As the amount of information recorded and stored electronically grows ever larger, it becomes increasingly useful, if not essential, to develop better and more efficient ways to summarize and extract information from these large, multivariate data sets. The field of classification does just that-investigates sets of "objects" to see if they can be summarized into a small number of classes comprising similar objects. Researchers have made great strides in the field over the last twenty years, and classification is no longer perceived as being concerned solely with exploratory analyses. The second edition of Classification incorporates many of the new and powerful methodologies developed since its first edition. Like its predecessor, this edition describes both clustering and graphical methods of representing data, and offers advice on how to decide which methods of analysis best apply to a particular data set. It goes even further, however, by providing critical overviews of recent developments not widely known, including efficient clustering algorithms, cluster validation, consensus classifications, and the classification of symbolic data. The author has taken an approach accessible to researchers in the wide variety of disciplines that can benefit from classification analysis and methods. He illustrates the methodologies by applying them to data sets-smaller sets given in the text, larger ones available through a Web site. Large multivariate data sets can be difficult to comprehend-the sheer volume and complexity can prove overwhelming. Classification methods provide efficient, accurate ways to make them less unwieldy and extract more information. Classification, Second Edition offers the ideal vehicle for gaining the background and learning the methodologies-and begin putting these techniques to use.

Catastrophe Theory was introduced in the 1960s by the renowned Fields Medal mathematician Rene' Thom as a part of the general theory of local singularities. Since then it has found applications across many areas, including biology, economics, and chemical kinetics. By investigating the phenomena of bifurcation and chaos, Catastrophe Theory proved t

Building upon the technical and organizational groundwork presented in the first edition, Risk Assessment and Decision Making in Business and Industry: A Practical Guide, Second Edition addresses the many aspects of risk/uncertainty (R/U) process implementation. This comprehensive volume covers four broad aspects of R/U: general concepts, implementation processes, technical aspects, and examples of application. Each section provides practical guidance, combining technical information with advice on how to implement R/U techniques and processes in real-world corporate environments. Following an examination of general principles involved in quantitatively assessing risks and their impact on value, the book describes the two main probabilistic measures of project value - Expected Value of Success (EVS) and the Expected Value for the Portfolio (EVP). The text clearly demonstrates how these metrics are used in individual-project and portfolio management. By presenting concepts in layman's terms and fully integrating advice related to technical and human characteristics of R/U-related corporate life, this book serves as a complete primer for professionals in any business environment. What's New in the Second Edition: Provides guidance for implementation of R/U processes in modern corporations Offers a crucial breakthrough by defining the terms "risk" and "uncertainty" in ways that can be applied in all aspects of science and business Explores real-world impediments to process change and implementation Addresses R/U from a corporate decision-maker's perspective, detailing how to employ R/U to set budgets, manage portfolios, value investments, and execute other critical tasks

The financial industry is swamped by credit products whose economic performance is linked to the performance of some underlying portfolio of credit-risky instruments, like loans, bonds, swaps, or asset-backed securities. Financial institutions continuously use these products for tailor-made long and short positions in credit risks. Based on a steadily growing market, there is a high demand for concepts and techniques applicable to the evaluation of structured credit products. Written from the perspective of practitioners who apply mathematical concepts to structured credit products, Structured Credit Portfolio Analysis, Baskets & CDOs starts with a brief wrap-up on basic concepts of credit risk modeling and then quickly moves on to more advanced topics such as the modeling and evaluation of basket products, credit-linked notes referenced to credit portfolios, collateralized debt obligations, and index tranches. The text is written in a self-contained style so readers with a basic understanding of probability will have no difficulties following it. In addition, many examples and calculations have been included to keep the discussion close to business applications. Practitioners as well as academics will find ideas and tools in the book that they can use for their daily work.

The Eighth International Conference on Difference Equations and Applications was held at Masaryk University in Brno, Czech Republic. This volume comprises refereed papers presented at this conference. These papers cover all important themes, conjectures, and open problems in the fields of discrete dynamical systems and ordinary and partial difference equations, classical and contemporary, theoretical and applied.

This book brings together diverse recent developments exploring the philosophy of mathematics in education. The unique combination of ethnomathematics, philosophy, history, education, statistics and mathematics offers a variety of different perspectives from which existing boundaries in mathematics education can be extended. The ten chapters in this book offer a balance between philosophy of and philosophy in mathematics education. Attention is paid to the implementation of a philosophy of mathematics within the mathematics curriculum.

As the theories and methods have evolved over the years, the mechanics of solid bodies has become unduly fragmented. Most books focus on specific aspects, such as the theories of elasticity or plasticity, the theories of shells, or the mechanics of materials. While a narrow focus serves immediate purposes, much is achieved by establishing the common foundations and providing a unified perspective of the discipline as a whole. Mechanics of Solids and Shells accomplishes these objectives. By emphasizing the underlying assumptions and the approximations that lead to the mathematical formulations, it offers a practical, unified presentation of the foundations of the mechanics of solids, the behavior of deformable bodies and thin shells, and the properties of finite elements. The initial chapters present the fundamental kinematics, dynamics, energetics, and behavior of materials that build the foundation for all of the subsequent developments. These are presented in full generality without the usual restrictions on the deformation. The general principles of work and energy form the basis for the consistent theories of shells and the approximations by finite elements. The final chapter views the latter as a means of approximation and builds a bridge between the mechanics of the continuum and the discrete assembly. Expressly written for engineers, Mechanics of Solids and Shells forms a reliable source for the tools of analysis and approximation. Its constructive presentation clearly reveals the origins, assumptions, and limitations of the methods described and provides a firm, practical basis for the use of those methods.

Wholeheartedly recommended to every student and user of mathematics, this is an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields studied in every university maths course, through Lie groups to cohomology and category theory, the author shows how the origins of each concept can be related to attempts to model phenomena in physics or in other branches of mathematics. Required reading for mathematicians, from beginners to experts.

The first two editions of An Introduction to Partial Differential Equations with MATLAB® gained popularity among instructors and students at various universities throughout the world. Plain mathematical language is used in a friendly manner to provide a basic introduction to partial differential equations (PDEs).

Suitable for a one- or two-semester introduction to PDEs and Fourier series, the book strives to provide physical, mathematical, and historical motivation for each topic. Equations are studied based on method of solution, rather than on type of equation.

This third edition of this popular textbook updates the structure of the book by increasing the role of the computational portion, compared to previous editions. The redesigned content will be extremely useful for students of mathematics, physics, and engineering who would like to focus on the practical aspects of the study of PDEs, without sacrificing mathematical rigor. The authors have maintained flexibility in the order of topics.

In addition, students will be able to use what they have learned in some later courses (for example, courses in numerical analysis, optimization, and PDE-based programming). Included in this new edition is a substantial amount of material on reviewing computational methods for solving ODEs (symbolically and numerically), visualizing solutions of PDEs, using MATLAB®'s symbolic programming toolbox, and applying various schemes from numerical analysis, along with suggestions for topics of course projects.

Students will use sample MATLAB® or Python codes available online for their practical experiments and for completing computational lab assignments and course projects.

Table of Contents

Chapter 1. Introduction

What are Partial Differential Equations?

PDEs We Can Already Solve

Initial and Boundary Conditions

Linear PDEs – Definitions

Linear PDEs – The Principle of Superposition

The Method of Characteristics I

The Method of Characteristics II

Separation of Variables for Linear, Homogeneous PDEs

Eigenvalue Problems

Chapter 2. The Big Three PDEs

Second-Order, Linear, Homogeneous PDEs with Constant Coefficients

The Heat Equation and Diffusion

The Wave Equation and the Vibrating String

Initial and Boundary Conditions for the Heat and Wave Equations

Laplace's Equation – The Potential Equation

D'Alembert's Solution for the Infinite String Problem

General Second-Order Linear PDEs and Characteristics

Using Separation of Variables to Solve the Big Three PDEs

Chapter 3. Using MATLAB for Solving Differential Equations and Visualizing Solutions

Visualizing Solutions of ODEs

Symbolic Math Toolbox for Solving ODEs

Solving BVPs Numerically Using bvp4(5)c

Solving PDEs Numerically Using pdepe

Exercises for Chapter 3

Lab Assignment #1: Review Chapters 1-3

Chapter 4. Fourier Series

Introduction

Properties of Sine and Cosine

The Fourier Series

The Fourier Series, Continued

Fourier Sine and Cosine Series

Chapter 5. Solving the Big Three PDEs on Finite Domains

Solving the Homogeneous Heat Equation for a Finite Rod

Solving the Homogeneous Wave Equation for a Finite String

Solving the Homogeneous Laplace’s Equation on a Rectangular Domain

Nonhomogeneous Problems

Chapter 6. Review of Numerical Methods for Solving ODEs

Approaches to Solving First-Order IVPs

Numerical Solutions Using Euler's Method

Numerical Solutions Using Runge–Kutta Methods

Solving Higher-Order ODEs Numerically

Implicit Approximations for BVPs

Exercises for Chapter 6

Chapter 7. Solving PDEs Using Finite Difference Approximations

Numerical Solutions for the Heat Equation

Explicit Scheme for the Wave Equation

Numerical Schemes for Laplace's Equation

Numerical Solution of First-Order PDEs

Exercises for Chapter 7

Lab Assignment #2: Review Chapters 6-7

Lab Assignment #3: Review Chapters 4-7

Chapter 8. Integral Transforms

The Laplace Transform for PDEs

Fourier Sine and Cosine Transforms

The Fourier Transform

The Infinite and Semi-Infinite Heat Equations

Other Integral Transforms and Integral Equations

Chapter 9. Using MATLAB's Symbolic Math Toolbox with Integral Transforms

Integral Transforms via Symbolic Programming

Solving ODEs Using the Laplace Transform in MATLAB

Symbolic Solution of PDEs Using the Laplace Transform

Symbolic Solution of PDEs Using the Fourier Transform

Exercises for Chapter 9

Lab Assignment #4: Review Chapters 8-9

Chapter 10. PDEs in Higher Dimensions

PDEs in Higher Dimensions: Examples and Derivations

The Heat and Wave Equations on a Rectangle; Multiple Fourier Series

Laplace's Equation in Polar Coordinates: Poisson's Integral Formula

Interlude 1: Bessel Functions

Interlude 2: The Legendre Polynomials

The Wave and Heat Equations in Polar Coordinates

Problems in Spherical Coordinates

The Infinite Wave Equation and Multiple Fourier Transforms

MATLAB Exercises for Chapter 10

Lab Assignment #5: Review Chapters 7 & 10

Chapter 11. Overview of Spectral, Finite Element, and Finite Volume Methods

Spectral Methods

Finite Element Methods

Finite Volume Methods

Exercises for Chapter 11

Appendix A: Important Definitions and Theorems

Appendix B: Bessel's Equation and the Method of Frobenius

Appendix C: A Menagerie of PDEs

Appendix D: Review of Math with MATLAB

Appendix E: Answers to Selected Exercises

References

Index

/

Representation Theory and Higher Algebraic K-Theory is the first book to present higher algebraic K-theory of orders and group rings as well as characterize higher algebraic K-theory as Mackey functors that lead to equivariant higher algebraic K-theory and their relative generalizations. Thus, this book makes computations of higher K-theory of group rings more accessible and provides novel techniques for the computations of higher K-theory of finite and some infinite groups. Authored by a premier authority in the field, the book begins with a careful review of classical K-theory, including clear definitions, examples, and important classical results. Emphasizing the practical value of the usually abstract topological constructions, the author systematically discusses higher algebraic K-theory of exact, symmetric monoidal, and Waldhausen categories with applications to orders and group rings and proves numerous results. He also defines profinite higher K- and G-theory of exact categories, orders, and group rings. Providing new insights into classical results and opening avenues for further applications, the book then uses representation-theoretic techniques-especially induction theory-to examine equivariant higher algebraic K-theory, their relative generalizations, and equivariant homology theories for discrete group actions. The final chapter unifies Farrell and Baum-Connes isomorphism conjectures through Davis-Luck assembly maps.

Developed by Jean-Paul Benzerci more than 30 years ago, correspondence analysis as a framework for analyzing data quickly found widespread popularity in Europe. The topicality and importance of correspondence analysis continue, and with the tremendous computing power now available and new fields of application emerging, its significance is greater than ever. Correspondence Analysis and Data Coding with Java and R clearly demonstrates why this technique remains important and in the eyes of many, unsurpassed as an analysis framework. After presenting some historical background, the author presents a theoretical overview of the mathematics and underlying algorithms of correspondence analysis and hierarchical clustering. The focus then shifts to data coding, with a survey of the widely varied possibilities correspondence analysis offers and introduction of the Java software for correspondence analysis, clustering, and interpretation tools. A chapter of case studies follows, wherein the author explores applications to areas such as shape analysis and time-evolving data. The final chapter reviews the wealth of studies on textual content as well as textual form, carried out by Benzecri and his research lab. These discussions show the importance of correspondence analysis to artificial intelligence as well as to stylometry and other fields. This book not only shows why correspondence analysis is important, but with a clear presentation replete with advice and guidance, also shows how to put this technique into practice. Downloadable software and data sets allow quick, hands-on exploration of innovative correspondence analysis applications.

This text offers an overview of the basic theories and techniques of functional analysis and its applications. It contains topics such as the fixed point theory starting from Ky Fan's KKM covering and quasi-Schwartz operators. It also includes over 200 exercises to reinforce important concepts.;The author explores three fundamental results on Banach spaces, together with Grothendieck's structure theorem for compact sets in Banach spaces (including new proofs for some standard theorems) and Helley's selection theorem. Vector topologies and vector bornologies are examined in parallel, and their internal and external relationships are studied. This volume also presents recent developments on compact and weakly compact operators and operator ideals; and discusses some applications to the important class of Schwartz spaces.;This text is designed for a two-term course on functional analysis for upper-level undergraduate and graduate students in mathematics, mathematical physics, economics and engineering. It may also be used as a self-study guide by researchers in these disciplines.

For many decades, Martin Gardner, the Grand Master of mathematical puzzles, has provided the tools and projects to furnish our all-too-sluggish minds with an athletic workout. Gardner's problems foster an agility of the mind as they entertain. This volume presents a new collection of problems and puzzles not previously published in book form. Martin Gardner has dedicated it to "all the underpaid teachers of mathematics everywhere, who love their subject and are able to communicate that love to their students."

This brilliant CGP book covers all the maths skills needed in AS and A-Level Psychology (the use of maths is required for up to 10% of the marks in the final exams and assessments). It explains Calculations, Graph Skills and Statistics, with clear study notes and step-by-step examples in the context of Psychology. And to make sure you've really got to grips with it all, there are practice questions for each topic - with answers included at the back of the book.

The second edition of a bestseller, Mathematical Techniques in GIS demystifies the mathematics used in the manipulation of spatially related data. The author takes a step-by-step approach through the basics of arithmetic, algebra, geometry, trigonometry and calculus that underpin the management of such data. He then explores the use of matrices, determinants and vectors in the handling of geographic information so that the data may be analyzed and displayed in two-dimensional form either in the visualization of the terrain or as map projections. See What's New in the Second Edition: Summaries at the end of each chapter Worked examples of techniques described Additional material on matrices and vectors Further material on map projections New material on spatial correlation A new section on global positioning systems Written for those who need to make use geographic information systems but have a limited mathematical background, this book introduces the basic statistical techniques commonly used in geographic information systems and explains best-fit solutions and the mathematics behind satellite positioning. By understanding the mathematics behind the gathering, processing, and display of information, you can better advise others on the integrity of results, the quality of the information, and the safety of using it.

powerful operations on them. An early step in this direction was the development of APl, and more recent examples have been SETl which enables a user to code in terms of mathematical enti ties such as sets and BDl which allows a user, presumably a businessman, to specify a computation in terms of a series of tabular forms and a series of processing paths through which data flows. The design and implementation of such languages are examined in chapters by P. GOLDBERG. Another extension to traditional methods is made possible by systems designed to automatically handle low level flow-of control decisions. All the above higher level languages do this implicitly with their built in operators. PROLOG is a language which does this with a theorem proving mechanism employing primarily unification and backtracking. The programmer specifies the problem to be solved with a set of formal logic statements including a theorem to be proved. The theorem proving system finds a way to combine the axioms to prove the theorem, and in the process, it completes the desired calculation. H. GAllAIRE has contributed a chapter describing PROLOG giving many examples of its usage."

In 1990, the National Science Foundation recommended that every college mathematics curriculum should include a second course in linear algebra. In answer to this recommendation, Matrix Theory: From Generalized Inverses to Jordan Form provides the material for a second semester of linear algebra that probes introductory linear algebra concepts while also exploring topics not typically covered in a sophomore-level class. Tailoring the material to advanced undergraduate and beginning graduate students, the authors offer instructors flexibility in choosing topics from the book. The text first focuses on the central problem of linear algebra: solving systems of linear equations. It then discusses LU factorization, derives Sylvester's rank formula, introduces full-rank factorization, and describes generalized inverses. After discussions on norms, QR factorization, and orthogonality, the authors prove the important spectral theorem. They also highlight the primary decomposition theorem, Schur's triangularization theorem, singular value decomposition, and the Jordan canonical form theorem. The book concludes with a chapter on multilinear algebra. With this classroom-tested text students can delve into elementary linear algebra ideas at a deeper level and prepare for further study in matrix theory and abstract algebra.

Why do so many learners, even those who are successful, feel that they are outsiders in the world of mathematics? Taking the central importance of language in the development of mathematical understanding as its starting point, Mathematical Literacy explores students' experiences of doing mathematics from primary school to university - what they think mathematics is, how it is presented to them, and what they feel about it. Building on a range of theory which focuses on community, knowledge, and identity, the author examines two particular issues: the relationship between language, learning, and mathematical knowledge, and the relationship between identity, equity, and processes of exclusion/inclusion. In this comprehensive and accessible book, the author extends our understanding of the process of gaining mathematical fluency, and provides tools for an exploration of mathematics learning across different groups in different social contexts. Mathematical Literacy's analysis of how learners develop particular relationships with the subject, and what we might do to promote equity through the development of positive relationships, is of interest across all sectors of education-to researchers, teacher educators, and university educators.

A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts.