Format: Paperback / softback, 418 pages, height x width: 235x155 mm, 7 Illustrations,
color; 24 Illustrations, black and white; X, 418 p. 31 illus., 7 illus. in color.
Pub. Date: 08-Jan-2023
ISBN-13: 9783031116155
The theory relating algebraic curves and Riemann surfaces exhibits the unity of mathematics: topology, complex analysis, algebra and geometry all interact in a deep way. This textbook offers an elementary introduction to this beautiful theory for an undergraduate audience. At the heart of the subject is the theory of elliptic functions and elliptic curves. A complex torus (or "donut") is both an abelian group and a Riemann surface. It is obtained by identifying points on the complex plane. At the same time, it can be viewed as a complex algebraic curve, with addition of points given by a geometric "chord-and-tangent" method. This book carefully develops all of the tools necessary to make sense of this isomorphism. The exposition is kept as elementary as possible and frequently draws on familiar notions in calculus and algebra to motivate new concepts. Based on a capstone course given to senior undergraduates, this book is intended as a textbook for courses at this level and includes a large number of class-tested exercises. The prerequisites for using the book are familiarity with abstract algebra, calculus and analysis, as covered in standard undergraduate courses.
1 Introduction.- Part I Algebraic curves.- 2 Algebra.- 3 Affine space.- 4 Projective space.- 5 Tangents.- 6 Bezout's theorem.- 7 The elliptic group.- Part II Riemann Surfaces.- 8 Quasi-Euclidean spaces.- 9 Connectedness, smooth and simple.- 10 Path integrals.- 11 Complex differentiation.- 12 Riemann surfaces.- Part III Curves and surfaces.- 13 Curves are surfaces.- 14 Elliptic functions and the isomorphism theorem.- 15 Puiseux theory.- 16 A brief history of elliptic functions.
Format: Paperback / softback, 94 pages, height x width: 235x155 mm, 15 Tables,
color; 1 Illustrations, black and white; X, 94 p. 1 illus.,
Series: SpringerBriefs in Mathematics
Pub. Date: 09-Dec-2022
ISBN-13: 9789811967306
This volume is devoted to the study of hyperbolic free boundary problems possessing variational structure. Such problems can be used to model, among others, oscillatory motion of a droplet on a surface or bouncing of an elastic body against a rigid obstacle. In the case of the droplet, for example, the membrane surrounding the fluid in general forms a positive contact angle with the obstacle, and therefore the second derivative is only a measure at the contact free boundary set. We will show how to derive the mathematical problem for a few physical systems starting from the action functional, discuss the mathematical theory, and introduce methods for its numerical solution. The mathematical theory and numerical methods depart from the classical approaches in that they are based on semi-discretization in time, which facilitates the application of the modern theory of calculus of variations.
Chapter 1. Introduction.
Chapter 2.Physical motivation.
Chapter 3.Discrete Morse flow.
Chapter 4. Discrete Morse flow with free boundary.-
Chapter 5.Energy-preserving discrete Morse flow.
Chapter 6.Numerical examples and applications.
Format: Paperback / softback, 302 pages, height x width: 235x155 mm, 77 Illustrations,
color; 43 Illustrations, black and white; XVIII, 302 p. 120 illus., 77 illus. in color.
Series: SUMS Readings
Pub. Date: 19-Dec-2022
ISBN-13: 9783031137822
This textbook introduces generalized trigonometric functions through the exploration of imperfect circles: curves defined by |x|p + |y|p = 1 where p 1. Grounded in visualization and computations, this accessible, modern perspective encompasses new and old results, casting a fresh light on duality, special functions, geometric curves, and differential equations. Projects and opportunities for research abound, as we explore how similar (or different) the trigonometric and squigonometric worlds might be. Comprised of many short chapters, the book begins with core definitions and techniques. Successive chapters cover inverse squigonometric functions, the many possible re-interpretations of , two deeper dives into parameterizing the squigonometric functions, and integration. Applications include a celebration of Piet Hein's work in design. From here, more technical pathways offer further exploration. Topics include infinite series; hyperbolic, exponential, and logarithmic functions; metrics and norms; and lemniscatic and elliptic functions. Illuminating illustrations accompany the text throughout, along with historical anecdotes, engaging exercises, and wry humor. Squigonometry: The Study of Imperfect Circles invites readers to extend familiar notions from trigonometry into a new setting. Ideal for an undergraduate reading course in mathematics or a senior capstone, this book offers scaffolding for active discovery. Knowledge of the trigonometric functions, single-variable calculus, and initial-value problems is assumed, while familiarity with multivariable calculus and linear algebra will allow additional insights into certain later material.
1. Introduction.-
2. Imperfection.-
3. A Squigonometry Introduction.-
4. p-metrics.-
5. Inverse squigonometric functions.-
6. The many values of Pi.-
7. Parametrizations.-
8. Arclength Parametrization.-
9. Integrating Squigonometric Functions.-
10. Three applications.-
11. Infinite series.-
12. Series and rational approximations.-
13. Alternate Coordinates.-
14. Hyperbolic Functions.-
15. Exponentials and Logarithms.-
16. Elliptic Integrals.-
17. Lemniscates and Ellipses.-
18. Geometry in the p-norm.-
19. Duality.-
20. Analytic Parametrizations.- A. Curve Menagerie.- B. Formulas and Integrals.- C. Parametrization Primer.- D. Proofs of Formulas and Theorems.- E. Alternate Pi Days.- F. Selected Exercise Hints and Solutions.
Format: Paperback / softback, 110 pages, height x width: 235x155 mm, 90 Tables, color; 90 Illustrations,
color; 18 Illustrations, black and white; XIII, 110 p. 108 illus., 90 illus. in color.
Series: SpringerBriefs in Statistics
Pub. Date: 01-Dec-2022
ISBN-13: 9783031163326
The book shows how risk, defined as the statistical expectation of loss, can be formally decomposed as the product of two terms: hazard probability and system vulnerability. This requires a specific definition of vulnerability that replaces the many fuzzy definitions abounding in the literature. The approach is expanded to more complex risk analysis with three components rather than two, and with various definitions of hazard. Equations are derived to quantify the uncertainty of each risk component and show how the approach relates to Bayesian decision theory. Intended for statisticians, environmental scientists and risk analysts interested in the theory and application of risk analysis, this book provides precise definitions, new theory, and many examples with full computer code. The approach is based on straightforward use of probability theory which brings rigour and clarity. Only a moderate knowledge and understanding of probability theory is expected from the reader.
Introduction to Probabilistic Risk Analysis (PRA).- Distribution-based single-threshold PRA.- Sampling-based single-threshold PRA.- Sampling-based single-threshold PRA: Uncertainty quantification (UQ).- Density estimation to move from sampling- to distribution-based PRA.- Copulas for distribution-based PRA.- Bayesian model-based PRA.- Sampling-based multi-threshold PRA: Gaussian linear example.- Distribution-based continuous PRA: Gaussian linear example.- Categorical PRA with other splits than for threshold-levels: spatio-temporal example.- Three-component PRA.- Introduction to Bayesian Decision Theory (BDT).- Implementation of BDT using Bayesian Networks.- A spatial example: forestry in Scotland.- Spatial BDT using model and emulator.- Linkages between PRA and BDT.- PRA vs. BDT in the spatial example.- Three-component PRA in the spatial example.- Discussion.
Format: Paperback / softback, 213 pages, height x width: 235x155 mm, 16 Tables,
color; 22 Illustrations, color; 85 Illustrations, black and white; XVI, 213 p. 107 illus., 22 illus. in color.
Series: Springer Undergraduate Mathematics Series
Pub. Date: 28-Dec-2022
ISBN-13: 9783031157646
This textbook provides a gentle overview of fundamental concepts related to one-variable calculus. The original approach is a result of the author's forty years of experience in teaching the subject at universities around the world. In this book, Dr. Zalduendo makes use of the history of mathematics and a friendly, conversational approach to attract the attention of the student, emphasizing what is more conceptually relevant and putting key notions in a historical perspective. Such an approach was conceived to help them to overcome potential difficulties in teaching and learning of this subject - caused, in many cases, by an excess of technicalities and computations. Besides covering the core of the discipline - real number, sequences and series, functions, derivatives, integrals, convexity and inequalities - the book is enriched by "side trips" to relevant subjects not usually seen in traditional calculus textbooks, touching on topics like curvature, the isoperimetric inequality, Riemann's rearrangement theorem, Snell's law, Buffon's needle problem, Gregory's series, random walk and the Gauss curve, and more. An insightful collection of exercises and applications completes this book, making it ideal as a supplementary textbook for a calculus course or the main textbook for an honors course on the subject.
Chapter.
1. The Real NumbersChapter.
2. Sequences and SeriesChapter.
3. FunctionsChapter.
4. The DerivativeChapter.
5. The IntegralChapter.
6. More DerivativesChapter.
7. Convexity and The Isoperimetric InequalityChapter.
8. More IntegralsChapter.
9. The Gamma Function
Format: Hardback, 930 pages, height x width: 235x155 mm, XVI, 930 p., 1 Hardback
Series: Developments in Mathematics 73
Pub. Date: 21-Dec-2022
ISBN-13: 9783031137174
This monograph is part of a larger program, materializing in five volumes, whose principal aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. Volume II is concerned with function spaces measuring size and/or smoothness, such as Hardy spaces, Besov spaces, Triebel-Lizorkin spaces, Sobolev spaces, Morrey spaces, Morrey-Campanato spaces, spaces of functions of Bounded Mean Oscillations, etc., in general geometric settings. Work here also highlights the close interplay between differentiability properties of functions and singular integral operators. The text is intended for researchers, graduate students, and industry professionals interested in harmonic analysis, functional analysis, geometric measure theory, and function space theory.
1 Preliminary Functional Analytic Matters.- 2 Abstract Fredholm Theory.- 3 Functions of Vanishing Mean Oscillations and Vanishing Hoelder Moduli.- 4 Hardy Spaces on Ahlfors Regular Sets.- 5 Banach Function Spaces, Extrapolation, and Orlicz Spaces.- 6 Morrey-Campanato Spaces, Morrey Spaces, and Their Pre-Duals on Ahlfors Regular Sets.- 7 Besov and Triebel-Lizorkin Spaces on Ahlfors Regular Sets.- 8 Boundary Traces from Weighted Sobolev Spaces into Besov Spaces.- 9 Besov and Triebel-Lizorkin Spaces in Open Sets.- 10 Strong and Weak Normal Boundary Traces of Vector Fields in Hardy and Morrey Spaces.- 11 Sobolev Spaces on the Geometric Measure Theoretic Boundary of Sets of Locally Finite Perimeter.- A. Terms and Notation Used in Volume II. References.- Index.
Format: Hardback, 406 pages, height x width: 235x155 mm, 1 Illustrations,
color; 5 Illustrations, black and white; XIV, 406 p. 6 illus., 1 illus. in color
Series: Graduate Texts in Mathematics 295
Pub. Date: 08-Dec-2022
ISBN-13: 9783031142048
This textbook introduces readers to the fundamental notions of modern probability theory. The only prerequisite is a working knowledge in real analysis. Highlighting the connections between martingales and Markov chains on one hand, and Brownian motion and harmonic functions on the other, this book provides an introduction to the rich interplay between probability and other areas of analysis. Arranged into three parts, the book begins with a rigorous treatment of measure theory, with applications to probability in mind. The second part of the book focuses on the basic concepts of probability theory such as random variables, independence, conditional expectation, and the different types of convergence of random variables. In the third part, in which all chapters can be read independently, the reader will encounter three important classes of stochastic processes: discrete-time martingales, countable state-space Markov chains, and Brownian motion. Each chapter ends with a selection of illuminating exercises of varying difficulty. Some basic facts from functional analysis, in particular on Hilbert and Banach spaces, are included in the appendix. Measure Theory, Probability, and Stochastic Processes is an ideal text for readers seeking a thorough understanding of basic probability theory. Students interested in learning more about Brownian motion, and other continuous-time stochastic processes, may continue reading the author's more advanced textbook in the same series (GTM 274).
Part I. Measure Theory.
Chapter 1. Measurable Spaces.
Chapter 2. Integration of Measurable Functions.
Chapter 3. Construction of Measures.-
Chapter 4. Lp Spaces.
Chapter 5. Product Measure.
Chapter 6. Signed Measures.
Chapter 7. Change of Variables.- Part II. Probability Theory.-
Chapter 8. Foundations of Probability Theory.
Chapter 9. Independence.-
Chapter 10. Convergence of Random Variables.
Chapter 11. Conditioning.- Part III. Stochastic Processes.
Chapter 12. Theory of Martingales.
Chapter 13. Markov Chains.
Chapter 14. Brownian Motion.
*
Format: Hardback, 329 pages, height x width: 235x155 mm, 48 Illustrations, black and white; X, 329 p. 48 illus.
Series: Studies in Universal Logic
Pub. Date: 12-Dec-2022
ISBN-13: 9783031148866
This book gives a comprehensive introduction to Universal Algebraic Logic. The three main themes are (i) universal logic and the question of what logic is, (ii) duality theories between the world of logics and the world of algebra, and (iii) Tarskian algebraic logic proper including algebras of relations of various ranks, cylindric algebras, relation algebras, polyadic algebras and other kinds of algebras of logic. One of the strengths of our approach is that it is directly applicable to a wide range of logics including not only propositional logics but also e.g. classical first order logic and other quantifier logics. Following the Tarskian tradition, besides the connections between logic and algebra, related logical connections with geometry and eventually spacetime geometry leading up to relativity are also part of the perspective of the book. Besides Tarskian algebraizations of logics, category theoretical perspectives are also touched upon. This book, apart from being a monograph containing state of the art results in algebraic logic, can be used as the basis for a number of different courses intended for both novices and more experienced students of logic, mathematics, or philosophy. For instance, the first two chapters can be used in their own right as a crash course in Universal Algebra.
Preface.- Acknowledgement.- 1 Notation, Elementary Concepts.-1.1 Sets, classes, tuples, simple operations on sets.-1.2 Binary relations, equivalence relations, functions.- 1.3 Orderings, ordinals, cardinals.- 1.4 Sequences.- 1.5 Direct product of families of sets.- 1.6 Relations of higher ranks.- 1.7 Closure systems.- 1.8 First order logic (FOL).- 2 Basics from Universal Algebra.-2.1 Examples for algebras.- 2.2 Building new algebras from old ones (operations on algebras).- 2.2.1 Subalgebra.- 2.2.2 Homomorphic image.- 2.2.3 A distinguished example: Lattices.- 2.2.4 Congruence relation.- 2.2.5 Cartesian product, direct decomposition.- 2.2.6 Subdirect decomposition.- 2.2.7 Ultraproduct, reduced product.- 2.3 Categories.- 2.4 Variety characterization, quasi-variety characterization.- 2.5 Free algebras.- 2.6 Boolean Algebras.- 2.7 Discriminator varieties.- 2.8 Boas and BAOs.- 3 General framework and algebraization.- 3.1 Defining the framework for studying logics.- 3.2 Concrete logics in the new framework.- 3.3 Algebraization.- 3.3.1 Having connectives, formula algebra.- 3.3.2 Compositionality, tautological formula algebra.- 3.3.3 Algebraic counterparts of a logic.- 3.3.4 Substitution properties.- 3.3.5 Filter property.- 3.3.6 General Logics.- 3.4 Connections with Abstract Algebraic Logic, Abstract Model Theory and Institutions.- 4 Bridge between logic and algebra.- 4.1 Algebraic characterization of compactness properties.- 4.2 Algebraic characterizations of completeness properties.- 4.2.1 Hilbert-type inference systems.- 4.2.2 Completeness and soundness.- 4.3 Algebraic characterization of definability properties.- 4.3.1 Syntactical Beth definability property.- 4.3.2 Beth definability property.- 4.3.3 Local Beth definability property.-4.3.4 Weak Beth definability property.- 4.4 Algebraic characterization of interpolation properties.- 4.4.1 Interpolation properties.- 4.4.2 Amalgamation and interpolation properties.- 4.5 Decidability.- 4.6 G odel's incompleteness property.- 5 Applying the machinery: Examples.- 5.1 Classical propositional logic LC.- 5.2 Arrow logic L_{REL}.- 5.3 Finite-variable fragments of first-order logic, with substituted atomic formulas, L'_n.- 5.4 n-variable fragment L_n of rst-order logic, for n \le \omega.- 5.5 First-order logic with nonstandard semantics, L^{a}_{n}.- 5.6 Variable-dependent first-order logic, L^{vd}_{n}.- 5.7 First-order logic, ranked version, L^{ranked}_{FOL}.- 5.8 First-order logic, rank-free (or type-less) version, L^{rf}_{FOL}.- 6 Generalizations and new kinds of logics.- 6.1 Generalizations.- 6.2 New kinds of logics.- 7 Appendix: Algebras of relations.- 7.1 Algebras of binary relations.- 7.2 Algebras of unitary relations.- 7.3 All unitary relations together.- Bibliography.- Index.- Index of symbols.