A key feature of this work is the historical flavor that is interwoven into the extensive and in-depth coverage of the subject. An interesting array of applications to combinatorics, graph theory, geometry, and intriguing mathematical puzzles is another highlight engaging the reader. The exposition is user-friendly, yet rigorous, so that a broad audience consisting of students, math teachers and instructors, computer scientists and other professionals, along with the mathematically curious will all benefit from this book.
Finally, Pell and Pell–Lucas Numbers provides enjoyment and excitement while sharpening the reader’s mathematical skills involving pattern recognition, proof-and-problem-solving techniques.
The math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In How Not to Be Wrong, Jordan Ellenberg shows us how terribly limiting this view is: Math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do—the whole world is shot through with it.
Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It’s a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does “public opinion” really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer?
How Not to Be Wrong presents the surprising revelations behind all of these questions and many more, using the mathematician’s method of analyzing life and exposing the hard-won insights of the academic community to the layman—minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia’s views on crime and punishment, the psychology of slime molds, what Facebook can and can’t figure out about you, and the existence of God.
Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is “an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. How Not to Be Wrong will show you how.
Modeling is one of the most effective, commonly used tools inengineering and the applied sciences. In this book, the authorsdeal with mathematical programming models both linear and nonlinearand across a wide range of practical applications.
Whereas other books concentrate on standard methods of analysis,the authors focus on the power of modeling methods for solvingpractical problems-clearly showing the connection between physicaland mathematical realities-while also describing and exploring themain concepts and tools at work. This highly computational coverageincludes:
* Discussion and implementation of the GAMS programmingsystem
* Unique coverage of compatibility
* Illustrative examples that showcase the connection between modeland reality
* Practical problems covering a wide range of scientificdisciplines, as well as hundreds of examples and end-of-chapterexercises
* Real-world applications to probability and statistics, electricalengineering, transportation systems, and more
Building and Solving Mathematical Programming Models in Engineeringand Science is practically suited for use as a professionalreference for mathematicians, engineers, and applied or industrialscientists, while also tutorial and illustrative enough foradvanced students in mathematics or engineering.
This book presents the mathematics of finite dimensionalconstrained optimization problems. It provides a basis for thefurther mathematical study of convexity, of more generaloptimization problems, and of numerical algorithms for the solutionof finite dimensional optimization problems. For readers who do nothave the requisite background in real analysis, the author providesa chapter covering this material. The text features abundantexercises and problems designed to lead the reader to a fundamentalunderstanding of the material.
Convexity and Optimization in Rn provides detailed discussionof:
* Requisite topics in real analysis
* Convex sets
* Convex functions
* Optimization problems
* Convex programming and duality
* The simplex method
A detailed bibliography is included for further study and an indexoffers quick reference. Suitable as a text for both graduate andundergraduate students in mathematics and engineering, thisaccessible text is written from extensively class-tested notes.
Elementary Number Theory with Applications 2e is ideally suited for undergraduate students and is especially appropriate for prospective and in-service math teachers at the high school and middle school levels.
* Loaded with pedagogical features including fully worked examples, graded exercises, chapter summaries, and computer exercises
* Covers crucial applications of theory like computer security, ISBNs, ZIP codes, and UPC bar codes
* Biographical sketches lay out the history of mathematics, emphasizing its roots in India and the Middle East
The official book behind the Academy Award-winning film The Imitation Game, starring Benedict Cumberbatch and Keira Knightley
It is only a slight exaggeration to say that the British mathematician Alan Turing (1912-1954) saved the Allies from the Nazis, invented the computer and artificial intelligence, and anticipated gay liberation by decades--all before his suicide at age forty-one. This New York Times–bestselling biography of the founder of computer science, with a new preface by the author that addresses Turing's royal pardon in 2013, is the definitive account of an extraordinary mind and life.
Capturing both the inner and outer drama of Turing’s life, Andrew Hodges tells how Turing’s revolutionary idea of 1936--the concept of a universal machine--laid the foundation for the modern computer and how Turing brought the idea to practical realization in 1945 with his electronic design. The book also tells how this work was directly related to Turing’s leading role in breaking the German Enigma ciphers during World War II, a scientific triumph that was critical to Allied victory in the Atlantic. At the same time, this is the tragic account of a man who, despite his wartime service, was eventually arrested, stripped of his security clearance, and forced to undergo a humiliating treatment program--all for trying to live honestly in a society that defined homosexuality as a crime.
The inspiration for a major motion picture starring Benedict Cumberbatch and Keira Knightley, Alan Turing: The Enigma is a gripping story of mathematics, computers, cryptography, and homosexual persecution.
Matrix computations lie at the heart of most scientific computational tasks. For any scientist or engineer doing large-scale simulations, an understanding of the topic is essential. Fundamentals of Matrix Computations, Second Edition explains matrix computations and the accompanying theory clearly and in detail, along with useful insights.
This Second Edition of a popular text has now been revised and improved to appeal to the needs of practicing scientists and graduate and advanced undergraduate students. New to this edition is the use of MATLAB for many of the exercises and examples, although the Fortran exercises in the First Edition have been kept for those who want to use them. This new edition includes:
* Numerous examples and exercises on applications including electrical circuits, elasticity (mass-spring systems), and simple partial differential equations
* Early introduction of the singular value decomposition
* A new chapter on iterative methods, including the powerful preconditioned conjugate-gradient method for solving symmetric, positive definite systems
* An introduction to new methods for solving large, sparse eigenvalue problems including the popular implicitly-restarted Arnoldi and Jacobi-Davidson methods
With in-depth discussions of such other topics as modern componentwise error analysis, reorthogonalization, and rank-one updates of the QR decomposition, Fundamentals of Matrix Computations, Second Edition will prove to be a versatile companion to novice and practicing mathematicians who seek mastery of matrix computation.
* Covers all recommended topics in a self-contained, comprehensive, and understandable format for students and new professionals
* Emphasizes problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof techniques, algorithm development and correctness, and numeric computations
* Weaves numerous applications into the text
* Helps students learn by doing with a wealth of examples and exercises:
- 560 examples worked out in detail
- More than 3,700 exercises
- More than 150 computer assignments
- More than 600 writing projects
* Includes chapter summaries of important vocabulary, formulas, and properties, plus the chapter review exercises
* Features interesting anecdotes and biographies of 60 mathematicians and computer scientists
* Instructor's Manual available for adopters
* Student Solutions Manual available separately for purchase (ISBN: 0124211828)
The field of differential equations is a keystone of scientific knowledge today, with broad applications in mathematics, engineering, physics, and other scientific fields. Encompassing both basic concepts and advanced results, Principles of Differential Equations is the definitive, hands-on introduction professionals and students need in order to gain a strong knowledge base applicable to the many different subfields of differential equations and dynamical systems.
Nelson Markley includes essential background from analysis and linear algebra, in a unified approach to ordinary differential equations that underscores how key theoretical ingredients interconnect. Opening with basic existence and uniqueness results, Principles of Differential Equations systematically illuminates the theory, progressing through linear systems to stable manifolds and bifurcation theory. Other vital topics covered include:Basic dynamical systems concepts Constant coefficients Stability The Poincaré return map Smooth vector fields
As a comprehensive resource with complete proofs and more than 200 exercises, Principles of Differential Equations is the ideal self-study reference for professionals, and an effective introduction and tutorial for students.
The Third Edition of this classic text offers a comprehensive guide to modeling, characterizing, and solving partial differential equations (PDEs). The author provides all the theory and tools necessary to solve problems via exact, approximate, and numerical methods. The Third Edition retains all the hallmarks of its previous editions, including an emphasis on practical applications, clear writing style and logical organization, and extensive use of real-world examples.
Among the new and revised material, the book features:
* A new section at the end of each original chapter, exhibiting the use of specially constructed Maple procedures that solve PDEs via many of the methods presented in the chapters. The results can be evaluated numerically or displayed graphically.
* Two new chapters that present finite difference and finite element methods for the solution of PDEs. Newly constructed Maple procedures are provided and used to carry out each of these methods. All the numerical results can be displayed graphically.
* A related FTP site that includes all the Maple code used in the text.
* New exercises in each chapter, and answers to many of the exercises are provided via the FTP site. A supplementary Instructor's Solutions Manual is available.
The book begins with a demonstration of how the three basic types of equations-parabolic, hyperbolic, and elliptic-can be derived from random walk models. It then covers an exceptionally broad range of topics, including questions of stability, analysis of singularities, transform methods, Green's functions, and perturbation and asymptotic treatments. Approximation methods for simplifying complicated problems and solutions are described, and linear and nonlinear problems not easily solved by standard methods are examined in depth. Examples from the fields of engineering and physical sciences are used liberally throughout the text to help illustrate how theory and techniques are applied to actual problems.
With its extensive use of examples and exercises, this text is recommended for advanced undergraduates and graduate students in engineering, science, and applied mathematics, as well as professionals in any of these fields. It is possible to use the text, as in the past, without use of the new Maple material.
A New York Times Notable Book.
The Babylonians invented it, the Greeks banned it, the Hindus worshiped it, and the Church used it to fend off heretics. Now it threatens the foundations of modern physics. For centuries the power of zero savored of the demonic; once harnessed, it became the most important tool in mathematics. For zero, infinity's twin, is not like other numbers. It is both nothing and everything.
In Zero, Science Journalist Charles Seife follows this innocent-looking number from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe, its rise and transcendence in the West, and its ever-present threat to modern physics. Here are the legendary thinkers—from Pythagoras to Newton to Heisenberg, from the Kabalists to today's astrophysicists—who have tried to understand it and whose clashes shook the foundations of philosophy, science, mathematics, and religion. Zero has pitted East against West and faith against reason, and its intransigence persists in the dark core of a black hole and the brilliant flash of the Big Bang. Today, zero lies at the heart of one of the biggest scientific controversies of all time: the quest for a theory of everything.
equations and modern finite element methods for their efficientnumerical solution
Partial Differential Equations and the Finite Element Methodprovides a much-needed, clear, and systematic introduction tomodern theory of partial differential equations (PDEs) and finiteelement methods (FEM). Both nodal and hierachic concepts of the FEMare examined. Reflecting the growing complexity and multiscalenature of current engineering and scientific problems, the authoremphasizes higher-order finite element methods such as the spectralor hp-FEM.
A solid introduction to the theory of PDEs and FEM contained inChapters 1-4 serves as the core and foundation of the publication.Chapter 5 is devoted to modern higher-order methods for thenumerical solution of ordinary differential equations (ODEs) thatarise in the semidiscretization of time-dependent PDEs by theMethod of Lines (MOL). Chapter 6 discusses fourth-order PDEs rootedin the bending of elastic beams and plates and approximates theirsolution by means of higher-order Hermite and Argyris elements.Finally, Chapter 7 introduces the reader to various PDEs governingcomputational electromagnetics and describes their finite elementapproximation, including modern higher-order edge elements forMaxwell's equations.
The understanding of many theoretical and practical aspects of bothPDEs and FEM requires a solid knowledge of linear algebra andelementary functional analysis, such as functions and linearoperators in the Lebesgue, Hilbert, and Sobolev spaces. Thesetopics are discussed with the help of many illustrative examples inAppendix A, which is provided as a service for those readers whoneed to gain the necessary background or require a refreshertutorial. Appendix B presents several finite element computationsrooted in practical engineering problems and demonstrates thebenefits of using higher-order FEM.
Numerous finite element algorithms are written out in detailalongside implementation discussions. Exercises, including manythat involve programming the FEM, are designed to assist the readerin solving typical problems in engineering and science.
Specifically designed as a coursebook, this student-testedpublication is geared to upper-level undergraduates and graduatestudents in all disciplines of computational engineeringandscience. It is also a practical problem-solving reference forresearchers, engineers, and physicists.
Successfully addressing the frustration many students experience asthey make the transition from computational mathematics to advancedcalculus and algebraic structures, Theorems, Corollaries, Lemmas,and Methods of Proof equips students with the tools needed tosucceed while providing a firm foundation in the axiomaticstructure of modern mathematics.
This essential book:
* Clearly explains the relationship between definitions,conjectures, theorems, corollaries, lemmas, and proofs
* Reinforces the foundations of calculus and algebra
* Explores how to use both a direct and indirect proof to prove atheorem
* Presents the basic properties of real numbers
* Discusses how to use mathematical induction to prove atheorem
* Identifies the different types of theorems
* Explains how to write a clear and understandable proof
* Covers the basic structure of modern mathematics and the keycomponents of modern mathematics
A complete chapter is dedicated to the different methods of proofsuch as forward direct proofs, proof by contrapositive, proof bycontradiction, mathematical induction, and existence proofs. Inaddition, the author has supplied many clear and detailedalgorithms that outline these proofs.
Theorems, Corollaries, Lemmas, and Methods of Proof uniquelyintroduces scratch work as an indispensable part of the proofprocess, encouraging students to use scratch work and creativethinking as the first steps in their attempt to prove a theorem.Once their scratch work successfully demonstrates the truth of thetheorem, the proof can be written in a clear and concise fashion.The basic structure of modern mathematics is discussed, and each ofthe key components of modern mathematics is defined. Numerousexercises are included in each chapter, covering a wide range oftopics with varied levels of difficulty.
Intended as a main text for mathematics courses such as Methods ofProof, Transitions to Advanced Mathematics, and Foundations ofMathematics, the book may also be used as a supplementary textbookin junior- and senior-level courses on advanced calculus, realanalysis, and modern algebra.
Topology: Point-Set and Geometric presents an introduction to topology that begins with the axiomatic definition of a topology on a set, rather than starting with metric spaces or the topology of subsets of Rn. This approach includes many more examples, allowing students to develop more sophisticated intuition and enabling them to learn how to write precise proofs in a brand-new context, which is an invaluable experience for math majors.
Along with the standard point-set topology topics—connected and path-connected spaces, compact spaces, separation axioms, and metric spaces—Topology covers the construction of spaces from other spaces, including products and quotient spaces. This innovative text culminates with topics from geometric and algebraic topology (the Classification Theorem for Surfaces and the fundamental group), which provide instructors with the opportunity to choose which "capstone" best suits his or her students.
Topology: Point-Set and Geometric features:A short introduction in each chapter designed to motivate the ideas and place them into an appropriate context Sections with exercise sets ranging in difficulty from easy to fairly challenging Exercises that are very creative in their approaches and work well in a classroom setting A supplemental Web site that contains complete and colorful illustrations of certain objects, several learning modules illustrating complicated topics, and animations of particularly complex proofs
Professor Kline begins with an overview, tracing the development of mathematics to the ancient Greeks, and following its evolution through the Middle Ages and the Renaissance to the present day. Subsequent chapters focus on specific subject areas, such as "Logic and Mathematics," "Number: The Fundamental Concept," "Parametric Equations and Curvilinear Motion," "The Differential Calculus," and "The Theory of Probability." Each of these sections offers a step-by-step explanation of concepts and then tests the student's understanding with exercises and problems. At the same time, these concepts are linked to pure and applied science, engineering, philosophy, the social sciences or even the arts.
In one section, Professor Kline discusses non-Euclidean geometry, ranking it with evolution as one of the "two concepts which have most profoundly revolutionized our intellectual development since the nineteenth century." His lucid treatment of this difficult subject starts in the 1800s with the pioneering work of Gauss, Lobachevsky, Bolyai and Riemann, and moves forward to the theory of relativity, explaining the mathematical, scientific and philosophical aspects of this pivotal breakthrough. Mathematics for the Nonmathematician exemplifies Morris Kline's rare ability to simplify complex subjects for the nonspecialist.
In an effort to fully appreciate the logic and structure of geometric proofs, Revolutions of Geometry places proofs into the context of geometry's history, helping readers to understand that proof writing is crucial to the job of a mathematician. Written for students and educators of mathematics alike, the book guides readers through the rich history and influential works, from ancient times to the present, behind the development of geometry. As a result, readers are successfully equipped with the necessary logic to develop a full understanding of geometric theorems.
Following a presentation of the geometry of ancient Egypt, Babylon, and China, the author addresses mathematical philosophy and logic within the context of works by Thales, Plato, and Aristotle. Next, the mathematics of the classical Greeks is discussed, incorporating the teachings of Pythagoras and his followers along with an overview of lower-level geometry using Euclid's Elements. Subsequent chapters explore the work of Archimedes, Viete's revolutionary contributions to algebra, Descartes' merging of algebra and geometry to solve the Pappus problem, and Desargues' development of projective geometry. The author also supplies an excursion into non-Euclidean geometry, including the three hypotheses of Saccheri and Lambert and the near simultaneous discoveries of Lobachevski and Bolyai. Finally, modern geometry is addressed within the study of manifolds and elliptic geometry inspired by Riemann's work, Poncelet's return to projective geometry, and Klein's use of group theory to characterize different geometries.
The book promotes the belief that in order to learn how to write proofs, one needs to read finished proofs, studying both their logic and grammar. Each chapter features a concise introduction to the presented topic, and chapter sections conclude with exercises that are designed to reinforce the material and provide readers with ample practice in writing proofs. In addition, the overall presentation of topics in the book is in chronological order, helping readers appreciate the relevance of geometry within the historical development of mathematics.
Well organized and clearly written, Revolutions of Geometry is a valuable book for courses on modern geometry and the history of mathematics at the upper-undergraduate level. It is also a valuable reference for educators in the field of mathematics.
"This book is an excellent introduction to the wide field ofboundary value problems."—Journal of EngineeringMathematics
"No doubt this textbook will be useful for both students andresearch workers."—Mathematical Reviews
A new edition of the highly-acclaimed guide to boundary valueproblems, now featuring modern computational methods andapproximation theory
Green's Functions and Boundary Value Problems, Third Editioncontinues the tradition of the two prior editions by providingmathematical techniques for the use of differential and integralequations to tackle important problems in applied mathematics, thephysical sciences, and engineering. This new edition presentsmathematical concepts and quantitative tools that are essential foreffective use of modern computational methods that play a key rolein the practical solution of boundary value problems. With acareful blend of theory and applications, the authors successfullybridge the gap between real analysis, functional analysis,nonlinear analysis, nonlinear partial differential equations,integral equations, approximation theory, and numerical analysis toprovide a comprehensive foundation for understanding and analyzingcore mathematical and computational modeling problems.
Thoroughly updated and revised to reflect recent developments,the book includes an extensive new chapter on the modern tools ofcomputational mathematics for boundary value problems. The ThirdEdition features numerous new topics, including:
Nonlinear analysis tools for Banach spaces
Finite element and related discretizations
Best and near-best approximation in Banach spaces
Iterative methods for discretized equations
Overview of Sobolev and Besov space linear
Methods for nonlinear equations
Applications to nonlinear elliptic equations
In addition, various topics have been substantially expanded,and new material on weak derivatives and Sobolev spaces, theHahn-Banach theorem, reflexive Banach spaces, the Banach Schauderand Banach-Steinhaus theorems, and the Lax-Milgram theorem has beenincorporated into the book. New and revised exercises foundthroughout allow readers to develop their own problem-solvingskills, and the updated bibliographies in each chapter provide anextensive resource for new and emerging research andapplications.
With its careful balance of mathematics and meaningfulapplications, Green's Functions and Boundary Value Problems, ThirdEdition is an excellent book for courses on applied analysis andboundary value problems in partial differential equations at thegraduate level. It is also a valuable reference for mathematicians,physicists, engineers, and scientists who use applied mathematicsin their everyday work.
Men of Mathematics provides a rich account of major mathematical milestones, from the geometry of the Greeks through Newton’s calculus, and on to the laws of probability, symbolic logic, and the fourth dimension. Bell breaks down this majestic history of ideas into a series of engrossing biographies of the great mathematicians who made progress possible—and who also led intriguing, complicated, and often surprisingly entertaining lives.
Never pedantic or dense, Bell writes with clarity and simplicity to distill great mathematical concepts into their most understandable forms for the curious everyday reader. Anyone with an interest in math may learn from these rich lessons, an advanced degree or extensive research is never necessary.
Numerical Analysis of Partial Differential Equations provides a comprehensive, self-contained treatment of the quantitative methods used to solve elliptic partial differential equations (PDEs), with a focus on the efficiency as well as the error of the presented methods. The author utilizes coverage of theoretical PDEs, along with the nu merical solution of linear systems and various examples and exercises, to supply readers with an introduction to the essential concepts in the numerical analysis of PDEs.
The book presents the three main discretization methods of elliptic PDEs: finite difference, finite elements, and spectral methods. Each topic has its own devoted chapters and is discussed alongside additional key topics, including:
The mathematical theory of elliptic PDEs
Numerical linear algebra
Multigrid and domain decomposition
PDEs posed on infinite domains
The book concludes with a discussion of the methods for nonlinear problems, such as Newton's method, and addresses the importance of hands-on work to facilitate learning. Each chapter concludes with a set of exercises, including theoretical and programming problems, that allows readers to test their understanding of the presented theories and techniques. In addition, the book discusses important nonlinear problems in many fields of science and engineering, providing information as to how they can serve as computing projects across various disciplines.
Requiring only a preliminary understanding of analysis, Numerical Analysis of Partial Differential Equations is suitable for courses on numerical PDEs at the upper-undergraduate and graduate levels. The book is also appropriate for students majoring in the mathematical sciences and engineering.
". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!"
—Monatshefte fur Mathematik
Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel’s theory of Abelian equations, casus irreducibili, and the Galois theory of origami.
In addition, this book features detailed treatments of several topics not covered in standard texts on Galois theory, including:The contributions of Lagrange, Galois, and Kronecker How to compute Galois groups Galois's results about irreducible polynomials of prime or prime-squared degree Abel's theorem about geometric constructions on the lemniscates Galois groups of quartic polynomials in all characteristics
Throughout the book, intriguing Mathematical Notes and Historical Notes sections clarify the discussed ideas and the historical context; numerous exercises and examples use Maple and Mathematica to showcase the computations related to Galois theory; and extensive references have been added to provide readers with additional resources for further study.
Galois Theory, Second Edition is an excellent book for courses on abstract algebra at the upper-undergraduate and graduate levels. The book also serves as an interesting reference for anyone with a general interest in Galois theory and its contributions to the field of mathematics.
Numerical Solution of Ordinary Differential Equations presents a complete and easy-to-follow introduction to classical topics in the numerical solution of ordinary differential equations. The book's approach not only explains the presented mathematics, but also helps readers understand how these numerical methods are used to solve real-world problems.
Unifying perspectives are provided throughout the text, bringing together and categorizing different types of problems in order to help readers comprehend the applications of ordinary differential equations. In addition, the authors' collective academic experience ensures a coherent and accessible discussion of key topics, including:
Taylor and Runge-Kutta methods
General error analysis for multi-step methods
Stiff differential equations
Differential algebraic equations
Two-point boundary value problems
Volterra integral equations
Each chapter features problem sets that enable readers to test and build their knowledge of the presented methods, and a related Web site features MATLAB® programs that facilitate the exploration of numerical methods in greater depth. Detailed references outline additional literature on both analytical and numerical aspects of ordinary differential equations for further exploration of individual topics.
Numerical Solution of Ordinary Differential Equations is an excellent textbook for courses on the numerical solution of differential equations at the upper-undergraduate and beginning graduate levels. It also serves as a valuable reference for researchers in the fields of mathematics and engineering.
"Arthur Benjamin ... joyfully shows you how to make nature's numbers dance."--Bill Nye (the science guy)
The Magic of Math is the math book you wish you had in school. Using a delightful assortment of examples-from ice-cream scoops and poker hands to measuring mountains and making magic squares-this book revels in key mathematical fields including arithmetic, algebra, geometry, and calculus, plus Fibonacci numbers, infinity, and, of course, mathematical magic tricks. Known throughout the world as the "mathemagician," Arthur Benjamin mixes mathematics and magic to make the subject fun, attractive, and easy to understand for math fan and math-phobic alike.
"A positively joyful exploration of mathematics."
-Publishers Weekly, starred review
"Each [trick] is more dazzling than the last."
Rich in both theory and application, Fourier Analysispresents a unique and thorough approach to a key topic in advancedcalculus. This pioneering resource tells the full story of Fourieranalysis, including its history and its impact on the developmentof modern mathematical analysis, and also discusses essentialconcepts and today's applications.
Written at a rigorous level, yet in an engaging style that doesnot dilute the material, Fourier Analysis brings twoprofound aspects of the discipline to the forefront: the wealth ofapplications of Fourier analysis in the natural sciences and theenormous impact Fourier analysis has had on the development ofmathematics as a whole. Systematic and comprehensive, the book:Presents material using a cause-and-effect approach,illustrating where ideas originated and what necessitated themIncludes material on wavelets, Lebesgue integration, L2 spaces,and related conceptsConveys information in a lucid, readable style, inspiringfurther reading and research on the subjectProvides exercises at the end of each section, as well asillustrations and worked examples throughout the text
Based upon the principle that theory and practice arefundamentally linked, Fourier Analysis is the ideal text andreference for students in mathematics, engineering, and physics, aswell as scientists and technicians in a broad range of disciplineswho use Fourier analysis in real-world situations.
". . . an enchanting book for those people in computer scienceor mathematics who are fascinated by the concept ofinfinity."—Computing Reviews
". . . a very well written introduction to set theory . . . easyto read and well suited for self-study . . . highlyrecommended."—Choice
The concept of infinity has fascinated and confused mankind forcenturies with theories and ideas that cause even seasonedmathematicians to wonder. The Mathematics of Infinity: A Guide toGreat Ideas, Second Edition uniquely explores how we can manipulatethese ideas when our common sense rebels at the conclusions we aredrawing.
Continuing to draw from his extensive work on the subject, theauthor provides a user-friendly presentation that avoidsunnecessary, in-depth mathematical rigor. This Second Editionprovides important coverage of logic and sets, elements andpredicates, cardinals as ordinals, and mathematical physics.Classic arguments and illustrative examples are provided throughoutthe book and are accompanied by a gradual progression ofsophisticated notions designed to stun readers' intuitive view ofthe world.
With an accessible and balanced treatment of both concepts andtheory, the book focuses on the following topics:
Logic, sets, and functions
Counting infinite sets
Well ordered sets
Logic and meta-mathematics
Inductions and numbers
Presenting an intriguing account of the notions of infinity, TheMathematics of Infinity: A Guide to Great Ideas, Second Edition isan insightful supplement for mathematics courses on set theory atthe undergraduate level. The book also serves as a fascinatingreference for mathematically inclined individuals who areinterested in learning about the world of counterintuitivemathematics.
Most people are familiar with history's great equations: Newton's Law of Gravity, for instance, or Einstein's theory of relativity. But the way these mathematical breakthroughs have contributed to human progress is seldom appreciated. In In Pursuit of the Unknown, celebrated mathematician Ian Stewart untangles the roots of our most important mathematical statements to show that equations have long been a driving force behind nearly every aspect of our lives.
Using seventeen of our most crucial equations--including the Wave Equation that allowed engineers to measure a building's response to earthquakes, saving countless lives, and the Black-Scholes model, used by bankers to track the price of financial derivatives over time--Stewart illustrates that many of the advances we now take for granted were made possible by mathematical discoveries.
An approachable, lively, and informative guide to the mathematical building blocks of modern life, In Pursuit of the Unknown is a penetrating exploration of how we have also used equations to make sense of, and in turn influence, our world.
“. . . the author’s style is totally lucid and veryeasy to read . . .the result is indeed a wonderful story.”—Mathematical Reviews
Written in a unique and accessible style for readers of variedmathematical backgrounds, the Second Edition of Primes ofthe Form p = x2+ ny2details the history behind how Pierre de Fermat’s workultimately gave birth to quadratic reciprocity and the genus theoryof quadratic forms. The book also illustrates how results of Eulerand Gauss can be fully understood only in the context of classfield theory, and in addition, explores a selection of themagnificent formulas of complex multiplication.
Primes of the Form p = x2 +ny2, Second Edition focuses onaddressing the question of when a prime p is of the formx2 + ny2,which serves as the basis for further discussion of variousmathematical topics. This updated edition has several new notablefeatures, including:
• A well-motivated introduction to the classicalformulation of class field theory
• Illustrations of explicit numerical examples todemonstrate the power of basic theorems in various situations
• An elementary treatment of quadratic forms and genustheory
• Simultaneous treatment of elementary and advancedaspects of number theory
• New coverage of the Shimura reciprocity law and aselection of recent work in an updated bibliography
Primes of the Form p = x2 +ny2, Second Edition is both a usefulreference for number theory theorists and an excellent text forundergraduate and graduate-level courses in number and Galoistheory.
". . . fills a considerable gap in the numerical analysis literature by providing a self-contained treatment . . . this is an important work written in a clear style . . . warmly recommended to any graduate student or researcher in the field of the numerical solution of partial differential equations."
Time-Dependent Problems and Difference Methods, Second Edition continues to provide guidance for the analysis of difference methods for computing approximate solutions to partial differential equations for time-dependent problems. The book treats differential equations and difference methods with a parallel development, thus achieving a more useful analysis of numerical methods.
The Second Edition presents hyperbolic equations in great detail as well as new coverage on second-order systems of wave equations including acoustic waves, elastic waves, and Einstein equations. Compared to first-order hyperbolic systems, initial-boundary value problems for such systems contain new properties that must be taken into account when analyzing stability. Featuring the latest material in partial differential equations with new theorems, examples, and illustrations,Time-Dependent Problems and Difference Methods, Second Edition also includes:High order methods on staggered grids Extended treatment of Summation By Parts operators and their application to second-order derivatives Simplified presentation of certain parts and proofs
Time-Dependent Problems and Difference Methods, Second Edition is an ideal reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs and to predict and investigate physical phenomena. The book is also excellent for graduate-level courses in applied mathematics and scientific computations.
The Golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order, beauty, and eternal mystery will always coexist.
Logic of Mathematics combines a full-scale introductory course inmathematical logic and model theory with a range of speciallyselected, more advanced theorems. Using a strict mathematicalapproach, this is the only book available that contains completeand precise proofs of all of these important theorems:
* Gödel's theorems of completeness and incompleteness
* The independence of Goodstein's theorem from Peanoarithmetic
* Tarski's theorem on real closed fields
* Matiyasevich's theorem on diophantine formulas
Logic of Mathematics also features:
* Full coverage of model theoretical topics such as definability,compactness, ultraproducts, realization, and omission oftypes
* Clear, concise explanations of all key concepts, from Booleanalgebras to Skolem-Löwenheim constructions and othertopics
* Carefully chosen exercises for each chapter, plus helpfulsolution hints
At last, here is a refreshingly clear, concise, and mathematicallyrigorous presentation of the basic concepts of mathematicallogic-requiring only a standard familiarity with abstract algebra.Employing a strict mathematical approach that emphasizes relationalstructures over logical language, this carefully organized text isdivided into two parts, which explain the essentials of the subjectin specific and straightforward terms.
Part I contains a thorough introduction to mathematical logic andmodel theory-including a full discussion of terms, formulas, andother fundamentals, plus detailed coverage of relational structuresand Boolean algebras, Gödel's completeness theorem, models ofPeano arithmetic, and much more.
Part II focuses on a number of advanced theorems that are centralto the field, such as Gödel's first and second theorems ofincompleteness, the independence proof of Goodstein's theorem fromPeano arithmetic, Tarski's theorem on real closed fields, andothers. No other text contains complete and precise proofs of allof these theorems.
With a solid and comprehensive program of exercises and selectedsolution hints, Logic of Mathematics is ideal for classroom use-theperfect textbook for advanced students of mathematics, computerscience, and logic.
Nobel Laureate Eugene Wigner once wondered about “the unreasonable effectiveness of mathematics” in the formulation of the laws of nature. Is God a Mathematician? investigates why mathematics is as powerful as it is. From ancient times to the present, scientists and philosophers have marveled at how such a seemingly abstract discipline could so perfectly explain the natural world. More than that—mathematics has often made predictions, for example, about subatomic particles or cosmic phenomena that were unknown at the time, but later were proven to be true. Is mathematics ultimately invented or discovered? If, as Einstein insisted, mathematics is “a product of human thought that is independent of experience,” how can it so accurately describe and even predict the world around us?
Physicist and author Mario Livio brilliantly explores mathematical ideas from Pythagoras to the present day as he shows us how intriguing questions and ingenious answers have led to ever deeper insights into our world. This fascinating book will interest anyone curious about the human mind, the scientific world, and the relationship between them.
This volume takes an altogether new approach to advanced algebra.Its intriguing title, inspired by the term postmodernism, denotes adeparture from van der Waerden's Modern Algebra--a book that hasdominated the field for nearly seventy years. Post-Modern Algebraoffers a truly up-to-date alternative to the standard approach,explaining topics from an applications-based perspective ratherthan by abstract principles alone. The book broadens the field ofstudy to include algebraic structures and methods used in currentand emerging mathematical research, and describes the powerful yetsubtle techniques of universal algebra and category theory.Classical algebraic areas of groups, rings, fields, and vectorspaces are bolstered by such topics as ordered sets, monoids,monoid actions, quasigroups, loops, lattices, Boolean algebras,categories, and Heyting algebras. The text features:
* A clear and concise treatment at an introductory level, tested inuniversity courses.
* A wealth of exercises illustrating concepts and their practicalapplication.
* Effective techniques for solving research problems in the realworld.
* Flexibility of presentation, making it easy to tailor material tospecific needs.
* Help with elementary proofs and algebraic notations for studentsof varying abilities.
Post-Modern Algebra is an excellent primary or supplementary textfor graduate-level algebra courses. It is also an extremely usefulresource for professionals and researchers in many areas who musttackle abstract, linear, or universal algebra in the course oftheir work.
The author, a leading authority in the field, shares with the reader many new results from his exhaustive research on the Hilbert transform of Schwartz distributions. He describes in detail how to use the Hilbert transform to solve theoretical and physical problems in a wide range of disciplines; these include aerofoil problems, dispersion relations, high-energy physics, potential theory problems, and others.
Innovative at every step, J. N. Pandey provides a new definition for the Hilbert transform of periodic functions, which is especially useful for those working in the area of signal processing for computational purposes. This definition could also form the basis for a unified theory of the Hilbert transform of periodic, as well as nonperiodic, functions.
The Hilbert transform and the approximate Hilbert transform of periodic functions are worked out in detail for the first time in book form and can be used to solve Laplace's equation with periodic boundary conditions. Among the many theoretical results proved in this book is a Paley-Wiener type theorem giving the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of Rn.
Placing a strong emphasis on easy application of theory and techniques, the book generalizes the Hilbert problem in higher dimensions and solves it in function spaces as well as in generalized function spaces. It simplifies the one-dimensional transform of distributions; provides solutions to the distributional Hilbert problems and singular integral equations; and covers the intrinsic definition of the testing function spaces and its topology.
The book includes exercises and review material for all major topics, and incorporates classical and distributional problems into the main text. Thorough and accessible, it explores new ways to use this important integral transform, and reinforces its value in both mathematical research and applied science.
The Hilbert transform made accessible with many new formulas and definitions
Written by today's foremost expert on the Hilbert transform of generalized functions, this combined text and reference covers the Hilbert transform of distributions and the space of periodic distributions. The author provides a consistently accessible treatment of this advanced-level subject and teaches techniques that can be easily applied to theoretical and physical problems encountered by mathematicians, applied scientists, and graduate students in mathematics and engineering.
Introducing many new inversion formulas that have been developed and applied by the author and his research associates, the book:
* Provides solutions to the distributional Hilbert problem and singular integral equations
* Focuses on the Hilbert transform of Schwartz distributions, giving intrinsic definitions of the space H(D) and its topology
* Covers the Paley-Wiener theorem and provides many important theoretical results of importance to research mathematicians
* Provides the characterization of functions and generalized functions whose Fourier transforms are supported in certain orthants of Rn
* Offers a new definition of the Hilbert transform of the periodic function that can be used for computational purposes in signal processing
* Develops the theory of the Hilbert transform of periodic distributions and the approximate Hilbert transform of periodic distributions
For more than forty years, A History of Mathematics has been the reference of choice for those looking to learn about the fascinating history of humankind’s relationship with numbers, shapes, and patterns. This revised edition features up-to-date coverage of topics such as Fermat’s Last Theorem and the Poincaré Conjecture, in addition to recent advances in areas such as finite group theory and computer-aided proofs.Distills thousands of years of mathematics into a single, approachable volume Covers mathematical discoveries, concepts, and thinkers, from Ancient Egypt to the present Includes up-to-date references and an extensive chronological table of mathematical and general historical developments.
Whether you're interested in the age of Plato and Aristotle or Poincaré and Hilbert, whether you want to know more about the Pythagorean theorem or the golden mean, A History of Mathematics is an essential reference that will help you explore the incredible history of mathematics and the men and women who created it.
Conformal Differential Geometry and Its Generalizations is thefirst and only text that systematically presents the foundationsand manifestations of conformal differential geometry. It offersthe first unified presentation of the subject, which wasestablished more than a century ago. The text is divided into sevenchapters, each containing figures, formulas, and historical andbibliographical notes, while numerous examples elucidate thenecessary theory.
Clear, focused, and expertly synthesized, Conformal DifferentialGeometry and Its Generalizations
* Develops the theory of hypersurfaces and submanifolds of anydimension of conformal and pseudoconformal spaces.
* Investigates conformal and pseudoconformal structures on amanifold of arbitrary dimension, derives their structure equations,and explores their tensor of conformal curvature.
* Analyzes the real theory of four-dimensional conformal structuresof all possible signatures.
* Considers the analytic and differential geometry of Grassmann andalmost Grassmann structures.
* Draws connections between almost Grassmann structures and webtheory.
Conformal differential geometry, a part of classical differentialgeometry, was founded at the turn of the century and gave rise tothe study of conformal and almost Grassmann structures in lateryears. Until now, no book has offered a systematic presentation ofthe multidimensional conformal differential geometry and theconformal and almost Grassmann structures.
After years of intense research at their respective universitiesand at the Soviet School of Differential Geometry, Maks A. Akivisand Vladislav V. Goldberg have written this well-conceived,expertly executed volume to fill a void in the literature. Dr.Akivis and Dr. Goldberg supply a deep foundation, applications,numerous examples, and recent developments in the field. Many ofthe findings that fill these pages are published here for the firsttime, and previously published results are reexamined in a unifiedcontext.
The geometry and theory of conformal and pseudoconformal spaces ofarbitrary dimension, as well as the theory of Grassmann and almostGrassmann structures, are discussed and analyzed in detail. Thetopics covered not only advance the subject itself, but poseimportant questions for future investigations. This exhaustive,groundbreaking text combines the classical results and recentdevelopments and findings.
This volume is intended for graduate students and researchers ofdifferential geometry. It can be especially useful to thosestudents and researchers who are interested in conformal andGrassmann differential geometry and their applications totheoretical physics.
"I have discovered a truly marvelous demonstration of this proposition which this margin is too narrow to contain."
With these words, the seventeenth-century French mathematician Pierre de Fermat threw down the gauntlet to future generations. What came to be known as Fermat's Last Theorem looked simple; proving it, however, became the Holy Grail of mathematics, baffling its finest minds for more than 350 years. In Fermat's Enigma--based on the author's award-winning documentary film, which aired on PBS's "Nova"--Simon Singh tells the astonishingly entertaining story of the pursuit of that grail, and the lives that were devoted to, sacrificed for, and saved by it. Here is a mesmerizing tale of heartbreak and mastery that will forever change your feelings about mathematics.
One of the few treatments available combining both group theory and character theory, Groups and Characters is an effective general textbook on these two fundamentally connected subjects. Presuming only a basic knowledge of abstract algebra as in a first-year graduate course, the text opens with a review of background material and then guides readers carefully through several of the most important aspects of groups and characters, concentrating mainly on finite groups.
Challenging yet accessible, Groups and Characters features:
* An extensive collection of examples surveying many different types of groups, including Sylow subgroups of symmetric groups, affine groups of fields, the Mathieu groups, and symplectic groups
* A thorough, easy-to-follow discussion of Polya-Redfield enumeration, with applications to combinatorics
* Inclusive explorations of the transfer function and normal complements, induction and restriction of characters, Clifford theory, characters of symmetric and alternating groups, Frobenius groups, and the Schur index
* Illuminating accounts of several computational aspects of group theory, such as the Schreier-Sims algorithm, Todd-Coxeter coset enumeration, and algorithms for generating character tables
As valuable as Groups and Characters will prove as a textbook for mathematicians, it has broader applications. With chapters suitable for use as independent review units, along with a full bibliography and index, it will be a dependable general reference for chemists, physicists, and crystallographers.
This book is designed to give the reader a solid understanding of Lebesgue measure and integration. It focuses on only the most fundamental concepts, namely Lebesgue measure for R and Lebesgue integration for extended real-valued functions on R. Starting with a thorough presentation of the preliminary concepts of undergraduate analysis, this book covers all the important topics, including measure theory, measurable functions, and integration. It offers an abundance of support materials, including helpful illustrations, examples, and problems. To further enhance the learning experience, the author provides a historical context that traces the struggle to define "area" and "area under a curve" that led eventually to Lebesgue measure and integration.
Lebesgue Measure and Integration is the ideal text for an advanced undergraduate analysis course or for a first-year graduate course in mathematics, statistics, probability, and other applied areas. It will also serve well as a supplement to courses in advanced measure theory and integration and as an invaluable reference long after course work has been completed.
With coverage spanning the foundations of origami construction and advanced methods using both paper and pencil and custom-built free software, Origami Design Secrets helps readers cultivate the intuition and skills necessary to develop their own designs. It takes them beyond merely following a recipe to crafting a work of art.
This volume presents the most up-to-date information available on aposteriori error estimation for finite element approximation inmechanics and mathematics. It emphasizes methods for ellipticboundary value problems and includes applications to incompressibleflow and nonlinear problems.
Recent years have seen an explosion in the study of a posteriorierror estimators due to their remarkable influence on improvingboth accuracy and reliability in scientific computing. In an effortto provide an accessible source, the authors have sought to presentkey ideas and common principles on a sound mathematicalfooting.
Topics covered in this timely reference include:
* Implicit and explicit a posteriori error estimators
* Recovery-based error estimators
* Estimators, indicators, and hierarchic bases
* The equilibrated residual method
* Methodology for the comparison of estimators
* Estimation of errors in quantities of interest
A Posteriori Error Estimation in Finite Element Analysis is a lucidand convenient resource for researchers in almost any field offinite element methods, and for applied mathematicians andengineers who have an interest in error estimation and/or finiteelements.
Departing from the standard methods of analysis, this unique book presents methodologies and algorithms based on the concept of orthogonality and demonstrates their application to both standard and novel problems in linear algebra. Covering basic theory of linear systems, linear inequalities, and linear programming, it focuses on elegant, computationally simple solutions to real-world physical, economic, and engineering problems. The authors clearly explain the reasons behind the analysis of different structures and concepts and use numerous illustrative examples to correlate the mathematical models to the reality they represent. Readers are given precise guidelines for:
* Checking the equivalence of two systems
* Solving a system in certain selected variables
* Modifying systems of equations
* Solving linear systems of inequalities
* Using the new exterior point method
* Modifying a linear programming problem
With few prerequisites, but with plenty of figures and tables, end-of-chapter exercises as well as Java and Mathematica programs available from the authors' Web site, this is an invaluable text/reference for mathematicians, engineers, applied scientists, and graduate students in mathematics.
The relative quadratic case was first settled by Hecke in 1923, then recast by Weil in 1964 into the language of unitary group representations. The analytic proof of the general n-th order case is still an open problem today, going back to the end of Hecke's famous treatise of 1923. The Fourier-Analytic Proof of Quadratic Reciprocity provides number theorists interested in analytic methods applied to reciprocity laws with a unique opportunity to explore the works of Hecke, Weil, and Kubota.
This work brings together for the first time in a single volume the three existing formulations of the Fourier-analytic proof of quadratic reciprocity. It shows how Weil's groundbreaking representation-theoretic treatment is in fact equivalent to Hecke's classical approach, then goes a step further, presenting Kubota's algebraic reformulation of the Hecke-Weil proof. Extensive commutative diagrams for comparing the Weil and Kubota architectures are also featured.
The author clearly demonstrates the value of the analytic approach, incorporating some of the most powerful tools of modern number theory, including adèles, metaplectric groups, and representations. Finally, he points out that the critical common factor among the three proofs is Poisson summation, whose generalization may ultimately provide the resolution for Hecke's open problem.
Dr. Michael Guillen, known to millions as the science editor of ABC's Good Morning America, tells the fascinating stories behind five mathematical equations.
As a regular contributor to daytime's most popular morning news show and an instructor at Harvard University, Dr. Michael Guillen has earned the respect of millions as a clear and entertaining guide to the exhilarating world of science and mathematics.
Now Dr. Guillen unravels the equations that have led to the inventions and events that characterize the modern world, one of which -- Albert Einstein's famous energy equation, E=mc2 -- enabled the creation of the nuclear bomb. Also revealed are the mathematical foundations for the moon landing, airplane travel, the electric generator -- and even life itself.
Praised by Publishers Weekly as "a wholly accessible, beautifully written exploration of the potent mathematical imagination," and named a Best Nonfiction Book of 1995, the stories behind The Five Equations That Changed the World, as told by Dr. Guillen, are not only chronicles of science, but also gripping dramas of jealousy, fame, war, and discovery.
This distinctive introduction to one of the most far-reaching and beautiful areas of mathematics focuses on Banach spaces as the milieu in which most of the fundamental concepts are presented. While occasionally using the more general topological vector space and locally convex space setting, it emphasizes the development of the reader's mathematical maturity and the ability to both understand and "do" mathematics. In so doing, Functional Analysis provides a strong springboard for further exploration on the wide range of topics the book presents, including:
* Weak topologies and applications
* Operators on Banach spaces
* Bases in Banach spaces
* Sequences, series, and geometry in Banach spaces
Stressing the general techniques underlying the proofs, Functional Analysis also features many exercises for immediate clarification of points under discussion. This thoughtful, well-organized synthesis of the work of those mathematicians who created the discipline of functional analysis as we know it today also provides a rich source of research topics and reference material.
In the twenty years since the first edition of Applied Functional Analysis was published, there has been an explosion in the number of books on functional analysis. Yet none of these offers the unique perspective of this new edition. Jean-Pierre Aubin updates his popular reference on functional analysis with new insights and recent discoveries-adding three new chapters on set-valued analysis and convex analysis, viability kernels and capture basins, and first-order partial differential equations. He presents, for the first time at an introductory level, the extension of differential calculus in the framework of both the theory of distributions and set-valued analysis, and discusses their application for studying boundary-value problems for elliptic and parabolic partial differential equations and for systems of first-order partial differential equations.
To keep the presentation concise and accessible, Jean-Pierre Aubin introduces functional analysis through the simple Hilbertian structure. He seamlessly blends pure mathematics with applied areas that illustrate the theory, incorporating a broad range of examples from numerical analysis, systems theory, calculus of variations, control and optimization theory, convex and nonsmooth analysis, and more. Finally, a summary of the essential theorems as well as exercises reinforcing key concepts are provided. Applied Functional Analysis, Second Edition is an excellent and timely resource for both pure and applied mathematicians.
One of the most often-encountered systems in nearly all areas of science and technology, positive linear systems is a specific but remarkable and fascinating class. Renowned scientists Lorenzo Farina and Sergio Rinaldi introduce readers to the world of positive linear systems in their rigorous but highly accessible book, rich in applications, examples, and figures.
This professional reference is divided into three main parts: The first part contains the definitions and basic properties of positive linear systems. The second part, following the theoretical exposition, reports the main conceptual results, considering applicable examples taken from a number of widely used models. The third part is devoted to the study of some classes of positive linear systems of particular relevance in applications (such as the Leontief model, the Leslie model, the Markov chains, the compartmental systems, and the queueing systems). Readers familiar with linear algebra and linear systems theory will appreciate the way arguments are treated and presented.
Extraordinarily comprehensive, Positive Linear Systems features:
* Applications from a variety of backgrounds including modeling, control engineering, computer science, demography, economics, bioengineering, chemistry, and ecology
* References and annotated bibliographies throughout the book
* Two appendices concerning linear algebra and linear systems theory for readers unfamiliar with the mathematics used
Farina and Rinaldi make no effort to hide their enthusiasm for the topics presented, making Positive Linear Systems: Theory and Applications an indispensable resource for researchers and professionals in a broad range of fields.
Beginning with the ancient Near East, the author traces the ideas and techniques developed in Egypt, Babylonia, China, and Arabia, looking into such manuscripts as the Egyptian Papyrus Rhind, the Ten Classics of China, and the Siddhantas of India. He considers Greek and Roman developments from their beginnings in Ionian rationalism to the fall of Constantinople; covers medieval European ideas and Renaissance trends; analyzes 17th- and 18th-century contributions; and offers an illuminating exposition of 19th century concepts. Every important figure in mathematical history is dealt with — Euclid, Archimedes, Diophantus, Omar Khayyam, Boethius, Fermat, Pascal, Newton, Leibniz, Fourier, Gauss, Riemann, Cantor, and many others. For this latest edition, Dr. Struik has both revised and updated the existing text, and also added a new chapter on the mathematics of the first half of the 20th century. Concise coverage is given to set theory, the influence of relativity and quantum theory, tensor calculus, the Lebesgue integral, the calculus of variations, and other important ideas and concepts. The book concludes with the beginnings of the computer era and the seminal work of von Neumann, Turing, Wiener, and others.
"The author's ability as a first-class historian as well as an able mathematician has enabled him to produce a work which is unquestionably one of the best." — Nature Magazine.
The year was 1915, and the young mathematician Emmy Noether had just settled into Göttingen University when Albert Einstein visited to lecture on his nearly finished general theory of relativity. Two leading mathematicians of the day, David Hilbert and Felix Klein, dug into the new theory with gusto, but had difficulty reconciling it with what was known about the conservation of energy. Knowing of her expertise in invariance theory, they requested Noether’s help. To solve the problem, she developed a novel theorem, applicable across all of physics, which relates conservation laws to continuous symmetries—one of the most important pieces of mathematical reasoning ever developed.
Noether’s "first" and "second" theorem was published in 1918. The first theorem relates symmetries under global spacetime transformations to the conservation of energy and momentum, and symmetry under global gauge transformations to charge conservation. In continuum mechanics and field theories, these conservation laws are expressed as equations of continuity. The second theorem, an extension of the first, allows transformations with local gauge invariance, and the equations of continuity acquire the covariant derivative characteristic of coupled matter-field systems. General relativity, it turns out, exhibits local gauge invariance. Noether’s theorem also laid the foundation for later generations to apply local gauge invariance to theories of elementary particle interactions.
In Dwight E. Neuenschwander’s new edition of Emmy Noether’s Wonderful Theorem, readers will encounter an updated explanation of Noether’s "first" theorem. The discussion of local gauge invariance has been expanded into a detailed presentation of the motivation, proof, and applications of the "second" theorem, including Noether’s resolution of concerns about general relativity. Other refinements in the new edition include an enlarged biography of Emmy Noether’s life and work, parallels drawn between the present approach and Noether’s original 1918 paper, and a summary of the logic behind Noether’s theorem.-- Cliff Chancey, University of Northern Iowa
Inspired by Albert Einstein's theory of relativity and Bertrand Russell and David Hilbert's pursuit of the fundamental rules of mathematics, some of the most brilliant minds of the generation came together in post-World War I Vienna to present the latest theories in mathematics, science, and philosophy and to build a strong foundation for scientific investigation. Composed of such luminaries as Kurt Gödel and Rudolf Carnap, and stimulated by the works of Ludwig Wittgenstein and Karl Popper, the Vienna Circle left an indelible mark on science.
Exact Thinking in Demented Times tells the often outrageous, sometimes tragic, and never boring stories of the men who transformed scientific thought. A revealing work of history, this landmark book pays tribute to those who dared to reinvent knowledge from the ground up.