Format: Hardback, 240 pages, height x width: 235x155 mm, 28 Illustrations,
black and white; XXII, 240 p. 28 illus., 1 Hardback
Series: Undergraduate Texts in Mathematics
Pub. Date: 16-Sep-2022
ISBN-13: 9783031056970
This textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction.
Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach.
Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.
1. Algebraic Preliminaries.-
2. Algebraic Numbers and Their Polynomials.-
3. Extending Fields.-
4. Irreducible Polynomials.-
5. Straightedge and Compass Constructions.-
6. Proofs of the Geometric Impossibilities.-
7. Zeros of Polynomials of Degrees 2, 3, and 4.-
8. Quintic Equations 1: Symmetric Polynomials.-
9. Quintic Equations II: The Abel-Ruffini Theorem.-
10. Transcendence of e and .-
11. An Algebraic Postscript.-
12. Other Impossibilities: Regular Polygons and Integration in Finite Terms.- References.- Index.
Format: Hardback, 400 pages, height x width: 235x155 mm, 20 Tables, color; 7 Illustrations,
color; 4 Illustrations, black and white; X, 400 p. 11 illus., 7 illus. in color.
Series: Behaviormetrics: Quantitative Approaches to Human Behavior 2
Pub. Date: 05-Sep-2022
ISBN-13: 9789811935244
This book provides expository derivations for moments of a family of pseudo distributions, which is an extended family of distributions including the pseudo normal (PN) distributions recently proposed by the author. The PN includes the skew normal (SN) derived by A. Azzalini and the closed skew normal (CSN) obtained by A. Dominguez-Molina, G. Gonzalez-Farias, and A. K. Gupta as special cases. It is known that the CSN includes the SN and other various distributions as special cases, which shows that the PN has a wider variety of distributions. The SN and CSN have symmetric and skewed asymmetric distributions. However, symmetric distributions are restricted to normal ones. On the other hand, symmetric distributions in the PN can be non-normal as well as normal. In this book, for the non-normal symmetric distributions, the term "kurtic normal (KN)" is used, where the coined word "kurtic" indicates "mesokurtic, leptokurtic, or platykurtic" used in statistics. The variety of the PN was made possible using stripe (tigerish) and sectional truncation in univariate and multivariate distributions, respectively. The proofs of the moments and associated results are not omitted and are often given in more than one method with their didactic explanations.
The Sectionally Truncated Normal Distribution.- Normal Moments Under Stripe Truncation and the Real-Valued Poisson Distribution.- The Basic Parabolic Cylinder Distribution and its Multivariate Extension.- The Pseudo-Normal (PN) Distribution.- The Kurtic-Normal (KN) Distribution.- The Normal-Normal (NN) Distribution.- The Decompositions of the PN and NN Distributed Variables.- The Truncated Pseudo-Normal (TPN) and Truncated Normal-Normal (TNN) Distributions.- The Student t- and Pseudo-t (PT) Distributions: Various Expressions of Mixtures.- Multivariate Measures of Skewness and Kurtosis.
Format: Hardback, 564 pages, height x width: 235x155 mm, 144 Illustrations,
color; 94 Illustrations, black and white; XVII, 564 p. 238 illus., 144 illus. in color.
Series: Mathematical Engineering
Pub. Date: 29-Aug-2022
ISBN-13: 9783031047282
This book provides qualitative and quantitative methods to analyze and better understand phenomena that change in space and time. An innovative approach is to incorporate ideas and methods from dynamical systems and equivariant bifurcation theory to model, analyze and predict the behavior of mathematical models. In addition, real-life data is incorporated in the derivation of certain models. For instance, the model for a fluxgate magnetometer includes experiments in support of the model.
The book is intended for interdisciplinary scientists in STEM fields, who might be interested in learning the skills to derive a mathematical representation for explaining the evolution of a real system. Overall, the book could be adapted in undergraduate- and postgraduate-level courses, with students from various STEM fields, including: mathematics, physics, engineering and biology.
Introduction- Algebraic Models.- Discrete Models.- Continuous Models.- Bifurcation Theory.- Network-Based Modeling.- Delay Models.- Spatial-Temporal Models.- Stochastic Models.- Model Reduction and Simplification.
Format: Hardback, 105 pages, height x width: 279x210 mm, 2 Illustrations,
color; 1 Illustrations, black and white; XI, 105 p. 3 illus., 2 illus. in color
Pub. Date: 13-Aug-2022
ISBN-13: 9783031079832
This study guide is designed for students taking courses in differential equations. The textbook includes examples, questions, and exercises that will help engineering students to review and sharpen their knowledge of the subject and enhance their performance in the classroom. Offering detailed solutions, multiple methods for solving problems, and clear explanations of concepts, this hands-on guide will improve student’s problem-solving skills and basic and advanced understanding of the topics covered in electric circuit analysis courses.
1) Problems: First-Order Differential Equations2) Solutions of Problems: First-Order Differential Equations3) Problems: Second-Order Differential Equations4) Solutions of Problems: Second-Order Differential Equations5) Problems: Series and their Applications in Solving Differential Equations6) Solutions of Problems: Series and their Applications in Solving Differential Equations7) Problems: Laplace Transform and its Applications in Solving Differential Equations8) Solutions of Problems: Laplace Transform and its Applications in Solving Differential Equations
Format: Hardback, 964 pages, height x width: 254x178 mm, weight: 2221 g, 600 Tables, color; 321 Illustrations,
color; 39 Illustrations, black and white; XVI, 964 p. 360 illus., 321 illus. in color.
Pub. Date: 29-Jun-2022
ISBN-13: 9783031053177
Mathematics and statistics are the bedrock of modern science. No matter which branch of science you plan to work in, you simply cannot avoid quantitative approaches. And while you won’t always need to know a great deal of theory, you will need to know how to apply mathematical and statistical methods in realistic scenarios. That is precisely what this book teaches. It covers the mathematical and statistical topics that are ubiquitous in early undergraduate courses, but does so in a way that is directly linked to science.
Beginning with the use of units and functions, this book covers key topics such as complex numbers, vectors and matrices, differentiation (both single and multivariable), integration, elementary differential equations, probability, random variables, inference and linear regression. Each topic is illustrated with widely-used scientific equations (such as the ideal gas law or the Nernst equation) and real scientific data, often taken directly from recent scientific papers. The emphasis throughout is on practical solutions, including the use of computational tools (such as Wolfram Alpha or R), not theoretical development. There is a large number of exercises, divided into mathematical drills and scientific applications, and full solutions to all the exercises are available to instructors.
Mathematics and Statistics for Science covers the core methods in mathematics and statistics necessary for a university degree in science, highlighting practical solutions and scientific applications. Its pragmatic approach is ideal for students who need to apply mathematics and statistics in a real scientific setting, whether in the physical sciences, life sciences or medicine.
Part I Units and Measurement.- 1 Units.- 2 Measurement, rounding and uncertainty.- Part II Functions and Complex Numbers.- 3 Functions.- 4 Exponential and log functions.- 5 Periodic functions.- 6 Linearising functions.- 7 Complex numbers.- Part III Vectors, Matrices and Linear Systems.- 8 Vectors.- 9 Matrices.- 10 Systems of linear equations.- 11 Solving systems of linear equations using matrices.- Part IV Differentiation: Functions of One Variable.- 12 Limits.- 13 Differentiation as a limit.-
14. Differentiation in practice.- 15 Numerical differentiation.- 16 Implicit differentiation.- 17 Maxima and minima.- Part V Differentiation: Functions of Multiple Variables.- 18 Functions of multiple variables.- 19 Partial derivatives.- 20 Extreme of functions of two (or more) variables.- Part VI Integration.- 21 The area under a curve.- 22 Calculating antiderivatives and areas.- 23 Integration techniques.- 24 Numerical integration.- Part VII Differential Equations.- 25 First-order ordinary differential equations.- 26 Numerical solutions of differential equations.- Part VIII Probability.- 27 Probability foundations.- 28 Random variables.- 29 Binomial distribution.- 30 Conditional probability.- 31 Total probability rule.- Part IX Statistical inference.- 32 Hypothesis test.- 33 Hypothesis testing in practice.- 34 Estimation and likelihood.- Part X Discrete Probability Distributions.- 35 Simulation and visualisation.- 36 Mean.- 37 Variance.- 38 Discrete probability models.- Part XI Continuous Probability Distributions.- 39 Continuous random variables.- 40 Common continuous probability models.- 41 Normal distribution and inference.- Part XII Linear Regression.- 42 Fitting linear functions: theory and practice.- 43 Quantifying relationships.- References.- Index.
Format: Hardback, 270 pages, height x width: 235x155 mm, 90 Illustrations,
color; 4 Illustrations, black and white; XV, 270 p. 94 illus., 90 illus. in color.
Series: Interdisciplinary Applied Mathematics 55
Pub. Date: 03-Sep-2022
ISBN-13: 9783031066917
Stochastic elasticity is a fast developing field that combines nonlinear elasticity and stochastic theories in order to significantly improve model predictions by accounting for uncertainties in the mechanical responses of materials. However, in contrast to the tremendous development of computational methods for large-scale problems, which have been proposed and implemented extensively in recent years, at the fundamental level, there is very little understanding of the uncertainties in the behaviour of elastic materials under large strains.
Based on the idea that every large-scale problem starts as a small-scale data problem, this book combines fundamental aspects of finite (large-strain) elasticity and probability theories, which are prerequisites for the quantification of uncertainties in the elastic responses of soft materials.
The problems treated in this book are drawn from the analytical continuum mechanics literature and incorporate random variables as basic concepts along with mechanical stresses and strains. Such problems are interesting in their own right but they are also meant to inspire further thinking about how stochastic extensions can be formulated before they can be applied to more complex physical systems.
1 Introduction.- 2 Finite elasticity as prior information.- 3 Are elastic materials like gambling machines?.- 4 Elastic instabilities.- 5 Oscillatory motions.- 6 Liquid crystal elastomers.- 7 Conclusion.- Appendix A Notation.- Appendix B Fundamental concepts.- Bibliography.- Index.
Format: Paperback / softback, 24 pages, height x width: 235x155 mm, 3 Tables, color; 3 Illustrations,
color; XIII, 24 p. 3 illus. in color
Series: Lecture Notes in Mathematics 2305
Pub. Date: 27-Aug-2022
ISBN-13: 9783031061851
The purpose of this monograph is to offer an accessible and essentially self-contained presentation of some mathematical aspects of the Feynman path integral in non-relativistic quantum mechanics. In spite of the primary role in the advancement of modern theoretical physics and the wide range of applications, path integrals are still a source of challenging problem for mathematicians. From this viewpoint, path integrals can be roughly described in terms of approximation formulas for an operator (usually the propagator of a Schroedinger-type evolution equation) involving a suitably designed sequence of operators. In keeping with the spirit of harmonic analysis, the guiding theme of the book is to illustrate how the powerful techniques of time-frequency analysis - based on the decomposition of functions and operators in terms of the so-called Gabor wave packets - can be successfully applied to mathematical path integrals, leading to remarkable results and paving the way to a fruitful interaction. This monograph intends to build a bridge between the communities of people working in time-frequency analysis and mathematical/theoretical physics, and to provide an exposition of the present novel approach along with its basic toolkit. Having in mind a researcher or a Ph.D. student as reader, we collected in Part I the necessary background, in the most suitable form for our purposes, following a smooth pedagogical pattern. Then Part II covers the analysis of path integrals, reflecting the topics addressed in the research activity of the authors in the last years.
- Itinerary - How Gabor Analysis met Feynman Path Integrals. - Part I Elements of Gabor Analysis. - Basic Facts of Classical Analysis. - The Gabor Analysis of Functions. - The Gabor Analysis of Operators. - Semiclassical Gabor Analysis. - Part II Analysis of Feynman Path Integrals. - Pointwise Convergence of the Integral Kernels. - Convergence in L(L2) - Potentials in the Sjoestrand Class. - Convergence in L(L2) - Potentials in Kato-Sobolev Spaces. - Convergence in the Lp Setting.