Format: Hardback, 319 pages, height x width: 235x155 mm, 5 Illustrations, color;
6 Illustrations, black and white; XXIX, 319 p. 11 illus., 5 illus. in color., 1 Hardback
Series: Computer Science Foundations and Applied Logic
Pub. Date: 23-May-2025
ISBN-13: 9783031853517
This unique textbook, in contrast to a standard logic text, provides the reader with a logic that can be used in practice to express and reason about mathematical ideas. The book is an introduction to simple type theory, a classical higher-order version of predicate logic that extends first-order logic.
It presents a practice-oriented logic called Alonzo that is based on Alonzo Church's formulation of simple type theory known as Church's type theory. Unlike traditional predicate logics, Alonzo admits undefined expressions. The book illustrates using Alonzo how simple type theory is suited ideally for reasoning about mathematical structures and constructing libraries of mathematical knowledge. For this second edition, more than 400 additions, corrections, and improvements have been made, including a new chapter on inductive sets and types.
E Offers the first book-length introduction to simple type theory as a predicate logic
E Provides the reader with a logic that is close to mathematical practice
E Includes a module system for building libraries of mathematical knowledge
E Employs two semantics, one for mathematics and one for logic
E Emphasizes the model-theoretic view of predicate logic
E Presents several important topics, such as definite description and theory morphisms, not usually found in standard logic textbooks
Aimed at students of mathematics and computing at the graduate or upper-undergraduate level, this book is well suited for mathematicians, computing professionals, engineers, and scientists who need a practical logic for expressing and reasoning about mathematical ideas.
Table of Contents
Chapter 1 Introduction.
Chapter 2 Answers to Readers Questions.-
Chapter 3 Preliminary Concepts.
Chapter 4 Syntax.
Chapter 5 Semantics.-
Chapter 6 Additional Notation.
Chapter 7 Beta-reduction and Substitution.-
Chapter 8 Proof Systems.
Chapter 9 Theories.
Chapter 10 Inductive Sets and Types.
Chapter 11 Sequences.
Chapter 12 Developments.
Chapter 13 Real Number Mathematics.
Chapter 14 Morphisms.
Chapter 15 Alonzo Variants.-
Chapter 16 Software Support.
Format: Hardback, 457 pages, height x width: 240x168 mm, 13 Illustrations, black and white; X, 457 p. 13 illus., 1 Hardback
Series: Classic Texts in the Sciences
Pub. Date: 08-May-2025
ISBN-13: 9783031854736
This book presents a historical account of Felix Klein's "Comparative Reflections on Recent Research in Geometry" (1872), better known as his "Erlangen Program.h Originally conceived and written when Klein was collaborating with Sophus Lie, this bold essay initially made little impression on contemporary researchers. Decades later, however, it eventually became a famous classic. Eminent mathematicians hailed Kleinfs main message ? the role of invariants of transformation groups in geometry ? as presaging major developments in mathematics and physics.
The first part of this book focuses on the prehistory surrounding Kleinfs gErlangen Program,h stressing the motivations that led Klein to write it. The core of the book (Part II) then presents a new translation of Klein's original text, followed by detailed textual analysis aimed at guiding the reader through its rather terse and opaque prose. Part III deals with its complicated reception history, treated in four periods spanning the years from 1872 to 1930. This culminated during Kleinfs lifetime with his efforts to promote the "Erlangen Programh as a framework for interpreting Einsteinfs theory of relativity. After his death in 1925, the viability of this framework became a contentious issue among leading differential geometers. Part IV looks back on the transformations in mathematics that led to a modernized interpretation of Kleinfs message. The book also explores in depth how the growing fame of the gErlangen Programh undermined Kleinfs friendship with Sophus Lie, leading to a dramatic public break between them in 1893.
Beyond the "Erlangen Programh itself, this book deals with many of Felix Kleinfs other works. As an introduction to a largely forgotten world of ideas, this study will appeal not only to experts but also to graduate students and all those with a serious interest in the history of modern mathematics.
Preface.- Introduction.- Part I. Prehistory of the Erlangen Program.-
1 Klein as a Young Geometer.-
2. Klein Encounters Sophus Lie.-
3. Klein on Cayleys Projective Metric.- Part II. Kleins Erlangen Program with Commentary.-
4. Kleins Erlangen Program.-
5. Textual Analysis of Kleins Erlangen Program.- Part III. Four Phases of Reception and Transformation.-
6. First Phase of Reception, 18731889.-
7. Second Phase of Reception, 18901899.-
8. Third Phase of Reception, 19001916.-
9. Fourth Phase of Reception, 19171930.- Part IV. Reconsiderations.-
10 Historical Reflections.-
Bibliography.-
Name Index.
Format: Hardback, 206 pages, height x width: 240x168 mm, 9 Illustrations, color;
1 Illustrations, black and white; XII, 206 p. 10 illus., 9 illus. in color., 1 Hardback
Series: Synthesis Lectures on Engineering, Science, and Technology
Pub. Date: 17-Jun-2025
ISBN-13: 9783031841392
This book discusses how relativistic quantum field theories must transform under strongly continuous unitary representations of the Poincare group. The focus is on the construction of the representations that provide the basis for the formulation of current relativistic quantum field theories of scalar fields, the Dirac field, and the electromagnetic field. Such construction is tied to the use of the methods of operator theory that also provide the basis for the formulation of quantum mechanics, up to the interpretation of the measurement process. In addition, since representation spaces of primary interest in quantum theory are infinite dimensional, the use of these methods is essential. Consequently, the book also calculates the generators of relevant strongly continuous one-parameter groups that are associated with the representations and, where appropriate, the corresponding spectrum. Part I of Quantum Spin and Representations of the Poincare Group specifically addresses: conventions; basic properties of SO(2) and SO(3); construction of a double cover of SO(3); SU(2) spinors; continuous unitary representation of SU(2); basic properties of the Lorentz Group; unitary representation of the restricted Lorentz Group; an extension to a strongly continuous representation of the restricted Poincare Group; and an extension to a unitary/anti-unitary representation of the Poincare Group.
Introduction.- Conventions.- Prerequisites.- Basic Properties of SO(2).-
Basic Properties of SO(3).- Construction of a Double Cover of SO(3).-
SU(2)-Spinors.- A Strongly Continuous Unitary Representation of SU(2).- Basic
Properties of the Lorentz Group.- Unitary Representation of the Restricted
Lorentz Group.- An Extension to a Strongly Continuous Representation of the
Restricted Poincare Group.- An Extension to a Unitary/Anti-unitary
Representation of the Poincare Group.-Appendix.- Bibliography.- Index of
Symbols.- Index.
Format: Hardback, 176 pages, height x width: 235x155 mm, XI, 176 p., 1 Hardback
Series: Springer Monographs in Mathematics
Pub. Date: 16-Jun-2025
ISBN-13: 9783031864605
This book provides a comprehensive coverage of the theory of conjugacy in finite classical groups. Given such a classical group G, the three fundamental problems considered are the following: to list a representative for each conjugacy class of G; to describe the centralizer of each representative, by giving its group structure and a generating set; and to solve the conjugacy problem in G?namely, given two elements of G, establish whether they are conjugate, and if so, find a conjugating element. The book presents comprehensive theoretical solutions to all three problems, and uses these solutions to formulate practical algorithms. In parallel to the theoretical work, implementations of these algorithms have been developed in Magma. These form a critical component of various general algorithms in computational group theory?for example, computing character tables and solving conjugacy problems in arbitrary finite groups.
1. Introduction and Background.-
2. General and Special Linear
Groups.-
3. Preliminaries on Classical Groups.-
4. Unipotent Classes in Good
Characteristic.-
5. Unipotent Classes in Bad Characteristic.-
6. Semisimple
Classes.-
7. General Conjugacy Classes.
Format: Hardback, 282 pages, height x width: 235x155 mm, 45 Illustrations, black and white; X, 282 p. 45 illus., 1 Hardback
Pub. Date: 20-Jun-2025
ISBN-13: 9783031865312
This textbook offers an introduction to ODEs that focuses on the qualitative behavior of differential equations rather than specialized methods for solving them. The book is organized around this approach with important topics, such as existence, uniqueness, qualitative behaviour, and stability, appearing in early chapters and explicit solution methods covered later. Proofs are included in an approachable manner, which are first motivated by describing the main ideas in a general sense before being written out in detail. A clear and accessible writing style is used, containing numerous examples and calculations throughout the text. Two appendices offer readers further material to explore, with the first using the orbits of the planets as an illustrative example and the second providing insightful historical notes. After reading this book, students will have a strong foundation for a course in PDEs or mathematical modeling.
Fundamentals of Ordinary Differential Equations is suitable for an undergraduate course for students who have taken basic calculus and linear algebra courses, and who are able to read and write basic proofs. Because of its detailed approach, it is also conducive to self-study.
What is an ordinary differential equation?.- First-order differential
equations.- Existence and uniqueness theorems.- Linear equations of higher
order.- Systems of differential equations.- The qualitative theory and the
phase plane.- Solution of differential equations by power series.- The
Laplace transform.- Appendix: The orbits of the planets.- Appendix:
Historical notes.
Format: Hardback, 556 pages, height x width: 235x155 mm, 384 Illustrations, color;
8 Illustrations, black and white; XX, 556 p. 392 illus., 384 illus. in color., 1 Hardback
Pub. Date: 18-Jun-2025
ISBN-13: 9783031869686
This textbook completes the authors series of books on solving complex math problems and is aimed at developing readers' geometric thinking to master the skills of solving solid geometry problems. Written in a friendly manner, it discusses many important and sometimes overlooked topics about polyhedra such as their cross sections, unfolding, inscribed and circumscribed solids, and figures of revolution. Over 350 unique problems with detailed solutions and hints are presented throughout the text, many of which are solved in multiple ways to aid readers with different mathematical backgrounds. If the problem is of historical significance or can be related to a similar problem solved in ancient times, its original solution, historical information about its creation and origin of its methods are also included.
Various applications of stereometry are also explored, including those to chemistry, molecular structures, and crystallography. For example, using Euler's formula for a convex polyhedron, the reader will learn how to explain the structure of various chemical compounds, such as how to predict the shape of the truncated icosahedron for the C60 fullerene molecule (the most powerful antioxidant known today) and to prove why the surface of any fullerene C2n consists of n -10 regular hexagons and always only 12 regular pentagons.
Demonstrating the connections between different areas of mathematics, Methods of Solving Solid Geometry Problems will be of interest to students who want to excel in math competitions and to those who aspire for greater mastery in linear algebra, analytic geometry, calculus, and more advanced topics. It can also be used by teachers to stimulate abstract thinking and bring out the originality of their students.
Introduction to Solid Geometry.- Polyhedra and their Properties.-
Cross-Section of Polyhedra.- Spheres, Cones, Cylinders, and More.- References.
Format: Paperback / softback, 294 pages, height x width: 240x168 mm, 32 Illustrations, color;
41 Illustrations, black and white; XII, 294 p. 73 illus., 32 illus. in color., 1 Paperback / softback
Series: Synthesis Lectures on Mathematics & Statistics
Pub. Date: 19-Jun-2025
ISBN-13: 9783031848278
This book was developed from lecture notes for an introductory graduate course and provides an essential introduction to chaotic maps in finite-dimensional spaces. Furthermore, the authors show how to apply this theory to infinite-dimensional systems corresponding to partial differential equations to study chaotic vibration of the wave equation subject to various types of nonlinear boundary conditions. The book provides background on chaos as a highly interesting nonlinear phenomenon and explains why it is one of the most important scientific findings of the past three decades. In addition, the book covers key topics including one-dimensional dynamical systems, bifurcations, general topological, symbolic dynamical systems, and fractals. The authors also show a class of infinite-dimensional nonlinear dynamical systems, which are reducible to interval maps, plus rapid fluctuations of chaotic maps. This second edition includes updated and expanded chapters as well as additional problems.
Simple Interval Maps and Their Iterations.- Total Variations of Iterates
of Maps.- Ordering among Periods: The Sharkovski Theorem.- Bifurcation
Theorems for Maps.- Homoclinicity. Lyapunoff Exponents.- Symbolic Dynamics,
Conjugacy and Shift Invariant Sets.- The Smale Horseshoe.- Fractals.- Rapid
Fluctuations of Chaotic Maps on RN.- Infinite-dimensional Systems Induced by
Continuous-Time Difference Equations.