Calculus and mathematical analysis Books

Book SynopsisPraise for the Second Edition This book should be an essential part of the personal library of every practicing statistician.Technometrics Thoroughly revised and updated, the new edition of Nonparametric Statistical Methods includes additional modern topics and procedures, more practical data sets, and new problems from real-life situations. The book continues to emphasize the importance of nonparametric methods as a significant branch of modern statistics and equips readers with the conceptual and technical skills necessary to select and apply the appropriate procedures for any given situation. Written by leading statisticians, Nonparametric Statistical Methods, Third Edition provides readers with crucial nonparametric techniques in a variety of settings, emphasizing the assumptions underlying the methods. The book provides an extensive array of examples that clearly illustrate how to use nonparametric approaches for handling one- or Table of ContentsPreface xiii 1. Introduction 1 1.1. Advantages of Nonparametric Methods 1 1.2. The Distribution-Free Property 2 1.3. Some Real-World Applications 3 1.4. Format and Organization 6 1.5. Computing with R 8 1.6. Historical Background 9 2. The Dichotomous Data Problem 11 Introduction 11 2.1. A Binomial Test 11 2.2. An Estimator for the Probability of Success 22 2.3. A Confidence Interval for the Probability of Success (Wilson) 24 2.4. Bayes Estimators for the Probability of Success 33 3. The One-Sample Location Problem 39 Introduction 39 Paired Replicates Analyses by Way of Signed Ranks 39 3.1. A Distribution-Free Signed Rank Test (Wilcoxon) 40 3.2. An Estimator Associated with Wilcoxon’s Signed Rank Statistic (Hodges–Lehmann) 56 3.3. A Distribution-Free Confidence Interval Based on Wilcoxon’s Signed Rank Test (Tukey) 59 Paired Replicates Analyses by Way of Signs 63 3.4. A Distribution-Free Sign Test (Fisher) 63 3.5. An Estimator Associated with the Sign Statistic (Hodges–Lehmann) 76 3.6. A Distribution-Free Confidence Interval Based on the Sign Test (Thompson, Savur) 80 One-Sample Data 84 3.7. Procedures Based on the Signed Rank Statistic 84 3.8. Procedures Based on the Sign Statistic 90 3.9. An Asymptotically Distribution-Free Test of Symmetry (Randles–Fligner–Policello–Wolfe, Davis–Quade) 94 Bivariate Data 102 3.10. A Distribution-Free Test for Bivariate Symmetry (Hollander) 102 3.11. Efficiencies of Paired Replicates and One-Sample Location Procedures 112 4. The Two-Sample Location Problem 115 Introduction 115 4.1. A Distribution-Free Rank Sum Test (Wilcoxon, Mann and Whitney) 115 4.2. An Estimator Associated with Wilcoxon’s Rank Sum Statistic (Hodges–Lehmann) 136 4.3. A Distribution-Free Confidence Interval Based on Wilcoxon’s Rank Sum Test (Moses) 142 4.4. A Robust Rank Test for the Behrens–Fisher Problem (Fligner–Policello) 145 4.5. Efficiencies of Two-Sample Location Procedures 149 5. The Two-Sample Dispersion Problem and Other Two-Sample Problems 151 Introduction 151 5.1. A Distribution-Free Rank Test for Dispersion–Medians Equal (Ansari–Bradley) 152 5.2. An Asymptotically Distribution-Free Test for Dispersion Based on the Jackknife–Medians Not Necessarily Equal (Miller) 169 5.3. A Distribution-Free Rank Test for Either Location or Dispersion (Lepage) 181 5.4. A Distribution-Free Test for General Differences in Two Populations (Kolmogorov–Smirnov) 190 5.5. Efficiencies of Two-Sample Dispersion and Broad Alternatives Procedures 200 6. The One-Way Layout 202 Introduction 202 6.1. A Distribution-Free Test for General Alternatives (Kruskal–Wallis) 204 6.2. A Distribution-Free Test for Ordered Alternatives (Jonckheere–Terpstra) 215 6.3. Distribution-Free Tests for Umbrella Alternatives (Mack–Wolfe) 225 6.3A. A Distribution-Free Test for Umbrella Alternatives, Peak Known (Mack–Wolfe) 226 6.3B. A Distribution-Free Test for Umbrella Alternatives, Peak Unknown (Mack–Wolfe) 241 6.4. A Distribution-Free Test for Treatments Versus a Control (Fligner–Wolfe) 249 Rationale For Multiple Comparison Procedures 255 6.5. Distribution-Free Two-Sided All-Treatments Multiple Comparisons Based on Pairwise Rankings–General Configuration (Dwass, Steel, and Critchlow–Fligner) 256 6.6. Distribution-Free One-Sided All-Treatments Multiple Comparisons Based on Pairwise Rankings-Ordered Treatment Effects (Hayter–Stone) 265 6.7. Distribution-Free One-Sided Treatments-Versus-Control Multiple Comparisons Based on Joint Rankings (Nemenyi, Damico–Wolfe) 271 6.8. Contrast Estimation Based on Hodges–Lehmann Two-Sample Estimators (Spjøtvoll) 278 6.9. Simultaneous Confidence Intervals for All Simple Contrasts (Critchlow–Fligner) 282 6.10. Efficiencies of One-Way Layout Procedures 287 7. The Two-Way Layout 289 Introduction 289 7.1. A Distribution-Free Test for General Alternatives in a Randomized Complete Block Design (Friedman, Kendall-Babington Smith) 292 7.2. A Distribution-Free Test for Ordered Alternatives in a Randomized Complete Block Design (Page) 304 Rationale for Multiple Comparison Procedures 315 7.3. Distribution-Free Two-Sided All-Treatments Multiple Comparisons Based on Friedman Rank Sums–General Configuration (Wilcoxon, Nemenyi, McDonald-Thompson) 316 7.4. Distribution-Free One-Sided Treatments Versus Control Multiple Comparisons Based on Friedman Rank Sums (Nemenyi, Wilcoxon-Wilcox, Miller) 322 7.5. Contrast Estimation Based on One-Sample Median Estimators (Doksum) 328 Incomplete Block Data–Two-Way Layout with Zero or One Observation Per Treatment–Block Combination 331 7.6. A Distribution-Free Test for General Alternatives in a Randomized Balanced Incomplete Block Design (BIBD) (Durbin–Skillings–Mack) 332 7.7. Asymptotically Distribution-Free Two-Sided All-Treatments Multiple Comparisons for Balanced Incomplete Block Designs (Skillings–Mack) 341 7.8. A Distribution-Free Test for General Alternatives for Data From an Arbitrary Incomplete Block Design (Skillings–Mack) 343 Replications–Two-Way Layout with at Least One Observation for Every Treatment–Block Combination 354 7.9. A Distribution-Free Test for General Alternatives in a Randomized Block Design with an Equal Number c(>1) of Replications Per Treatment–Block Combination (Mack–Skillings) 354 7.10. Asymptotically Distribution-Free Two-Sided All-Treatments Multiple Comparisons for a Two-Way Layout with an Equal Number of Replications in Each Treatment–Block Combination (Mack–Skillings) 367 Analyses Associated with Signed Ranks 370 7.11. A Test Based on Wilcoxon Signed Ranks for General Alternatives in a Randomized Complete Block Design (Doksum) 370 7.12. A Test Based on Wilcoxon Signed Ranks for Ordered Alternatives in a Randomized Complete Block Design (Hollander) 376 7.13. Approximate Two-Sided All-Treatments Multiple Comparisons Based on Signed Ranks (Nemenyi) 379 7.14. Approximate One-Sided Treatments-Versus-Control Multiple Comparisons Based on Signed Ranks (Hollander) 382 7.15. Contrast Estimation Based on the One-Sample Hodges–Lehmann Estimators (Lehmann) 386 7.16. Efficiencies of Two-Way Layout Procedures 390 8. The Independence Problem 393 Introduction 393 8.1. A Distribution-Free Test for Independence Based on Signs (Kendall) 393 8.2. An Estimator Associated with the Kendall Statistic (Kendall) 413 8.3. An Asymptotically Distribution-Free Confidence Interval Based on the Kendall Statistic (Samara-Randles, Fligner–Rust, Noether) 415 8.4. An Asymptotically Distribution-Free Confidence Interval Based on Efron’s Bootstrap 420 8.5. A Distribution-Free Test for Independence Based on Ranks (Spearman) 427 8.6. A Distribution-Free Test for Independence Against Broad Alternatives (Hoeffding) 442 8.7. Efficiencies of Independence Procedures 450 9. Regression Problems 451 Introduction 451 One Regression Line 452 9.1. A Distribution-Free Test for the Slope of the Regression Line (Theil) 452 9.2. A Slope Estimator Associated with the Theil Statistic (Theil) 458 9.3. A Distribution-Free Confidence Interval Associated with the Theil Test (Theil) 460 9.4. An Intercept Estimator Associated with the Theil Statistic and Use of the Estimated Linear Relationship for Prediction (Hettmansperger–McKean–Sheather) 463 k(≥2) Regression Lines 466 9.5. An Asymptotically Distribution-Free Test for the Parallelism of Several Regression Lines (Sen, Adichie) 466 General Multiple Linear Regression 475 9.6. Asymptotically Distribution-Free Rank-Based Tests for General Multiple Linear Regression (Jaeckel, Hettmansperger–McKean) 475 Nonparametric Regression Analysis 490 9.7. An Introduction to Non-Rank-Based Approaches to Nonparametric Regression Analysis 490 9.8. Efficiencies of Regression Procedures 494 10. Comparing Two Success Probabilities 495 Introduction 495 10.1. Approximate Tests and Confidence Intervals for the Difference between Two Success Probabilities (Pearson) 496 10.2. An Exact Test for the Difference between Two Success Probabilities (Fisher) 511 10.3. Inference for the Odds Ratio (Fisher, Cornfield) 515 10.4. Inference for k Strata of 2 × 2 Tables (Mantel and Haenszel) 522 10.5. Efficiencies 534 11. Life Distributions and Survival Analysis 535 Introduction 535 11.1. A Test of Exponentiality Versus IFR Alternatives (Epstein) 536 11.2. A Test of Exponentiality Versus NBU Alternatives (Hollander–Proschan) 545 11.3. A Test of Exponentiality Versus DMRL Alternatives (Hollander–Proschan) 555 11.4. A Test of Exponentiality Versus a Trend Change in Mean Residual Life (Guess–Hollander–Proschan) 563 11.5. A Confidence Band for the Distribution Function (Kolmogorov) 568 11.6. An Estimator of the Distribution Function When the Data are Censored (Kaplan–Meier) 578 11.7. A Two-Sample Test for Censored Data (Mantel) 594 11.8. Efficiencies 605 12. Density Estimation 609 Introduction 609 12.1. Density Functions and Histograms 609 12.2. Kernel Density Estimation 617 12.3. Bandwidth Selection 624 12.4. Other Methods 628 13. Wavelets 629 Introduction 629 13.1. Wavelet Representation of a Function 630 13.2. Wavelet Thresholding 644 13.3. Other Uses of Wavelets in Statistics 655 14. Smoothing 656 Introduction 656 14.1. Local Averaging (Friedman) 657 14.2. Local Regression (Cleveland) 662 14.3. Kernel Smoothing 667 14.4. Other Methods of Smoothing 675 15. Ranked Set Sampling 676 Introduction 676 15.1. Rationale and Historical Development 676 15.2. Collecting a Ranked Set Sample 677 15.3. Ranked Set Sampling Estimation of a Population Mean 685 15.4. Ranked Set Sample Analogs of the Mann–Whitney–Wilcoxon Two-Sample Procedures (Bohn–Wolfe) 717 15.5. Other Important Issues for Ranked Set Sampling 737 15.6. Extensions and Related Approaches 742 16. An Introduction to Bayesian Nonparametric Statistics via the Dirichlet Process 744 Introduction 744 16.1. Ferguson’s Dirichlet Process 745 16.2. A Bayes Estimator of the Distribution Function (Ferguson) 749 16.3. Rank Order Estimation (Campbell and Hollander) 752 16.4. A Bayes Estimator of the Distribution When the Data are Right-Censored (Susarla and Van Ryzin) 755 16.5. Other Bayesian Approaches 759 Bibliography 763 R Program Index 791 Author Index 799 Subject Index 809

Read more less

Book SynopsisThis new edition continues to serve as a comprehensive guide to modern and classical methods of statistical computing. The book is comprised of four main parts spanning the field: Optimization Integration and Simulation Bootstrapping Density Estimation and Smoothing Within these sections,each chapter includes a comprehensive introduction and step-by-step implementation summaries to accompany the explanations of key methods. The new edition includes updated coverage and existing topics as well as new topics such as adaptive MCMC and bootstrapping for correlated data. The book website now includes comprehensive R code for the entire book. There are extensive exercises, real examples, and helpful insights about how to use the methods in practice.Table of ContentsPREFACE xv ACKNOWLEDGMENTS xvii 1 REVIEW 1 1.1 Mathematical Notation 1 1.2 Taylor’s Theorem and Mathematical Limit Theory 2 1.3 Statistical Notation and Probability Distributions 4 1.4 Likelihood Inference 9 1.5 Bayesian Inference 11 1.6 Statistical Limit Theory 13 1.7 Markov Chains 14 1.8 Computing 17 PART I OPTIMIZATION 2 OPTIMIZATION AND SOLVING NONLINEAR EQUATIONS 21 2.1 Univariate Problems 22 2.2 Multivariate Problems 34 Problems 54 3 COMBINATORIAL OPTIMIZATION 59 3.1 Hard Problems and NP-Completeness 59 3.2 Local Search 65 3.3 Simulated Annealing 68 3.4 Genetic Algorithms 75 3.5 Tabu Algorithms 85 Problems 92 4 EM OPTIMIZATION METHODS 97 4.1 Missing Data, Marginalization, and Notation 97 4.2 The EM Algorithm 98 4.3 EM Variants 111 Problems 121 PART II INTEGRATION AND SIMULATION 5 NUMERICAL INTEGRATION 129 5.1 Newton–Côtes Quadrature 129 5.2 Romberg Integration 139 5.3 Gaussian Quadrature 142 5.4 Frequently Encountered Problems 146 Problems 148 6 SIMULATION AND MONTE CARLO INTEGRATION 151 6.1 Introduction to the Monte Carlo Method 151 6.2 Exact Simulation 152 6.3 Approximate Simulation 163 6.4 Variance Reduction Techniques 180 Problems 195 7 MARKOV CHAIN MONTE CARLO 201 7.1 Metropolis–Hastings Algorithm 202 7.2 Gibbs Sampling 209 7.3 Implementation 218 Problems 230 8 ADVANCED TOPICS IN MCMC 237 8.1 Adaptive MCMC 237 8.2 Reversible Jump MCMC 250 8.3 Auxiliary Variable Methods 256 8.4 Other Metropolis–Hastings Algorithms 260 8.5 Perfect Sampling 264 8.6 Markov Chain Maximum Likelihood 268 8.7 Example: MCMC for Markov Random Fields 269 Problems 279 PART III BOOTSTRAPPING 9 BOOTSTRAPPING 287 9.1 The Bootstrap Principle 287 9.2 Basic Methods 288 9.3 Bootstrap Inference 292 9.4 Reducing Monte Carlo Error 302 9.5 Bootstrapping Dependent Data 303 9.6 Bootstrap Performance 315 9.7 Other Uses of the Bootstrap 316 9.8 Permutation Tests 317 Problems 319 PART IV DENSITY ESTIMATION AND SMOOTHING 10 NONPARAMETRIC DENSITY ESTIMATION 325 10.1 Measures of Performance 326 10.2 Kernel Density Estimation 327 10.3 Nonkernel Methods 341 10.4 Multivariate Methods 345 Problems 359 11 BIVARIATE SMOOTHING 363 11.1 Predictor–Response Data 363 11.2 Linear Smoothers 365 11.3 Comparison of Linear Smoothers 377 11.4 Nonlinear Smoothers 379 11.5 Confidence Bands 384 11.6 General Bivariate Data 388 Problems 389 12 MULTIVARIATE SMOOTHING 393 12.1 Predictor–Response Data 393 12.2 General Multivariate Data 413 Problems 416 DATA ACKNOWLEDGMENTS 421 REFERENCES 423 INDEX 457

Read more less

Book SynopsisIntroducing several important and useful methods for analyzing planar transmission line structures, this text discusses such topics as the theory and applications of Green's functions, the conformal mapping method, spectral domain methods, variational methods.Trade Review"...this book introduces the most commonly used techniques for analyzing microwave planar transmission live structures." (SciTech Book News, Vol. 25, No. 2, June 2001) "All important fundamental concepts and principles are covered as far as is possible with in a text of reasonable size...addresses student of electromagnetic theory...also...the engineer who is need of knowledge and practical, easy-to-apply formulas for the various line systems." (Measurement Science & Technology, Vol. 12, No. 10, October 2001) "...covers the analysis methods...from basics to advanced levels. All important fundamental concepts and principles are covered as far as is possible within a text of reasonable size." (Measurement Science & Technology, Vol. 12, No. 10, October 2001)Table of ContentsFundamentals of Electromagnetic Theory. Green's Function. Planar Transmission Lines. Conformal Mapping. Variational Methods. Spectral-Domain Method. Mode-Matching Method. Index.

Read more less

Book SynopsisContains 36 lectures solely on Fourier analysis and the FFT. Time and frequency domains, representation of waveforms in terms of complex exponentials and sinusoids, convolution, impulse response and the frequency transfer function, modulation and demodulation are among the topics covered. The text is linked to a complete FFT system on the accompanying disk where almost all of the exercises can be either carried out or verified. End-of-chapter exercises have been carefully constructed to serve as a development and consolidation of concepts discussed in the text.Table of ContentsCONTINUOUS FOURIER ANALYSIS. Background. Fourier Series for Periodic Functions. The Fourier Integral. Fourier Transforms of Some Important Functions. The Method of Successive Differentiation. Frequency-Domain Analysis. Time-Domain Analysis. The Properties. The Sampling Theorems. DISCRETE FOURIER ANALYSIS. The Discrete Fourier Transform. Inside the Fast Fourier Transform. The Discrete Fourier Transform as an Estimator. The Errors in Fast Fourier Transform Estimation. The Four Kinds of Convolution. Emulating Dirac Deltas and Differentiation on the Fast FourierTransform. THE USER'S MANUAL FOR THE ACCOMPANYING DISKS. Appendices. Answers to the Exercises. Index.

Read more less

Book SynopsisThe standard topics for a one-term undergraduate real analysis course are covered in this book. In addition, examples are given that show the ways in which real analysis is used in ordinary and partial differential equations, probability theory, numerical analysis, and number theory.Table of ContentsPreface Chapter 1 Preliminaries 1 The Real Numbers 1 Sets and Functions 6 Cardinality 15 Methods of Proof 20 Chapter 2 Sequences 27 Convergence 27 Limit Theorems 35 Two-state Markov Chains 40 Cauchy Sequences 44 Supremum and Infimum 52 The Bolzano-Weierstrass Theorem 55 The Quadratic Map 60 Projects 68 Chapter 3 The Riemann Integral 73 Continuity 73 Continuous Functions on Closed Intervals 80 The Riemann Integral 87 Numerical Methods 95 Discontinuities 103 Improper Integrals 113 Projects 119 Chapter 4 Differentiation 121 Differentiable Functions 121 The Fundamental Theorem of Calculus 129 Taylor’s Theorem 134 Newton’s Method 140 Inverse Functions 147 Functions of Two Variables 151 Projects 159 Chapter 5 Sequences of Functions 163 Pointwise and Uniform Convergence 163 Limit Theorems 169 The Supremum Norm 175 Integral Equations 183 The Calculus of Variations 188 Metric Spaces 196 The Contraction Mapping Principle 203 Normed Linear Spaces 210 Projects 219 Chapter 6 Series of Functions 223 Lim sup and Lim inf 223 Series of Real Constants 228 The Weierstrass M-test 238 Power Series 245 Complex Numbers 252 Infinite Products and Prime Numbers 260 Projects 270 Chapter 7 Differential Equations 273 Local Existence 273 Global Existence 283 The Error Estimate for Euler’s Method 289 Projects 296 Chapter 8 Complex Analysis 299 Analytic Functions 299 Integration on Paths 305 Cauchy's Theorem 312 Projects 320 Chapter 9 Fourier Series 323 The Heat Equation 323 Definitions and Examples 331 Pointwise Convergence 337 Mean-square Convergence 345 Projects 355 Chapter 10 Probability Theory 359 Discrete Random Variables 359 Coding Theory 368 Continuous Random Variables 376 The Variation Metric 386 Projects 398 Bibliography 403 Symbol Index 406 Index 409

Read more less

Book SynopsisEngineers must make decisions regarding the distribution of expensive resources in a manner that will be economically beneficial. This problem can be realistically formulated and logically analyzed with optimization theory. This book shows engineers how to use optimization theory to solve complex problems.Table of ContentsLinear Spaces. Hilbert Space. Least-Squares Estimation. Dual Spaces. Linear Operators and Adjoints. Optimization of Functionals. Global Theory of Constrained Optimization. Local Theory of Constrained Optimization. Iterative Methods of Optimization. Indexes.

Read more less

Book SynopsisThe authors are pioneering a new approach to classifying existing chemometric techniques for data analysis in one and two dimensions, using a practical applications approach to illustrating chemical examples and problems. Written in a simple, balanced, applications-based style, the book will appeal to both theorists and non-mathematicians.Trade Review"Statisticians, biochemists, engineers, and health researchers will benefit a lot from this wonderful book." (Journal of Statistical Computation and Simulation, November 2005) "...quite useful for persons who apply signal processing methods in chemistry." (Technometrics, May 2005) "…my overall impression of the text is favorable…I would recommend this book to chemists who are interested in using wavelets in their research and to faculty…" (Journal of the American Chemical Society, February 23, 2005) "I recommend this book to chemists who are interested in using wavelets in their research and to faculty who would like to teach graduate students about signal processing..." (Analytical Chemistry, February 1, 2005) "The presentation of information makes it easy for reader to find the relevant information. The text is well-written and understandable." (E-STREAMS, October 2004)Table of ContentsPreface xiii Chapter 1 Introduction 1 1.1. Modern Analytical Chemistry 1 1.1.1. Developments in Modern Chemistry 1 1.1.2. Modern Analytical Chemistry 2 1.1.3. Multidimensional Dataset 3 1.2. Chemometrics 5 1.2.1. Introduction to Chemometrics 5 1.2.2. Instrumental Response and Data Processing 8 1.2.3. White, Black, and Gray Systems 9 1.3. Chemometrics-Based Signal Processing Techniques 10 1.3.1. Common Methods for Processing Chemical Data 10 1.3.2. Wavelets in Chemistry 11 1.4. Resources Available on Chemometrics and Wavelet Transform 12 1.4.1. Books 12 1.4.2. Online Resources 14 1.4.3. Mathematics Software 15 Chapter 2 One-dimensional Signal Processing Techniques in Chemistry 23 2.1. Digital Smoothing and Filtering Methods 23 2.1.1. Moving-Window Average Smoothing Method 24 2.1.2. Savitsky-Golay Filter 25 2.1.3. Kalman Filtering 32 2.1.4. Spline Smoothing 36 2.2. Transformation Methods of Analytical Signals 39 2.2.1. Physical Meaning of the Convolution Algorithm 39 2.2.2. Multichannel Advantage in Spectroscopy and Hadamard Transformation 41 2.2.3. Fourier Transformation 44 2.2.3.1. Discrete Fourier Transformation and Spectral Multiplex Advantage 45 2.2.3.2. Fast Fourier Transformation 48 2.2.3.3. Fourier Transformation as Applied to Smooth Analytical Signals 50 2.2.3.4. Fourier Transformation as Applied to Convolution and Deconvolution 52 2.3. Numerical Differentiation 54 2.3.1. Simple Difference Method 54 2.3.2. Moving-Window Polynomial Least-Squares Fitting Method 55 2.4. Data Compression 57 2.4.1. Data Compression Based on B-Spline Curve Fitting 57 2.4.2. Data Compression Based on Fourier Transformation 64 2.4.3. Data Compression Based on Principal-Component Analysis 64 Chapter 3 Two-dimensional Signal Processing Techniques in Chemistry 69 3.1. General Features of Two-Dimensional Data 69 3.2. Some Basic Concepts for Two-Dimensional Data from Hyphenated Instrumentation 70 3.2.1. Chemical Rank and Principal-Component Analysis (PCA) 71 3.2.2. Zero-Component Regions and Estimation of Noise Level and Background 75 3.3. Double-Centering Technique for Background Correction 77 3.4. Congruence Analysis and Least-Squares Fitting 78 3.5. Differentiation Methods for Two-Dimensional Data 80 3.6 Resolution Methods for Two-Dimensional Data 81 3.6.1. Local Principal-Component Analysis and Rankmap 83 3.6.2. Self-Modeling Curve Resolution and Evolving Resolution Methods 85 3.6.2.1. Evolving Factor Analysis (EFA) 88 3.6.2.2. Window Factor Analysis (WFA) 90 3.6.2.3. Heuristic Evolving Latent Projections (HELP) 94 Chapter 4 Fundamentals of Wavelet Transform 99 4.1. Introduction to Wavelet Transform and Wavelet Packet Transform 100 4.1.1. A Simple Example: Haar Wavelet 103 4.1.2. Multiresolution Signal Decomposition 108 4.1.3. Basic Properties of Wavelet Function 112 4.2. Wavelet Function Examples 113 4.2.1. Meyer Wavelet 113 4.2.2. B-Spline (Battle--Lemarié) Wavelets 114 4.2.3. Daubechies Wavelets 116 4.2.4. Coiflet Functions 117 4.3. Fast Wavelet Algorithm and Packet Algorithm 118 4.3.1. Fast Wavelet Transform 119 4.3.2. Inverse Fast Wavelet Transform 122 4.3.3. Finite Discrete Signal Handling with Wavelet Transform 125 4.3.4. Packet Wavelet Transform 132 4.4. Biorthogonal Wavelet Transform 134 4.4.1. Multiresolution Signal Decomposition of Biorthogonal Wavelet 134 4.4.2. Biorthogonal Spline Wavelets 136 4.4.3. A Computing Example 137 4.5. Two-Dimensional Wavelet Transform 140 4.5.1. Multidimensional Wavelet Analysis 140 4.5.2. Implementation of Two-Dimensional Wavelet Transform 141 Chapter 5 Application of Wavelet Transform In Chemistry 147 5.1. Data Compression 148 5.1.1. Principle and Algorithm 149 5.1.2. Data Compression Using Wavelet Packet Transform 155 5.1.3. Best-Basis Selection and Criteria for Coefficient Selection 158 5.2. Data Denoising and Smoothing 166 5.2.1. Denoising 167 5.2.2. Smoothing 173 5.2.3. Denoising and Smoothing Using Wavelet Packet Transform 179 5.2.4. Comparison between Wavelet Transform and Conventional Methods 182 5.3. Baseline/Background Removal 183 5.3.1. Principle and Algorithm 184 5.3.2. Background Removal 185 5.3.3. Baseline Correction 191 5.3.4. Background Removal Using Continuous Wavelet Transform 191 5.3.5. Background Removal of Two-Dimensional Signals 196 5.4. Resolution Enhancement 199 5.4.1. Numerical Differentiation Using Discrete Wavelet Transform 200 5.4.2. Numerical Differentiation Using Continuous Wavelet Transform 205 5.4.3. Comparison between Wavelet Transform and other Numerical Differentiation Methods 210 5.4.4. Resolution Enhancement 212 5.4.5. Resolution Enhancement by Using Wavelet Packet Transform 220 5.4.6. Comparison between Wavelet Transform and Fast Fourier Transform for Resolution Enhancement 221 5.5. Combined Techniques 225 5.5.1. Combined Method for Regression and Calibration 225 5.5.2. Combined Method for Classification and Pattern Recognition 227 5.5.3. Combined Method of Wavelet Transform and Chemical Factor Analysis 228 5.5.4. Wavelet Neural Network 230 5.6. An Overview of the Applications in Chemistry 232 5.6.1. Flow Injection Analysis 233 5.6.2. Chromatography and Capillary Electrophoresis 234 5.6.3. Spectroscopy 238 5.6.4. Electrochemistry 244 5.6.5. Mass Spectrometry 246 5.6.6. Chemical Physics and Quantum Chemistry 248 5.6.7. Conclusion 249 Appendix Vector and Matrix Operations and Elementary MATLAB 257 A.1. Elementary Knowledge in Linear Algebra 257 A.1.1. Vectors and Matrices in Analytical Chemistry 257 A.1.2. Column and Row Vectors 259 A.1.3. Addition and Subtraction of Vectors 259 A.1.4. Vector Direction and Length 260 A.1.5. Scalar Multiplication of Vectors 261 A.1.6. Inner and Outer Products between Vectors 262 A.1.7. The Matrix and Its Operations 263 A.1.8. Matrix Addition and Subtraction 264 A.1.9. Matrix Multiplication 264 A.1.10. Zero Matrix and Identity Matrix 264 A.1.11. Transpose of a Matrix 265 A.1.12. Determinant of a Matrix 265 A.1.13. Inverse of a Matrix 266 A.1.14. Orthogonal Matrix 266 A.1.15. Trace of a Square Matrix 267 A.1.16. Rank of a Matrix 268 A.1.17. Eigenvalues and Eigenvectors of a Matrix 268 A.1.18. Singular-Value Decomposition 269 A.1.19. Generalized Inverse 270 A.1.20. Derivative of a Matrix 271 A.1.21. Derivative of a Function with Vector as Variable 271 A.2. Elementary Knowledge of MATLAB 273 A.2.1. Matrix Construction 275 A.2.2. Matrix Manipulation 275 A.2.3. Basic Mathematical Functions 276 A.2.4. Methods for Generating Vectors and Matrices 278 A.2.5. Matrix Subscript System 280 A.2.6. Matrix Decomposition 286 A.2.6.1. Singular-Value Decomposition (SVD) 286 A.2.6.2. Eigenvalues and Eigenvectors (eig) 287 A.2.7. Graphic Functions 288 Index 293

Read more less

Book SynopsisA timely book on a topic that has witnessed a surge of interest over the last decade, owing in part to several novel applications, most notably in data compression and computational molecular biology. It describes methods employed in average case analysis of algorithms, combining both analytical and probabilistic tools in a single volume.Trade Review"Surveying the major techniques of average case analysis, this graduate textbook presents both analytical methods used for well-structured algorithms and probabilistic methods used for more structurally complex algorithms." (SciTech Book News, Vol. 25, No. 3, September 2001) "...contains a comprehensive treatment on probabilistic, combinatorial, and analytical techniques and methods...treatment is clear, rigorous, self-contained, with many examples and exercises." (Zentralblatt MATH Vol. 968, 2001/18) "This well-organized book...is certainly useful...It is a valuable source for a deeper and more precise understanding of the behaviors of algorithms on sequences." (Mathematical Reviews, 2002f)Table of ContentsForeword. Preface. Acknowledgments. PROBLEMS ON WORDS. Data Structures and Algorithms on Words. Probabilistic and Analytical Models. PROBABILISTIC AND COMBINATORIAL TECHNIQUES. Inclusion-Exclusion Principle. The First and Second Moment Methods. Subadditive Ergodic Theorem and Large Deviations. Elements of Information Theory. ANALYTIC TECHNIQUES. Generating Functions. Complex Asymptotic Methods. Mellin Transform and Its Applications. Analytic Poissonization and Depoissonization. Bibliography. Index.

Read more less

Book SynopsisAn accessible, clearly organized survey of the basic topics of measure theory for students and researchers in mathematics, statistics, and physics In order to fully understand and appreciate advanced probability, analysis, and advanced mathematical statistics, a rudimentary knowledge of measure theory and like subjects must first be obtained.Trade Review"…an excellent read…I was impressed with the wealth of information and the amount of flawless detail." (Journal of the American Statistical Association, March 2006) “…contains many really good exercises…the style is clear and the notation appropriate…” (Zentralbaltt MATH, May 2005)Table of ContentsPreface. Acknowledgments. 1. Set Systems. 2. Measures. 3. Extensions of Measures. 4. Lebesgue Measure. 5. Measurable Functions. 6. The Lebesgue Integral. 7. Integrals Relative to Lebesgue Measure. 8. The Lp Spaces. 9. The Radon–Nikodym Theorem. 10. Products of Two Measure Spaces. 11. Arbitrary Products of Measure Spaces. References. Index.

Read more less

Book SynopsisDiscusses techniques that have important applications to wireless engineering.Table of ContentsPreface xv 1 Notations and Mathematical Preliminaries 1 1.1 Notations and Abbreviations 1 1.2 Mathematical Preliminaries 2 1.2.1 Functions and Integration 2 1.2.2 The Fourier Transform 4 1.2.3 Regularity 4 1.2.4 Linear Spaces 7 1.2.5 Functional Spaces 8 1.2.6 Sobolev Spaces 10 1.2.7 Bases in Hilbert Space H 11 1.2.8 Linear Operators 12 Bibliography 14 2 Intuitive Introduction to Wavelets 15 2.1 Technical History and Background 15 2.1.1 Historical Development 15 2.1.2 When Do Wavelets Work? 16 2.1.3 A Wave Is a Wave but What Is a Wavelet? 17 2.2 What Can Wavelets Do in Electromagnetics and Device Modeling? 18 2.2.1 Potential Benefits of Using Wavelets 18 2.2.2 Limitations and Future Direction of Wavelets 19 2.3 The Haar Wavelets and Multiresolution Analysis 20 2.4 How Do Wavelets Work? 23 Bibliography 28 3 Basic Orthogonal Wavelet Theory 30 3.1 Multiresolution Analysis 30 3.2 Construction of Scalets 3.2.1 Franklin Scalet 32 3.2.2 Battle-Lemarie Scalets 39 3.2.3 Preliminary Properties of Scalets 40 3.3 Wavelet ^ ( r ) 42 3.4 Franklin Wavelet 48 3.5 Properties of Scalets (p(co) 51 3.6 Daubechies Wavelets 56 3.7 Coifman Wavelets (Coiflets) 64 3.8 Constructing Wavelets by Recursion and Iteration 69 3.8.1 Construction of Scalets 69 3.8.2 Construction of Wavelets 74 3.9 Meyer Wavelets 75 3.9.1 Basic Properties of Meyer Wavelets 75 3.9.2 Meyer Wavelet Family 83 3.9.3 Other Examples of Meyer Wavelets 92 3.10 Mallat's Decomposition and Reconstruction 92 3.10.1 Reconstruction 92 3.10.2 Decomposition 93 3.11 Problems 95 3.11.1 Exercise 1 95 3.11.2 Exercise 2 95 3.11.3 Exercise 3 97 3.11.4 Exercise 4 97 Bibliography 98 4 Wavelets in Boundary Integral Equations 100 4.1 Wavelets in Electromagnetics 100 4.2 Linear Operators 102 4.3 Method of Moments (MoM) 103 4.4 Functional Expansion of a Given Function 107 4.5 Operator Expansion: Nonstandard Form 110 4.5.1 Operator Expansion in Haar Wavelets 111 4.5.2 Operator Expansion in General Wavelet Systems 113 4.5.3 Numerical Example 114 4.6 Periodic Wavelets 120 4.6.1 Construction of Periodic Wavelets 120 4.6.2 Properties of Periodic Wavelets 123 4.6.3 Expansion of a Function in Periodic Wavelets 127 4.7 Application of Periodic Wavelets: 2D Scattering 128 4.8 Fast Wavelet Transform (FWT) 133 4.8.1 Discretization of Operation Equations 133 4.8.2 Fast Algorithm 134 4.8.3 Matrix Sparsification Using FWT 135 4.9 Applications of the FWT 140 4.9.1 Formulation 140 4.9.2 Circuit Parameters 141 4.9.3 Integral Equations and Wavelet Expansion 143 4.9.4 Numerical Results 144 4.10 Intervallic Coifman Wavelets 144 4.10.1 Intervallic Scalets 145 4.10.2 Intervallic Wavelets on [0, 1] 154 4.11 Lifting Scheme and Lazy Wavelets 156 4.11.1 Lazy Wavelets 156 4.11.2 Lifting Scheme Algorithm 157 4.11.3 Cascade Algorithm 159 4.12 Green's Scalets and Sampling Series 159 4.12.1 Ordinary Differential Equations (ODEs) 160 4.12.2 Partial Differential Equations (PDEs) 166 4.13 Appendix: Derivation of Intervallic Wavelets on [0, 1] 172 4.14 Problems 185 4.14.1 Exercise 5 185 4.14.2 Exercise 6 185 4.14.3 Exercise 7 185 4.14.4 Exercise 8 186 4.14.5 Project 1 187 Bibliography 187 5 Sampling Biorthogonal Time Domain Method (SBTD) 189 5.1 Basis FDTD Formulation 189 5.2 Stability Analysis for the FDTD 194 5.3 FDTD as Maxwell's Equations with Haar Expansion 198 5.4 FDTD with Battle-Lemarie Wavelets 201 5.5 Positive Sampling and Biorthogonal Testing Functions 205 5.6 Sampling Biorthogonal Time Domain Method 215 5.6.1 SBTD versus MRTD 215 5.6.2 Formulation 215 5.7 Stability Conditions for Wavelet-Based Methods 219 5.7.1 Dispersion Relation and Stability Analysis 219 5.7.2 Stability Analysis for the SBTD 222 5.8 Convergence Analysis and Numerical Dispersion 223 5.8.1 Numerical Dispersion 223 5.8.2 Convergence Analysis 225 5.9 Numerical Examples 228 5.10 Appendix: Operator Form of the MRTD 233 5.11 Problems 236 5.11.1 Exercise 9 236 5.11.2 Exercise 10 237 5.11.3 Project 2 237 Bibliography 238 6 Canonical Multiwavelets 240 6.1 Vector-Matrix Dilation Equation 240 6.2 Time Domain Approach 242 6.3 Construction of Multiscalets 245 6.4 Orthogonal Multiwavelets yjr(t) 255 6.5 Intervallic Multiwavelets xj/(t) 258 6.6 Multiwavelet Expansion 261 6.7 Intervallic Dual Multiwavelets \j/(t) 264 6.8 Working Examples 269 6.9 Multiscalet-Based ID Finite Element Method (FEM) 276 6.10 Multiscalet-Based Edge Element Method 280 6.11 Spurious Modes 285 6.12 Appendix 287 6.13 Problems 296 6.13.1 Exercise 11 296 Bibliography 297 7 Wavelets in Scattering and Radiation 299 7.1 Scattering from a 2D Groove 299 7.1.1 Method of Moments (MoM) Formulation 300 7.1.2 Coiflet-Based MoM 304 7.1.3 Bi-CGSTAB Algorithm 305 7.1.4 Numerical Results 305 7.2 2D and 3D Scattering Using Intervallic Coiflets 309 7.2.1 Intervallic Scalets on [0,1] 309 7.2.2 Expansion in Coifman Intervallic Wavelets 312 7.2.3 Numerical Integration and Error Estimate 313 7.2.4 Fast Construction of Impedance Matrix 317 7.2.5 Conducting Cylinders, TM Case 319 7.2.6 Conducting Cylinders with Thin Magnetic Coating 322 7.2.7 Perfect Electrically Conducting (PEC) Spheroids 324 7.3 Scattering and Radiation of Curved Thin Wires 329 7.3.1 Integral Equation for Curved Thin-Wire Scatterers and Antennae 330 7.3.2 Numerical Examples 331 7.4 Smooth Local Cosine (SLC) Method 340 7.4.1 Construction of Smooth Local Cosine Basis 341 7.4.2 Formulation of 2D Scattering Problems 344 7.4.3 SLC-Based Galerkin Procedure and Numerical Results 347 7.4.4 Application of the SLC to Thin-Wire Scatterers and Antennas 355 7.5 Microstrip Antenna Arrays 357 7.5.1 Impedance Matched Source 358 7.5.2 Far-Zone Fields and Antenna Patterns 360 Bibliography 363 8 Wavelets in Rough Surface Scattering 366 8.1 Scattering of EM Waves from Randomly Rough Surfaces 366 8.2 Generation of Random Surfaces 368 8.2.1 Autocorrelation Method 370 8.2.2 Spectral Domain Method 373 8.3 2D Rough Surface Scattering 376 8.3.1 Moment Method Formulation of 2D Scattering 376 8.3.2 Wavelet-Based Galerkin Method for 2D Scattering 380 8.3.3 Numerical Results of 2D Scattering 381 8.4 3D Rough Surface Scattering 387 8.4.1 Tapered Wave of Incidence 388 8.4.2 Formulation of 3D Rough Surface Scattering Using Wavelets 391 8.4.3 Numerical Results of 3D Scattering 394 Bibliography 399 9 Wavelets in Packaging, Interconnects, and EMC 401 9.1 Quasi-static Spatial Formulation 402 9.1.1 What Is Quasi-static? 402 9.1.2 Formulation 403 9.1.3 Orthogonal Wavelets in L2([0, 1]) 406 9.1.4 Boundary Element Method and Wavelet Expansion 408 9.1.5 Numerical Examples 412 9.2 Spatial Domain Layered Green's Functions 415 9.2.1 Formulation 417 9.2.2 Prony's Method 423 9.2.3 Implementation of the Coifman Wavelets 424 9.2.4 Numerical Examples 426 9.3 Skin-Effect Resistance and Total Inductance 429 9.3.1 Formulation 431 9.3.2 Moment Method Solution of Coupled Integral Equations 433 9.3.3 Circuit Parameter Extraction 435 9.3.4 Wavelet Implementation 437 9.3.5 Measurement and Simulation Results 438 9.4 Spectral Domain Green's Function-Based Full-Wave Analysis 440 9.4.1 Basic Formulation 440 9.4.2 Wavelet Expansion and Matrix Equation 444 9.4.3 Evaluation of Sommerfeld-Type Integrals 447 9.4.4 Numerical Results and Sparsity of Impedance Matrix 451 9.4.5 Further Improvements 455 9.5 Full-Wave Edge Element Method for 3D Lossy Structures 455 9.5.1 Formulation of Asymmetric Functionals with Truncation Conditions 456 9.5.2 Edge Element Procedure 460 9.5.3 Excess Capacitance and Inductance 464 9.5.4 Numerical Examples 466 Bibliography 469 10 Wavelets in Nonlinear Semiconductor Devices 474 10.1 Physical Models and Computational Efforts 474 10.2 An Interpolating Subdivision Scheme 476 10.3 The Sparse Point Representation (SPR) 478 10.4 Interpolation Wavelets in the FDM 479 10.4.1 ID Example of the SPR Application 480 10.4.2 2D Example of the SPR Application 481 10.5 The Drift-Diffusion Model 484 10.5.1 Scaling 486 10.5.2 Discretization 487 10.5.3 Transient Solution 489 10.5.4 Grid Adaptation and Interpolating Wavelets 490 10.5.5 Numerical Results 492 10.6 Multiwavelet Based Drift-Diffusion Model 498 10.6.1 Precision and Stability versus Reynolds 499 10.6.2 MWFEM-Based ID Simulation 502 10.7 The Boltzmann Transport Equation (BTE) Model 504 10.7.1 Why BTE? 505 10.7.2 Spherical Harmonic Expansion of the BTE 505 10.7.3 Arbitrary Order Expansion and Galerkin's Procedure 509 10.7.4 The Coupled Boltzmann-Poisson System 515 10.7.5 Numerical Results 517 Bibliography 524 Index 527

Read more less

Book SynopsisFactor analysis, a mathematical technique for studying matrices of data, has long been used in the behavioural sciences. This new edition of a work on its application to chemical problems has been thoroughly revised and includes an added chapter on special methods of factor analysis.Table of ContentsMain Steps; Mathematical Formulation of Target Factor Analysis; Effects of Experimental Error on Target Factor Analysis; Numerical Examples of Target Factor Analysis; Special Methods of Factor Analysis; Component Analysis; Nuclear Magnetic Resonance; Chromatography; Additional Applications; Appendices; Bibliography; Author Index; Subject Index.

Read more less

Book SynopsisA comprehensive solutions manual for students using the Vector Calculus text This book gives a comprehensive and thorough introduction to ideas and major results of the theory of functions of several variables and of modern vector calculus in two and three dimensions. Clear and easy-to-follow writing style, carefully crafted examples, wide spectrum of applications and numerous illustrations, diagrams, and graphs invite students to use the textbook actively, helping them to both enforce their understanding of the material and to brush up on necessary technical and computational skills. The Student Solutions Manual to Accompany Vector Calculus also pays particular attention to material that some students find challenging, such as the chain rule, Implicit Function Theorem, parametrizations, or the Change of Variables Theorem.

Read more less

Book SynopsisThe purpose of this book is to bridge the gap between differential geometry of Euclidean space of three dimensions and the more advanced work on differential geometry of generalised space.Table of Contents1. Some Preliminaries; 2. Coordinates, Vectors , Tensors; 3. Riemannian Metric; 4. Christoffel's Three-Index Symbols. Covariant Differentiation; 5. Curvature of a Curve. Geodeics, Parallelism of Vectors; 6. Congruences and Orthogonal Ennuples; 7. Riemann Symbols. Curvature of a Riemannian Space; 8. Hypersurfaces; 9. Hypersurfaces in Euclidean Space. Spaces of Constant Curvature; 10. Subspaces of a Riemannian Space.

Read more less

Book SynopsisIn this book, Professor Copson gives a rigorous account of the theory of partial differential equations of the first orderTable of ContentsPreface; 1. Partial differential equations of the first order; 2. Characteristics of equations of the second order; 3. Boundary value and initial value problems; 4. Equations of hyperbolic type; 5. Reimann's method; 6. The equation of wave motions; 7. Marcel Riesz's method; 8. Potential theory in the plane; 9. Subharmonic functions and the problem of Dirichlet; 10. Equations of elliptic type in the plane; 11. Equations of elliptic type in space; 12. The equation of heat; Appendix; Books for further reading; Index.

Read more less

Book SynopsisThe theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph in which each site is connected to k close neighbors, but also has long-range connections. At a similar time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers have led to an explosion of research. The purpose of this book is to use a wide variety of mathematical argument to obtain insights into the properties of these graphs. A unique feature is the interest in the dynamics of process taking place on the graph in addition to Trade Review'A very valuable addition to the growing field of random graphs, providing a systematic coverage of these novel models.' Michael Krivelevich, Mathematical Reviews'The book is written in a friendly, chatty style, making it easy to read; I very much like that. In summary, Random Graph Dynamics is a nice contribution to the area of random graphs and a source of valuable insights.' Malwina J. Luczak, Journal of the American Statistical AssociationTable of Contents1. Overview; 2. Erdos–Renyi random graphs; 3. Fixed degree distributions; 4. Power laws; 5. Small worlds; 6. Random walks; 7. CHKNS model.

Read more less

Book SynopsisThe theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approachTrade Review' … this is a very valuable new reference for the subject, incorporating many modern results and perspectives that are not present in earlier texts on this topic … this book would serve as an excellent foundation with which to begin studying other aspects of random matrix theory.' Terence Tao, Mathematical Reviews'… the book aims to introduce some of the modern techniques of random matrix theory in a comprehensive and rigorous way. It has a broad range of topics and most of them are fairly accessible. The focus is on introducing and explaining the main techniques, rather than obtaining the most general results. Additional references are given for the reader who wants to continue the study of a certain topic. The writing style is careful and the book is mostly self-contained with complete proofs. This is an excellent new contribution to random matrix theory.' Journal of Approximation TheoryTable of ContentsPreface; 1. Introduction; 2. Real and complex Wigner matrices; 3. Hermite polynomials, spacings, and limit distributions for the Gaussian ensembles; 4. Some generalities; 5. Free probability; Appendices; Bibliography; General conventions; Glossary; Index.

Read more less

Book SynopsisFor the second edition of this very successful text, Professor Binmore has written two chapters on analysis in vector spaces. The discussion extends to the notion of the derivative of a vector function as a matrix and the use of second derivatives in classifying stationary points.Table of ContentsPreface to the first edition; Preface to the second edition; 1. Real numbers; 2. Continuum property; 3. Natural numbers; 4. Convergent sequences; 5. Subsequences; 6. Series; 7. Functions; 8. Limits of functions; 9. Continuity; 10. Differentiation; 11. Mean value theorems; 12. Monotone functions; 13. Integration; 14. Exponential and logarithm; 15. Power series; 16. Trigonometric functions; 17. The gamma function; 18. Vectors; 19. Vector derivatives; 20. Appendix; Solutions to exercises; Further problems; Suggested further reading; Notation; Index.

Read more less

Book SynopsisThis textbook is an introduction to the theory of Hilbert space and its applications. The notion of Hilbert space is central in functional analysis and is used in numerous branches of pure and applied mathematics. Some basic familiarity with real analysis, linear algebra and metric spaces is assumed, but otherwise the book is self-contained.Trade Review' … the author's style is a delight. Each topic is carefully motivated and succinctly presented, and the exposition is enthusiastic and limpid … Young has done a really fine job in presenting a subject of great mathematical elegance as well as genuine utility, and I recommend it heartily.' The Times Higher Education SupplementTable of ContentsForeword; Introduction; 1. Inner product spaces; 2. Normed spaces; 3. Hilbert and Banach spaces; 4. Orthogonal expansions; 5. Classical Fourier series; 6. Dual spaces; 7. Linear operators; 8. Compact operators; 9. Sturm-Liouville systems; 10. Green's functions; 11. Eigenfunction expansions; 12. Positive operators and contractions; 13. Hardy spaces; 14. Interlude: complex analysis and operators in engineering; 15. Approximation by analytic functions; 16. Approximation by meromorphic functions; Appendix; References; Answers to selected problems; Afterword; Index of notation; Subject index.

Read more less

Book SynopsisThis classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and exhaustively both the statement and proof of all the standard inequalities of analysis. The authors were well known for their powers of exposition and were able here to make the subject accessible to a wide audience of mathematicians.Trade Review'In retrospect one sees that 'Hardy, Littlewood and Pólya' has been one of the most important books in analysis in the last few decades. It had an impact on the trend of research and is still influencing it. In looking through the book now one realises how little one would like to change the existing text.' A. Zygmund, Bulletin of the AMSTable of Contents1. Introduction; 2. Elementary mean values; 3. Mean values with an arbitrary function and the theory of convex functions; 4. Various applications of the calculus; 5. Infinite series; 6. Integrals; 7. Some applications of the calculus of variations; 8. Some theorems concerning bilinear and multilinear forms; 9. Hilbert's inequality and its analogues and extensions; 10. Rearrangements; Appendices; Bibliography.

Read more less

Book SynopsisFirst published in 1941, this book, by one of the foremost geometers of his day, rapidly became a classic. In its original form the book constituted a section of Hodge's essay for which the Adam's prize of 1936 was awarded, but the author substantially revised and rewrote it.Table of Contents1. Reimannian Manifolds; 2. Integrals and their periods; 3. Harmonic Integrals; 4. Applications to algebraic varieties; 5. Applications to the theory of continuous groups.

Read more less

Book SynopsisA textbook presenting the theory and underlying techniques of perturbation methods in a manner suitable for senior undergraduates from a broad range of disciplines.Trade Review'A nice and readable introduction.' Monatshefte für MathematikTable of ContentsPreface; 1. Algebraic equations; 2. Asymptotic expansions; 3. Integrals; 4. Regular problems in PDEs; 5. Matched asymptotic expansions; 6. Method of strained coordinates; 7. Method of multiple scales; 8. Improved convergence; Bibliography; Index.

Read more less

Book SynopsisMathematics education in schools has seen a revolution in recent years. Students everywhere expect the subject to be well-motivated, relevant and practical. When such students reach higher education the traditional development of analysis, often rather divorced from the calculus which they learnt at school, seems highly inappropriate. Shouldn't every step in a first course in analysis arise naturally from the student's experience of functions and calculus at school? And shouldn't such a course take every opportunity to endorse and extend the student's basic knowledge of functions? In Yet Another Introduction to Analysis the author steers a simple and well-motivated path through the central ideas of real analysis. Each concept is introduced only after its need has become clear and after it has already been used informally. Wherever appropriate the new ideas are related to school topics and are used to extend the reader's understanding of those topics. A first course in analysis at colleTrade Review"Bryant's style is extremely leisurely, copiously illustrated, often intuitively appealing, chatty and unintimidating, in contrast to other treatments of similar material..." ChoiceTable of ContentsPreface; 1. Firm foundations; 2. Gradually getting there; 3. A functional approach; 4. Calculus at last!; 5. An integrated conclusion; Solutions to exercises; Index.

Read more less

Book SynopsisThis text on analysis of Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. It is aimed at students who have had a first course in differentiable manifolds, and the Riemannian geometry used is developed from the beginning.Trade Review"The book is well written.... This book provides a very readable introduction to heat kernal methods and it can be strongly recommended for graduate students of mathematics looking for a thorough introduction to the topic." Friedbert PrÜfer, Mathematical ReviewsTable of ContentsIntroduction; 1. The Laplacian on a Riemannian manifold; 2. Elements of differential geometry; 3. The construction of the heat kernel; 4. The heat equation approach to the Atiyah-Singer index theorem; 5. Zeta functions of Laplacians; Bibliography; Index.

Read more less

Book SynopsisThis 1994 book introduces the tools of modern differential geometry, exterior calculus, manifolds, vector bundles and connections, to advanced undergraduate and beginning graduate students in mathematics, physics and engineering. There are nearly 200 exercises, making the book ideal for both classroom use and self-study.Trade Review"...Darling's exegesis is clear and easy to understand, and his frequent use of examples is beneficial to the reader. There are many exercises that serve to reinforce the concepts." D.P. Turner, Choice"...easy on the eyes; some nice exercises..." American Mathematical Monthly"The exposition is clear and, in the American textbook style, has many exercises, both theoretical and computational. In summary, this text provides a worthwhile elementary introduction to anyone who wants to understand the basic mathematical ingredients of Differential Geometry and its interactions with Physics." F.E. Burstall, Contemporary Physics"...a good introduction to differential geometry and its applications to physics by using the calculus of differential forms...Nearly 200 exercises and many examples will help the reader's understanding...this book can be recommended as a good textbook for advanced undergraduate and beginning graduate students in mathematics, physics, and engineering." Akira Asada, Mathematical ReviewsTable of ContentsPreface; 1. Exterior algebra; 2. Exterior calculus on Euclidean space; 3. Submanifolds of Euclidean spaces; 4. Surface theory using moving frames; 5. Differential manifolds; 6. Vector bundles; 7. Frame fields, forms and metrics; 8. Integration on oriented manifolds; 9. Connections on vector bundles; 10. Applications to gauge field theory; Bibliography; Index.

Read more less

Book SynopsisThe late Professor G. N. Watson wrote his monumental 1995 treatise on the theory of Bessel functions with two objects in view. The first was the development of applications of the fundamental processes of the theory of complex variables; and the second was the compilation of a collection of results of value to mathematicians and physicists, who encounter Bessel functions in the course of their researches. The completeness of his theoretical account, combined with the wide scope of the practical examples and the extensive numerical tables, have resulted in a book which is indispensable to pure and applied mathematicians, as well as to physicists.Trade Review'In Professor Watson's treatise, which is a monument of erudition and … clear exposition, we have a rigorous mathematical treatment of all types of Bessel functions.' L. M. Milne-Thomson, Nature'A veritable mine of information … indispensable to all those who have occasion to use Bessel functions.' S. Chandrasekhar, The Astrophysical JournalTable of Contents1. Bessel functions before 1826; 2. The Bessel coefficients; 3. Bessel functions; 4. Differential equations; 5. Miscellaneous properties of Bessel functions; 6. Integral representations of Bessel functions; 7. Asymptotic expansions of Bessel functions; 8. Bessel functions of large order; 9. Polynomials associated with Bessel functions; 10. Functions associated with Bessel functions; 11. Addition theorems; 12. Definite integrals; 13. Infinitive integrals; 14. Multiple integrals; 15. The zeros of Bessel functions; 16. Neumann series and Lommel's functions of two variables; 17. Kapteyn series; 18. Series of Fourier–Bessel and Dini; 19. Schlömlich series; 20. The tabulation of Bessel functions; Tables of Bessel functions; Bibliography; Indices.

Read more less

Book SynopsisGregory's Classical Mechanics is a thorough, self-contained and highly readable account of a subject many students find difficult. The author's clear and systematic style promotes a good understanding of the subject: each concept is motivated and illustrated by worked examples, while problem sets provide ample practice for understanding and technique.Trade Review'The writing here is a picture of clarity and directness … The exercises include plenty of interesting and challenging problems … an attractive and well-written exposition of classical mechanics. I wish it had been my textbook when I was a student.' Mathematical Association of AmericaTable of ContentsPart I. Newtonian Mechanics of a Single Particle: 1. The algebra and calculus of vectors; 2. Velocity, acceleration and scalar angular velocity; 3. Newton's laws of motion and the law of gravitation; 4. Problems in particle dynamics; 5. Linear oscillations; 6. Energy conservation; 7. Orbits in a central field; 8. Non-linear oscillations and phase space; Part II. Multi-particle Systems: 9. The energy principle; 10. The linear momentum principle; 11. The angular momentum principle; Part III. Analytical mechanics: 12. Lagrange's equations and conservation principle; 13. The calculus of variations and Hamilton's principle; 14. Hamilton's equations and phase space; Part IV. Further Topics: 15. The general theory of small oscillations; 16. Vector angular velocity and rigid body kinematics; 17. Rotating reference frames; 18. Tensor algebra and the inertia tensor; 19. Problems in rigid body dynamics; Appendix: centres of mass and moments of inertia; Answers to the problems; Bibliography; Index.

Read more less

Book SynopsisNow available in a fully revised and updated second edition, this well established textbook provides a straightforward introduction to the theory of probability. The presentation is entertaining without any sacrifice of rigour; important notions are covered with the clarity that the subject demands. Topics covered include conditional probability, independence, discrete and continuous random variables, basic combinatorics, generating functions and limit theorems, and an introduction to Markov chains. The text is accessible to undergraduate students and provides numerous worked examples and exercises to help build the important skills necessary for problem solving.Trade Review'… this book is a superb resource of theory and application, which should be on every lecturer's shelves, and those of many students. You may never need to buy another book on probability.' Keith Hirst, The Mathematical Gazette'Excellent! A vast number of well-chosen worked examples and exercises guide the reader through the basic theory of probability at the elementary level … an excellent text which I am sure will give a lot of pleasure to students and teachers alike.' International Statistics Institute'… would make a fine addition to an undergraduate library. A student with a solid background in calculus, linear algebra, and set theory will find many useful tools of elementary probability here.' Phil Gilbert, The Mathematics Teacher'Stirzaker does an excellent job of developing problem-solving skills in an introductory probability text. Numerous examples and practice exercises are provided that only serve to enhance a student's problem solving abilities … Highly recommended.' D. J. Gougeon, Choice'The book would make an excellent text for the properly prepared class, a solid instructor's reference for both probability applications and problems, as well as a fine work for purposes of self-study.' J. Philip Smith, School Science and Mathematics'This book is a valuable resource for anyone teaching probability-and, accordingly, I am glad to have it.' The Journal of the Royal Statistical Society'The strong feature of the textbook is a choice of good examples … the book is suitable for a first university course in probability and very useful for self-study.' EMS Newsletter'As the first edition, the book is very well written, in a clear, detailed and readable style. It is accessible for undergraduate students and would make a good textbook for first courses on probability or for self-study.' Zentralblatt MATHTable of Contents1. Probability; 2. Conditional probability and independence; 3. Counting; 4. Random variables: distribution and expectation; 5. Random vectors: independence and dependence; 6. Generating functions and their applications; 7. Continuous random variables; 8. Jointly continuous random variables; 9. Markov chains; Appendix.

Read more less

Book SynopsisUsing the Cauchy-Schwarz inequality as a guide, this 2004 book presents a fascinating collection of problems related to inequalities and coaches readers through solutions. Undergraduate and beginning graduate students in mathematics, theoretical computer science, statistics, engineering, and economics will find the book perfect for self-study or as a supplement to probability and analysis courses.Trade Review'This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics.' Zentralblatt MATH'… pleaseant reading for everyone with a solid real analysis background at undergraduate level, even before reading Pólya-Szegö. In fact, even researchers working on topics close to those in this book can find much to add to their repertoire.' Tamás Erdélyi, Department of Mathematics, Texas A&M University'The book is special … A large mathematics department with a functional graduate program could easily consider to offer a course based on this book.' Tamas Erdelyi, Journal of Approximation TheoryTable of Contents1. Starting with Cauchy; 2. The AM-GM inequality; 3. Lagrange's identity and Minkowski's conjecture; 4. On geometry and sums of squares; 5. Consequences of order; 6. Convexity - the third pillar; 7. Integral intermezzo; 8. The ladder of power means; 9. Hölder's inequality; 10. Hilbert's inequality and compensating difficulties; 11. Hardy's inequality and the flop; 12. Symmetric sums; 13. Majorization and Schur convexity; 14. Cancellation and aggregation; Solutions to the exercises; Notes; References.

Read more less

Book SynopsisThis book is a text on mathematical analysis suitable for graduate students and advanced undergraduates. It provides an extensive introduction to proof and to rigorous mathematical thinking. It contains many remarks and examples and 500 exercises designed to provide motivation, test understanding, help practice mathematical writing and explore additional topics.Trade Review'The self-contained text, suitable for advanced undergraduates, provides an extensive introduction into mathematical analysis, from the fundamentals to more advanced material.' Zentralblatt fur Didaktik der MathematikTable of Contents1. Introduction; 2. The real and complex numbers; 3. Real and complex sequences; 4. Series; 5. Power series; 6. Metric spaces; 7. Continuous functions; 8. Calculus; 9. Some special functions; 10. Lebesgue measure on the line; 11. Lebesgue integration on the line; 12. Function spaces; 13. Fourier series; 14. Applications of Fourier series; 15. Ordinary differential equations; Appendix: the Banach-Tarski paradox; Hints for some exercises.

Read more less

Book SynopsisThe subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This 1997 book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.Trade Review'The book is a welcome extension of the existing literature about this important topic … It is recommended to students of mathematics and physics interested in the applications of the theory and the theory itself.' European Mathematical Society'With an easy mind the reviewer can recommend this book to those who want to become acquainted with the subject and to those who look for a book which can serve as guide for a course on the subject … the exemplary way in which Elliptic Curves is written, made reviewing a pleasure.' Niew Archief voor WiskundeTable of Contents1. First ideas: complex manifolds, Riemann surfaces, and projective curves; 2. Elliptic functions and elliptic integrals; 3. Theta functions; 4. Modular groups and molecular functions; 5. Ikosaeder and the quintic; 6. Imaginary quadratic fields; 7. The arithmetic of elliptic fields.

Read more less

Book SynopsisMathematical Analysis (often called Advanced Calculus) is generally found by students to be one of their hardest courses in Mathematics. This text uses the so-called sequential approach to continuity, differentiability and integration to make it easier to understand the subject.Topics that are generally glossed over in the standard Calculus courses are given careful study here. For example, what exactly is a 'continuous' function? And how exactly can one give a careful definition of 'integral'? The latter question is often one of the mysterious points in a Calculus course - and it is quite difficult to give a rigorous treatment of integration! The text has a large number of diagrams and helpful margin notes; and uses many graded examples and exercises, often with complete solutions, to guide students through the tricky points. It is suitable for self-study or use in parallel with a standard university course on the subject.Table of ContentsPreface; Introduction: calculus and analysis; 1. Numbers; 2. Sequences; 3. Series; 4. Continuity; 5. Limits and continuity; 6. Differentiation; 7. Integration; 8. Power series; Appendix 1. Sets, functions and proofs; Appendix 2. Standard derivatives and primitives; Appendix 3. The first 1,000 decimal places of the square root of 2, e and pi; Appendix 4. Solutions to the problems; Index.

Read more less

Book SynopsisA self-contained and elementary presentation of Lie group theory, concentrating on analysis on Lie groups. The author describes in detail many interesting examples with topics ranging from Haar measure to harmonic functions. With numerous exercises and worked examples, it's ideal for a graduate course on analysis on Lie groups.Trade Review"The main themes are carefully explained and illustrated by well-chosen examples. He succeeds in putting a remarkable wealth of material into a 300-page book which will certainly serve as a basis for many courses on the subject." Joachim Hilgert, Mathematical ReviewsTable of ContentsPreface; 1. The linear group; 2. The exponential map; 3. Linear Lie groups; 4. Lie algebras; 5. Haar measure; 6. Representations of compact groups; 7. The groups SU(2) and SO(3), Haar measure; 8. Analysis on the group SU(2); 9. Analysis on the sphere; 10. Analysis on the spaces of symmetric and Hermitian matrices; 11. Irreducible representations of the unitary group; 12. Analysis on the unitary group; Bibliography; Index.

Read more less

Book SynopsisIntegration has a long history: its roots can be traced as far back as the ancient Greeks. The first genuinely rigorous definition of an integral was that given by Riemann, and further (more general, and so more useful) definitions have since been given by Lebesgue, Denjoy, Perron, Kurzweil and Henstock, and this culminated in the work of McShane. This textbook provides an introduction to this theory, and it presents a unified yet elementary approach that is suitable for beginning graduate and final year undergraduate students.Trade Review'… already it is worthy of a place in our standard curriculum … The book of Lee and Vyborny serves well as an introduction and reference for anyone interested in this topic.' J. Alan Alewine and Eric Schechter, American Mathematical Monthly'… the authors do an excellent job of presenting their material. The book is written with clarity and enthusiasm.' Brian Jefferies'This is a valuable addition to the literature …'. Jean Mawhin, Bulletin of the Belgian Mathematical SocietyTable of ContentsPreface; 1. Introduction; 2. Basic theory; 3. Theory development; 4. The SL-integral; 5. Generalized AC function; 6. Integration in several dimensions; 7. Some applications; 8. List of symbols; Appendices.

Read more less

Book SynopsisSpecial functions, natural generalizations of the elementary functions, have been studied for centuries. The greatest mathematicians, among them Euler, Gauss, Legendre, Eisenstein, Riemann, and Ramanujan, have laid the foundations for this beautiful and useful area of mathematics. This treatise presents an overview of special functions, focusing primarily on hypergeometric functions and the associated hypergeometric series, including Bessel functions and classical orthogonal polynomials, using the basic building block of the gamma function. In addition to relatively new work on gamma and beta functions, such as Selberg's multidimensional integrals, many important but relatively unknown nineteenth century results are included. Other topics include q-extensions of beta integrals and of hypergeometric series, Bailey chains, spherical harmonics, and applications to combinatorial problems. The authors provide organizing ideas, motivation, and historical background for the study and applicaTrade Review'Occasionally there is published a mathematics book that one is compelled to describe as, well, let us say, special. Special Functions is certainly one of those rare books. … this treatise … should become a classic. Every student, user, and researcher in analysis will want to have it close at hand as she/he works.' The Mathematical Intelligencer' … the material is written in an excellent manner … I recommend this book warmly as a rich source of information to everybody who is interested in 'Special Functions'.' Zentralblatt MATH' … this book contains a wealth of fascinating material which is presented in a user-friendly way. If you want to extend your knowledge of special functions, this is a good place to start. Even if your interests are in number theory or combinatorics, there is something for you too … the book can be warmly recommended and should be in all good libraries.' Adam McBride, The Mathematical Gazette' … it comes into the range of affordable books that you want to (and probably should have on your desk'. Jean Mawhin, Bulletin of the Belgian Mathematical Society'The book is full of beautiful and interesting formulae, as was always the case with mathematics centred around special functions. It is written in the spirit of the old masters, with mathemtics developed in terms of formulas. There are many historical comments in the book. It can be recommended as a very useful reference.' European Mathematical Society'… full of beautiful and interesting formulae … It can be recommended as a very useful reference.' EMS Newsletter'a very erudite text and reference in special functions' Allen Stenger, MAA ReviewsTable of Contents1. The Gamma and Beta functions; 2. The hypergeometric functions; 3. Hypergeometric transformations and identities; 4. Bessel functions and confluent hypergeometric functions; 5. Orthogonal polynomials; 6. Special orthogonal transformations; 7. Topics in orthogonal polynomials; 8. The Selberg integral and its applications; 9. Spherical harmonics; 10. Introduction to q-series; 11. Partitions; 12. Bailey chains; Appendix 1. Infinite products; Appendix 2. Summability and fractional integration; Appendix 3. Asymptotic expansions; Appendix 4. Euler-Maclaurin summation formula; Appendix 5. Lagrange inversion formula; Appendix 6. Series solutions of differential equations.

Read more less

Book SynopsisThe problem of evaluating integrals is well known to every student who has had a year of calculus. It was an especially important subject in 19th century analysis and it has now been revived with the appearance of symbolic languages. In this book, the authors use the problem of exact evaluation of definite integrals as a starting point for exploring many areas of mathematics. The questions discussed in this book, first published in 2004, are as old as calculus itself. In presenting the combination of methods required for the evaluation of most integrals, the authors take the most interesting, rather than the shortest, path to the results. Along the way, they illuminate connections with many subjects, including analysis, number theory, algebra and combinatorics. This will be a guided tour of exciting discovery for undergraduates and their teachers in mathematics, computer science, physics, and engineering.Trade Review'I recommend this book highly as a source of rewarding projects for undergraduates (and others) to home their analytic skills and gain an appreciation for this area of mathematics. The authors clearly had great love for the material and their enthusiasm comes through in an infectious manner.' SIAM Review'The authors have managed to write a very readable account about integrals, accessible even to advanced undergraduates. Some of the topics of the book could be used for undergraduate reading and research projects. This way the book could serve as a 'springboard to many unexpected investigations and discoveries in mathematics.' Zentralblatt MATHTable of Contents1. Introduction; 2. Factorials and binomial coefficients; 3. The method of partial fractions; 4. A simple rational function; 5. A review of power series; 6. The exponential and logarithm functions; 7. The trigonometric functions and pi; 8. A quartic integral; 9. The normal integral; 10. Euler's constant; 11. Eulerian integrals: the Gamma and Beta functions; 12. The Riemann zeta function; 13. Logarithmic integrals; 14. A master formula; 15. Appendix: the revolutionary WZ method.

Read more less

Book SynopsisThe author presents an accessible and self-contained introduction to harmonic map theory and its analytical aspects, covering recent developments in the regularity theory of weakly harmonic maps. The book begins by introducing these concepts, stressing the interplay between geometry, the role of symmetries and weak solutions. The reader is then presented with a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. A self-contained presentation of 'exotic' functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. The importance of conservation laws is stressed and the concept of a 'Coulomb moving frame' is explained in detail. The book ends with further applications and illustrations of Coulomb moving frames to the theory of surfaces.Trade Review'The book is very well written and it contains truly beautiful geometrical analysis. It also contains a quick, direct introduction to the current research.' EMS NewsletterTable of ContentsPreface; Introduction; Acknowledgements; Notations; 1. Geometric and analytic setting; 2. Harmonic maps with symmetries; 3. Compensations and exotic function spaces; 4. Harmonic maps without symmetries; 5. Surfaces with mean curvature in L2; References.

Read more less

Book SynopsisThis book, first published in 2003, is a self-contained introduction to the principal results and ideas in the theories of completely positive maps, completely bounded maps, dilation theory, operator spaces and operator algebras, plus some of their main applications. An indispensable introduction to the theory of operator spaces for graduate students and experts alike.Trade Review'This book has been written by one of the leading figures in the field. the choice of the presented material has been done in a masterly manner … an excellent introduction to this theory for graduate students. It should also provide a valuable reference source for researchers in the field.' Zentralblatt für Mathematik'The book is carefully written, proofs are often accompanied with notes helping to explain the situation.' EMS Newsletter'Paulsen's book has the advantage of still being concise and staying close to the origins of the theory … the subject of operator spaces is now very well covered and has been made accessible to both the newcomer to the subject, and the specialist looking for concise references, alike. In conclusion, we quote from the cover text of [2]: 'This will be an indispensable introduction to the theory of operator spaces for all who want to know more.' We add: you surely will want to know more.' Martin Mathieu, Queen's University BelfastTable of Contents1. Introduction; 2. Positive maps; 3. Completely positive maps; 4. Dilation theorems; 5. Commuting contractions; 6. Completely positive maps into Mn; 7. Arveson's extension theorems; 8. Completely bounded maps; 9. Completely bounded homomorphisms; 10. Polynomially bounded operators; 11. Applications to K-spectral sets; 12. Tensor products and joint spectral sets; 13. Operator systems and operator spaces; 14. An operator space bestiary; 15. Injective envelopes; 16. Multipliers and operator algebras; 17. Completely bounded multilinear maps; 18. Applications of operator algebras; 19. Similarity and factorization degree.

Read more less

Book SynopsisThis revised and expanded new edition will continue to meet the needs for an authoritative, up-to-date, self contained and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series. Some sections and exercises have been added and the bibliography has been revised to maintain its comprehensiveness.Trade Review'I love this book! It is great! This really is a book you can learn the subject from. The plentiful exercises vary from elementary to challenging with lots of each. Congratulations and thanks are due the authors.' George Andrews, American Math. Monthly'The book is remarkable in many ways. It is comprehensive, at least, comprehensive to date. As is typical of most works on the subject, it is clearly and carefully written. While no book can conceivably incorporate all the important results, particularly those obtained in the last decade, many of them are included as exercises. And this is the feature all other books on the subject lack: a set of exercises. Each chapter is topped off by a challenging series of problems which lead the reader to recreate recent discoveries. Anyone who works even a small percentage of them will soon be an expert. A generous series of historical notes concludes each chapter. The book is user friendly in every respect. The book has two excellent Appendices which summarize the identities and summation formulas derived in the text, an exhaustive 25 page list of references, and a nontrivial index. Now anyone working in combinatorics, group representation theory, coding theory, and related fields will want to own it. Many physicists will find it bears directly on matters of interest to them. Computer scientists may find the book increasingly timely. Those who have refrained from entering the field because of the tortuous notation can now have untroubled access to its mysteries. I say, come in, the water's fine.' Jet Wimp, SIAM Review'This is an excellent and very informative book on the subject. After a gentle introduction to basic series and some special cases (such as the 'q'-binomial theorem) the authors bring the reader up to the latest results on the general theory and its extensions, many such results are due to them. The exercises are utilized to include results that found no room in the detailed treatment. In addition to these exercises, notes at the end of each chapter point the reader to related topics. This alone makes the book an invaluable reference to those who are interested in basic series.' Waleed A. Al-Salam, Math. Reviews'Thus the present book, devoted to 'q'-hypergeometric series, appears at a very timely moment. The result is excellent. The first chapter presents a clear and elementary introduction to the subject. At the end of the book there are excellent indices and compendia of formulas.' Tom H. Koornwinder, Bulletin of London Mathematical Society'… a very modern, self-contained, comprehensive and successful monograph, interesting and useful, for physicists as well as for mathematicians from various branches, who wish to learn about the subject.' European Mathematical Society NewsletterTable of ContentsForeword; Preface; 1. Basic hypergeometric series; 2. Summation, transformation, and expansion formulas; 3. Additional summation, transformation, and expansion formulas; 4. Basic contour integrals; 5. Bilateral basic hypergeometric series; 6. The Askey-Wilson q-beta integral and some associated formulas; 7. Applications to orthogonal polynomials; 8. Further applications; 9. Linear and bilinear generating functions for basic orthogonal polynomials; 10. q-series in two or more variables; 11. Elliptic, modular, and theta hypergeometric series; Appendices; References; Author index; Subject index; Symbol index.

Read more less

Using the Cauchy-Schwarz inequality as a guide, this 2004 book presents a fascinating collection of problems related to inequalities and coaches readers through solutions. Undergraduate and beginning graduate students in mathematics, theoretical computer science, statistics, engineering, and economics will find the book perfect for self-study or as a supplement to probability and analysis courses.

Read more less

Book SynopsisThis authoritative text presents a broad view of the spectral theory of non-self-adjoint linear operators and contains many illustrative examples and exercises. Topics discussed include Fredholm theory, Hilbert-Schmidt and trace class operators, one-parameter semigroups, perturbations of their generators and a thorough account of the new theory of pseudospectra.Trade Review'One will look in vain for the notions of pseudospectrum, hull and numerical range in standard functional analysis texts, so Davies has done us a great service by explaining them through beautiful theorems and examples. More generally, his book is the first to offer a comprehensive survey of the spectral theory of non-self-adjoint operators, including both 'classical' and 'cutting edge' results, showing that this theory holds as much promise as the self-adjoint theory in both foundations and application. The scope of the book is truly enormous and is only partly reflected by listing the chapter titles … [a] beautiful volume, which has no competitors.' The Mathematical IntelligencerTable of ContentsPreface; 1. Elementary operator theory; 2. Function spaces; 3. Fourier transforms and bases; 4. Intermediate operator theory; 5. Operators on Hilbert space; 6. One-parameter semigroups; 7. Special classes of semigroup; 8. Resolvents and generators; 9. Quantitative bounds on operators; 10. Quantitative bounds on semigroups; 11. Perturbation theory; 12. Markov chains and graphs; 13. Positive semigroups; 14. NSA Schrödinger operators.

Read more less