Encyclopedia of Physical Science and Technology - Mathematics [3 ed.]

Table of contents :
Glossary......Page 1
Functions......Page 2
Order Relations......Page 3
Blockmodels......Page 4
Generators and Relations......Page 5
Binary Relations and Boolean Matrices......Page 6
Finite State Machines......Page 7
Mathematical Linguistics......Page 8
Fundamental Homomorphism Theorems......Page 9
Cyclic Groups......Page 10
Orbits......Page 11
Enumeration......Page 12
Basis and Dimension......Page 13
Matrices and Determinants......Page 14
Boolean Vectors and Matrices......Page 15
Ring of Integers......Page 16
Euclidean Domains......Page 17
Ideals and Congruences......Page 18
Simple and Semisimple Rings......Page 19
Modules......Page 20
Fields and Extensions......Page 21
Applications to Solvability and Constructibility......Page 22
Finite Fields......Page 23
Applications to Latin Squares......Page 24
Applications to Projective Planes......Page 25
Nonassociative Algebras and Higher Order Algebras......Page 26
Categories and Topoi......Page 27
Algebraic Topology......Page 28
Current and Recent Research......Page 29
References......Page 30
Affine Space......Page 31
Algebraic Sets Defining Ideals......Page 32
Regular and Rational Functions......Page 33
Examples......Page 34
Projective Algebraic Sets......Page 35
Singularities......Page 36
Elliptic and Hyperelliptic Curves......Page 37
Surfaces and Higher Dimensions......Page 38
Computation......Page 39
Interpolation......Page 40
References......Page 41
Glossary......Page 42
Background......Page 43
Polynomial and Rational Approximation......Page 44
Best Local Approximation......Page 45
Incomplete Polynomials......Page 46
Chebyshev Center......Page 47
n Widths......Page 48
Bivariate Spline Functions......Page 49
Compact Operators and M Ideals......Page 51
Constrained Approximations......Page 52
Factorization of Biinfinite Matrices......Page 54
Interpolation......Page 55
Multivariate Polyhedral Splines......Page 59
Quadratures......Page 61
Gauss Quadrature......Page 62
Chebyshev Quadratures......Page 63
Smoothing by Spline Functions......Page 64
Wavelets......Page 67
References......Page 68
Glossary......Page 69
Examples......Page 70
Elementary Properties......Page 71
Subalgebras......Page 72
Homomorphisms......Page 73
The Finite Case......Page 74
General Representation Theory......Page 75
Completeness and Higher Distributivity......Page 76
Switching Functions and Electronics......Page 77
Other Applications......Page 78
References......Page 79
Glossary......Page 80
Two Basic Problems in Calculus......Page 81
Algebraic Expressions......Page 82
Trigonometric Expressions......Page 83
The Derivative Defined......Page 84
Some Derivatives Calculated......Page 85
The Derivative and Graphing......Page 86
Maxima and Minima......Page 88
The Integral Defined......Page 89
Evaluation of Integrals......Page 90
Volumes......Page 91
Normal Distribution......Page 92
Some Consequences......Page 93
Infinite Series......Page 94
References......Page 96
Glossary......Page 97
Curves......Page 98
The Cauchy–Riemann Equations and Harmonic Functions......Page 99
Line Integrals and Winding Numbers......Page 100
Geometric Properties of Analytic Functions......Page 101
Analytic Continuation......Page 102
Poles and Meromorphic Functions......Page 103
The Cauchy Residue Theorem......Page 104
Infinite Products, Partial Fractions, and Approximations......Page 105
Riemann Mapping Theorem......Page 106
Riemann Surfaces......Page 107
References......Page 108
Glossary......Page 109
Enumerative Proof......Page 110
Computer Programming......Page 111
Four-Color Conjecture......Page 112
Projective Plane of Order 10......Page 113
Estimation......Page 114
Optimization......Page 115
Future Directions......Page 116
References......Page 117
Glossary......Page 118
What Can Be Done......Page 119
What Is a Hypergeometric Series?......Page 120
The Chu–Vandermonde Identity......Page 121
Sister Celine’s Technique......Page 122
Wilf and Zeilberger......Page 123
Conclusions......Page 124
References......Page 125
Glossary......Page 126
Definitions......Page 127
Examples......Page 129
Descriptions of Convex Sets......Page 130
The Polar of a Convex Set......Page 132
Algebraic Operations on Convex Sets......Page 133
Symmetrizing Operations......Page 134
The Hausdorff Metric......Page 136
Separation and Support Theorems......Page 137
Volumes and Mixed Volumes......Page 138
Inequalities......Page 140
Polytopes......Page 142
Zonoids......Page 143
Simplices......Page 144
Sets of Constant Width......Page 145
References......Page 146
Glossary......Page 147
Two Problems......Page 148
The Curse of Dimensionality......Page 149
Massive Datasets......Page 150
Logistic Regression......Page 151
Neural Nets......Page 152
Boosting......Page 153
Clustering Strategies......Page 154
Racing......Page 155
Alternating Conditional Expectations (ACE)......Page 156
Neural Nets (NN)......Page 157
Multivariate Adaptive Regression Splines (MARS)......Page 158
Comparisons......Page 159
References......Page 160
Error-Correcting Codes......Page 162
Combinatorial Designs......Page 164
Perfect Codes......Page 165
Constant Weight Codes......Page 166
Maximum Distance Separable Codes......Page 167
Convolutional Codes......Page 168
References......Page 169
Glossary......Page 170
First-Order Ordinary Differential Equations......Page 171
nth-Order Ordinary Differential Equations......Page 172
Examples of Initial-Value Problems......Page 173
Existence of Solutions......Page 174
Uniqueness of Solutions......Page 176
Continuity of Solutions with Respect to Parameters......Page 177
Comparison Theory......Page 178
Linear Homogeneous and Nonhomogeneous Systems......Page 179
Linear Systems with Constant Coefficients......Page 182
Linear Systems With Periodic Coefficients......Page 185
Linear nth-Order Ordinary Differential Equations......Page 186
The Concept of an Equilibrium Point......Page 189
Definitions of Stability and Boundedness......Page 190
Linear Systems......Page 192
Two-Dimensional Linear Systems......Page 193
Lyapunov Functions......Page 194
Lyapunov Stability and Instability Results: Motivation......Page 197
Principal Lyapunov Stability and Instability Theorems......Page 198
Converse Theorems......Page 200
Lyapunov’s First Method......Page 201
References......Page 202
Importance......Page 203
The Wave Equation......Page 204
Linear Equations......Page 205
Problems associated with Partial Differential Equations......Page 206
Mixed Problems......Page 207
Separation of Variables......Page 208
Fourier Transforms......Page 209
Hilbert Space Methods......Page 210
Iterations......Page 211
Variational Methods......Page 212
Critical Point Theory......Page 213
Periodic Solutions......Page 214
References......Page 216
Nature of Combinatorics......Page 217
Permutations and Combinations......Page 218
Binomial Coefficients......Page 219
Discrete Probability......Page 220
Recurrence Relations and Counting Problems......Page 221
Generating Functions and Counting Problems......Page 222
Inclusion–Exclusion Principle......Page 223
Pigeonhole Principle......Page 225
Combinatorial Designs......Page 226
References......Page 227
Glossary......Page 228
Examples of Physical Systems......Page 229
Linear Systems......Page 230
Semilinear Systems......Page 231
Existence of Solutions for Linear Evolution Equations and Semigroups......Page 232
Stability......Page 234
System Identification......Page 235
Controllability......Page 236
Existence of Optimal Controls......Page 237
Necessary Conditions of Optimality......Page 239
Stochastic Evolution Equations......Page 241
Existence of Optimal Controls......Page 242
Stability, Identification, and Controllability......Page 247
Necessary Conditions of Optimality......Page 248
Nonlinear Stochastic Evolution Equations......Page 249
m-Times Integrated Semigroups......Page 250
Measure Solution......Page 251
Impulsive Systems......Page 252
References......Page 253
Glossary......Page 255
Problem Formulation......Page 256
Examples......Page 258
A Longest-Route Problem......Page 259
A Resource Allocation Problem......Page 260
A Stochastic Inventory Control Problem......Page 261
Functional Equation of Dynamic Programming......Page 263
Backward Induction and the Principle of Optimality......Page 264
A Resource Allocation Problem......Page 266
The Inventory Example......Page 267
Introduction......Page 268
Basic Results......Page 269
Policy Iteration......Page 270
Modified Policy Iteration......Page 271
Policy Iteration......Page 272
Modified Policy Iteration......Page 273
Bounds on the Optimal Total Expected Discounted Reward......Page 274
The Average Expected Reward Criteria......Page 275
Partially Observable Models......Page 276
Continuous-State, Discrete-Time Models......Page 277
References......Page 278
Glossary......Page 279
Definition of Fourier Series......Page 280
Convergence of Fourier Series......Page 283
Convergence in Norm......Page 286
Summability of Fourier Series......Page 287
Generalized Fourier Series......Page 289
Discrete Fourier Series......Page 292
References......Page 295
Glossary......Page 296
Initiator and generator......Page 297
Quadratic Julia sets.......Page 298
Circle inversion limit sets.......Page 299
Wiener brownian motion......Page 300
Fractional Brownian motion......Page 301
Self-affine cartoons with mild to wild randomness......Page 302
Diffusion-limited aggregation (DLA; Vicsek, 1992)......Page 303
Similarity Dimension......Page 304
Mass Dimension......Page 305
Algebra of Dimensions and Latent Dimensions......Page 306
Subordination and Products of Dimension......Page 307
Latent Dimensions That Exceed That of the Embedding Space......Page 308
Thermodynamic Formalism......Page 309
The Crosscuts Structure......Page 310
Weierstrass–Mandelbrot Functions......Page 311
Fractals and Diffefrential or Partial Differential Equations......Page 312
Partial Differential Equations on Domains with Fractal Boundaries (“Can One Hear the Shape of a Fractal Drum?”)......Page 313
The Large-Scale Distribution of Galaxies......Page 314
Fractals in the Arts and in Teaching......Page 315
References......Page 316
Glossary......Page 319
Linear Spaces......Page 320
Normed Linear Spaces......Page 321
Hilbert Spaces......Page 322
Linear Operators......Page 324
Bounded Operators......Page 325
Compact Operators......Page 326
Unbounded Operators......Page 327
Projection and Decomposition......Page 328
The Spectral Theorem......Page 329
Operator Equations......Page 330
Inversion......Page 331
Weak Solutions of Poisson’s Equation......Page 332
Two-Point Boundary Value Problems......Page 333
Heisenberg’s Principle......Page 334
References......Page 335
Glossary......Page 336
Distributions......Page 337
Algebraic Operations on Distributions......Page 338
Analytic Operations on Distributions......Page 339
Pseudo-Function, Hadamard Finite Part and Regularization......Page 340
Distributional Derivatives of Discontinuous Functions......Page 341
Direct Product and Convolution of Distributions......Page 343
Fourier Transform......Page 344
Poisson Summation Formula......Page 346
Asymptotic Evaluation of Integrals: A Distributional Approach......Page 347
See also the Following Articles......Page 348
References......Page 349
Glossary......Page 350
Introduction......Page 351
Connectedness......Page 353
Trees......Page 354
Eulerian and Hamiltonian Graphs......Page 355
Colorings of Graphs......Page 356
Planar Graphs......Page 358
Factorization Theory......Page 359
Graph Reconstruction......Page 361
Extremal Theory......Page 362
Directed Graphs......Page 363
Random Graphs......Page 365
References......Page 366
Glossary......Page 367
Univariate Guidelines......Page 369
Quantile Plots......Page 370
Quantile–Quantile Plots......Page 371
Direct Comparison of Two Distributions and the Mean Difference Plot......Page 372
Multivariate Visualization......Page 374
Communication Objectives......Page 375
Distance Judgments and 3-D Scatterplots......Page 376
Scatterplot Matrices, Parallel Coordinates, and Stereo Ray Glyphs......Page 377
Glyphs and Layouts......Page 378
References......Page 379
Glossary......Page 381
Fundamental Concepts......Page 382
Important Examples......Page 384
Basic Constructions......Page 386
Permutation Groups and Geometric Groups......Page 388
Group Representations and Linear Groups......Page 390
Finite Simple Groups......Page 394
Other Topics......Page 396
See also the Following Articles......Page 397
References......Page 398
Glossary......Page 399
Group Representation Theory......Page 400
Representations and Characters......Page 402
Construction of Representations......Page 405
Continuous Lie Groups......Page 407
SU(2) Covering Group of the Rotation Group......Page 408
Four-Dimensional Rotation Group and Homogeneous Lorentz Group......Page 409
Symmetry of the Hydrogen Atom......Page 410
Isotropic Harmonic Oscillator......Page 411
Splitting of Atomic Levels......Page 413
Selection Rules in Atomic Spectra......Page 414
Applications in Nuclear Physics......Page 416
The Elliott Model......Page 417
Normal Modes of Vibration in Molecules......Page 418
Brillouin Zones and Compatibility Relations......Page 420
Application of Lie Groups in Electrical Engineering......Page 421
Unitary Irreducible Representations of SU(2)......Page 423
Glebsch–Gordan Coefficients for SU(2) and Isospin......Page 424
Isospin Clebsch–Gordan Coefficients......Page 425
SU(3) and Particle Physics......Page 426
Gell-Mann–Okubo Mass Formula......Page 428
SU(6) and the Quark Model......Page 429
Electroweak Theory......Page 432
SU(5) Grand Unification......Page 433
Renormalization Group Equations and Evolutions of Gauge Couplings......Page 434
Supersymmetry......Page 435
Applications in Geometrical Optics......Page 436
The Renormalization Group......Page 437
References......Page 439
Glossary......Page 441
The Method of Successive Approximations......Page 442
The Fredholm Alternative......Page 444
The Fredholm Operator......Page 446
Hermitian Kernels and the Hilbert–Schmidt Theory......Page 447
Proof......Page 448
Proof......Page 449
Proof......Page 450
The Cauchy Integral Equation......Page 453
Singular Integral Equations with a Logarithmic Kernel......Page 454
The Cauchy Kernel and the Riemann–Hilbert Problem......Page 455
Wiener–Hopf Integral Equation......Page 458
Nonlinear Integral Equations......Page 459
A Taylor Expansion Technique......Page 460
References......Page 461
Glossary......Page 462
Introduction......Page 463
Knots as Analog Computers......Page 464
Invariants of Knots and Links, a First Pass......Page 465
The Quandle and the Determinant of a Knot......Page 469
The Jones Polynomial......Page 475
Conjecture.......Page 476
The Bracket State Sum......Page 478
First Steps in Bracketology......Page 480
Framing Philosophy, Twist and Writhe......Page 484
Mirror Mirror......Page 486
Vassiliev Invariants......Page 487
Vassiliev Invariants and Lie Algebras......Page 489
A Quick Review of Quantum Mechanics......Page 495
Schrodinger’s Equation......Page 498
Dirac Brackets......Page 499
Knot Amplitudes......Page 500
Topological Quantum Field Theory, First Steps......Page 505
Links and the Wilson Loop......Page 506
Graph Invariants and Vassiliev Invariants......Page 507
References......Page 508
Glossary......Page 509
Historical Background......Page 510
A Simplex Iteration......Page 512
Finiteness of the Simplex Method......Page 514
Duality Results......Page 515
The Dual Simplex Method......Page 516
The Criss-Cross Method......Page 517
Interior-Point Methods......Page 518
Reduction to Canonical Form......Page 519
Embedding into Self-Dual Model......Page 520
Central Path......Page 521
Using Approximate Centers......Page 522
Complexity Analysis......Page 523
Finding the Optimal Partition......Page 524
Other Interior-Point Methods......Page 525
Branch-and-Bound Methods......Page 526
Sensitivity Analysis......Page 527
References......Page 528
Definitions......Page 529
Central Extensions......Page 530
Differential-Geometric Properties......Page 531
Stratifications......Page 532
The Fundamental Representation and The Spin Representation......Page 533
Kac Character Formula......Page 534
Quivers......Page 535
References......Page 536
Glossary......Page 537
Notations......Page 538
Geometrical Spaces......Page 539
Euclidean Spaces......Page 540
Manifolds......Page 541
Bundles......Page 542
Combining Vector Bundles......Page 543
Module of Sections......Page 545
Tangent Vector Fields......Page 546
Cotangent Vector Fields......Page 547
Products of Fields......Page 548
Exterior Derivative......Page 549
Derivatives of Smooth Maps......Page 550
Using Cohomology Data......Page 551
Volume Forms......Page 552
Metric Tensors......Page 553
Linear Connection......Page 555
Parallel Transport......Page 556
Radial geodesics......Page 557
Metric Connection......Page 558
Torsion of a Connection......Page 559
Curvature of a Connection......Page 560
b-Incompleteness......Page 561
Topology, Geometry, and Physics......Page 562
References......Page 563
Glossary......Page 565
Propositional Semantics......Page 566
Propositional Inference......Page 567
First-Order Syntax......Page 568
First-Order Semantics......Page 569
Databases......Page 570
First-Order Inference......Page 571
Proof Theory......Page 572
Coding (Godel numbering)......Page 573
Turing Machines......Page 574
Unsolvable Problems......Page 575
Undecidable Theories......Page 576
Hierarchies......Page 577
Recursive Equations......Page 578
Modal and Temporal Logic......Page 579
Intuitionistic Logic......Page 580
Cardinal Arithmetic......Page 581
The Paradoxes......Page 582
Zermelo-Fraenkel Set Theory......Page 583
References......Page 585
Glossary......Page 586
Mathematical Modeling......Page 587
Classification of Mathematical Models......Page 589
Conventional Modeling (Direct Modeling)......Page 591
System Identification and Parameter Estimation (Inverse Modeling)......Page 592
Gradient Techniques......Page 593
Model Validation......Page 594
Modeling with Neural Networks......Page 595
References......Page 596
Introduction......Page 598
Lebesgue Measure......Page 599
The Lebesgue Integral......Page 600
The Lp Spaces and Inequalities for Integrals......Page 604
Differentiation and Integration......Page 605
Product Spaces and Product Measures......Page 606
General Measures......Page 608
The Radon–Nikod´Ym Theorem and Signed Measures......Page 610
Extensions of Measures and Lebesgue–Stieltjes Integrals......Page 612
The Radon–Nikod´Ym Property for Banach Spaces......Page 613
Measure and Fractals......Page 615
References......Page 616
Optimization and Nonlinear Programming......Page 617
The Solution of a Nonlinear Program......Page 618
Special Types of Nonlinear Programs......Page 619
The Geometry......Page 620
First-Order Conditions......Page 621
Stability and Duality......Page 624
Basic Concepts......Page 625
Unconstrained Optimization Algorithms......Page 626
The Sequential Quadratic Programming Algorithm......Page 627
Interior Point Methods......Page 628
Statistical Applications......Page 629
References......Page 630
Glossary......Page 631
Algebraic Number Fields......Page 632
Diophantine Equations......Page 633
Fermat’s “Last Theorem”......Page 634
Waring’s Problem......Page 635
Elliptic Curves......Page 636
Transcendental Numbers......Page 638
Prime Number Theory......Page 639
The Prime Number Theorem (PNT)......Page 640
Partitions......Page 641
Computational Number Theory......Page 642
References......Page 643
Glossary......Page 644
Number Theory versus Numerology......Page 645
Pythagorean Triples......Page 646
The Fundamental Theorem of Arithmetic......Page 647
Greatest Common Divisor......Page 648
Divisibility Rules......Page 649
Perfect Numbers......Page 650
Diophantine Equations......Page 651
Magic Squares......Page 652
Indeterminate Problems......Page 653
Congruences......Page 654
Chinese Remainder Theorem......Page 655
Wilson’s Theorem......Page 656
Quadratic Residues......Page 657
Formulas for Primes......Page 658
Representation of Numbers in Certain Forms......Page 659
Factorization Methods......Page 660
Fibonacci Numbers......Page 661
Modern Directions......Page 662
Algebraic Number Theory......Page 663
Conjectures......Page 664
Fermat’s Last Theorem......Page 665
References......Page 666
Glossary......Page 668
The Numerical Approach to Problem Solution......Page 669
Errors, Their Sources, and Propagation......Page 670
Condition and Numerical Stability......Page 671
Number Representation......Page 672
Computer-Representable Numbers......Page 673
Roundoff Error......Page 674
Background......Page 676
Roots of Polynomials......Page 677
Systems of Linear Equations......Page 678
Direct Methods......Page 680
Iteration Methods......Page 682
Systems of Nonlinear Equations......Page 683
Background......Page 684
Symmetric Matrices......Page 685
Nonsymmetric Matrices......Page 686
Polynomial Representation and Interpolation......Page 687
Piecewise Polynomials and Splines......Page 688
Differentiation and Finite Differences......Page 689
Integration......Page 690
Ordinary Differential Equations—Initial-Value Problems......Page 692
Numerical Instability and Stiffness......Page 694
Ordinary Differential Equations— Boundary-Value Problems......Page 695
Partial Differential Equations......Page 696
Large, Sparse Matrices......Page 697
New Approximation Methods......Page 698
References......Page 699
Glossary......Page 700
Historical Perspective......Page 701
The Education and Training of the OR Practitioner/Researcher......Page 702
Linear Programming (LP)......Page 703
Integer Programming (IP), Graphs, and Combinatorial Optimization......Page 704
Variational Inequality and Complementarity Problems (VI & CP)......Page 705
Nonlinear Programming (NLP)......Page 706
Dynamic Programming (DP)......Page 707
Entropy Optimization (EO)......Page 708
Stochastic Processes......Page 709
Stationarity......Page 711
Multi-Server Queues......Page 712
Simulation......Page 713
Decision Analysis......Page 714
Representation of a Game......Page 715
Two-Person Non-Zero-Sum Games......Page 716
The Fuzzy Paradigm......Page 717
Fuzzy Arithmetic......Page 718
Fuzzy Logic......Page 719
Aggregate Production Planning......Page 720
Inventory Management......Page 721
Facilities Location and Layout......Page 722
Selected Fields of Application......Page 723
References......Page 724
Glossary......Page 726
Introduction......Page 727
Simple Static Properties of Percolation......Page 728
Fractals, Scaling, and Renormalization......Page 730
Kinetic Aspects of Percolation......Page 735
Other Developments......Page 739
References......Page 740
Glossary......Page 741
Elementary Illustrative Example......Page 742
Same Problem, Different Approach......Page 743
Summation Theory......Page 744
Unusual Ways to Insert the Perturbation Parameter......Page 745
Regular Versus Singular Perturbation Theory......Page 746
Boundary-Layer Theory......Page 747
Wkb Theory......Page 749
Perturbative Calculation of Eigenvalues......Page 750
Multiple-Scale Perturbation Theory......Page 751
References......Page 752
Glossary......Page 754
Historical Evolution......Page 755
Basic Notions of Modern Probability......Page 758
Proposition 1 (Cebysev’s Inequality)......Page 760
Example 1......Page 762
Strong Law of Large Numbers and its Uses......Page 763
Conditional Probability and Dependence......Page 765
Example 1......Page 766
Markov Dependence......Page 767
Martingales......Page 769
Kac-Slepian Paradox......Page 770
Theorem......Page 771
Stationary and Harmonizable Processes and Spectra......Page 772
Theorem (Berry–Ess´een)......Page 774
Statistical Inference Problems and Comments......Page 775
Reference......Page 776
Glossary......Page 777
Queueing Systems......Page 778
Notation......Page 779
Queue Equations......Page 780
Queues with Time-Dependent Parameters......Page 781
Queues with Balking and Reneging......Page 782
A Random Walk with Partially Reflecting Barriers......Page 783
Double-Ended Queues with Bulk Service......Page 784
Imbedded Markov Chain Queueing Systems......Page 785
Typical Telephone Booth Problem......Page 787
Queueing Problem Involving Traffic Lights......Page 788
Dynamic Flows in Communication Network Models......Page 789
Boundary-Value Problems in Queueing Theory......Page 790
References......Page 791
Glossary......Page 792
Preliminaries......Page 793
Tests for the Rank of the Regression Matrix......Page 794
Rank of Regression Matrix by Model Selection Methods......Page 796
Reduction of Dimensionality Under the Fanova Model......Page 797
Rank of the Covariance Matrix of Random Effects......Page 798
Variable Selection in Discriminant Analysis......Page 799
Tests for Rank of Canonical Correlation Matrix......Page 802
Variable Selection in Canonical Correlation Analysis......Page 804
Dimensionality and Dependence in Two-way Contingency Table......Page 805
Nonexact Test for the Two-sample Problem......Page 806
Multivariate Discrimination Analysis......Page 807
References......Page 808
The Beginnings: Galilei and Cantor......Page 811
Ordinal Numbers......Page 812
The Cumulative Hierarchy......Page 813
Borel Sets and Projective Sets......Page 814
Exponentiation of Singular Cardinals......Page 815
The Measure Problem......Page 816
Lebesgue Measurability of Projective Sets of Reals......Page 817
Well-Ordering the Real Line......Page 818
Descriptive Set Theory, Large Cardinals and Determinacy......Page 819
Combinatorial Set Theory and the Duality Program......Page 820
References......Page 821
Introduction......Page 822
Important Developments in Soliton Theory......Page 825
Solitons......Page 827
Dromions......Page 828
The Inverse Spectral Method......Page 829
Transform Methods for Linear PDEs......Page 831
A New Method......Page 832
References......Page 833
Introduction......Page 834
The Fundamental Problem......Page 835
Optimal Statistical Objective Analysis......Page 836
Kriging......Page 837
Empirical Interpolation Techniques......Page 838
References......Page 839
Glossary......Page 840
The Sphere Packing Problem—Its Statement and History......Page 841
Packing of Convex Bodies......Page 845
Sphere Packings and Codes......Page 847
References......Page 848
Glossary......Page 849
Robustness to the Prior Distribution......Page 850
Precise Measurement......Page 852
Fiducial Argument......Page 853
Procedural Robustness......Page 856
Robustness to Likelihood Function......Page 858
Examples of NonRobustness......Page 859
References......Page 860
Introduction......Page 862
Decision Making......Page 865
Criticism......Page 866
References......Page 869
Glossary......Page 870
Probability Model......Page 871
Statistical Model......Page 872
Foundations......Page 873
Conditioning Principle......Page 874
References......Page 875
Glossary......Page 877
Multivariate Probability Theory......Page 878
Inference for the Linear Model......Page 880
Point Estimation......Page 881
Confidence Regions......Page 882
Bayesian Inference......Page 883
The Bootstrap......Page 884
Cross-Validation......Page 885
Missing and Censored Data......Page 886
Regression Analysis......Page 887
Correlation Analysis......Page 888
Residuals......Page 889
Stepwise Selection......Page 890
Generalized Additive Models......Page 891
Projection Pursuit Regression, MARS, and ACE......Page 892
Experimental Design......Page 893
Univariate Response......Page 894
Split-plot designs......Page 895
Analysis of Variance Tables......Page 896
Simultaneous Inference......Page 898
Multivariate Response......Page 899
Cluster Analysis......Page 900
Cluster Analysis of Variables......Page 901
Principal Components Analysis......Page 902
Factor Analysis......Page 903
Some Basic Notation and Results......Page 904
Applying the Basic Theory to Log-Linear Models for Three-Way Tables......Page 905
Discriminant Analysis......Page 907
CART Analysis......Page 908
Detrending Data......Page 910
Time-Domain Models......Page 911
Statistical Software Packages......Page 912
References......Page 913
Glossary......Page 916
Two-Sample Location Problems......Page 917
Estimation......Page 918
Example: Cholesterol Data......Page 919
Tests......Page 920
Example: Darwin Data......Page 921
One-Sample Location Models......Page 922
Two-Sample Scale Problem......Page 923
Wilcoxon Regression Estimates......Page 924
Tests of Linear Hypotheses......Page 925
Example: Telephone Data......Page 926
Estimates of the Scale Parameters τ and τS......Page 927
Studentized Residuals......Page 928
High Breakdown Robust Estimates......Page 929
Diagnostics To Differentiate between HBR and Wilcoxon Fits......Page 931
Example: Stars Data......Page 932
Optimal Rank-Based Analyses......Page 933
One-Way Designs......Page 934
Pseudo-Observations......Page 935
Two-Way Designs......Page 936
Friedman’s Test......Page 937
Kendall’s τ......Page 938
References......Page 939
Physical Setting......Page 940
History of the Problem......Page 941
Mushy Zones......Page 942
Alloy Solidification......Page 943
Large-Scale Simulation......Page 944
References......Page 945
Basics of Stochastic Processes......Page 946
Gaussian Processes......Page 948
Brownian Motion......Page 949
Weakly Stationary Stochastic Processes......Page 950
Harmonizable Class......Page 951
Periodically Correlated Class......Page 952
Integrability......Page 953
Karhunen–Lo`eve Expansion......Page 954
Series Representation......Page 955
Stochastic Fields......Page 956
References......Page 957
Glossary......Page 958
Preliminary Concepts......Page 959
Tilings by Polytopes......Page 961
Monohedral Tilings......Page 962
The Symmetry Group of a Tiling......Page 963
General Considerations......Page 965
Euler’s Theorem......Page 966
Metrical Symmetry......Page 968
Lattice Tilings......Page 969
Space Fillers in Dimension d......Page 970
Aperiodicity......Page 972
The Penrose Tilings......Page 974
The Projection Method......Page 975
References......Page 977
Glossary......Page 978
Basic Notions of Topology and Examples......Page 979
Nets......Page 981
Continuous, Open Maps and Homeomorphisms......Page 982
Upper (or Lower) Semicontinuous Functions......Page 983
Quotient Topology......Page 984
Complete Regularity......Page 985
Metric Spaces and Metrization......Page 986
Compact Spaces......Page 988
Hemicompact Spaces......Page 989
Perfect and Proper Maps......Page 990
Uniform Spaces and Uniformization......Page 991
Proximity Spaces......Page 992
Extensions and Embeddings......Page 993
Fixed Points......Page 994
Marriage of Topology and Algebra......Page 995
Topological Vector Spaces, Banach Spaces, and Hilbert Spaces......Page 996
Closed-Graph Theorem......Page 997
Topological Algebras......Page 998
Algebraic Topology......Page 999
Jordan Curves......Page 1000
Homology......Page 1001
References......Page 1002
Glossary......Page 1003
Historical Perspective......Page 1004
Analytic Foundations......Page 1005
Necessary Conditions......Page 1006
Sufficient Conditions; Convexity......Page 1007
Isoperimetric Constraints......Page 1008
Lagrangian Constraints......Page 1009
Optimal Control Problems......Page 1010
Minimization Theory......Page 1011
Hamiltonian Contributions......Page 1012
Dirichlet’s Principle......Page 1013
Plateau’s Problem......Page 1014
Variational Calculus in the Large......Page 1015
References......Page 1016
Glossary......Page 1017
Time–Frequency Analysis......Page 1018
Resolution of Identity......Page 1019
Multiresolution Analysis......Page 1020
Discrete Wavelet Transforms......Page 1022
Filters and Quadrature Filters......Page 1023
Littlewood–Paley Wavelets Shannon’s Wavelets......Page 1024
Daubechies’ Wavelets......Page 1025
Wavelet Packets......Page 1026
Multiscale Wavelet Packets......Page 1027
Wavelet Density Estimation......Page 1028
Wavelet Shrinkage......Page 1030
Bayesian Wavelet Shrinkage......Page 1031
The Bayesian and Empirical Bayesian Approach......Page 1032
Cycle Spinning......Page 1033
Two-Dimensional Wavelet Algorithm and Image Processing......Page 1034
References......Page 1035
Glossary......Page 1036
Basic Techniques......Page 1037
Thresholding and Compression......Page 1039
Noise Removal......Page 1040
The Haar Family of Wavelets......Page 1041
Dilation Equations and Operators......Page 1043
Multiresolution Analysis......Page 1044
Dilation Equation for Daubechies Wavelets......Page 1045
The Cascade Algorithm......Page 1046
Analysis of Polynomials by Daubechies Wavelets......Page 1047
Analysis of Finite Signals......Page 1049
A Final Example......Page 1050
References......Page 1051