Mathematics as Problem Solving

Alexander Soifer, "Mathematics as Problem Solving"
Sp,.ger | 2009 | ISBN: 0387746463 | 106 pages | PDF | 1,7 MB

Retelling the best solutions and sharing the secrets of discovery are part of the process of teaching problem solving. Ideally, this process is characterized by mathematical skill, good taste, and wit. It is a characteristically personal process and the best such teachers have surely left their personal marks on students and readers. Alexander Soifer is a teacher of problem solving and his book, Mathematics as Problem Solving, is designed to introduce problem solving to the next generation.

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Geometry and Topology of Manifolds

Geometry and Topology of Manifolds By Hans U. Boden, Ian Hambleton, Andrew J. Nicas, B. Doug Park
Publisher: A M S 2005 | 347 Pages | ISBN: 0821837249 | DJVU | 3 MB

This status report features the most recent developments in the field, spanning a wide range of topical areas in the computer simulation of condensed matter/materials physics. Both established and new topics are included, ranging from the statistical mechanics of classical magnetic spin models to electronic structure calculations, quantum simulations, and simulations of soft condensed matter. The book presents new physical results as well as novel methods of simulation and data analysis. Highlights of this volume include various aspects of non-equilibrium statistical mechanics, studies of properties of real materials using both classical model simulations and electronic structure calculations, and the use of computer simulations in teaching.

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Introduction to the Mathematics for Computer Graphics

John A. Vince, "Introduction to the Mathematics for Computer Graphics"
Sp r inger | 2010 | ISBN: 1849960224 | 304 pages | PDF | 1,5 MB

1.3 Who Should Read This Book?
I have written this book as a reference for anyone intending to study computer
graphics, computer animation, computer games or virtual reality, especially for
people who want to understand the technical aspects. Although it is possible to
study these topics without requiring the support of mathematics, increasingly, there
are situations and projects that require animators, programmers and technical directors
to resort to mathematics to resolve unforeseen technical problems. This may be
in the form of a script or an extra piece of program code.

1.4 Aims and Objectives of This Book
One of the aims of this book is to bring together a range of useful mathematical
topics that are relevant to computer graphics. And the real objective is to provide
programmers and animators with an understanding of mathematics so that they can
solve all sorts of problems with confidence.
I have attempted to do this by exploring a range of mathematical topics without
intimidating the reader with mathematical symbols and abstract ideas. Hopefully, I
will be able to explain each topic in a simple and practical manner, with a variety of
practical examples.
This is far from an exhaustive study of the mathematics associated with computer
graphics. Each chapter introduces the reader to a new topic, and should leave the
reader confident and capable of studying more advanced books.

1.5 Assumptions Made in This Book
I suppose that I do expect that readers will have some understanding of arithmetic
and the general knowledge of the principles of mathematics, such as the ideas of
algebra. But, apart from that, each subject will be introduced as though it were the
first time it had been discovered.
In the chapter on curves and surfaces I have used a little calculus. Readers who
have not studied this subject should not be concerned about missing some vital piece
of information. I only included it to keep the explanation complete.

1.6 How to Use This Book
I would advise starting at the beginning and proceeding chapter by chapter. Where
a subject seems familiar, just jump ahead until a challenge is discovered. Once you
have read the book, keep it handy so that you can refer to it when the occasion arises.
Although I have tried to maintain a sequence to the mathematical ideas, so that
one idea leads to another, in some cases this has proved impossible. For example,
determinants are referred to in the chapter on vectors, but they are described in detail
in the next chapter on transforms. Similarly, the later chapter on analytic geometry
contains some basic ideas of geometry, but its position was dictated by its use of
vectors. Consequently, on some occasions, the reader will have to move between
chapters to read about related topics.

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Algebra, Volume II

Algebra, Volume II By B.L. van der Waerden
Publisher: Springer | 2003 | 284 Pages | ISBN: 0387406255 | PDF | 15 MB

This widely known textbook, formally titled Modern Algebra, by the noted Dutch mathematician van der Waerden is now back in print. Algebra originated from notes taken by the author from Emil Artin's lectures. The author extended the scope of these notes to include research of Emmy Noether and her students. The first German edition appeared in 1930-1931, with subsequent editions having been brought up to date. "The basic notions of algebra, groups, rings, modules, fields, and the main theories pertaining to these notions are treated in the classical two volume textbook of van der Waerden. Although more than half a century has elapsed since the appearance of this remarkable book, it is in no way dated, and for the majority of the questions it treats, no better source can be found even today.

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Algebra: Volume I

Algebra: Volume I
Springer | 2003 | ISBN: 0387406247 | 265 pages | PDF | 14,9 MB

This widely known textbook, formally titled Modern Algebra, by the noted Dutch mathematician van der Waerden is now back in print. Algebra originated from notes taken by the author from Emil Artin's lectures. The author extended the scope of these notes to include research of Emmy Noether and her students. The first German edition appeared in 1930-1931, with subsequent editions having been brought up to date. "The basic notions of algebra, groups, rings, modules, fields, and the main theories pertaining to these notions are treated in the classical two volume textbook of van der Waerden. Although more than half a century has elapsed since the appearance of this remarkable book, it is in no way dated, and for the majority of the questions it treats, no better source can be found even today." (I.R. Shafarevich: Encyclopaedia of Mathematical Sciences, Volume 11.1990)

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Collins Dictionary of Mathematics

Collins Dictionary of Mathematics By E. J. and J. M. Borwein Borowski
Publisher: Collins 2005 | 529 Pages | ISBN: 0007800800 | DJVU | 7 MB

Some dictionaries of mathematics lack reach. This one has high aspirations, and it succeeds. It is written with the undergraduate mathematics student in mind, which is to say: for anyone with a serious interest in mathematics, formal student or not. Here's an example of an entry. Fully capitalized terms are entries themselves.
"Galois theory, n. the algebraic study of GROUPS of AUTOMORPHISMS of FIELDS in which one associates an EXTENSION FIELD with a given ALGEBRAIC EQUATION. The theory grew out of Galois' highly original study of the solubility of equations, devised in part to prove the impossibility of SOLUTION BY RADICALS of the general QUINTIC, Abel's proof of which was unknown to him. See also CARDANO'S FORMULA."
The entry for Cardano's formula takes up half a page. For more examples, entries concerned with terminology involving 'limit' fill two pages, including entries on limit, limit inferior or lower limit, limit of integration, limit ordinal, limit point (which refers directly without definition to the entry on cluster point), limit superior or upper limit. There are many examples of this kind of usefulness. The entry on exponential function takes up three-quarters of a page. Whenever I browse a mathematics dictionary in a bookstore, the first term I look for is CATEGORY. Then I check for FUNCTOR. This dictionary includes them both. In fact, the entry on category takes up a third of the page, and the entry on functor takes up about the same amount. There is also an entry on COMMUTATIVE DIAGRAM, of nearly the same length. This is no mere "dictionary", with terse, unhelpful definitions or descriptions. This is a reference dictionary of real value.

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Approximation of Large-Scale Dynamical Systems (Advances in Design and Control)

 

Approximation of Large-Scale Dynamical Systems (Advances in Design and Control) (Advances in Design and Control)
By Athanasios C. Antoulas
Publisher Society for Industrial and Applied Mathematic | ISBN: 0898715296 | edition 2005 | PDF | 481 pages | 22,61 mb

Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational, accuracy, and storage capabilities, model reduction (system approximation) is often necessary. Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications. This book is for anyone interested in model reduction. Graduate students and researchers in the fields of system and control theory, numerical analysis, and the theory of partial differential equations/computational fluid dynamics will find it an excellent reference. Contents List of Figures; Foreword; Preface; How to Use this Book; Part I: Introduction. Chapter 1: Introduction; Chapter 2: Motivating Examples; Part II: Preliminaries. Chapter 3: Tools from Matrix Theory; Chapter 4: Linear Dynamical Systems: Part 1; Chapter 5: Linear Dynamical Systems: Part 2; Chapter 6: Sylvester and Lyapunov equations; Part III: SVD-based Approximation Methods. Chapter 7: Balancing and balanced approximations; Chapter 8: Hankel-norm Approximation; Chapter 9: Special topics in SVD-based approximation methods; Part IV: Krylov-based Approximation Methods; Chapter 10: Eigenvalue Computations; Chapter 11: Model Reduction Using Krylov Methods; Part V: SVD–Krylov Methods and Case Studies. Chapter 12: SVD–Krylov Methods; Chapter 13: Case Studies; Chapter 14: Epilogue; Chapter 15: Problems; Bibliography; Index.

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Discrete Convex Analysis (Monographs on Discrete Math and Applications)

Discrete Convex Analysis (Monographs on Discrete Math and Applications) By Kazuo Murota
Publisher: Society for Industrial and Applied Mathematic 2003 | 389 Pages | ISBN: 0898715407 | PDF | 17 MB

Discrete Convex Analysis is a novel paradigm for discrete optimization that combines the ideas in continuous optimization (convex analysis) and combinatorial optimization (matroid/submodular function theory) to establish a unified theoretical framework for nonlinear discrete optimization. The study of this theory is expanding with the development of efficient algorithms and applications to a number of diverse disciplines like matrix theory, operations research, and economics. This self-contained book is designed to provide a novel insight into optimization on discrete structures and should reveal unexpected links among different disciplines. It is the first and only English-language monograph on the theory and applications of discrete convex analysis. Discrete Convex Analysis provides the information that professionals in optimization will need to “catch up” with this new theoretical development. It also presents an unexpected connection between matroid theory and mathematical economics and expounds a deeper connection between matrices and matroids than most standard textbooks

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Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems

Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer Series in Computational Mathematic) By E. Hairer, G. Wanner
Publisher: Springer-Verlag 1991 | 601 Pages | ISBN: 0387537759 | PDF | 8.7 MB

The subject of this book is the solution of stiff differential equations and of differential-algebraic systems (differential equations with constraints). There is a chapter on one-step and extrapolation methods for stiff problems, another one on multistep methods and general linear methods for stiff problems, and a third one on the treatment of singular perturbation problems and differential-algebraic systems. The beginning of each chapter is of an introductory nature, followed by practical applications, the discusssion of numerical results, theoretical investigations on the order of accuracy, linear and non-linear stability convergence and asymptotic expansions. Stiff and differential-algebraic problems arise everywhere in scientific computations ( e.g., in physics, chemistry, biology, control engineering, electrical network analysis, mechanical sytems). Many applications as well as computer programmes are presented.

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Discontinuous Galerkin Methods For Solving Elliptic And parabolic Equations: Theory and Implementation

Beatrice Riviere, "Discontinuous Galerkin Methods For Solving Elliptic And parabolic Equations: Theory and Implementation "
Society for Industrial and Applied Mathematic | 2008-07-14 | ISBN: 089871656X | 212 pages | PDF | 3,3 MB

Discontinuous Galerkin (DG) methods for solving partial differential equations, developed in the late 1990s, have become popular among computational scientists. This book covers both theory and computation as it focuses on three primal DG methods--the symmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and nonsymmetric interior penalty Galerkin which are variations of interior penalty methods. The author provides the basic tools for analysis and discusses coding issues, including data structure, construction of local matrices, and assembling of the global matrix. Computational examples and applications to important engineering problems are also included.
Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation is divided into three parts: Part I focuses on the application of DG methods to second order elliptic problems in one dimension and in higher dimensions. Part II presents the time-dependent parabolic problems without and with convection. Part III contains applications of DG methods to solid mechanics (linear elasticity), fluid dynamics (Stokes and Navier Stokes), and porous media flow (two-phase and miscible displacement).
Appendices contain proofs and MATLAB® code for one-dimensional problems for elliptic equations and routines written in C that correspond to algorithms for the implementation of DG methods in two or three dimensions.

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Elementary Calculus of Financial Mathematics

A. J. Roberts, “Elementary Calculus of Financial Mathematics”
Society for Industrial and Applied Mathematic | 2008 | ISBN: 0898716675 | 140 pages | PDF | 1,7 MB

Modern financial mathematics relies on the theory of random processes in time, reflecting the erratic fluctuations in financial markets.This book introduces the fascinating area of financial mathematics and its calculus in an accessible manner geared toward undergraduate students. Using little high-level mathematics, the author presents the basic methods for evaluating financial options and building financial simulations.
By emphasizing relevant applications and illustrating concepts with color graphics, Elementary Calculus of Financial Mathematics presents the crucial concepts needed to understand financial options among these fluctuations. Among the topics covered are the binomial lattice model for evaluating financial options, the Black Scholes and Fokker Planck equations, and the interpretation of Ito s formula in financial applications. Each chapter includes exercises for student practice and the appendices offer MATLAB® and SCILAB code as well as alternate proofs of the Fokker Planck equation and Kolmogorov backward equation.
Audience: This book will be useful to teachers and undergraduate students of mathematics or finance.

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Stochastic Processes, Estimation, and Control

Jason Lee Speyer, "Stochastic Processes, Estimation, and Control "
Society for Industrial and Applied Mathematic | 2008-10 | ISBN: 0898716551 | 400 pages | PDF | 2,4 MB

A comprehensive treatment of stochastic systems beginning with the foundations of probability and ending with stochastic optimal control. The book divides into three interrelated topics. First, the concepts of probability theory, random variables and stochastic processes are presented, which leads easily to expectation, conditional expectation, and discrete time estimation and the Kalman filter. With this background, stochastic calculus and continuous-time estimation are introduced. Finally, dynamic programming for both discrete-time and continuous-time systems leads to the solution of optimal stochastic control problems resulting in controllers with significant practical application. This book will be valuable to first year graduate students studying systems and control, as well as professionals in this field.

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Introduction to the Numerical Analysis of Incompressible Viscous Flows

William Layton, "Introduction to the Numerical Analysis of Incompressible Viscous Flows "
Society for Industrial and Applied Mathematic | 2008-08-07 | ISBN: 0898716578 | 233 pages | PDF | 2,9 MB

This book treats the numerical analysis of finite element computational fluid dynamics. Assuming minimal background, the text covers finite element methods; the derivation, behavior, analysis, and numerical analysis of Navier Stokes equations; and turbulence and turbulence models used in simulations. Each chapter on theory is followed by a numerical analysis chapter that expands on the theory. The chapters contain numerous exercises.
Introduction to the Numerical Analysis of Incompressible Viscous Flows provides the foundation for understanding the interconnection of the physics, mathematics, and numerics of the incompressible case, which is essential for progressing to the more complex flows not addressed in this book (e.g., viscoelasticity, plasmas, compressible flows, coating flows, flows of mixtures of fluids, and bubbly flows). With mathematical rigor and physical clarity, the book progresses from the mathematical preliminaries of energy and stress to finite element computational fluid dynamics in a format manageable in one semester.

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A Unified Approach to Boundary Value Problems

A Unified Approach to Boundary Value Problems

Publisher: Society for Industrial and Applied Mathematic | Pages: 352 | 2008-08-28 | ISBN 0898716519 | PDF | 3 MB

This book presents a new approach to analyzing initial-boundary value problems for integrable partial differential equations (PDEs) in two dimensions, a method that the author first introduced in 1997 and which is based on ideas of the inverse scattering transform. This method is unique in also yielding novel integral representations for the explicit solution of linear boundary value problems, which include such classical problems as the heat equation on a finite interval and the Helmholtz equation in the interior of an equilateral triangle. The author s thorough introduction allows the interested reader to quickly assimilate the essential results of the book, avoiding many computational details. Several new developments are addressed in the book, including a new transform method for linear evolution equations on the half-line and on the finite interval; analytical inversion of certain integrals such as the attenuated radon transform and the Dirichlet-to-Neumann map for a moving boundary; analytical and numerical methods for elliptic PDEs in a convex polygon; and integrable nonlinear PDEs. An epilogue provides a list of problems on which the author s new approach has been used, offers open problems, and gives a glimpse into how the method might be applied to problems in three dimensions. Audience: A Unified Approach to Boundary Value Problems is appropriate for courses in boundary value problems at the advanced undergraduate and first-year graduate levels. Applied mathematicians, engineers, theoretical physicists, mathematical biologists, and other scholars who use PDEs will also find the book valuable. Contents: Preface; Introduction; Chapter 1: Evolution Equations on the Half-Line; Chapter 2: Evolution Equations on the Finite Interval; Chapter 3: Asymptotics and a Novel Numerical Technique; Chapter 4: From PDEs to Classical Transforms; Chapter 5: Riemann Hilbert and d-Bar Problems; Chapter 6: The Fourier Transform and Its Variations; Chapter 7: The Inversion of the Attenuated Radon Transform and Medical Imaging; Chapter 8: The Dirichlet to Neumann Map for a Moving Boundary; Chapter 9: Divergence Formulation, the Global Relation, and Lax Pairs; Chapter 10: Rederivation of the Integral Representations on the Half-Line and the Finite Interval; Chapter 11: The Basic Elliptic PDEs in a Polygonal Domain; Chapter 12: The New Transform Method for Elliptic PDEs in Simple Polygonal Domains; Chapter 13: Formulation of Riemann Hilbert Problems; Chapter 14: A Collocation Method in the Fourier Plane; Chapter 15: From Linear to Integrable Nonlinear PDEs; Chapter 16: Nonlinear Integrable PDEs on the Half-Line; Chapter 17: Linearizable Boundary Conditions; Chapter 18: The Generalized Dirichlet to Neumann Map; Chapter 19: Asymptotics of Oscillatory Riemann Hilbert Problems; Epilogue; Bibliography; Index.

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Accuracy and Reliability in Scientific Computing

Edited by Bo Einarsson “Accuracy and Reliability in Scientific Computing"
Society for Industrial and Applied Mathematic | 2005-07-01 | ISBN: 0898715849 | 364 pages | PDF | 17,2 MB

Numerical software is used to test scientific theories, design airplanes and bridges, operate manufacturing lines, control power plants and refineries, analyze financial derivatives, identify genomes, and provide the understanding necessary to derive and analyze cancer treatments. Because of the high stakes involved, it is essential that results computed using software be accurate, reliable, and robust. Unfortunately, developing accurate and reliable scientific software is notoriously difficult. This book investigates some of the difficulties related to scientific computing and provides insight into how to overcome them and obtain dependable results. The tools to assess existing scientific applications are described, and a variety of techniques that can improve the accuracy and reliability of newly developed applications is discussed. Accuracy and Reliability in Scientific Computing can be considered a handbook for improving the quality of scientific computing. It will help computer scientists address the problems that affect software in general as well as the particular challenges of numerical computation: approximations occurring at all levels, continuous functions replaced by discretized versions, infinite processes replaced by finite ones, and real numbers replaced by finite precision numbers. Divided into three parts, it starts by illustrating some of the difficulties in producing robust and reliable scientific software. Well-known cases of failure are reviewed and the what and why of numerical computations are considered. The second section describes diagnostic tools that can be used to assess the accuracy and reliability of existing scientific applications. In the last section, the authors describe a variety of techniques that can be employed to improve the accuracy and reliability of newly developed scientific applications. The authors of the individual chapters are international experts, many of them members of the IFIP Working Group on Numerical Software. Accuracy and Reliability in Scientific Computing contains condensed information on the main features of six major programming languages — Ada, C, C++, Fortran, Java, and Python — and the INTLAB toolbox of the MATLABâ software and the PRECISE toolbox of Fortran are discussed in detail. This book has an accompanying website, www.nsc.liu.se/wg25/book/, with codes, links, color versions of some illustrations, and additional material. The book will be of interest to any scientist, engineer, or physicist who wants to improve the reliability and accuracy of computed results, especially when the computations are critical or large. It will be of interest to practitioners who use numerical software for real applications and want to avoid potential difficulties.

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Design Sensitivity Analysis

Lisa Gayle Stanley, Davis Stewart “Design Sensitivity Analysis"
Society for Industrial and Applied Mathematic | 2002-10-01 | ISBN: 0898715245 | 139 pages | PDF | 8,4 MB

This book provides an understandable introduction to one approach to design sensitivity computation and illustrates some of the important mathematical and computational issues inherent in using the sensitivity equation method (SEM) for partial differential equations. The authors use basic models to illustrate the computational issues that one might encounter when applying the SEM in a laboratory or research setting, while providing an overview of applications and computational issues regarding sensitivity calculations performed by way of continuous sensitivity equation methods (CSEM). Here they focus on the construction and analysis of algorithms for computing sensitivities. For readers already acquainted with the concept of a sensitivity equation, the authors include mathematical background for a deeper understanding of their approach. Finally, the book explores the use of SEMs for applications in the area of computational fluid dynamics, demonstrating that the early examples readers encounter in the book can be observed in the context of a more realistic physical setting. Several colour figures are included.

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Computational Methods for Option Pricing

Yves Achdou, Olivier Pironneau “Computational Methods for Option Pricing"
Society for Industrial and Applied Mathematic | 2005-07-01 | ISBN: 0898715733 | 297 pages | PDF | 11,2 MB

This book is a must for becoming better acquainted with the modern tools of numerical analysis for several significant computational problems arising in finance. Important aspects of finance modeling are reviewed, involving partial differential equations and numerical algorithms for the fast and accurate pricing of financial derivatives and the calibration of parameters. The best numerical algorithms are fully explored and discussed, from their mathematical analysis up to their implementation in C++ with efficient numerical libraries. This is one of the few books that thoroughly covers the following topics: mathematical results and efficient algorithms for pricing American options; modern algorithms with adaptive mesh refinement for European and American options; regularity and error estimates are derived and give strong support to the mesh adaptivity, an essential tool for speeding up the numerical implementations; calibration of volatility with European and American options; the use of automatic differentiation of computer codes for computing greeks.

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Topics in Mathematic Modelling of Composite Materials

R. Kohn "Topics in Mathematic Modelling of Composite Materials"
Birkhäuser Boston | English | 1997-06-26 | ISBN: 0817636625 | 336 pages | DJVU | 9,1 MB

Contains English translation of certain fundamental papers by the French and Russian schools, on the macroscopic behavior of microscopically heterogeneous materials. DLC: Composite materials - Mathematical models.

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Morse Theory (Annals of Mathematic Studies AM-51)

Morse Theory (Annals of Mathematic Studies AM-51) By John Milnor
Publisher: Princeton University Press 1963 | 160 Pages | ISBN: 0691080089 | PDF | 3 MB

One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory.

Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master."

Milnor has published five books with Princeton University Press.

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