mm, 614 Illustrations, color; 6 Illustrations,
black and white; XI, 676 p. 620 illus., 614 illus. in color., 1 Hardback
Pub. Date: 23-Oct-2024
ISBN-13: 9783031741524
This undergraduate textbook provides a comprehensive treatment of Euclidean and transformational geometries, supplemented by substantial discussions of topics from various non-Euclidean and less commonly taught geometries, making it ideal for both mathematics majors and pre-service teachers. Emphasis is placed on developing students' deductive reasoning skills as they are guided through proofs, constructions, and solutions to problems. The text frequently emphasizes strategies and heuristics of problem solving including constructing proofs (Where to begin? How to proceed? Which approach is more promising? Are there multiple solutions/proofs? etc.). This approach aims not only to enable students to successfully solve unfamiliar problems on their own, but also to impart a lasting appreciation for mathematics.
The text first explores, at a higher level and in much greater depth, topics that are normally taught in high school geometry courses: definitions and axioms, congruence, circles and related concepts, area and the Pythagorean theorem, similarity, isometries and size transformations, and composition of transformations. Constructions and the use of transformations to carry out constructions are emphasized. The text then introduces more advanced topics dealing with non-Euclidean and less commonly taught topics such as inversive, hyperbolic, elliptic, taxicab, fractal, and solid geometries. By examining what happens when one or more of the building blocks of Euclidean geometry are altered, students will gain a deeper understanding of and appreciation for Euclidean concepts.
To accommodate students with different levels of experience in the subject, the basic definitions and axioms that form the foundation of Euclidean geometry are covered in Chapter 1. Problem sets are provided after every section in each chapter and include nonroutine problems that students will enjoy exploring. While not necessarily required, the appropriate use of freely available dynamic geometry software and other specialized software referenced in the text is strongly encouraged; this is especially important for visual learners and for forming conjectures and testing hypotheses.
Preface.- Surprising Results and Basic Notions.- Congruence, Constructions, and the Parallel Postulate.- Circles.- Area and the Pythagorean Theorem.- Similarity.- Isometries and Size Transformations.- Composition of Transformations.- More Recent Discoveries.- Inversion.- Hyperbolic Geometry.- Elliptic Geometries.- Projective Geometry.- Taxicab Geometry.- Fractal Geometry.- Solid Geometry.
Format: Hardback, 182 pages, height x width: 235x155 mm, 43 Illustrations, color; 19 Illustrations,
black and white; XVI, 182 p. 62 illus., 43 illus. in color., 1 Hardback
Series: Studies in Big Data 167
Pub. Date: 18-Mar-2025
ISBN-13: 9783031771446
This book explores the dynamically developing areas of quantum computing and quantum cryptography. The book offers an in-depth examination of the possibilities and difficulties presented by these revolutionary technologies, with the goal of connecting abstract ideas with real-world applications. The book is an extremely helpful resource in the context of the upcoming quantum age. This highlights the importance of creating cryptographic techniques that can withstand the power of quantum computers to protect digital communications and vital infrastructures. This work makes a substantial contribution to the topic of cybersecurity by doing a comprehensive analysis of classical and quantum cryptography approaches, as well as actual implementations and performance evaluations. The book plays a vital role in providing valuable guidance to researchers, practitioners, and policymakers. It offers valuable insights that are necessary for effectively managing the shift towards quantum-secure technology and safeguarding the future security of digital information.
Introduction.- Classical and Modern Cryptography.- Quantum Algorithms.-
Quantum Cryptography.- Quantum Computing Applications.- Future Directions and
Open Challenges.- Conclusion.
Format: Paperback / softback, 287 pages, height x width: 240x168 mm, 1 Illustrations, color; 32 Illustrations,
black and white; XV, 287 p. 33 illus., 1 illus. in color.
Series: Frontiers in the History of Science
Pub. Date: 18-Mar-2025
ISBN-13: 9783031772580
This monograph presents an interpretive essay on the foundations of Leibnizfs calculus, accompanied by key texts in English translation. The essay examines Leibniz's evolving views on infinitesimals and infinite numbers, tracing their development from his early metaphysical ideas to his mature justifications of the calculus.
Leibniz first proposed treating infinitesimals as fictions in the 1670s, in line with the mathematical practices of his time, where abstract concepts could be used in calculations without implying their existence. By 1676, he rejected their status as quantities, yet continued to refine his arguments on this topic into the 1690s.
The essay concludes with an analysis of Leibnizfs defense of his calculus in the early 18th century, showing how his later works naturally extended from earlier insights. This monograph will be a valuable resource for scholars and students of Leibniz and the history of science.
- Part I Interpretive Essay
.- Chapter 1. Introduction
.- Chapter 2. On the Metaphysics of the Continuum (1669-1676).
- Chapter 3. Mathematical Fictions.
- Chapter 4. De Quadratura Arithmetica (DQA).
- Chapter 5. Infinitesimals and Existence after 1676.
- Chapter 6. Leibnizs Mature Justifications of the Calculus
.- Chapter 7. Conclusion.- Part II A selection of translations of key texts.
- Chapter 8: Texts for chapter 2, On the Metaphysics of the Continuum (1669-1676).
- Chapter 9: Texts for chapter 3, Mathematical Fictions.
- Chapter 10: Texts for chapter 4, De Quadratura Arithmetica (DQA).
- Chapter 11: Texts for chapter 5, Infinitesimals and Existence after 1676.
- Chapter 12: Texts for chapter 6, Leibnizs Mature Justifications of the Calculus.
Format: Hardback, 422 pages, height x width: 235x155 mm, XVI, 422 p., 1 Hardback
Series: Springer Monographs in Mathematics
Pub. Date: 18-Mar-2025
ISBN-13: 9789819604296
This comprehensive book explores the intricate realm of fine potential theory. Delving into the real theory, it navigates through harmonic and subharmonic functions, addressing the famed Dirichlet problem within finely open sets of R^n. These sets are defined relative to the coarsest topology on R^n, ensuring the continuity of all subharmonic functions. This theory underwent extensive scrutiny starting from the 1970s, particularly by Fuglede, within the classical or axiomatic framework of harmonic functions. The use of methods from fine potential theory has led to solutions of important classical problems and has allowed the discovery of elegant results for extension of classical holomorphic function to wider classes of gdomainsh. Moreover, this book extends its reach to the notion of plurisubharmonic and holomorphic functions within plurifinely open sets of C^n and its applications to pluripotential theory. These open sets are defined by coarsest topology that renders all plurisubharmonic functions continuous on C^n.
The presentation is meticulously crafted to be largely self-contained, ensuring accessibility for readers at various levels of familiarity with the subject matter. Whether delving into the fundamentals or seeking advanced insights, this book is an indispensable reference for anyone intrigued by potential theory and its myriad applications. Organized into five chapters, the first four unravel the intricacies of fine potential theory, while the fifth chapter delves into plurifine pluripotential theory.
Background in Potential Theory.- Fundamentals of Fine Potential Theory.- Further Developments.- Fine Complex Potential Theory.
Format: Hardback, 695 pages, height x width: 235x155 mm, 24 Illustrations, color; 4 Illustrations,
black and white; XXIII, 695 p. 28 illus., 24 illus. in color., 1 Hardback
Series: Springer Undergraduate Texts in Mathematics and Technology
Pub. Date: 18-Mar-2025
ISBN-13: 9783031784231
This text combines a compact linear algebra course with a serious dip into various physical applications. It may be used as a primary text for a course in linear algebra or as a supplementary text for courses in applied math, scientific computation, mathematical physics, or engineering.
The text is divided into two parts.
Part 1 comprises a fairly standard presentation of linear algebra. Chapters 1?3 contain the core mathematical concepts typical for an introductory course while Chapter 4 contains numerous "short" applications. Chapter 5 is a repository of standard facts about matrix factorization and quadratic forms together with the "connective tissue" of topics needed for a coherent discussion, including the singular value decomposition, the Jordan normal form, Sylvester's law of inertia and the Witt theorems. Part I contains around 300 exercises, found throughout the text, and are an integral part of the presentation.
Part 2 features deeper applications. Each of these "large" applications require no more than linear algebra to discuss, though the style and arrangement of results would be challenging to a beginning student and more appropriate for a second or later course. Chapter 6 provides an introduction to the discrete Fourier transform, including the fast Fourier algorithm. Chapter 7 is a thorough introduction to isometries and some of the classical groups, and how these groups have come to be important in physics. Chapter 8 is a fairly detailed look at real algebras and completes a presentation of the classical Lie groups and algebras. Chapter 9 is a careful discussion of tensors on a finite-dimensional vector space, finishing with the Hodge Star operator and the Grassmann algebra. Finally, Chapter 10 gives an introduction to classical mechanics including Noether's first theorem and emphasizes how the classical Lie groups, discussed in earlier chapters, become important in this setting.
The Chapters of Part 2 are intended to give a sense of the ubiquity, of the indispensable utility, of linear algebra in modern science and mathematics and some feel for way it is actually used in disparate subject areas. Twelve appendices are included. The last seven refer to MATLABR code which, though not required and rarely mentioned in the text, can be used to augment understanding. For example, fifty-five MATLAB functions implement every tensor operation from Chapter 9. A zipped file of all code is available for download from the author's website.
Preface.- Part I Linear Algebra.- 1 Basics.- 2 Vector Spaces.- 3
Bilinear and Sesquilinear Forms.- 4 Shorter Applications.- 5 Factorization.-
Part II Extended Applications.- 6 The Discrete Fourier Transform.- 7 Geometry
Associated with Real Bilinear Forms.- Algebras.- 9 Tensors.- 10 Mechanics:
Newton, Lagrange, Hamilton, Noether.- A Permutation and the Signum Function.-
B Groups.- C Polynomial Roots are Continuous in the Coefficients.- D The
Fundamental Theorem of Algebra.- E There is a Basis for Every Vector Space.-
F The GNU Linear Programming Kit (GLPK).- G MATLAB Code for Adjunct
Matrices.- H MATLAB Code for Rotations and Euler Angles.- I. MATLAB Code for
Lorentz Matrices.- J MATLAB Code to Illustrate DFT Facts.- K MATLAB Code for
Quaternions, Octonions and Gamma Matrices.- L MATLAB Code for Tensor
Operations.- Suggested Further Reading.- References.- Index.
Format: Hardback, 647 pages, height x width: 235x155 mm, 124 Illustrations, color; 36 Illustrations,
black and white; XII, 647 p. 160 illus., 124 illus. in color., 1 Hardback
Series: Springer Series in Computational Mathematics 61
Pub. Date: 25-Feb-2025
ISBN-13: 9783031806834
This book offers a comprehensive presentation of numerical methods for elliptic boundary value problems that satisfy discrete maximum principles (DMPs). The satisfaction of DMPs ensures that numerical solutions possess physically admissible values, which is of utmost importance in numerous applications. A general framework for the proofs of monotonicity and discrete maximum principles is developed for both linear and nonlinear discretizations. Starting with the Poisson problem, the focus is on convection-diffusion-reaction problems with dominant convection, a situation which leads to a numerical problem with multi-scale character. The emphasis of this book is on finite element methods, where classical (usually linear) and modern nonlinear discretizations are presented in a unified way. In addition, popular finite difference and finite volume methods are discussed. Besides DMPs, other important properties of the methods, like convergence, are studied. Proofs are presented step by step, allowing readers to understand the analytic techniques more easily. Numerical examples illustrate the behavior of the methods.
Introduction. - Convection-Di usion-Reaction Problems and Maximum Principles.- Discrete Maximum Principles.- Partitions of the Domain.- Finite Element Methods.- Finite Element Methods for Diffusion Problems.- Finite Element Methods for Reaction-Diffusion Problems.- Linear Finite Element Methods for Convection-Diffusion-Reaction Problems.- Nonlinear Finite Element Methods for Convection-Diffusion-Reaction Problems: Discretizations Based on Modi ed Variational Forms.- Nonlinear Finite Element Methods for Convection-Diffusion-Reaction Problems: Algebraically Stabilized Methods.- Finite Difference Methods.- Finite Volume Methods.- A Numerical Study for a Problem with Different Regimes.- Outlook.