Mathematics Books

Book SynopsisThe revised and updated new edition of the popular optimization book for engineers The thoroughly revised and updated fifth edition ofEngineering Optimization: Theory and Practiceoffers engineers a guide to the important optimization methods that are commonly used in a wide range of industries. The authora noted expert on the topicpresents both the classical and most recent optimizations approaches. The book introduces the basic methods and includes information on more advanced principles and applications. The fifth edition presents four new chapters: Solution of Optimization Problems Using MATLAB; Metaheuristic Optimization Methods; Multi-Objective Optimization Methods; and Practical Implementation of Optimization. All of the book''s topics are designed to be self-contained units with the concepts described in detail with derivations presented. The author puts the emphasis on computational aspects of optimization and includes design examples and problemsTable of ContentsPreface xvii Acknowledgment xxi About the Author xxiii 1 Introduction to Optimization 1 1.1 Introduction 1 1.2 Historical Development 3 1.2.1 Modern Methods of Optimization 4 1.3 Engineering Applications of Optimization 5 1.4 Statement of an Optimization Problem 6 1.4.1 Design Vector 6 1.4.2 Design Constraints 7 1.4.3 Constraint Surface 7 1.4.4 Objective Function 8 1.4.5 Objective Function Surfaces 9 1.5 Classification of Optimization Problems 14 1.5.1 Classification Based on the Existence of Constraints 14 1.5.2 Classification Based on the Nature of the Design Variables 14 1.5.3 Classification Based on the Physical Structure of the Problem 15 1.5.4 Classification Based on the Nature of the Equations Involved 18 1.5.5 Classification Based on the Permissible Values of the Design Variables 27 1.5.6 Classification Based on the Deterministic Nature of the Variables 28 1.5.7 Classification Based on the Separability of the Functions 29 1.5.8 Classification Based on the Number of Objective Functions 31 1.6 Optimization Techniques 33 1.7 Engineering Optimization Literature 34 1.8 Solutions Using MATLAB 34 References and Bibliography 34 Review Questions 40 Problems 41 2 Classical Optimization Techniques 57 2.1 Introduction 57 2.2 Single-Variable Optimization 57 2.3 Multivariable Optimization with no Constraints 62 2.3.1 Definition: rth Differential of f 62 2.3.2 Semidefinite Case 67 2.3.3 Saddle Point 67 2.4 Multivariable Optimization with Equality Constraints 69 2.4.1 Solution by Direct Substitution 69 2.4.2 Solution by the Method of Constrained Variation 71 2.4.3 Solution by the Method of Lagrange Multipliers 77 2.5 Multivariable Optimization with Inequality Constraints 85 2.5.1 Kuhn–Tucker Conditions 90 2.5.2 Constraint Qualification 90 2.6 Convex Programming Problem 96 References and Bibliography 96 Review Questions 97 Problems 98 3 Linear Programming I: Simplex Method 109 3.1 Introduction 109 3.2 Applications of Linear Programming 110 3.3 Standard form of a Linear Programming Problem 112 3.3.1 Scalar Form 112 3.3.2 Matrix Form 112 3.4 Geometry of Linear Programming Problems 114 3.5 Definitions and Theorems 117 3.5.1 Definitions 117 3.5.2 Theorems 120 3.6 Solution of a System of Linear Simultaneous Equations 122 3.7 Pivotal Reduction of a General System of Equations 123 3.8 Motivation of the Simplex Method 127 3.9 Simplex Algorithm 128 3.9.1 Identifying an Optimal Point 128 3.9.2 Improving a Nonoptimal Basic Feasible Solution 129 3.10 Two Phases of the Simplex Method 137 3.11 Solutions Using MATLAB 143 References and Bibliography 143 Review Questions 143 Problems 145 4 Linear Programming II: Additional Topics and Extensions 159 4.1 Introduction 159 4.2 Revised Simplex Method 159 4.3 Duality in Linear Programming 173 4.3.1 Symmetric Primal–Dual Relations 173 4.3.2 General Primal–Dual Relations 174 4.3.3 Primal–Dual Relations when the Primal Is in Standard Form 175 4.3.4 Duality Theorems 176 4.3.5 Dual Simplex Method 176 4.4 Decomposition Principle 180 4.5 Sensitivity or Postoptimality Analysis 187 4.5.1 Changes in the Right-Hand-Side Constants bi 188 4.5.2 Changes in the Cost Coefficients cj 192 4.5.3 Addition of New Variables 194 4.5.4 Changes in the Constraint Coefficients aij 195 4.5.5 Addition of Constraints 197 4.6 Transportation Problem 199 4.7 Karmarkar’s Interior Method 202 4.7.1 Statement of the Problem 203 4.7.2 Conversion of an LP Problem into the Required Form 203 4.7.3 Algorithm 205 4.8 Quadratic Programming 208 4.9 Solutions Using Matlab 214 References and Bibliography 214 Review Questions 215 Problems 216 5 Nonlinear Programming I: One-Dimensional Minimization Methods 225 5.1 Introduction 225 5.2 Unimodal Function 230 Elimination Methods 231 5.3 Unrestricted Search 231 5.3.1 Search with Fixed Step Size 231 5.3.2 Search with Accelerated Step Size 232 5.4 Exhaustive Search 232 5.5 Dichotomous Search 234 5.6 Interval Halving Method 236 5.7 Fibonacci Method 238 5.8 Golden Section Method 243 5.9 Comparison of Elimination Methods 246 Interpolation Methods 247 5.10 Quadratic Interpolation Method 248 5.11 Cubic Interpolation Method 253 5.12 Direct Root Methods 259 5.12.1 Newton Method 259 5.12.2 Quasi-Newton Method 261 5.12.3 Secant Method 263 5.13 Practical Considerations 265 5.13.1 How to Make the Methods Efficient and More Reliable 265 5.13.2 Implementation in Multivariable Optimization Problems 266 5.13.3 Comparison of Methods 266 5.14 Solutions Using MATLAB 267 References and Bibliography 267 Review Questions 267 Problems 268 6 Nonlinear Programming II: Unconstrained Optimization Techniques 273 6.1 Introduction 273 6.1.1 Classification of Unconstrained Minimization Methods 276 6.1.2 General Approach 276 6.1.3 Rate of Convergence 276 6.1.4 Scaling of Design Variables 277 Direct Search Methods 280 6.2 Random Search Methods 280 6.2.1 Random Jumping Method 280 6.2.2 Random Walk Method 282 6.2.3 Random Walk Method with Direction Exploitation 283 6.2.4 Advantages of Random Search Methods 284 6.3 Grid Search Method 285 6.4 Univariate Method 285 6.5 Pattern Directions 288 6.6 Powell’s Method 289 6.6.1 Conjugate Directions 289 6.6.2 Algorithm 293 6.7 Simplex Method 298 6.7.1 Reflection 298 6.7.2 Expansion 301 6.7.3 Contraction 301 Indirect Search (Descent) Methods 304 6.8 Gradient of a Function 304 6.8.1 Evaluation of the Gradient 306 6.8.2 Rate of Change of a Function Along a Direction 307 6.9 Steepest Descent (Cauchy) Method 308 6.10 Conjugate Gradient (Fletcher–Reeves) Method 310 6.10.1 Development of the Fletcher–Reeves Method 310 6.10.2 Fletcher–Reeves Method 311 6.11 Newton’s Method 313 6.12 Marquardt Method 316 6.13 Quasi-Newton Methods 317 6.13.1 Computation of [Bi] 318 6.13.2 Rank 1 Updates 319 6.13.3 Rank 2 Updates 320 6.14 Davidon–Fletcher–Powell Method 321 6.15 Broyden–Fletcher–Goldfarb–Shanno Method 327 6.16 Test Functions 330 6.17 Solutions Using Matlab 332 References and Bibliography 333 Review Questions 334 Problems 336 7 Nonlinear Programming III: Constrained Optimization Techniques 347 7.1 Introduction 347 7.2 Characteristics of a Constrained Problem 347 Direct Methods 350 7.3 Random Search Methods 350 7.4 Complex Method 351 7.5 Sequential Linear Programming 353 7.6 Basic Approach in the Methods of Feasible Directions 360 7.7 Zoutendijk’s Method of Feasible Directions 360 7.7.1 Direction-Finding Problem 362 7.7.2 Determination of Step Length 364 7.7.3 Termination Criteria 367 7.8 Rosen’s Gradient Projection Method 369 7.8.1 Determination of Step Length 372 7.9 Generalized Reduced Gradient Method 377 7.10 Sequential Quadratic Programming 386 7.10.1 Derivation 386 7.10.2 Solution Procedure 389 Indirect Methods 392 7.11 Transformation Techniques 392 7.12 Basic Approach of the Penalty Function Method 394 7.13 Interior Penalty Function Method 396 7.14 Convex Programming Problem 405 7.15 Exterior Penalty Function Method 406 7.16 Extrapolation Techniques in the Interior Penalty Function Method 410 7.16.1 Extrapolation of the Design Vector X 410 7.16.2 Extrapolation of the Function f 412 7.17 Extended Interior Penalty Function Methods 414 7.17.1 Linear Extended Penalty Function Method 414 7.17.2 Quadratic Extended Penalty Function Method 415 7.18 Penalty Function Method for Problems with Mixed Equality and Inequality Constraints 416 7.18.1 Interior Penalty Function Method 416 7.18.2 Exterior Penalty Function Method 418 7.19 Penalty Function Method for Parametric Constraints 418 7.19.1 Parametric Constraint 418 7.19.2 Handling Parametric Constraints 420 7.20 Augmented Lagrange Multiplier Method 422 7.20.1 Equality-Constrained Problems 422 7.20.2 Inequality-Constrained Problems 423 7.20.3 Mixed Equality–Inequality-Constrained Problems 425 7.21 Checking the Convergence of Constrained Optimization Problems 426 7.21.1 Perturbing the Design Vector 427 7.21.2 Testing the Kuhn–Tucker Conditions 427 7.22 Test Problems 428 7.22.1 Design of a Three-Bar Truss 429 7.22.2 Design of a Twenty-Five-Bar Space Truss 430 7.22.3 Welded Beam Design 431 7.22.4 Speed Reducer (Gear Train) Design 433 7.22.5 Heat Exchanger Design [7.42] 435 7.23 Solutions Using MATLAB 435 References and Bibliography 435 Review Questions 437 Problems 439 8 Geometric Programming 449 8.1 Introduction 449 8.2 Posynomial 449 8.3 Unconstrained Minimization Problem 450 8.4 Solution of an Unconstrained Geometric Programming Program using Differential Calculus 450 8.4.1 Degree of Difficulty 453 8.4.2 Sufficiency Condition 453 8.4.3 Finding the Optimal Values of Design Variables 453 8.5 Solution of an Unconstrained Geometric Programming Problem Using Arithmetic–Geometric Inequality 457 8.6 Primal–dual Relationship and Sufficiency Conditions in the Unconstrained Case 458 8.6.1 Primal and Dual Problems 461 8.6.2 Computational Procedure 461 8.7 Constrained Minimization 464 8.8 Solution of a Constrained Geometric Programming Problem 465 8.8.1 Optimum Design Variables 466 8.9 Primal and Dual Programs in the Case of Less-than Inequalities 466 8.10 Geometric Programming with Mixed Inequality Constraints 473 8.11 Complementary Geometric Programming 475 8.11.1 Solution Procedure 477 8.11.2 Degree of Difficulty 478 8.12 Applications of Geometric Programming 480 References and Bibliography 491 Review Questions 493 Problems 493 9 Dynamic Programming 497 9.1 Introduction 497 9.2 Multistage Decision Processes 498 9.2.1 Definition and Examples 498 9.2.2 Representation of a Multistage Decision Process 499 9.2.3 Conversion of a Nonserial System to a Serial System 500 9.2.4 Types of Multistage Decision Problems 501 9.3 Concept of Suboptimization and Principle of Optimality 501 9.4 Computational Procedure in Dynamic Programming 505 9.5 Example Illustrating the Calculus Method of Solution 507 9.6 Example Illustrating the Tabular Method of Solution 512 9.6.1 Suboptimization of Stage 1 (Component 1) 514 9.6.2 Suboptimization of Stages 2 and 1 (Components 2 and 1) 514 9.6.3 Suboptimization of Stages 3, 2, and 1 (Components 3, 2, and 1) 515 9.7 Conversion of a Final Value Problem into an Initial Value Problem 517 9.8 Linear Programming as a Case of Dynamic Programming 519 9.9 Continuous Dynamic Programming 523 9.10 Additional Applications 526 9.10.1 Design of Continuous Beams 526 9.10.2 Optimal Layout (Geometry) of a Truss 527 9.10.3 Optimal Design of a Gear Train 528 9.10.4 Design of a Minimum-Cost Drainage System 529 References and Bibliography 530 Review Questions 531 Problems 532 10 Integer Programming 537 10.1 Introduction 537 Integer Linear Programming 538 10.2 Graphical Representation 538 10.3 Gomory’s Cutting Plane Method 540 10.3.1 Concept of a Cutting Plane 540 10.3.2 Gomory’s Method for All-Integer Programming Problems 541 10.3.3 Gomory’s Method for Mixed-Integer Programming Problems 547 10.4 Balas’ Algorithm for Zero–One Programming Problems 551 Integer Nonlinear Programming 553 10.5 Integer Polynomial Programming 553 10.5.1 Representation of an Integer Variable by an Equivalent System of Binary Variables 553 10.5.2 Conversion of a Zero–One Polynomial Programming Problem into a Zero–One LP Problem 555 10.6 Branch-and-Bound Method 556 10.7 Sequential Linear Discrete Programming 561 10.8 Generalized Penalty Function Method 564 10.9 Solutions Using MATLAB 569 References and Bibliography 569 Review Questions 570 Problems 571 11 Stochastic Programming 575 11.1 Introduction 575 11.2 Basic Concepts of Probability Theory 575 11.2.1 Definition of Probability 575 11.2.2 Random Variables and Probability Density Functions 576 11.2.3 Mean and Standard Deviation 578 11.2.4 Function of a Random Variable 580 11.2.5 Jointly Distributed Random Variables 581 11.2.6 Covariance and Correlation 583 11.2.7 Functions of Several Random Variables 583 11.2.8 Probability Distributions 585 11.2.9 Central Limit Theorem 589 11.3 Stochastic Linear Programming 589 11.4 Stochastic Nonlinear Programming 594 11.4.1 Objective Function 594 11.4.2 Constraints 595 11.5 Stochastic Geometric Programming 600 References and Bibliography 602 Review Questions 603 Problems 604 12 Optimal Control and Optimality Criteria Methods 609 12.1 Introduction 609 12.2 Calculus of Variations 609 12.2.1 Introduction 609 12.2.2 Problem of Calculus of Variations 610 12.2.3 Lagrange Multipliers and Constraints 615 12.2.4 Generalization 618 12.3 Optimal Control Theory 619 12.3.1 Necessary Conditions for Optimal Control 619 12.3.2 Necessary Conditions for a General Problem 621 12.4 Optimality Criteria Methods 622 12.4.1 Optimality Criteria with a Single Displacement Constraint 623 12.4.2 Optimality Criteria with Multiple Displacement Constraints 624 12.4.3 Reciprocal Approximations 625 References and Bibliography 628 Review Questions 628 Problems 629 13 Modern Methods of Optimization 633 13.1 Introduction 633 13.2 Genetic Algorithms 633 13.2.1 Introduction 633 13.2.2 Representation of Design Variables 634 13.2.3 Representation of Objective Function and Constraints 635 13.2.4 Genetic Operators 636 13.2.5 Algorithm 640 13.2.6 Numerical Results 641 13.3 Simulated Annealing 641 13.3.1 Introduction 641 13.3.2 Procedure 642 13.3.3 Algorithm 643 13.3.4 Features of the Method 644 13.3.5 Numerical Results 644 13.4 Particle Swarm Optimization 647 13.4.1 Introduction 647 13.4.2 Computational Implementation of PSO 648 13.4.3 Improvement to the Particle Swarm Optimization Method 649 13.4.4 Solution of the Constrained Optimization Problem 649 13.5 Ant Colony Optimization 652 13.5.1 Basic Concept 652 13.5.2 Ant Searching Behavior 653 13.5.3 Path Retracing and Pheromone Updating 654 13.5.4 Pheromone Trail Evaporation 654 13.5.5 Algorithm 655 13.6 Optimization of Fuzzy Systems 660 13.6.1 Fuzzy Set Theory 660 13.6.2 Optimization of Fuzzy Systems 662 13.6.3 Computational Procedure 663 13.6.4 Numerical Results 664 13.7 Neural-Network-Based Optimization 665 References and Bibliography 667 Review Questions 669 Problems 671 14 Metaheuristic Optimization Methods 673 14.1 Definitions 673 14.2 Metaphors Associated with Metaheuristic Optimization Methods 673 14.3 Details of Representative Metaheuristic Algorithms 680 14.3.1 Crow Search Algorithm 680 14.3.2 Firefly Optimization Algorithm (FA) 681 14.3.3 Harmony Search Algorithm 684 14.3.4 Teaching-Learning-Based Optimization (TLBO) 687 14.3.5 Honey Bee Swarm Optimization Algorithm 689 References and Bibliography 692 Review Questions 694 15 Practical Aspects of Optimization 697 15.1 Introduction 697 15.2 Reduction of Size of an Optimization Problem 697 15.2.1 Reduced Basis Technique 697 15.2.2 Design Variable Linking Technique 698 15.3 Fast Reanalysis Techniques 700 15.3.1 Incremental Response Approach 700 15.3.2 Basis Vector Approach 704 15.4 Derivatives of Static Displacements and Stresses 705 15.5 Derivatives of Eigenvalues and Eigenvectors 707 15.5.1 Derivatives of ;;i 707 15.5.2 Derivatives of Yi 708 15.6 Derivatives of Transient Response 709 15.7 Sensitivity of Optimum Solution to Problem Parameters 712 15.7.1 Sensitivity Equations Using Kuhn–Tucker Conditions 712 15.7.2 Sensitivity Equations Using the Concept of Feasible Direction 714 References and Bibliography 715 Review Questions 716 Problems 716 16 Multilevel and Multiobjective Optimization 721 16.1 Introduction 721 16.2 Multilevel Optimization 721 16.2.1 Basic Idea 721 16.2.2 Method 722 16.3 Parallel Processing 726 16.4 Multiobjective Optimization 729 16.4.1 Utility Function Method 730 16.4.2 Inverted Utility Function Method 730 16.4.3 Global Criterion Method 730 16.4.4 Bounded Objective Function Method 730 16.4.5 Lexicographic Method 731 16.4.6 Goal Programming Method 732 16.4.7 Goal Attainment Method 732 16.4.8 Game Theory Approach 733 16.5 Solutions Using MATLAB 735 References and Bibliography 735 Review Questions 736 Problems 737 17 Solution of Optimization Problems Using MATLAB 739 17.1 Introduction 739 17.2 Solution of General Nonlinear Programming Problems 740 17.3 Solution of Linear Programming Problems 742 17.4 Solution of LP Problems Using Interior Point Method 743 17.5 Solution of Quadratic Programming Problems 745 17.6 Solution of One-Dimensional Minimization Problems 746 17.7 Solution of Unconstrained Optimization Problems 746 17.8 Solution of Constrained Optimization Problems 747 17.9 Solution of Binary Programming Problems 750 17.10 Solution of Multiobjective Problems 751 References and Bibliography 755 Problems 755 A Convex and Concave Functions 761 B Some Computational Aspects of Optimization 767 B.1 Choice of Method 767 B.2 Comparison of Unconstrained Methods 767 B.3 Comparison of Constrained Methods 768 B.4 Availability of Computer Programs 769 B.5 Scaling of Design Variables and Constraints 770 B.6 Computer Programs for Modern Methods of Optimization 771 References and Bibliography 772 C Introduction to MATLAB® 773 C.1 Features and Special Characters 773 C.2 Defining Matrices in MATLAB 774 C.3 Creating m-Files 775 C.4 Optimization Toolbox 775 Answers to Selected Problems 777 Index 787

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Book SynopsisDieses vierfarbige Lehrbuch wendet sich an Studierende der Mathematik in Bachelor- und Lehramts-Studiengängen. Es bietet in einem Band ein lebendiges Bild der mathematischen Inhalte, die üblicherweise im ersten Studienjahr behandelt werden (und etliches mehr). Mathematik-Studierende finden wichtige Begriffe, Sätze und Beweise ausführlich und mit vielen Beispielen erklärt und werden an grundlegende Konzepte und Methoden herangeführt.Im Mittelpunkt stehen das Verständnis der mathematischen Zusammenhänge und des Aufbaus der Theorie sowie die Strukturen und Ideen wichtiger Sätze und Beweise. Es wird nicht nur ein in sich geschlossenes Theoriengebäude dargestellt, sondern auch verdeutlicht, wie es entsteht und wozu die Inhalte später benötigt werden.Herausragende Merkmale sind:- durchgängig vierfarbiges Layout mit mehr als 600 Abbildungen- prägnant formulierte Kerngedanken bilden die Abschnittsüberschriften- Selbsttests in kurzen Abständen ermöglichen Lernkontrollen während des Lesens- farbige Merkkästen heben das Wichtigste hervor- „Unter-der-Lupe“-Boxen zoomen in Beweise hinein, motivieren und erklären Details- „Hintergrund-und-Ausblick“-Boxen stellen Zusammenhänge zu anderen Gebieten und weiterführenden Themen her- Zusammenfassungen zu jedem Kapitel sowie Übersichtsboxen- mehr als 400 Verständnisfragen, Rechenaufgaben und Aufgaben zu Beweisen- deutsch-englisches Symbol- und Begriffsglossar Der inhaltliche Schwerpunkt liegt auf den Themen der Vorlesungen Analysis 1 und 2 sowie Linearer Algebra 1 und 2. Behandelt werden darüber hinaus Inhalte und Methodenkompetenzen, die vielerorts im ersten Studienjahr der Mathematikausbildung vermittelt werden.Hinweise, Lösungswege und Ergebnisse zu allen Aufgaben des Buchs stehen als PDF-Dateien auf http://sn.pub/extras in dem Ordner für das Werk Arens et al, „Mathematik“, Copyrightjahr 2018 zur Verfügung. Das Buch wird allen Studierenden der Mathematik vom Beginn des Studiums bis in höhere Semester hinein ein verlässlicher Begleiter sein.Für die 2. Auflage ist es vollständig durchgesehen, an zahlreichen Stellen didaktisch weiter verbessert und um einige Themen ergänzt worden.Stimme zur ersten Auflage:„Besonders gut gefallen mir die Übersichtlichkeit und die Verständlichkeit, besonders aber die Sichtbarmachung der Verbindung von Analysis und linearer Algebra, die in den Erstsemestervorlesungen oft zu kurz kommt.” Sylvia Prinz, Institut für Mathematikdidaktik, Universität zu KölnTable of ContentsVorwort.- 1 Was ist Mathematik und was tun Mathematiker?- 2 Logik, Mengen, Abbildungen − die Sprache der Mathematik.- 2.1 Junktoren und Quantoren.- 2.2 Grundbegriffe aus der Mengenlehre.- 2.3 Abbildungen.- 2.4 Relationen.- Zusammenfassung.- Aufgaben.- 3 Algebraische Strukturen − ein Blick hinter die Rechenregeln.- 3.1 Gruppen.- 3.2 Homomorphismen.- 3.3 Körper.- 3.4 Ringe.- Zusammenfassung.- Aufgaben.- 4 Zahlbereiche − Basis nicht nur der Analysis.- 4.1 Reelle Zahlen.- 4.2 Körperaxiome für die reellen Zahlen.- 4.3 Anordnungsaxiome für die reellen Zahlen.- 4.4 Ein Vollständigkeitsaxiom für die reellen Zahlen.- 4.5 Natürliche Zahlen und vollständige Induktion.- 4.6 Ganze Zahlen und rationale Zahlen.- 4.7 Komplexe Zahlen: Ihre Arithmetik und Geometrie.- Zusammenfassung.- Aufgaben.- 5 Lineare Gleichungssysteme − ein Tor zur linearen Algebra.- 5.1 Erste Lösungsversuche.- 5.2 Das Lösungsverfahren von Gauß und Jordan.- 5.3 Das Lösungskriterium und die Struktur der Lösung.- Zusammenfassung.- Aufgaben.- 6 Vektorräume − von Basen und Dimensionen.- 6.1 Der Vektorraumbegriff.- 6.2 Beispiele von Vektorräumen.- 6.3 Untervektorräume.- 6.4 Basis und Dimension.- 6.5 Summe und Durchschnitt von Untervektorräumen.- Zusammenfassung.- Aufgaben.- 7 Analytische Geometrie − Rechnen statt Zeichnen.- 7.1 Punkte und Vektoren im Anschauungsraum.- 7.2 Das Skalarprodukt im Anschauungsraum.- 7.3 Weitere Produkte von Vektoren im Anschauungsraum.- 7.4 Abstände zwischen Punkten, Geraden und Ebenen.- 7.5 Wechsel zwischen kartesischen Koordinatensystemen.- Zusammenfassung.- Aufgaben.- 8 Folgen − der Weg ins Unendliche.- 8.1 Der Begriff einer Folge.- 8.2 Konvergenz.- 8.3 Häufungspunkte und Cauchy-Folgen.- Zusammenfassung.- Aufgaben.- 9 Funktionen und Stetigkeit − ε trifft auf δ.- 9.1 Grundlegendes zu Funktionen.- 9.2 Beschränkte und monotone Funktionen.- 9.3 Grenzwerte für Funktionen und die Stetigkeit.- 9.4 Abgeschlossene, offene, kompakte Mengen.- 9.5 Stetige Funktionen mit kompaktem Definitionsbereich, Zwischenwertsatz.- Zusammenfassung.- Aufgaben.- 10 Reihen − Summieren bis zum Letzten.- 10.1 Motivation und Definition.- 10.2 Kriterien für Konvergenz.- 10.3 Absolute Konvergenz.- 10.4 Kriterien für absolute Konvergenz.- Zusammenfassung.- Aufgaben.- 11 Potenzreihen − Alleskönner unter den Funktionen.- 11.1 Definition und Grundlagen.- 11.2 Die Darstellung von Funktionen durch Potenzreihen.- 11.3 Die Exponentialfunktion.- 11.4 Trigonometrische Funktionen.- 11.5 Der Logarithmus.- Zusammenfassung.- Aufgaben.- 12 Lineare Abbildungen und Matrizen − Brücken zwischen Vektorräumen.- 12.1 Definition und Beispiele.- 12.2 Verknüpfungen von linearen Abbildungen.- 12.3 Kern, Bild und die Dimensionsformel.- 12.4 Darstellungsmatrizen.- 12.5 Das Produkt von Matrizen.- 12.6 Das Invertieren von Matrizen.- 12.7 Elementarmatrizen.- 12.8 Basistransformation.- 12.9 Der Dualraum.- Zusammenfassung.- Aufgaben.- <13 Determinanten − Kenngrößen von Matrizen.- 13.1 Die Definition der Determinante.- 13.2 Determinanten von Endomorphismen.- 13.3 Berechnung der Determinante.- 13.4 Anwendungen der Determinante.- Zusammenfassung.- Aufgaben.- 14 Normalformen − Diagonalisieren und Triangulieren.- 14.1 Diagonalisierbarkeit.- 14.2 Eigenwerte und Eigenvektoren.- 14.3 Berechnung der Eigenwerte und Eigenvektoren.- 14.4 Algebraische und geometrische Vielfachheit.- 14.5 Die Exponentialfunktion für Matrizen.- 14.6 Das Triangulieren von Endomorphismen.- 14.7 Die Jordan-Normalform.- 14.8 Die Berechnung einer Jordan-Normalform und Jordan-Basis.- Zusammenfassung.- Aufgaben.- 15 Differenzialrechnung − die Linearisierung von Funktionen.- 15.1 Die Ableitung.- 15.2 Differenziationsregeln.- 15.3 Der Mittelwertsatz.- 15.4 Verhalten differenzierbarer Funktionen.- 15.5 Taylorreihen.- Zusammenfassung.- Aufgaben.- 16 Integrale − von lokal zu global.- 16.1 Integration von Treppenfunktionen.- 16.2 Das Lebesgue-Integral.- 16.3 Stammfunktionen.- 16.4 Integrationstechniken.- 16.5 Integration über unbeschränkte Intervalle oder Funktionen.- 16.6 Parameterabhängige Integrale.- 16.7 Weitere Integrationsbegriffe.- Zusammenfassung.- Aufgaben.- 17 Euklidische und unitäre Vektorräume − orthogonales Diagonalisieren.- 17.1 Euklidische Vektorräume.- 17.2 Norm, Abstand, Winkel, Orthogonalität.- 17.3 Orthonormalbasen und orthogonale Komplemente.- 17.4 Unitäre Vektorräume.- 17.5 Orthogonale und unitäre Endomorphismen.- 17.6 Selbstadjungierte Endomorphismen.- 17.7 Normale Endomorphismen.- Zusammenfassung.- Aufgaben.- 18 Quadriken − vielseitig nutzbare Punktmengen.- 18.1 Symmetrische Bilinearformen.- 18.2 Hermitesche Sesquilinearformen.- 18.3 Quadriken und ihre Hauptachsentransformation.- 18.4 Die Singulärwertzerlegung.- 18.5 Die Pseudoinverse einer linearen Abbildung.- Zusammenfassung.- Aufgaben.- 19 Funktionenräume − Analysis und lineare Algebra Hand in Hand.- 19.1 Metrische Räume und ihre Topologie, normierte Räume.- 19.2 Konvergenz und Stetigkeit in metrischen Räumen.- 19.3 Kompaktheit.- 19.4 Zusammenhangsbegriffe.- 19.5 Vollständigkeit.- 19.6 Banach- und Hilberträume.- Zusammenfassung.- Aufgaben.- 20 Differenzialgleichungen − Funktionen sind gesucht.- 20.1 Begriffsbildungen.- 20.2 Elementare analytische Techniken.- 20.3 Existenz und Eindeutigkeit.- 20.4 Grundlegende numerische Verfahren.- Zusammenfassung.- Aufgaben .- 21 Funktionen mehrerer Variablen − Differenzieren im Raum.- 21.1 Einführung.- 21.2 Differenzierbarkeitsbegriffe: Totale und partielle Differenzierbarkeit.- 21.3 Differenziationsregeln.- 21.4 Mittelwertsätze und Schranksätze.- 21.5 Höhere partielle Ableitungen und der der Vertauschungssatz von H. A. Schwarz.- 21.6 Taylor-Formel und lokale Extrema.- 21.7 Der Lokale Umkehrsatz.- 21.8 Der Satz über implizite Funktionen.- Zusammenfassung.- Aufgaben.- 22 Gebietsintegrale − das Ausmessen von Mengen.- 22.1 Definition und Eigenschaften.- 22.2 Die Berechnung von Integralen.- 22.3 Die Transformationsformel.- 22.4 Wichtige Koordinatensysteme.- Zusammenfassung.- Aufgaben.- 23 Vektoranalysis − im Zentrum steht der Gauß'sche Satz.- 23.1 Kurven und Kurvenintegrale.- 23.2 Flächen und Flächenintegrale.- 23.3 Der Gauß’sche Satz.- Zusammenfassung.- Aufgaben.- 24 Optimierung − ein sehr generelles Problem.- 24.1 Lineare Optimierung.- 24.2 Das Simplex-Verfahren.- 24.3 Dualitätstheorie.- Zusammenfassung.- Aufgaben.- 25 Elementare Zahlentheorie − Teiler und Vielfache.- 25.1 Teilbarkeit.- 25.2 Der euklidische Algorithmus.- 25.3 Der Fundamentalsatz der Arithmetik.- 25.4 ggT und kgV.- 25.5 Zahlentheoretische Funktionen.- 25.6 Rechnen mit Kongruenzen.- Zusammenfassung.- Aufgaben.- 26 Elemente der diskreten Mathematik − die Kunst des Zählens.- 26.1 Einführung in die Graphentheorie.- 26.2 Einführung in die Kombinatorik.- 26.3 Erzeugende Funktionen.- Zusammenfassung.- Aufgaben.- Hinweise zu den Aufgaben.- Lösungen zu den Aufgaben.- Symbolglossar.- Index.

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Book SynopsisRich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.Trade Review“The aim of this book is to survey the basic notions and results of general algebra; also, it is a detailed and self-contained introduction to general algebra from the point of view of categories and functors. … The author takes care in writing full proofs throughout the book and he shows also ways of possible applications. The text contains a wealth material and should serve as a textbook for readers interested in this field.” (Danica Jakubíková-Studenovská, zbMATH 1317.08001, 2015)Table of Contents1 About the course, and these notes.- Part I: Motivation and Examples.- 2 Making Some Things Precise.- 3 Free Groups.- 4 A Cook's Tour.- Part II: Basic Tools and Concepts.- 5 Ordered Sets, Induction, and the Axiom of Choice.- 6 Lattices, Closure Operators, and Galois Connections.- 7 Categories and Functors.- 8 Universal Constructions.- 9 Varieties of Algebras.- Part III: More on Adjunctions.- 10 Algebras, Coalgebras, and Adjunctions.- References.- List of Exercises.- Symbol Index.- Word and Phrase Index.

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Book SynopsisThis book and CD-ROM package has been developed in a lab-oriented course taught at Cal Tech in 1995 and 1996, and simultaneously augmented by artificial life research conducted there.Table of Contents1 Flavors of Artificial Life.- 1.1 Whither a Theory of the Living State?.- 1.2 Emulation and Simulation.- 1.3 Carbon-Based Artificial Life.- 1.4 Turing and von Neumann Automata.- 1.5 Cellular Automata.- 1.6 Overview.- 2 Artificial Chemistry and Self-Replicating Code.- 2.1 Virtual Machines and Self-Reproducing CA.- 2.2 Viruses and Core Worlds.- 2.3 The tierra System.- 2.4 avida, amoeba, and the Origin of Life.- 2.5 Overview.- 3 Introduction to Information Theory.- 3.1 Information Theory and Life.- 3.2 Channels and Coding.- 3.3 Uncertainty and Shannon Entropy.- 3.4 Joint and Conditional Uncertainty.- 3.5 Information.- 3.6 Noiseless Coding.- 3.7 Channel Capacity and Fundamental Theorem.- 3.8 Information Transmission Capacity for Genomes.- 3.9 Overview.- 4 Statistical Mechanics and Thermodynamics.- 4.1 Phase Space and Statistical Distribution Function.- 4.2 Averages, Ergodicity, and the Ergodic Theorem.- 4.3 Thermodynamical Equilibrium, Relaxation.- 4.4 Energy.- 4.5 Entropy.- 4.6 Second Law of Thermodynamics.- 4.7 Tèmperature.- 4.8 The Gibbs Distribution.- 4.9 Nonequilibrium Thermodynamics.- 4.10 First-Order Phase Transitions.- 4.11 Overview.- 5 Complexity of Simple Living Systems.- 5.1 Complexity and Information.- 5.2 The Maxwell Demon.- 5.3 Kolmogorov Complexity.- 5.4 Physical Complexity and the Natural Maxwell Demon.- 5.5 Complexity of tRNA.- 5.6 Complexity in Artificial Life.- 5.7 Overview.- 6 Self-Organization to Criticality.- 6.1 Self-Organization and Sandpiles.- 6.2 SOC in Forest Fires.- 6.3 SOC in the Living State.- 6.4 Theories of SOC.- 6.5 Overview.- 7 Percolation.- 7.1 Site Percolation.- 7.2 Cluster Size Distribution.- 7.3 Percolation in 1D.- 7.4 Higher-Dimensional Euclidean Lattices.- 7.5 Percolation on the Bethe Lattice.- 7.6 Scaling Theory.- 7.7 Percolation and Evolution.- 7.8 Overview.- 8 Fitness Landscapes.- 8.1 Theoretical Formulation.- 8.2 Example Landscapes.- 8.3 Fractal Landscapes.- 8.4 Diffusive and Nondiffusive Processes.- 8.5 RNA Landscapes.- 8.6 Fitness Landscape in avida.- 8.7 Overview.- 9 Experiments with avida.- 9.1 Choice of Chemistry.- 9.2 A Simple Experiment.- 9.3 Experiments in Adaptation.- 9.4 Experiments with Species and Genetic Distance.- 9.5 Overview.- 10 Propagation of Information.- 10.1 Information Transport and Equilibrium.- 10.2 The Artificial Life System sanda.- 10.3 Diffusion and Waves.- 10.4 Comparison: Theory and Experiment.- 10.5 Overview.- 11 Adaptive Learning at the Error Threshold.- 11.1 Information Processing at the Edge of Chaos.- 11.2 Adaptation to Computation in avida.- 11.3 Eigen’s Error Threshold.- 11.4 Molecular Evolution as an Ising Model.- 11.5 The Race to the Error Threshold.- 11.6 Approach to Error Threshold in avida.- 11.7 Overview.- A The avida User’s Manual.- A.1 Introduction.- A.2 A Beginner’s Guide to avida.- A.3 Time Slicing and the Fitness Landscape.- A.4 Reproduction.- A.5 The Virtual Computer.- A.6 Mutations.- A.7 Installing avida.- A.8 The Text Interface.- A.9 Configuring avida Runs.- A.10 Guide to Output Files.- A.11 Summary of Variables.- A.12 Glossary.- References.

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Book SynopsisA book for anyone who wishes to illustrate their mathematical ideas. It is organised by material, rather than by subject area, and purposefully emphasizes the process of creating things, including discussions of failures that occurred along the way.Table of Contents Drawings Paper & fiber arts Laser cutting Graphics Video & virtual reality 3D printing Mechanical constructions and other materials Multiple ways to illustrate the same thing Acknowledgments Image credits Index.

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Book SynopsisEveryone who's had to get to grips with chance knows how tricky even its simplest manifestations can be. Its workings are a constant challenge to common sense: a run of luck goes bad just when you trust it; expert predictions of everything from the weather to elections prove hopelessly unreliable; proven health advice turns out to be anything but. Award-winning scientist and writer Robert Matthews shows us how we can cut through the conundrums of chance. He gives us access to some of the most potent intellectual tools ever developed, and explains how we can use them to guide our judgements and decisions. By the end of the book you'll know: -The secret to predicting coincidences; -The golden rule of professional gamblers; -How to tell when insurance is a waste of money; -When to heed health and diet warnings - and when to ignore them; -How to tell when forecasts are worth taking seriously; -How to make better choices in the face of uncertainty. Using a host of real-life examples, this groundbreaking book shows how the laws of probability can sharpen your decisions, make the most of your luck - and quite possibly transform your life.Trade ReviewIt takes an extraordinary writer to animate this driest of subjects for a general audience. That writer is Matthews ... At a time when mathematics needs charismatic ambassadors more than ever, Matthews has written a book of great significance. -- Oliver Moody * Times *Beguiling ... Matthews has the knack of explaining things clearly for the nonspecialist, leavening the formulae with intriguing snippets of history and biography ... his enthusiasm contributes to a lively and fascinating narrative. -- Ian Critchley * Sunday Times *Praise for Why Don't Spiders Stick to Their Webs: "Matthews gives us his wisdom like a beneficent and well-read uncle, entertaining his guests at the dinner table." -- Brian Clegg * Popular Science Books *Praise for 25 Big Ideas: "Robert Matthews has a gift for finding the simple, fascinating stories at the heart of concepts transforming the modern world" -- John Rennie, former Editor * Scientific American *

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Book SynopsisInspire and engage your students with this fully updated Lower Secondary Maths course from Collins offering comprehensive coverage of the curriculum framework and Thinking and Working Mathematically skills. Written by an experienced team, each Stage (7–9) comprises a comprehensive Student’s Book, extensive Workbook and supportive Teacher’s Guide.

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Book SynopsisWhile Eugenio Calabi is best known for his contributions to the theory of Calabi-Yau manifolds, this Steele-Prize-winning geometer’s fundamental contributions to mathematics have been far broader and more diverse than might be guessed from this one aspect of his work. His works have deep influence and lasting impact in global differential geometry, mathematical physics and beyond. By bringing together 47 of Calabi’s important articles in a single volume, this book provides a comprehensive overview of his mathematical oeuvre, and includes papers on complex manifolds, algebraic geometry, Kähler metrics, affine geometry, partial differential equations, several complex variables, group actions and topology. The volume also includes essays on Calabi’s mathematics by several of his mathematical admirers, including S.K. Donaldson, B. Lawson and S.-T. Yau, Marcel Berger; and Jean Pierre Bourguignon. This book is intended for mathematicians and graduate students around the world. Calabi’s visionary contributions will certainly continue to shape the course of this subject far into the future.Trade Review“In my case, I spent several happy hours learning about affine differential geometry, something that would certainly never have happened if I had not picked up this volume. … The collected works of Eugenio Calabi are worthy of a place on the bookshelf of any person with a serious interest in differential geometry.” (Joel Fine, EMS Magazine, May 11, 2023)Table of ContentsPreface.- J.-P. Bourguignon, Eugenio Calabi’s Short Biography.- Bibliographic List of Works.- S.-T. Yau, An Essay on Eugenio Calabi.- Part I: Commentaries on Calabi’s Life and Work: B. Lawson, Reflections on the Early Work of Eugenio Calabi.- M. Berger, Encounter with a Geometer: Eugenio Calabi.- J.-P. Bourguignon, Eugenio Calabi and Kähler Metrics.- C. LeBrun, Eugenio Calabi and the Curvature of Kähler Manifolds.- X. Chen, S. Donaldson, Calabi’s Work on Affine Differential Geometry and Results of Bernstein Type.- Part II: Collected Works: E. Calabi ,Ar. Dvoretzky, Convergence- and Sum-Factors for Series of Complex Numbers (1951).- E. Calabi, D. C. Spencer, Completely Integrable Almost Complex Manifolds (1951).- E. Calabi, Metric Riemann Surfaces (1953).- E. Calabi, M. Rosenlicht, Complex Analytic Manifolds Without Countable Base (1953).- E. Calabi, B. Eckmann, A Class of Compact, Complex Manifolds Which Are Not Algebraic (1953).- E. Calabi, Isometric Imbedding of Complex Manifolds (1953).- E. Calabi, The Space of Kähler Metrics (1954).- E. Calabi, The Variation of Kähler Metrics I. The Structure of the Space (1954).- E. Calabi, The Variation of Kähler Metrics II. A Minimum Problem (1954).- E. Calabi, On Kähler Manifolds With Vanishing Canonical Class (1957).- E. Calabi, Construction and Properties of Some 6-Dimensional Almost Complex Manifolds (1958).- E. Calabi, Improper Affine Hyperspheres of Convex Type and a Generalization of a Theorem by K. Jörgens (1958).- E. Calabi, An Extension of E. Hopf’s Maximum Principle with an Application to Riemannian Geometry (1958).- E. Calabi, Errata: An Extension of E. Hopf’s Maximum Principle with an Application to Riemannian Geometry (1959).- E. Calabi, E. Vesentini, Sur les variétés complexes compactes localement symétriques (1959).- E. Calabi, E. Vesentini, On Compact, Locally Symmetric Kähler Manifolds (1960).- E. Calabi, On Compact, Riemannian Manifolds with Constant Curvature I. (1961).- E. Calabi, L. Markus Relativistic Space Forms (1962).- E. Calabi, Linear Systems of Real Quadratic Forms (1964).- E. Calabi, Quasi-Surjective Mappings and a Generalization of Morse Theory (1966).- E. Calabi, Minimal Immersions of Surfaces in Euclidean Spheres (1967).- E. Calabi, On Ricci Curvature and Geodesics (1967).- E. Calabi, On Differentiable Actions of Compact Lie Groups on Compact Manifolds (1968).- E. Calabi, An Intrinsic Characterization of Harmonic One-Forms (1969).- E. Calabi, On the Group of Automorphisms of a Symplectic Manifold (1970).- E. Calabi, P. Hartman, On the Smoothness of Isometries (1970).- E. Calabi, Examples of Bernstein Problems for Some Nonlinear Equations (1970).- E. Calabi, Über singuläre symplektische Strukturen (1971).- E. Calabi, Complete Affine Hyperspheres I (1972).- E. Calabi, A Construction of Nonhomogeneous Einstein Metrics (1975).- E. Calabi, H. S. Wilf, On the Sequential and Random Selection of Subspaces Over a Finite Field (1977).- E. Calabi, Métriques kählériennes et fibrés holomorphes (1978).- E. Calabi, Isometric Families of Kähler Structures (1980).- E. Calabi, Géométrie différentielle affine des hypersurfaces (1981).- E. Calabi, Linear Systems of Real Quadratic Forms II (1982).- E. Calabi, Extremal Kähler Metrics (1982).- E. Calabi, Hypersurfaces with Maximal Affinely Invariant Area (1982).- E. Calabi, Extremal Kähler Metrics II (1985).- E. Calabi, Convex Affine Maximal Surfaces (1988).- E. Calabi, Affine Differential Geometry and Holomorphic Curves (1990).- E. Calabi, J. Cao Simple Closed Geodesics on Convex Surfaces (1992).- F. Beukers, J. A. C. Kolk and E. Calabi, Sums of Generalized Harmonic Series and Volumes (1993).- E. Calabi and H. Gluck, What are the Best Almost-Complex Structures on the 6-Sphere? (1993).- E. Calabi, Extremal Isosystolic Metrics for Compact Surfaces (1996).- E. Calabi, P. J. Olver, A. Tannenbaum, Affine Geometry, Curve Flows, and Invariant Numerical Approximations (1996).- J.-P. Bourguignon, E. Calabi, J. Eells, O. Garcia-Prada, M. Gromov, Where Does Geometry Go? A Research and Education Perspective (2001).- E. Calabi, X. Chen, The Space of Kähler Metrics II (2002).- Acknowledgements.

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Book SynopsisWhy is 7 such a lucky number and 13 so unlucky? Why does a jury traditionally have `12 good men and true', and why are there 24 hours in the day and 60 seconds in a minute? This fascinating new book explores the world of numbers from pin numbers to book titles, and from the sixfold shape of snowflakes to the way our roads, houses and telephone numbers are designated in fact and fiction. Using the numbers themselves as its starting point it investigates everything from the origins and meaning of counting in early civilizations to numbers in proverbs, myths and nursery rhymes and the ancient `science' of numerology. It also focuses on the quirks of odds and evens, primes, on numbers in popular sports - and much, much more. So whether you've ever wondered why Heinz has 57 varieties, why 999 is the UK's emergency phone number but 911 is used in America, why Coco Chanel chose No. 5 for her iconic perfume, or how the title Catch 22 was chosen, then this is the book for you. Dip in anywhere and you'll find that numbers are not just for adding and measuring but can be hugely entertaining and informative whether you're buying a diamond or choosing dinner from the menu.Table of Contents1 Introduction 8 2 Numbers of many sorts 10 3 Numbers and counting 12 4 1 - The number of unity 17 5 2 - The duality 19 6 3 - The trinity of perfection 21 7 4 - Truth and justice 23 8 5 - The number of nature 26 9 6 - Without a fault Six and the snowflake 29 10 7 - The symbol of fortune 32 11 The seven wonders of the world 35 12 Puzzles to solve 37 13 8 - Harmony and balance 38 14 9 - Unbounded 40 15 The nine muses 43 16 10 - On our fingers and toes 44 17 11 - The final hour 46 18 12 - A number in time 47 19 The twelve labours of Hercules 49 20 13 - And other teens 53 21 0 - The story of zero 54 22 Lucky - and unlucky - numbers 56 23 Odds and evens 59 24 Many favourite numbers 60 25 The secrets of numerology 62 26 Big numbers 66 27 Small numbers 68 28 The appeal of primes 69 29 Making shapes with numbers 72 30 Numbers - more different types 74 31 - the most famous number 75 32 Fibonacci - the brilliant number sequence 77 33 The golden ratio 79 34 Numbers in use 81 35 The world we live in 84 36 Our planet earth 86 37 Lines on the map 89 38 Measuring the world 94 39 A matter of weight 98 40 By volume 101 41 More about money 104 42 Inventing the calendar 106 43 Time and the circle 109 44 The living world 110 45 The human body 114 46 A number for your home 120 47 Addresses in fiction 121 48 A dark history 123 49 The streets of power 124 50 Numbers for the post 126 51 Roads to take - navigating by numbers 127 52 The number to call 130 53 PIN - what's your number? 132 54 Edible connections 134 55 Sizing up our drinks 137 56 Of yarns, fabrics and clothes 139 57 All that glitters 140 58 Beauty by numbers 142 59 For our leisure and entertainment 144 60 Proverbs and sayings 146 61 Bingo lingo 147 62 Books with numbers 149 63 In the film title 156 64 Counting in song 161 65 Jazz numbers 163 66 A musical miscellany 165 67 Poetry's secrets revealed 167 68 The beautiful game 170 69 The oval ball game - rugby football 174 70 The game of golf 176 71 Throwing darts 178 72 Cricket - bat on ball 179 73 On court - the game of tennis 182 74 Snooker and other cue games 185 75 The game of baseball 187 76 Chancing your luck 189 77 Throwing dice 192 78 Playing dominoes 194 79 Solving the square 195 80 Index 200 81 About the author 208

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Book SynopsisWenn auch Sie Ihre kleinen Problemchen mit medizinischer Statistik haben, sind Sie hier genau richtig. Mit viel Witz bringen Ihnen die Autoren Geraldine Rauch, Konrad Neumann, Ulrike Grittner, Carolin Herrmann und Jochen Kruppa die Prinzipien der Biostatistik näher. In diesem Buch lernen Sie alles, was Sie benötigen, um Statistik im medizinischen Bereich erfolgreich einzusetzen. Angefangen bei der Begriffskunde und den Grundlagen, erfahren Sie alles von Studientypen über deskriptive Verfahren, Verteilungen, Schätzungen oder Korrelation und Regression bis hin zur Ereigniszeitanalyse, diagnostischen Tests und multiplem Testen. Die Autoren bringen Ihnen das theoretisch vermittelte Wissen mit vielen anschaulichen Beispielen näher. So schaffen Sie die nächste Klausur mit Links!Table of ContentsÜber die Autoren 7 Einleitung 17 Über dieses Buch 17 Was Sie nicht lesen müssen 18 Konventionen in diesem Buch 18 Törichte Annahmen über den Leser 19 Wie dieses Buch aufgebaut ist 19 Teil I: Medizinische Statistik – Gel(i)ebte Daten 19 Teil II: Keine Forschung ohne Fundament - Grundlagen für einen gelungenen Start 19 Teil III: Was Sie unbedingt brauchen - Theorie trifft Praxis 20 Teil IV: Blick über den Tellerrand - weiterführende Methoden 20 Symbole, die in diesem Buch verwendet werden 20 Teil I: Medizinische Statistik - Gel(i)ebte Daten 23 Kapitel 1 Statistik und Medizin – wie passt das zusammen? 25 Medizinische Statistik, Biostatistik, medizinische Biometrie: Eine Begriffsbestimmung 26 Wo wird medizinische Statistik gebraucht? 26 Aufgabenbereiche der medizinischen Statistik 27 Anwendung statistischer Methoden in medizinischen Forschungsprojekten 27 Kommunikation mit medizinischen Anwendern 28 Entwicklung neuer statistischer Methoden 28 Literatur 29 Kapitel 2 Besser beraten lassen - Hilfe holen erwünscht 31 Wo finden Sie biometrische Unterstützung? 31 Vorbereitung auf einen Beratungstermin 33 Welche Software brauchen Sie? 34 SPSS - gut für Anwender 34 R Project - Am Puls der neusten biometrischen Methoden 36 SAS und STATA - validiert für die Industrie 36 Literatur 37 Teil II: Keine Forschung ohne Fundament - Grundlagen 39 Kapitel 3 Grundbegriffe und Studientypen 41 Grundlegende Begriffe empirischer Forschung in der Medizin 42 Grundgesamtheit und Stichprobe 42 Validität und Reliabilität 45 Endpunkte 47 Störgrößen und Verzerrung 49 Verschiedene Studientypen unterscheiden 50 Unterscheidung nach Zielsetzung - konfirmatorische versus explorative Studien 50 Unterscheidung nach Blickrichtung - prospektive versus retrospective Studien 53 Studientypen in der Primär- und Sekundärforschung 54 Klinische Studie konzipieren - das Studiendesign 56 Kontrolle ist besser - die Kontrollgruppe 57 Zufällige Zuteilung - Randomisierung 59 Denn Sie wissen nicht, was sie tun - Verblindung 65 Noch einmal in Kürze 66 Literatur 67 Kapitel 4 Modelle für die Wirklichkeit 69 Was sind Wahrscheinlichkeiten? 70 Modellannahmen, Verteilungen und Schätzung 72 Merkmale und Verteilungen 72 Zufallsfehler und Bias 78 Gängige Verteilungsannahmen 80 Die First Lady der Verteilungen – die Normalverteilung 80 Die Binomialverteilung 85 Weitere Verteilungen 88 Literatur 91 Teil III: Was Sie unbedingt brauchen - Theorie trifft Praxis 93 Kapitel 5 Die Kunst der Beschreibung - Deskriptive Statistik 95 Was ist das eigentlich - deskriptive Statistik? 96 Wo brauchen Sie deskriptive Statistik? 97 Merkmale unterscheiden - Skalenniveaus 99 Methoden der Deskription 104 Beschreibung kategorieller Merkmale 105 Kennzahlen, Tabellen und Lagemaße 105 Grafische Darstellung - Torten und Balken 108 Beschreibung ordinalskalierter Merkmale 111 Lage und Streuung - Median und Quartile 111 Grafische Darstellung - der Boxplot 114 Beschreibung intervall- und verhältnisskalierter Merkmale 117 Lage und Streuung - Mittelwert und Standardabweichung 117 Grafische Darstellung - Histogramme 122 Wichtiges hervorheben, Unwichtiges weglassen 125 Literatur 126 Kapitel 6 Nachweis durch Kontrolle des Zufalls – Konfirmatorische Statistik 127 Konfirmatorisch, induktiv, schließend - eine Begriffsbestimmung 128 Idee des statistischen Tests - der konfirmatorische Umweg 128 Die sechs Schritte des statistischen Tests 130 Von der Fragestellung zur Hypothese 130 Die Formulierung der Fragestellung 131 Formulierung der Null- und Alternativhypothese 132 Einseitig und zweiseitig formulierte Hypothesen 134 Was sagen die Daten? Von den Daten zur Testentscheidung 135 Was ist extrem? Wahl einer geeigneten Teststatistik 136 Verteilung der Teststatistik und kritischer Wert 138 p-Wert und Signifikanzniveau 142 Interpretation des Testergebnisses - nichts ist bewiesen 145 Fehlentscheidungen und Fehlerwahrscheinlichkeiten 145 Literatur 152 Kapitel 7 t-Test & Co: Die Klassiker unter den Tests 153 Statistische Tests zum Vergleich von Erwartungswerten und anderen Lagemaßen 154 Der t-Test für zwei unverbundene Stichproben 154 Der t-Test für zwei unverbundene Stichproben mit unterschiedlichen Standardabweichungen 161 Der t-Test für zwei verbundene Stichproben 163 Die Varianzanalyse (ANOVA) für mehr als zwei unverbundene Stichproben 168 Der U-Test für zwei unverbundene Stichproben 174 Der Vorzeichen-Rang-Test nach Wilcoxon für zwei verbundene Stichproben 182 Der Kruskal-Wallis-Test für mehr als zwei unverbundene Stichproben 187 Statistische Tests zum Vergleich von Anteilen und Wahrscheinlichkeiten 190 Der Chiquadrat-Test für zwei unverbundene Stichproben 191 Der Chiquadrat-Test für allgemeine Kreuztabellen 197 Der Binomialtest für eine Stichprobe 201 Der McNemar-Test für zwei verbundene Stichproben 205 Literatur 209 Kapitel 8 Den Behandlungseffekt quantifizieren – Punktschätzer und Konfidenzintervalle 211 Quantifizierung des Effekts - der Punktschätzer 212 Die Größe des Effekts - das Konfidenzintervall 214 Signifikanz versus Relevanz - nicht zu verwechseln 222 Punktschätzer und Konfidenzintervalle für verschiedene Datensituationen 225 Punktschätzer und Konfidenzintervall für Erwartungswerte 227 Punktschätzer und Konfidenzintervall für die Differenz zweier Erwartungswerte 229 Punktschätzer und Konfidenzintervall für einen Anteil 232 Punktschätzer und Konfidenzintervall für die Differenz zweier Anteile 236 Literatur 238 Kapitel 9 Was sonst noch wichtig ist - Vor und nach dem statistischen Test 239 Kontrolle des Fehlers 2 Art? Grundprinzip der Fallzahlplanung 240 Mehr als eine Fragestellung - multiples Testen 252 Adjustierung für multiples Testen 255 Das Ergebnis einer Studie berichten 264 Literatur 265 Kapitel 10 Zusammenhänge und Vorhersage – Korrelation und Regression 267 Wie stark ist die Verbindung - Maße des Zusammenhangs 268 Der Korrelationskoeffizient nach Pearson 270 Der Korrelationskoeffizient nach Spearman 280 Kendalls τ 284 Der φ-Koeffizient 285 Regressionsmodelle 287 Die lineare Regression 289 Erweiterte (lineare) Regressionsmodelle 295 Die logistische Regression 300 Literatur 305 Teil IV: Blick über den Tellerrand – weiterführende Methoden 307 Kapitel 11 Wer lebt länger? Analyse von Ereigniszeiten 309 Was sind Ereigniszeitdaten? Zeiten, Ereignisse und Zensierungen 309 Schätzung von Ereigniswahrscheinlichkeiten - Kaplan-Meier zeigt, wie es geht 313 Gruppenvergleich - Überlebensfunktion, Hazards und Hazard Ratios 321 Logrank-Test und Cox-Regression 323 Literatur 325 Kapitel 12 Methoden zur Bewertung der Diagnostik und Übereinstimmung 327 Diagnostische Studien 327 Goldstandard und Referenzdiagnostik 328 Güte von diagnostischen Tests - Sensitivität und Spezifität 329 Prädiktive Werte und Satz von Bayes 333 Die ROC-Kurve 337 Literatur 344 Kapitel 13 Ausgewählte Methoden epidemiologischer Studien 345 Verzerrungen vermeiden durch Matching 346 Verschiedene Arten des Matchings 349 Auswertung gematchter Daten 353 Löcher in den Daten - Vom Umgang mit fehlenden Werten 353 Fehlen die Daten zufällig? Mechanismen fehlender Werte 354 Fehlende Werte ersetzen - Imputation 355 Literatur 356 Kapitel 14 Methodik von systematischen Reviews und Metaanalysen 357 Systematische Reviews und Metaanalysen in der Medizin 357 Ablauf von systematischen Reviews und Metaanalysen 359 Vom systematischen Review zur Metaanalyse – Gepoolte Effektschätzer 362 Grafische Darstellung einer Metaanalyse - der Forest-Plot 365 Homogenität und Heterogenität 366 Publication Bias und Funnel-Plot 368 Vor- und Nachteile von systematischen Reviews und Metaanalysen 370 Literatur 371 Teil V: Der Top-Ten-Teil 373 Zehn statistische Irrtümer 375 Irrtum 1: Statistische Tests liefern wertvollere Ergebnisse als descriptive Datenauswertungen 375 Irrtum 2: Ein nicht-signifikantes Testergebnis deutet auf die Gültigkeit der Nullhypothese hin 376 Irrtum 3: Der p-Wert ist die Wahrscheinlichkeit, dass die Nullhypothese richtig ist 376 Irrtum 4: Ein kleiner p-Wert spricht immer für einen großen Effekt 377 Irrtum 5: Bei Verwendung eines parametrischen Tests müssen die Daten normalverteilt sein 377 Irrtum 6: Signifikante Ergebnisse sind immer auch klinisch relevant 377 Irrtum 7: Alle p-Werte unter 5% sind signifikante Ergebnisse 378 Irrtum 8: Für eine Fallzahlplanung werden nur ein paar Werte in eine Eingabemaske eingegeben 378 Irrtum 9: Ein multivariates Regressionsmodell dient der Vorhersage einer Zielgröße aus mehreren Einflussgrößen 379 Irrtum 10: Nur Studien mit signifikanten Ergebnissen sollten publiziert werden 379 Stichwortverzeichnis 381

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Book SynopsisTrade Review"I greatly enjoyed Richeson's Tales of Impossibility. It deserves to become a classic and can be highly recommended."---Robin Wilson, Times Higher Education"Even if you never read a single proof through to its conclusion, you’ll enjoy the many entertaining side trips into a geometry far beyond what you learned in high school."---Jim Stein, New Books in Mathematics"The whole book, both informative and amusing, is a highly recommended read."---Adhemar Bulteel, European Mathematical Society"This book was a pleasure to read and I would recommend it for anybody who wants a lovely overview of many areas of the history of mathematics, with a focus on some very easy to understand problems."---Jonathan Shock, Mathemafrica"Richeson clearly explains what it means to be impossible to solve a problem, cites other impossibility results, goes into detail about geometric constructions with various instruments, and discusses the defective proofs and the cranks that have turned up along the way." * Mathematics Magazine *"This fascinating text will appeal to all those interested in the history of mathematics, not leasy because of its helpful notes on each chapter and its two dozen pages of references for further reading"---Laurence E. Nicholas CMath FIMA, Mathematics Today"A fact-filled, insightful, panoramic view of how mathematics developed to what it is today transformed by folks thinking both inside and outside of G so as to resolve the impossible."---Andrew J. Simoson, Mathematical Intelligencer

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Book SynopsisA witty, fast-paced thriller with a dash of mathematics and a large dose of danger Life is not going smoothly for Tom Winscombe. His girlfriend Dorothy has vanished, taking with her all the equipment and money of the company she ran with her friend Ali. Now Tom and Ali are forced to eke out an awkward shared bedsit existence while they try to work out what she is up to. Meanwhile, Tom has other things on his mind, including how to untangle his father from a cryptocurrency scam, how to break into a hospital in order to interrogate an old acquaintance and what is the significance of the messages he’s been receiving from Rufus Fairbanks’s LinkedIn account. Tom and Ali’s investigations lead them in a host of unexpected and frankly dangerous directions, involving a pet python, an offshore stag do and an improbable application of the Fibonacci sequence. But at the end of it all, will they find Dorothy – and will she ever be able to explain just exactly what is going on?Trade ReviewPraise for Jonathan Pinnock: ‘Lovely stuff’ Ian Rankin'A series of humorous, riotous mathematical mysteries' David Nicholls‘He makes funny and self-deprecating company’ The Herald‘Jonathan Pinnock writes compelling tales with a deliciously wicked glint in his eye’ Ian Skillicorn, National Short Story Week ‘Jonathan Pinnock is Roald Dahl’s natural successor’ Vanessa Gebbie‘Funny, clever, and sometimes brilliantly daft. A comedy that I am sure would have made Pythagoras, Archimedes and Douglas Adams all laugh out loud’ Scott Pack on The Truth About Archie and Pye

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Book SynopsisQuantitative genetics is the study of continuously varying traits which make up the majority of biological attributes of evolutionary and commercial interest. This book provides a much-needed up-to-date, in-depth yet accessible text for the field. In lucid language, the author guides readers through the main concepts of population and quantitative genetics and their applications. It is written to be approachable to even those without a strong mathematical background, including applied examples, a glossary of key terms, and problems and solutions to support students in grasping important theoretical developments and their relevance to real-world biology. An engaging, must-have textbook for advanced undergraduate and postgraduate students. Given its applied focus, it also equips researchers in genetics, genomics, evolutionary biology, animal and plant breeding, and conservation genetics with the understanding and tools for genetic improvement, comprehension of the genetic basis of human Trade Review'Quantitative genetics as a scientific discipline isn't dead just yet, despite predictions of its demise over many decades. In fact, it is very much alive in the genomics era, across a wide range of disciplines, including plant and animal breeding, evolutionary genetics and human (medical) genetics. Armando Caballero's timely textbook, a translation and update from his Spanish version, combines a description of the theory and methods underlying quantitative trait variation in populations with data examples and applications from modern genome technologies. It is an excellent introduction to the field, and demonstrates once again how population and quantitative genetics theory has stood the test of time and is highly relevant today.' Peter M. Visscher, University of Queensland'Armando Caballero's work is a masterful tour through both evolutionary and applied quantitative genetics. It provides a fruitful and unusual blend of population and quantitative genetics, and it will be extremely useful for anyone who wants to learn more about either of these fields.' Michael Whitlock, University of British Columbia'As the field within genetics having arguably the deepest history, quantitative genetics continues as a lively endeavour advancing understanding of the inheritance and change of traits that are continuous in their distributions and complex in the genetic and environmental influences on them. I welcome Caballero's text for new generations of students coming up to speed in this important and challenging field. The problems and questions concluding each chapter will especially aid them in testing their growing understanding. This text will also serve as a valuable resource for established practitioners of quantitative genetics.' Ruth G. Shaw, University of MinnesotaTable of ContentsPreface; Preface to the Spanish version; 1. Continuous variation; 2. Forces of change in the allele frequencies; 3. Components of phenotypic values and variances; 4. Inbreeding and coancestry; 5. Effective population size; 6. Estimation of genetic values, variances and covariances; 7. Mutation; 8. Consequences of inbreeding; 9. Artificial selection; 10. Natural selection; 11. Genomic analysis of quantitative traits; Solution to the problems and self-assessment questions; Glossary; References; Index.

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Book SynopsisThe Bellman function, a powerful tool originating in control theory, can be used successfully in a large class of difficult harmonic analysis problems and has produced some notable results over the last thirty years. This book by two leading experts is the first devoted to the Bellman function method and its applications to various topics in probability and harmonic analysis. Beginning with basic concepts, the theory is introduced step-by-step starting with many examples of gradually increasing sophistication, culminating with CalderónZygmund operators and end-point estimates. All necessary techniques are explained in generality, making this book accessible to readers without specialized training in non-linear PDEs or stochastic optimal control. Graduate students and researchers in harmonic analysis, PDEs, functional analysis, and probability will find this to be an incisive reference, and can use it as the basis of a graduate course.Trade Review'I first encountered Bellman functions about 35 years ago when advising engineers striving to minimize the expenditure of diamond chips in silicon grinding. Fifteen years later I was amused to learn that Nazarov, Treil, and Volberg successfully applied similar ideas to a variety of problems in harmonic analysis. Together with Vasyunin (and other analysts), they developed these techniques into a powerful tool which is carefully explained in the present book. The book is written on a level accessible to graduate students and I recommend it to everyone who wishes to join the Bellman functions club.' Mikhail Sodin, Tel Aviv UniversityTable of ContentsIntroduction; 1. Examples of Bellman functions; 2. What you always wanted to know about Stochastic Optimal Control, but were afraid to ask; 3. Conformal martingales models. Stochastic and classical Ahlfors-Beurling operators; 4. Dyadic models. Application of Bellman technique to upper estimates of singular integrals; 5. Application of Bellman technique to the end-point estimates of singular integrals.

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Book SynopsisThis handbook provides practitioners with a collection of original ideas on foreign exchange rates, and provides the necessary background on relevant concepts, risks, and policies for working in today's international economic climate.Table of ContentsPreface xxiii Contributors xxvii part one Overview 1 Foreign Exchange Market Structure, Players, and Evolution 3 1.1 Introduction, 3 1.2 Geography and Composition of Currency Trading, 4 1.2.1 Which Currencies are Traded? 6 1.2.2 What Instruments are Traded? 9 1.2.3 How is Trading Regulated? 9 1.3 Players and Information in FX Markets, 11 1.3.1 Who Needs Liquidity? 12 1.3.2 Who Provides Liquidity? 15 1.3.3 Asymmetric Information and Exchange Rate Determination, 19 1.4 Electronic Trading Revolution in FX Markets, 21 1.4.1 The Telephone Era, 22 1.4.2 The Rise of the Computer, 22 1.4.3 Recent Developments in Electronic Trading, 30 1.5 Survey of Multibank FX Platforms, 35 1.6 Summary, 38 Glossary, 39 Acknowledgments, 41 References, 42 2 Macro Approaches to Foreign Exchange Determination 45 2.1 Introduction, 45 2.2 Models of the Nominal Exchange Rate, 46 2.2.1 The Monetary Model, 46 2.2.2 Portfolio Balance Models, 49 2.2.3 Empirical Evidence, 51 2.3 Real Models of the Real Exchange Rate, 54 2.3.1 Purchasing Power Parity, 55 2.3.2 Balassa–Samuelson and Productivity-Based Models, 56 2.3.3 Two-Good Models, 59 2.4 New Directions in Exchange-Rate Modeling, 60 2.4.1 Taking Reaction Functions Seriously, 60 2.4.2 The Impact of Financial Globalization, 63 2.4.3 The Risk Premium and Order Flow, 64 2.5 Conclusions, 65 Acknowledgments, 65 References, 66 3 Micro Approaches to Foreign Exchange Determination 73 3.1 Introduction, 73 3.2 Perspectives on Spot-Rate Dynamics, 74 3.2.1 Decomposition of Depreciation Rates, 74 3.2.2 Macro- and Microperspectives, 77 3.3 Currency Trading Models and their Implications, 80 3.3.1 The Portfolio Shifts Model, 81 3.3.2 Empirical Implications, 88 3.4 Exchange Rates, Order Flows, and the Macro Economy, 95 3.4.1 A Micro-Based Macro model, 96 3.4.2 Empirical Implications, 100 3.5 Conclusion, 105 Appendix, 105 3.6 Acknowledgment, 108 References, 108 4 The Exchange Rate in a Behavioral Finance Framework 111 4.1 Introduction, 111 4.1.1 Mainstream Exchange Rate Models, 111 4.1.2 Away from the Mainstream, 113 4.2 Exchange Rate Puzzles, 114 4.2.1 Disconnect Puzzle and Excess Volatility Puzzle, 114 4.2.2 Unit Root Property, 115 4.2.3 Volatility Clustering, 118 4.2.4 Fat-Tailed Distributed Exchange Rate Returns, 119 4.3 A Prototype Behavioral Model of the Foreign Exchange Market, 122 4.4 Conclusion, 127 References, 129 5 The Evolution of Exchange Rate Regimes and Some Future Perspectives 133 5.1 Introduction, 133 5.2 A Brief History of Currency Regimes, 135 5.3 Performance of the Laisser-Faire Exchange Rate System, 1973–2010, 138 5.3.1 Market Discipline, 139 5.3.2 Economic Policy Coordination, 140 5.3.3 Integration of Emerging Market Countries into the Global Economy, 140 5.4 Trends in Currency Use, 141 5.4.1 Global Imbalances and the Financial Crisis of 2007–2009, 143 5.5 Prospects for the Future, 144 5.5.1 The Current System, 144 5.5.2 Toward a more Managed International Monetary System? 146 5.5.3 How and When Will Reform Occur? 150 5.5.4 A Global Nominal Anchor? 151 5.6 Concluding Comments, 153 Appendix A: A Formal Test of Hollowing Out, 154 References, 156 part two Exchange Rate Models and Methods 6 Purchasing Power Parity in Economic History 161 6.1 Introduction, 161 6.2 Categorization of Purchasing-Power-Parity Theories, 162 6.3 Historical Application of PPP: Premodern Periods, 163 6.3.1 Ancient Period, 163 6.3.2 Medieval Period, 164 6.3.3 Sixteenth-Century Spain, 165 6.4 Techniques of Testing PPP Theory in Economic-History Literature, 165 6.4.1 Comparative-Static Computation, 165 6.4.2 Regression Analysis, 165 6.4.3 Testing for Causality, 165 6.4.4 Nonstationarity and Spurious Regression, 166 6.4.5 Testing for Stationarity, 167 6.4.6 Cointegration Analysis, 167 6.5 Price Variable in PPP Computations, 168 6.6 Modern Period: Testing of PPP, 169 6.6.1 Early North America, 169 6.6.2 Bullionist Periods, 170 6.6.3 Floating Rates—Second-Half of Nineteenth Century, 171 6.6.4 Classic Metallic Standards, 172 6.6.5 World War I, 172 6.6.6 Floating Rates—1920s, 173 6.6.7 1930s, 175 6.6.8 Interwar Period, 175 6.6.9 Spain—Long Term, 176 6.6.10 Guatemala—Long Term, 176 6.7 Analysis of U.S. Return to Gold Standard in 1879, 177 6.8 Establishment and Assessment of a Fixed Exchange Rate in Interwar Period, 177 6.8.1 United Kingdom, 177 6.8.2 France, 179 6.9 Conclusions, 180 References, 181 7 Purchasing Power Parity in Tradable Goods 189 7.1 Introduction, 189 7.2 The LOP and Price Indices, 190 7.3 Empirical Evidence on the LOP, 194 7.3.1 Early Tests of the LOP, 194 7.3.2 The Border Effect, 194 7.3.3 Barriers to Arbitrage and Nonlinearities, 195 7.3.4 The Tradable Versus Nontradable Goods Dichotomy, 198 7.3.5 The Aggregation Bias and Micro Price Studies, 199 7.4 Purchasing Power Parity, 200 7.4.1 Transitory and Structural Disparities from Parity, 203 7.5 Aggregating from the LOP to PPP: What Can We Infer? 205 7.5.1 An Eyeball Analysis of PPP, 207 7.6 Conclusion and Implications, 213 Appendix: TAR Modeling, 214 Acknowledgments, 215 References, 215 8 Statistical and Economic Methods for Evaluating Exchange Rate Predictability 221 8.1 Introduction, 221 8.2 Models for Exchange Rate Predictability, 224 8.2.1 A Present Value Model for Exchange Rates, 224 8.2.2 Predictive Regressions, 226 8.3 Statistical Evaluation of Exchange Rate Predictability, 228 8.4 Economic Evaluation of Exchange Rate Predictability, 231 8.4.1 The Dynamic FX Strategy, 231 8.4.2 Mean-Variance Dynamic Asset Allocation, 231 8.4.3 Performance Measures, 232 8.4.4 Transaction Costs, 234 8.5 Combined Forecasts, 235 8.6 Empirical Results, 237 8.6.1 Data on Exchange Rates and Economic Fundamentals, 237 8.6.2 Predictive Regressions, 242 8.6.3 Statistical Evaluation, 244 8.6.4 Economic Evaluation, 249 8.7 Conclusion, 256 Appendix A: The Bootstrap Algorithm, 259 Acknowledgments, 260 References, 260 9 When Are Pooled Panel-Data Regression Forecasts of Exchange Rates More Accurate than the Time-Series Regression Forecasts? 265 9.1 Introduction, 265 9.2 Panel Data Exchange Rate Determination Studies, 267 9.3 Asymptotic Consequences of Pooling, 268 9.3.1 Predictive Regression Estimated on Full Sample, 268 9.3.2 Out-of-Sample Prediction, 271 9.4 Monte Carlo Study, 272 9.5 An Illustration with Data, 275 9.6 Conclusions, 278 References, 279 10 Carry Trades and Risk 283 10.1 Introduction, 283 10.2 The Carry Trade: Basic Facts, 285 10.2.1 What is a Carry Trade? 285 10.2.2 Measuring the Returns to the Carry Trade, 286 10.3 Pricing the Returns to the Carry Trade, 290 10.4 Empirical Findings, 293 10.4.1 Traditional Risk Factors, 293 10.4.2 Factors Derived from Currency Returns, 299 10.5 Time-Varying Risk and Rare Events, 308 10.6 Conclusion, 311 Acknowledgments, 311 References, 311 11 Currency Fair Value Models 313 11.1 Introduction, 313 11.2 Models/Taxonomy, 315 11.2.1 ‘‘Adjusted PPP’’: Harrod-Balassa-Samuelson and Penn Effects, 315 11.2.2 The Behavioral Equilibrium Exchange Rate Family of Models, 316 11.2.3 The Underlying Balance (UB) Approach, 320 11.2.4 External Sustainability (ES) Approach, 324 11.2.5 The Natural Real Exchange Rate (NATREX), 325 11.2.6 The Indirect Fair Value (IFV), 325 11.3 Implementation Choices and Model Characteristics, 328 11.3.1 Horizon/Frequency, 329 11.3.2 Direct Econometric Estimation Versus ‘‘Methods of Calculation’’, 331 11.3.3 Treatment of External Imbalances , 332 11.3.4 Real Versus Nominal Exchange Rates, 333 11.3.5 Bilateral Versus Effective Exchange Rate, 333 11.3.6 Time Series Versus Cross Section or Panel, 336 11.3.7 Model Maintenance, 336 11.4 Conclusion, 337 Acknowledgments, 338 References, 339 12 Technical Analysis in the Foreign Exchange Market 343 12.1 Introduction, 343 12.2 The Practice of Technical Analysis, 345 12.2.1 The Philosophy of Technical Analysis, 345 12.2.2 Types of Technical Analysis, 346 12.3 Studies of Technical Analysis in the Foreign Exchange Market, 350 12.3.1 Why Study Technical Analysis? 350 12.3.2 Survey Evidence on the Practice of Technical Analysis, 350 12.3.3 Computing Signals and Returns, 351 12.3.4 Early Studies: Skepticism before the Tide Turns, 353 12.3.5 Pattern Recognition, Intraday Data, and Other Exchange Rates, 353 12.4 Explaining The Success of Technical Analysis, 355 12.4.1 Data Snooping, Publication Bias, and Data Mining, 355 12.4.2 Temporal Variation in Trading Rule Returns, 357 12.4.3 Do Technical Trading Returns Compensate Investors for Bearing Risk? 359 12.4.4 Does Foreign Exchange Intervention Create Trading Rule Profits? 361 12.4.5 Do Cognitive Biases Create Trading Rule Profits? 363 12.4.6 Do Markets Adapt to Arbitrage Away Trading Rule Profits? 365 12.5 The Future of Research on Technical Analysis, 366 12.6 Conclusion, 367 Acknowledgments, 368 References, 368 13 Modeling Exchange Rates with Incomplete Information 375 13.1 Introduction, 375 13.2 Basic Monetary Model, 376 13.3 Information Heterogeneity, 379 13.4 Model Uncertainty, 381 13.5 Infrequent Decision Making, 385 13.6 Conclusion, 388 Acknowledgments, 388 References, 389 14 Exchange Rates in a Stochastic Discount Factor Framework 391 14.1 Introduction, 391 14.2 Exchange Rates and Stochastic Discount Factors, 392 14.2.1 Stochastic Discount Factors, 392 14.2.2 Real Exchange Rates and Currency Risk Premia, 395 14.3 Empirical Evidence, 398 14.3.1 From UIP Regressions to Currency Portfolios, 398 14.3.2 Annual Currency Excess Returns and Aggregate Risk, 399 14.3.3 Monthly Currency Excess Returns, 403 14.3.4 Implications for Stochastic Discount Factors, 403 14.3.5 Predictability of Currency Excess Returns, 405 14.4 Models, 407 14.4.1 Habits, 407 14.4.2 Long-Run Risk, 411 14.4.3 Disaster Risk, 414 14.5 Conclusion, 417 References, 417 15 Volatility and Correlation Timing in Active Currency Management 421 15.1 Introduction, 421 15.2 Dynamic Models for Volatility and Correlation, 424 15.2.1 The Set of Multivariate Models, 425 15.2.2 The Set of Univariate Models for Volatility Timing, 427 15.2.3 Pairwise Model Comparisons, 427 15.2.4 Estimation and Forecasting, 427 15.3 The Economic Value of Volatility and Correlation Timing, 428 15.3.1 The Dynamic Strategy, 428 15.3.2 Dynamic Asset Allocation with CRRA Utility, 428 15.3.3 Performance Measures, 429 15.3.4 Transaction Costs, 430 15.4 Parameter Uncertainty in Bayesian Asset Allocation, 430 15.5 Model Uncertainty, 431 15.5.1 The BMA Strategy, 432 15.5.2 The BMW Strategy, 432 15.6 Empirical Results, 432 15.6.1 Data and Descriptive Statistics, 432 15.6.2 Bayesian Estimation, 433 15.6.3 Evaluating Volatility and Correlation Timing, 434 15.7 Conclusion, 440 Appendix A: Univariate Models for Volatility Timing, 442 Appendix B: Parameter Uncertainty and the Predictive Density, 443 Acknowledgments, 444 References, 444 part three FX Markets and Products 16 Active Currency Management Part I: Is There a Premium for Currency Investing (Beta) 453 16.1 Introduction, 453 16.2 Beta in the Foreign Exchange Markets, 455 16.2.1 Understanding the FX Carry Trade, 455 16.2.2 FX Carry as a Broader Strategy, 456 16.2.3 FX Trend-Based Strategies, 458 16.2.4 Value-Based Strategies Within FX, 460 16.2.5 USD Directional Trade, 461 16.2.6 Correlation between these FX Strategies and Other Forms of Beta, 462 16.2.7 Weighted Portfolio of FX Strategies, 463 16.3 Multiple Forms of FX Beta, 465 16.4 Carry FX Indices from Banks, 465 16.5 Trend-Following FX Indices from Banks, 467 16.6 Conclusion, 468 References, 469 17 Active Currency Management Part II: Is There Skill or Alpha in Currency Investing? 471 17.1 Introduction, 471 17.2 Alternative Currency Management Mandates, 473 17.2.1 Features of a Currency Mandate, 473 17.2.2 Structural and Operational Choices, 476 17.2.3 The Alpha Continuum and Implications of Active Currency Mandates, 477 17.3 Benchmarks for Currency Fund Management, 477 17.3.1 A Basic Factor Model for Currency Returns, 479 17.4 Empirical Evidence with the Barclay Currency Traders Index and Individual Fund Managers, 481 17.4.1 Empirical Evidence with the Barclay Currency Traders Index, 481 17.4.2 Individual Currency Manager Returns, 485 17.4.3 Alternative Information Ratio, 493 17.5 Empirical Evidence: Fund Managers on the DB FX Select Platform, 496 17.5.1 Grouping Managers into a Fund of Funds, 496 17.6 Conclusions and Investment Implications, 498 References, 499 18 Currency Hedging for International Bond and Equity Investors 503 18.1 Introduction, 503 18.2 Overview of Empirical Hedging Studies, 504 18.3 Return and Volatility Impact of Currency Hedging, 506 18.3.1 Theoretical Background, 506 18.3.2 Methodology, 508 18.3.3 Summary of Findings on the Return and Volatility Impact of Currency Hedging, 525 18.4 Hedge Instruments—Currency Forwards versus Options, 526 18.4.1 Why Do Hedge Cash Flows Matter? 526 18.4.2 Historical Performance of Hedging with Options, 527 18.4.3 Summary of Findings on Hedging with Options Versus Forwards, 532 18.5 Managing Tracking Error in Forward Hedges, 533 18.5.1 How Often to Rebalance? 533 18.5.2 Trigger-Based Versus Regular Rebalancing, 539 18.5.3 Summary of Findings on Hedge Rebalancing, 539 18.6 Conclusions, 541 References, 543 19 FX Reserve Management 545 19.1 FX Reserve Management, 545 19.2 FX Reserve Uses, 545 19.3 FX Reserve Sources, 546 19.4 Objectives of Reserves Management, 547 19.5 Techniques of Reserve Management, 547 19.6 Historical Perspective, 548 19.7 What Assets Do Central Banks Hold? 549 19.8 Constraints, 550 19.9 External Managers, 551 19.10 Costs of Accumulation and Holding of Reserves, 551 19.11 Diversification, 552 19.12 Challenges to Diversification and Size of Reserves, 552 19.13 Changing Role of the Dollar as the International Reserve Currency, 554 19.14 Reserve Management if the Dollar is Replaced as the Reserve Currency, 557 19.15 Conclusion, 559 Acknowledgments, 559 References, 559 20 High Frequency Finance: Using Scaling Laws to Build Trading Models 563 20.1 Introduction, 563 20.2 The Intrinsic Time Framework, 565 20.3 Scaling Laws, 567 20.3.1 The New Scaling Laws, 568 20.3.2 The Coastline, 573 20.4 The Scale of Market Quakes, 574 20.5 Trading Models, 577 20.5.1 Overview, 577 20.5.2 Coastline Trader, 578 20.5.3 Monthly Statistics, 580 20.6 Conclusion, 582 Acknowledgments, 582 References, 582 21 Algorithmic Execution in Foreign Exchange 585 21.1 Introduction, 585 21.1.1 Drawing from the Equity Market, 586 21.1.2 What is Going to Work for Foreign Exchange? 587 21.2 Key Components of an Algorithmic Execution Framework, 589 21.2.1 Smart Order Routing (SOR), 589 21.2.2 Intelligence, 590 21.2.3 Speed, 591 21.3 Types of Algorithms, 592 21.3.1 Time Slicers, 592 21.3.2 Sweeper, 592 21.3.3 Iceberg, 592 21.3.4 Opportunistic, 592 21.3.5 Participators, 594 21.3.6 Internalization Strategies, 594 21.3.7 Dynamic Algorithms, 595 21.4 What Execution Strategies are Most Effective? 595 21.4.1 Measuring Performance, 596 21.5 Looking Forward, 596 Appendix A, 596 References, 597 22 Foreign Exchange Strategy Based Products 599 22.1 Introduction, 599 22.2 Evolution of the Foreign Exchange Market, 600 22.2.1 Disappointing Early Years, 600 22.2.2 Emergence of ‘‘Puzzles’’ in FX, 601 22.2.3 Growth of FX Market Turnover and Currency Managers, 602 22.3 Foreign Exchange Investable Indices and Strategy-Based Products, 606 22.3.1 Why Profit Opportunities Exist? 606 22.3.2 Beta and Alpha in Foreign Exchange, 607 22.3.3 Why is FX Attractive? 613 22.3.4 Why use Strategy-Based FX Products? 619 22.4 Conclusion, 620 References, 620 23 Foreign Exchange Futures, Forwards, and Swaps 623 23.1 Introduction, 623 23.2 Market Basics and Size, 625 23.2.1 FX Outright Forwards and Futures, 625 23.2.2 FX Swaps and Cross-Currency Swaps, 628 23.2.3 Market Size, 635 23.3 Dislocations of the FX and Cross-Currency Swap Markets under Financial Crises, 637 23.3.1 Japan Premium Case in the Late 1990s, 637 23.3.2 The Global Financial Crisis from 2007, 639 23.4 Conclusion, 643 Acknowledgments, 643 References, 643 24 FX Options and Volatility Derivatives: An Overview from the Buy-Side Perspective 647 24.1 Introduction, 647 24.2 Why Would One Bother with an Option? 648 24.2.1 History, 648 24.2.2 FX Options, 649 24.3 Market for FX Options, 655 24.3.1 Overview, 655 24.3.2 Players, 656 24.3.3 Setting the Price, 658 24.4 Volatility, 660 24.4.1 Overview of Models, 660 24.4.2 Some Stylized Facts and Implied Moments, 664 24.4.3 Is Volatility an Asset Class? 666 24.4.4 Anti-Black Swan Strategies, 674 24.4.5 Black Swan Strategies, 676 24.5 FX Options from the Buy-Side Perspective, 683 24.5.1 Strike versus Leverage, 683 24.5.2 Implied Distribution, 685 24.5.3 Long-Dated Options versus Short-Dated Option, 689 24.5.4 Black Swan Fund, 692 24.5.5 Currency Hedging of Illiquid Assets, 693 Acknowledgment, 695 References, 695 part four FX Markets and Policy 25 A Common Framework for Thinking about Currency Crises 699 25.1 Introduction, 699 25.2 The KFG Model, 701 25.3 Extensions, 706 25.3.1 Attack-Conditional Monetary Policy, 706 25.3.2 Devaluation, 707 25.3.3 Sterilization and Interest Rate Defense, 709 25.3.4 Lender of Last Resort and Currency Crises, 711 25.4 Empirical Work, 713 25.5 Conclusion, 714 References, 715 26 Official Intervention in the Foreign Exchange Market 717 26.1 Introduction, 717 26.2 Official FX Interventions and Reserve Accumulation: Stylized Facts, Motives, and Effects, 721 26.3 Empirical Evidence on the Effectiveness of Official FX Interventions, 725 26.3.1 A Simple Conceptual Framework, 726 26.3.2 Time-Series Approach: Evidence on Effectiveness and Channels, 728 26.3.3 Event-Study Approach: Evidence on Longer-Term Effectiveness, 739 26.4 Conclusions, 746 26.5 Acknowledgements, 746 References, 747 27 Exchange Rate Misalignment—The Case of the Chinese Renminbi 751 27.1 Introduction, 751 27.2 Background, 752 27.3 Undervalued or Overvalued, 754 27.3.1 The FEER Misalignment Estimate, 754 27.3.2 The Penn Effect Regression, 757 27.3.3 Data Revision, 759 27.4 Concluding Remarks, 762 Acknowledgments, 763 References, 763 28 Choosing an Exchange Rate Regime 767 28.1 Five Advantages of Fixed Exchange Rates , 768 28.2 Econometric Evidence on the Bilateral Trade Effects of Currency Regimes, 770 28.2.1 Time-Series Dimension, 771 28.2.2 Omitted Variables, 772 28.2.3 Endogeneity of the Currency Decision, 773 28.2.4 Implausible Magnitude of the Estimate, 774 28.2.5 Country Size, 775 28.3 Five Advantages of Floating Exchange Rates, 775 28.4 How to Weigh Up the Advantages of Fixing Versus Floating, 777 28.5 Country Characteristics That Should Help Determine the Choice of Regime, 778 28.6 Alternative Nominal Anchors, 780 References, 781 Index 785

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Book SynopsisCollins International Primary Maths supports best practice in primary maths teaching, whilst encouraging teacher professionalism and autonomy. A wealth of supporting digital assets are provided for every lesson, including slideshows, tools and games to ensure they are rich, lively and engaging.

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Book SynopsisHelps you solve word problems in algebra one step at a time. This guide offers complete directions for solving problems that involve time, money distance, work, and more.Table of ContentsHow to Work Word Problems. Numbers. Time, Rate, and Distance. Mixtures. Coins. Age. Levers. Finance. Work. Plane Geometric Figures. Digits.

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Book SynopsisThe ideal review for your tensor calculus courseMore than 40 million students have trusted Schaumâs Outlines for their expert knowledge and helpful solved problems. Written by renowned experts in their respective fields, Schaumâs Outlines cover everything from math to science, nursing to language. The main feature for all these books is the solved problems. Step-by-step, authors walk readers through coming up with solutions to exercises in their topic of choice. 300 solved problems Coverage of all course fundamentals Effective problem-solving techniques Complements or supplements the major logic textbooks Supports all the major textbooks for tensor calculus courses Table of Contents1. The Einstein Summation Convention. 2. Basic Linear Algebra for Tensors. 3. General Tensors. 4. Tensor Operations. 5. Tests for Tensor Character. 6. The Metric Tensor. 7. The Derivative of a Tensor. 8. Further Riemannian Geometry. 9. Riemannian Curvature. 10. Spaces of Zero Curvature. 11. Tensors in Differential Geometry. 12. Tensors in Mechanics. 13. Tensors in Special Relativity. 14. Tensors Without Coordinates. 15. Introduction to Tensor Manifolds.

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Book SynopsisNot Even Wrong is a fascinating exploration of our attempts to come to grips with perhaps the most intellectually demanding puzzle of all: how does the universe work at its most fundamnetal level?The book begins with an historical survey of the experimental and theoretical developments that led to the creation of the phenomenally successful ''Standard Model'' of particle physics around 1975. Despite its successes, the Standard Model does not answer all the key questions and physicists continuing search for answers led to the development of superstring theory. However, after twenty years, superstring theory has failed to advance beyond the Standard Model. The absence of experimental evidence is at the core of this controversial situation which means that it is impossible to prove that superstring theory is either right or wrong. To date, only the arguments of the theory''s advocates have received much publicity. Not Even Wrong provides readers with anothTrade ReviewHighly readable, accessible and powerfully persuasive -- John Cornwell * Sunday Times *Will embolden other string critics to speak up and encourage talented young physicists to pursue other lines of research -- John Horgan * Prospect *Compulsive reading -- Roger PenroseIt's a call to arms * New Scientist *

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Book SynopsisTable of ContentsManifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorenz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity. Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds. Fundamental Groups and Covering Manifolds. Lie Groups. Newtonian Gravitation.

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Book SynopsisChaos exists in systems all around us. Even the simplest system of cause and effect can be subject to chaos, denying us accurate predictions of its behaviour, and sometimes giving rise to astonishing structures of large-scale order. Our growing understanding of Chaos Theory is having fascinating applications in the real world - from technology to global warming, politics, human behaviour, and even gambling on the stock market.Leonard Smith shows that we all have an intuitive understanding of chaotic systems. He uses accessible maths and physics (replacing complex equations with simple examples like pendulums, railway lines, and tossing coins) to explain the theory, and points to numerous examples in philosophy and literature (Edgar Allen Poe, Chang-Tzu, Arthur Conan Doyle) that illuminate the problems. The beauty of fractal patterns and their relation to chaos, as well as the history of chaos, and its uses in the real world and implications for the philosophy of science are all discussed in this Very Short Introduction.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.Trade ReviewLeonard Smith's Chaos (part of the Oxford Very Short Introduction series) will give you the clearest (but not too painful idea) of the maths involved... There's a lot packed into this little book, and for such a technical exploration it's surprisingly readble and enjoyable - I really wanted to keep turning the pages. Smith also has some excellent words of wisdom about common misunderstandings of chaos theory... One of the best books so far in this useful and informative series. * popularscience.co.uk *Table of ContentsPreface ; 1. The Emergence of Chaos ; 2. Exponential Growth, Nonlinearity, Common Sense ; 3. Chaos in Context: Determinism Randomness and Noise ; 4. Chaos in Mathematical Models ; 5. Fractals, Strange Attractors, and Dimension(s) ; 6. Quantifying the Dynamics of Uncertainty ; 7. Real numbers, Real Observations and Computers ; 8. Sorry, Wrong Number: Statistics and Chaos ; 9. Predictability: Does Chaos Constrain Our Forecasts? ; 10. Applied Chaos: Can We See Through Our Models? ; 11. Philosophy in Chaos ; Glossary ; Further Reading

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Book SynopsisGottfried Wilhelm Leibniz (1646-1716) was a man of extraordinary intellectual creativity who lived an exceptionally rich and varied intellectual life in troubled times. More than anything else, he was a man who wanted to improve the life of his fellow human beings through the advancement of all the sciences and the establishment of a stable and just political order. In this Very Short Introduction Maria Rosa Antognazza outlines the central features of Leibniz''s philosophy in the context of his overarching intellectual vision and aspirations. Against the backdrop of Leibniz''s encompassing scientific ambitions, she introduces the fundamental principles of Leibniz''s thought, as well as his theory of truth and theory of knowledge. Exploring Leibniz''s contributions to logic, mathematics, physics, and metaphysics, she considers how his theories sat alongside his concerns with politics, diplomacy, and a broad range of practical reforms: juridical, economic, administrative, technological, medical, and ecclesiastical. Discussing Leinbniz''s theories of possible worlds, she concludes by looking at what is ultimately real in this actual world that we experience, the good and evil there is in it, and Leibniz''s response to the problem of evil through his theodicy. ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.Table of ContentsCONCLUSION; REFERENCES; FURTHER READING; INDEX

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Book SynopsisMeasurement is a fundamental concept that underpins almost every aspect of the modern world. It is central to the sciences, social sciences, medicine, and economics, but it affects everyday life. We measure everything - from the distance of far-off galaxies to the temperature of the air, levels of risk, political majorities, taxes, blood pressure, IQ, and weight. The history of measurement goes back to the ancient world, and its story has been one of gradual standardization. Today there are different types of measurement, levels of accuracy, and systems of units, applied in different contexts. Measurement involves notions of variability, accuracy, reliability, and error, and challenges such as the measurement of extreme values.In this Very Short Introduction, David Hand explains the common mathematical framework underlying all measurement, the main approaches to measurement, and the challenges involved. Following a brief historical account of measurement, he discusses measurement as used in the physical sciences and engineering, the life sciences and medicine, the social and behavioural sciences, economics, business, and public policy.ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.Table of ContentsREFERENCES; FURTHER READING; INDEX

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Book SynopsisFinally a self-contained, one volume, graduate-level algebra text that is readable by the average graduate student and flexible enough to accommodate a wide variety of instructors and course contents. Therefore it stresses clarity rather than brevity and contains an extraordinarily large number of illustrative exercises.Trade ReviewFrom the book reviews:“This is a text for a first-year graduate course in abstract algebra. It covers all the standard topics and has more than enough material for a year course.” (Allen Stenger, MAA Reviews, September, 2014)Thomas W. HungerfordAlgebra"An excellent text from which to teach the beginning graduate survey course in algebra and I would recommend it to anyone considering a text for such a course."—LINEAR AND MULTILINEAR ALGEBRATable of ContentsIntroduction. Prerequisites and Preliminaries; 1. Groups; 2. The Structure of Groups; 3. Rings; 4. Modules; 5. Fields and Galois Theory; 6. The Structure of Fields; 7. Linear Algebra; 8. Commutative Rings and Modules; 9. The Structure of Rings; 10. Categories; List of Symbols; Bibliography; Index

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Book SynopsisDescriptive set theory has been one of the main areas of research in set theory for almost a century. It includes a wide variety of examples, more than 400 exercises, and applications, in order to illustrate the general concepts and results of the theory.Table of ContentsI Polish Spaces.- 1. Topological and Metric Spaces.- 1.A Topological Spaces.- 1.B Metric Spaces.- 2. Trees.- 2.A Basic Concepts.- 2.B Trees and Closed Sets.- 2.C Trees on Produtcs.- 2.D Leftmost Branches.- 2.E Well-founded Trees and Rank.- 2.F The Well-founded Part of a Tree.- 2.G The Kleene-Brouwer Ordering.- 3. Polish Spaces.- 3.A Definitions and Examples.- 3.B Extensions of Continuous Functions and Homeomorphisms.- 3.C Polish Subspaces of Polish Spaces.- 4. Compact Metrizable Spaces.- 4.A Basic Facts.- 4.B Examples.- 4.C A Universality Property of the Hilbert Cube.- 4.D Continuous Images of the Cantor Space.- 4.E The Space of Continuous Functions on a Compact Space.- 4.F The Hyperspace of Compact Sets.- 5. Locally Compact Spaces.- 6. Perfect Polish Spaces.- 6.A Embedding the Cantor Space in Perfect Polish Spaces.- 6.B The Cantor-Bendixson Theorem.- 6.C Cantor-Bendixson Derivatives and Ranks.- 7.Zero-dimensional Spaces.- 7.A Basic Facts.- 7.B A Topological Characterization of the Cantor Space.- 7.C A Topological Characterization of the Baire Space.- 7.D Zero-dimensional Spaces aa Subspaces of the Baire Space.- 7.F Polish Spaces as Continuous Images of the Baire Space.- 7.F Closed Subsets Homcomorphic to the Baire Space.- 8. Baire Category.- 8.A Meager Sets.- 8.B Baire Spaces.- 8.C Choquet Games and Spaces.- 8.D Strong Choquet Games and Spaces.- 8.E A Characterization of Polish Spaces.- 8.F Sets with the Baire Property.- 8.G Localization.- 8.H The Banach-Mazur Game.- 8.I Baire Measurable Functions.- 8.J Category Quantifiers.- 8.K The Kuratowski-Ulam Theorem.- 8.L Some Applications.- 8.M Separate and Joint Continuity.- 9. Polish Groups.- 9.A Metrizable and Polish Groups.- 9.B Examples of Polish Groups.- 9.C Basic Facts about Baire Groups and Their Actions.- 9.D Universal Polish Groups.- II Borel Sets.- 10. Measurable Spaces and Functions.- 10.A Sigma-Algebras and Their Generators.- 10.B Measurable Spaces and Functions.- 11. Borel Sets and Functions.- 11.A Borel Sets in Topological Spaces.- 11.B The Borel Hierarchy.- 11.C Borel Functions.- 12. Standard Borel Spaces.- 12.A Borel Sets and Functions in Separable Metrizable Spaces.- 12.B Standard Borel Spaces.- 12.C The Effros Borel Space.- 12.D An Application to Selectors.- 12.E Further Examples.- 12.F Standard Borel Groups.- 13. Borel Sets as Clopen Sets.- 13.A Turning Borel into Clopen Sets.- 13.B Other Representations of Borel Sets.- 13.C Turning Borel into Continuous Functions.- 14. Analytic Sets and the Separation Theorem.- 14.A Basic Facts about Analytic Sets.- 14.B The Lusin Separation Theorem.- 14.C Sousliri’s Theorem.- 15. Borel Injections and Isomorphisms.- 15.A Borel Injective Images of Borel Sets.- 15.B The Isomorphism Theorem.- 15.C Homomorphisms of Sigma-Algebras Induced by Point Maps.- 15.D Some Applications to Group Actions.- 16. Borel Sets and Baire Category.- 16.A Borel Definability of Category Notions.- 16.B The Vaught Transforms.- 16.C Connections with Model Theory.- 16.D Connections with Cohen’s Forcing Method.- 17. Borel Sets and Measures.- 17.A General Facts on Measures.- 17.B Borel Measures.- 17.C Regularity and Tightness of Measures.- 17.D Lusin’s Theorem on Measurable Functions.- 17.E The Space of Probability Borel Measures.- 17.F The Isomorphism Theorem for Measures.- 18. Uniformization Theorems.- 18.A The Jankov, von Neumann Uniformization Theorem.- 18.B “Large Section” Uniformization Results.- 18.C “Small Section” Uniformization Results.- 18.D Selectors and Transversals.- 19. Partition Theorems.- 19.A Partitions with a Comeager or Non-meager Piece.- 19.B A Ramsey Theorem for Polish Spaces.- 19.C The Galvin-Prikry Theorem.- 19.D Ramsey Sets and the Ellentuck Topology.- 19.E An Application to Banach Space Theory.- 20. Borel Determinacy.- 20.A Infinite Games.- 20.B Determinacy of Closed Games.- 20.C Borel Determinacy.- 20.D Game Quantifiers.- 21. Games People Play.- 21.A The *-Games.- 21.B Unfolding.- 21.C The Banach-Mazur or **-Games.- 21.D The General Unfolded Banach-Mazur Games.- 21.E Wadge Games.- 21.F Separation Games and Hurewicz’s Theorem.- 21.G Turing Degrees.- 22. The Borel Hierarchy.- 22. A Universal Sets.- 22.B The Borel versus the Wadge Hierarchy.- 22.C Structural Properties.- 22.D Additional Results.- 22.E The Difference Hierarchy.- 23. Some Examples.- 23.A Combinatorial Examples.- 23.B Classes of Compact Sets.- 23.C Sequence Spaces.- 23.D Classes of Continuous Functions.- 23.E Uniformly Convergent Sequences.- 23.F Some Universal Sets.- 23.G Further Examples.- 24. The Baire Hierarchy.- 24.A The Baire Classes of Functions.- 24.B Functions of Baire Class 1.- III Analytic Sets.- 25. Representations of Analytic Sets.- 25.A Review.- 25.B Analytic Sets in the Baire Space.- 25.C The Souslin Operation.- 25.D Wellordered Unions and Intersections of Borel Sets.- 25. E Analytic Sets as Open Sets in Strong Choquet Spaces.- 26. Universal and Complete Sets.- 26.A Universal Analytic Sets.- 26.B Analytic Determinacy.- 26.C Complete Analytic Sets.- 26.D Classification up to Borel Isomorphism.- 27. Examples.- 27.A The Class of Ill-founded Trees.- 27.B Classes of Closed Sets.- 27.C Classes of Structures in Model Theory.- 27.D Isomorphism.- 27.E Some Universal Sets.- 27.F Miscellanea.- 28. Separation Theorems.- 28.A The Lusin Separation Theorem Revisited.- 28.B The Novilcov Separation Theorem.- 28.C Borel Sets with Open or Closed Sections.- 28.D Some Special Separation Theorems.- 28.E “Hurewicz-Type” Separation Theorems.- 29. Regularity Properties.- 29.A The Perfect Set Property.- 29.B Measure. Category, and Ramsey.- 29.C A Closure Property for the Souslin Operation.- 29.D The Class of C-Sets.- 29.E Analyticity of “Largeness” Conditions on Analytic Sets.- 30. Capacities.- 30.A The Basic Concept.- 30.B Examples.- 30.C The Choquet Capacitability Theorem.- 31. Analytic Well-founded Relations.- 31.A Bounds on Ranks of Analytic Well-founded Relations.- 31.B The Kunen-Martin Theorem.- IV Co-Analytic Sets.- 32. Review.- 32.A Basic Facts.- 32.B Representations of Co-Analytic Sets.- 32.C Regularity Properties.- 33. Examples.- 33.A Well-founded Trees and Wellorderings.- 33.B Classes of Closed Sets.- 33.C Sigma-ldoals of Compact Sets.- 33.D Differentiable Functions.- 33.E Everywhere Convergence.- 33.F Parametrizing Baire Class 1 Functions.- 33.G A Method for Proving Completeness.- 33.H Singular Functions.- 33.I Topological Examples.- 33.J Homeomorphisms of Compact Spaces.- 33.K Classes of Separable Banach Spaces.- 33.L Other Examples.- 34. Co-Analytic Ranks.- 34.A Ranks and Prewellorderings.- 34.B Ranked Classes.- 34.C Co-Analytic Ranks.- 34.D Derivatives.- 34.E Co-Analytic Ranks Associated with Borel Derivatives.- 34.F Examples.- 35. Rank Theory.- 35.A Basic Properties of Ranked Classes.- 35.B Parametrizing Bi-Analytic and Borel Sets.- 35.C Reflection Theorems.- 35.D Boundedness Properties of Ranks.- 35.E The Rank Method.- 35.F The Strategic Uniformization Theorem.- 35.G Co-Analytic Families of Closed Sets and Their Sigma-Ideals.- 35.H Borel Sots with F? and K? Sections.- 36. Scales and Uniformiiatiou.- 36.A Kappa-Souslin Sets.- 36.B Scales.- 36.C Sealed Classes and Urniformization.- 36.D The Novikov-Kondô Uniformization Theorem.- 36.E Regularity Properties of Uniformizing Functions.- 36.F Uniforniizing Co-Analytic Sets with Large Sections.- 36.G Examples of Co-Analytic Scales.- V Projective Sets.- 37. The Projective Hierarchy.- 37.A Basic Facts.- 37.B Examples.- 38. Projective Determinacy.- 38.A The Second Level of the Projective Hierarchy.- 38.B Projective Determinacy.- 38.C Regularity Properties.- 39. The Periodicity Theorems.- 39.A Periodicity in the Projective Hierarchy.- 39.B The First Periodicity Theorem.- 39.C The Second Periodicity Theorem.- 39.D The Third Periodicity Theorem.- 40. Epilogue.- 40.A Extensions of the Projective Hierarchy.- 40.B Effective Descriptive Set Theory.- 40.C Large Cardinals.- 40.D Connections to Other Areas of Mathematics.- Appendix A. Ordinals and Cardinals.- Appendix B. Well-founded Relations.- Appendix C. On Logical Notation.- Notes and Hints.- References.- Symbols and Abbreviations.

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Book SynopsisFirst Part.- I The Complex Plane and Elementary Functions.- II Analytic Functions.- III Line Integrals and Harmonic Functions.- IV Complex Integration and Analyticity.- V Power Series.- VI Laurent Series and Isolated Singularities.- VII The Residue Calculus.- Second Part.- VIII The Logarithmic Integral.- IX The Schwarz Lemma and Hyperbolic Geometry.- X Harmonic Functions and the Reflection Principle.- XI Conformal Mapping.- Third Part.- XII Compact Families of Meromorphic Functions.- XIII Approximation Theorems.- XIV Some Special Functions.- XV The Dirichlet Problem.- XVI Riemann Surfaces.- Hints and Solutions for Selected Exercises.- References.- List of Symbols.Table of Contents* The Complex Plane and Elementary Functions * Analytic Functions * Line Integrals and Harmonic Functions * Complex Integration and Analyticity * Power Series * Laurent Series and Isolated Singularities * The Residue Calculus * The Logarithmic Integral * The Schwarz Lemma and Hyperbolic Geometry * Harmonic Functions and the Reflection Principle * Conformal Mapping * Compact Families of Meromorphic Functions * Approximation Theorems * Some Special Functions * The Dirichlet Problem * Riemann Surfaces

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Book SynopsisWritten by two of the most prominent figures in the field of graph theory, this comprehensive text provides a remarkably student-friendly approach. Geared toward undergraduates taking a first course in graph theory, its sound yet accessible treatment emphasizes the history of graph theory and offers unique examples and lucid proofs. 2004 edition.

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Combinatorics on words appears in many different areas of mathematics and theoretical computer science. With background material, full proofs whenever possible, and a discussion of further developments, this 2002 book is both a comprehensive introduction to the subject and a valuable reference source for researchers.

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Book SynopsisIn 1931 Kurt Godel published his fundamental paper, On Formally Undecidable Propositions of Principia Mathematica and Related Systems. This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. The authors provide an explanation of the main ideas and broad implications of Godel's discovery.Trade ReviewA little masterpiece of exegesis. * Nature *An excellent nontechnical account of the substance of Gödel's celebrated paper. -- American Mathematical SocietyTable of ContentsContents Foreword to the New Edition by Douglas R. Hofstadter ix Acknowledgments xxiii i Introduction 1 ii The Problem of Consistency 7 iii Absolute Proofs of Consistency 25 iv The Systematic Codification of Formal Logic 37 v An Example of a Successful Absolute Proof of Consistency 45 vi The Idea of Mapping and Its Use in Mathematics 57 vii Godel's Proofs 68 a Godel numbering 68 b The arithmetization of meta-mathematics 80 c The heart of Godel's argument 92 viii Concluding Reflections 109 Appendix: Notes 114 Brief Bibliography 125 Index 127

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Book Synopsis

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