Mathematics Books

Book SynopsisA Tour of Data Science: Learn R and Python in Parallel covers the fundamentals of data science, including programming, statistics, optimization, and machine learning in a single short book. It does not cover everything, but rather, teaches the key concepts and topics in Data Science. It also covers two of the most popular programming languages used in Data Science, R and Python, in one source.Key features: Allows you to learn R and Python in parallel Cover statistics, programming, optimization and predictive modelling, and the popular data manipulation tools – data.table and pandas Provides a concise and accessible presentation Includes machine learning algorithms implemented from scratch, linear regression, lasso, ridge, logistic regression, gradient boosting trees, etc. Appealing to data scientists, statisticians, quantitative analysts, and others who want to learn progrTable of ContentsAssumptions about the reader’s backgroundBook overview Introduction to R/Python Programming Calculator Variable and TypeFunctions Control flowsSome built-in data structures Revisit of variables Object-oriented programming (OOP) in R/Python Miscellaneous More on R/Python Programming Work with R/Python scripts Debugging in R/Python Benchmarking Vectorization Embarrassingly parallelism in R/Python Evaluation strategySpeed up with C/C++ in R/PythonA first impression of functional programming Miscellaneous data.table and pandasSQL Get started with data.table and pandas Indexing & selecting data Add/Remove/UpdateGroup by Join Random Variables, Distributions & Linear Regression A refresher on distributions Inversion sampling & rejection sampling Joint distribution & copula Fit a distribution Confidence intervalHypothesis testing Basics of linear regression Ridge regression Optimization in PracticeConvexity Gradient descent Root-finding General purpose minimization tools in R/Python Linear programming Miscellaneous Machine Learning - A gentle introduction Supervised learning Gradient boosting machine Unsupervised learning Reinforcement learning Deep Q-Networks Computational differentiation Miscellaneous

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Book SynopsisGraduate mathematics students will find this book an easy-to-follow, step-by-step guide to the subject. Rotman’s book gives a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology.Trade ReviewFrom the reviews of the second edition:"Joseph J. Rotman is a renowned textbook author in contemporary mathematics. Over the past four decades, he has published numerous successful texts of introductory character, mainly in the field of modern abstract algebra and its related disciplines. … Now, in the current second edition, the author has reworked the original text considerably. While the first edition covered exclusively aspects of the homological algebra of groups, rings, and modules, that is, topics from its first period of development, the new edition includes some additional material from the second period, together with numerous other, more recent results from the homological algebra of groups, rings, and modules. The new edition has almost doubled in size and represents a substantial updating of the classic original. … All together, a popular classic has been turned into a new, much more topical and comprehensive textbook on homological algebra, with all the great features that once distinguished the original, very much to the belief [of its] new generation of readers." (Werner Kleinert, Zentralblatt)"The new expanded second edition … attempts to cover more ground, basically going from the (concrete) category of modules over a given ring, as in the first edition, to an abelian category and to treat the important example of the category of sheaves on a topological space. … the exercise at the end of every section, plenty of examples and motivation for the many new concepts set this book apart and make it an ideal textbook for a course on the subject." (Felipe Zaldivar, MAA Online, December, 2008)"This is the second edition of Rotman’s introduction to the more classical aspects of homological algebra … . The book is mainly concerned with homological algebra in module categories … . The book is full of illustrative examples and exercises. It contains many references for further study and also to original sources. All this makes Rotman’s book very convenient for beginners in homological algebra as well as a reference book." (Fernando Muro, Mathematical Reviews, Issue 2009 i)Table of ContentsHom and Tensor.- Special Modules.- Specific Rings.- Setting the Stage.- Homology.- Tor and Ext.- Homology and Rings.- Homology and Groups.- Spectral Sequences.

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Book SynopsisProblems.- Arithmetic Functions.- Primes in Arithmetic Progressions.- The Prime Number Theorem.- The Method of Contour Integration.- Functional Equations.- Hadamard Products.- Explicit Formulas.- The Selberg Class.- Sieve Methods.- p-adic Methods.- Equidistribution.- Solutions.- Arithmetic Functions.- Primes in Arithmetic Progressions.- The Prime Number Theorem.- The Method of Contour Integration.- Functional Equations.- Hadamard Products.- Explicit Formulas.- The Selberg Class.- Sieve Methods.- p-adic Methods.- Equidistribution.Trade ReviewM.R. MurtyProblems in Analytic Number Theory"The reviewer strongly approves of the problem-based approach to learning, and recommends this book to any student of analytic number theory."—MATHEMATICAL REVIEWSFrom the reviews of the second edition:“This expanded and corrected second edition of this useful and interesting book has a new chapter on the topic of equidistribution. … this monograph gives important results and techniques for specific topics, together with many exercises. … I do enjoy this book … and I imagine when I take the graduate course in the subject that it will be of a greater benefit, which is why I offered such a high rating.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, July, 2013)"The second edition of the book has eleven chapters … . the book can be used both as a problem book (as its title shows) and also as a textbook (as the series in which the book is published shows). … is ideal as a text for a first course in analytic number theory, either at the senior undergraduate or the graduate level. … I believe that this book will be very useful for students, researchers and professors. It is well written … ." (Mehdi Hassani, MathDL, April, 2008)Table of ContentsProblems.- Arithmetic Functions.- Primes in Arithmetic Progressions.- The Prime Number Theorem.- The Method of Contour Integration.- Functional Equations.- Hadamard Products.- Explicit Formulas.- The Selberg Class.- Sieve Methods.- p-adic Methods.- Equidistribution.- Solutions.- Arithmetic Functions.- Primes in Arithmetic Progressions.- The Prime Number Theorem.- The Method of Contour Integration.- Functional Equations.- Hadamard Products.- Explicit Formulas.- The Selberg Class.- Sieve Methods.- p-adic Methods.- Equidistribution.

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Book SynopsisIt then proceeds to a discussion of modules, emphasizing a comparison with vector spaces, and presents a thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory, culminating in the finite dimensional spectral theorem for normal operators.Trade ReviewFrom the reviews of the first edition:"… The book is very well written and has a good set of exercises. It is a suitable choice as a graduate textbook as well as a reference book." A.A. Jafarian for ZentralblattMATHFrom the reviews of the second edition:"In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. … As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. … the exercises are rewritten and expanded. … Overall, I found the book a very useful one. … It is a suitable choice as a graduate text or as a reference book." (Ali-Akbar Jafarian, Zentralblatt MATH, Vol. 1085, 2006)"This is a formidable volume, a compendium of linear algebra theory, classical and modern … . The development of the subject is elegant … . The proofs are neat … . The exercise sets are good, with occasional hints given for the solution of trickier problems. … It represents linear algebra and does so comprehensively." (Henry Ricardo, MathDL, May, 2005)From the reviews of the third edition:“This is the 3rd edition of a well written graduate book on linear algebra. … The list of references has been enlarged considerably. The book is suitable for a second course on linear algebra and/or a graduate text, as well as a reference text.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, May, 2014)"This is the 3rd edition of a well written graduate book on linear algebra. … The book covers a wide range of topics in a moderate length and careful manner. … The list of references has been enlarged considerably. … is suitable for a second course on linear algebra and/or a graduate text, as well as a reference text." (A. Arvanitoyeorgos, Zentralblatt MATH, Vol. 1132 (10), 2008)Table of Contents* Vector Spaces * Linear Transformations * The Isomorphism Theorems * Modules I: Basic Properties * Modules II: Free and Noetherian Modules * Modules over a Principal Ideal Domain * The Structure of a Linear Operator * Eigenvalues and Eigenvectors * Real and Complex Inner Product Spaces * Structure Theory for Normal Operators * Metric Vector Spaces: The Theory of Bilinear Forms * Metric Spaces * Hilbert Spaces * Tensor Products * Positive Solutions to Linear Systems: Convexity and Separation * Affine Geometry * Operator Factorizations: QR and Singular Value * The Umbral Calculus * References * Index

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Book SynopsisAs a text for an undergraduate mathematics course for nonmajors, Mathematics and Politics requires no prerequisites in either area while the underlying philosophy involves minimizing algebraic computations and focusing instead on some conceptual aspects of mathematics in the context of important real-world questions in political science.Trade ReviewFrom the reviews of the second edition:“Mathematics and Politics is the fruit of undergraduate mathematics courses taught by the authors. The primary audience is political and social science majors. … The writing style is appropriate for the intended audience with the understanding that the students/readers have some familiarity with political science economics or sociology. … Overall the book serves as a useful quantitative introduction to several of the covered topics. … Faculty in the social sciences should strongly consider Mathematics and Politics as a resource/reference.” (J. Douglas Barrett, Technometrics, Vol. 53 (1), February, 2011)“It is intended to serve as a text for social science and humanities students that will highlight the power and utility of mathematics. … if you are considering a course as described above, this textbook deserves to be the one that will entice you into taking the plunge. And if you simply want to educate yourself in areas of social science mathematics that have only recently started to get the attention they deserve, Mathematics and Politics deserves your strong consideration.” (Edward W. Packel, SIAM Review, Vol. 52 (4), 2010)Table of ContentsSocial Choice.- Yes–No Voting.- Political Power.- Conflict.- Fairness.- Escalation.- More Social Choice.- More Yes–No Voting.- More Political Power.- More Conflict.- More Fairness.- More Escalation.

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Book Synopsis1. The Sixteenth and Early Seventeenth Centuries.- 1.1. Introduction.- 1.2. Napier and Logarithms.- 1.3. Briggs and His Logarithms.- 1.4. Bürgi and His Antilogarithms.- 1.5. Interpolation.- 1.6. Vieta and Briggs.- 1.7. Kepler.- 2. The Age of Newton.- 2.1. Introduction.- 2.2. Logarithms and Finite Differences.- 2.3. Trigonometric Tables.- 2.4. The Newton-Raphson and Other Iterative Methods.- 2.5. Finite Differences and Interpolation.- 2.6. Maclaurin on the Euler-Maclaurin Formula.- 2.7. Stirling.- 2.8. Leibniz.- 3. Euler and Lagrange.- 3.1. Introduction.- 3.2. Summation of Series.- 3.3. Euler on the Euler-Maclaurin Formula.- 3.4. Applications of the Summation Formula.- 3.5. Euler on Interpolation.- 3.6. Lunar Theory.- 3.7. Lagrange on Difference Equations.- 3.8. Lagrange on Functional Equations.- 3.9. Lagrange on Fourier Series.- 3.10. Lagrange on Partial Difference Equations.- 3.11. Lagrange on Finite Differences and Interpolation.- 3.12. Lagrange on Hidden Periodicities.- 3.13. LagranTable of Contents1. The Sixteenth and Early Seventeenth Centuries.- 1.1. Introduction.- 1.2. Napier and Logarithms.- 1.3. Briggs and His Logarithms.- 1.4. Bürgi and His Antilogarithms.- 1.5. Interpolation.- 1.6. Vieta and Briggs.- 1.7. Kepler.- 2. The Age of Newton.- 2.1. Introduction.- 2.2. Logarithms and Finite Differences.- 2.3. Trigonometric Tables.- 2.4. The Newton-Raphson and Other Iterative Methods.- 2.5. Finite Differences and Interpolation.- 2.6. Maclaurin on the Euler-Maclaurin Formula.- 2.7. Stirling.- 2.8. Leibniz.- 3. Euler and Lagrange.- 3.1. Introduction.- 3.2. Summation of Series.- 3.3. Euler on the Euler-Maclaurin Formula.- 3.4. Applications of the Summation Formula.- 3.5. Euler on Interpolation.- 3.6. Lunar Theory.- 3.7. Lagrange on Difference Equations.- 3.8. Lagrange on Functional Equations.- 3.9. Lagrange on Fourier Series.- 3.10. Lagrange on Partial Difference Equations.- 3.11. Lagrange on Finite Differences and Interpolation.- 3.12. Lagrange on Hidden Periodicities.- 3.13. Lagrange on Trigonometric Interpolation.- 4. Laplace, Legendre, and Gauss.- 4.1. Introduction.- 4.2. Laplace on Interpolation.- 4.3. Laplace on Finite Differences.- 4.4. Laplace Summation Formula.- 4.5. Laplace on Functional Equations.- 4.6. Laplace on Finite Sums and Integrals.- 4.7. Laplace on Difference Equations.- 4.8. Laplace Transforms.- 4.9. Method of Least Squares.- 4.10. Gauss on Least Squares.- 4.11. Gauss on Numerical Integration.- 4.12. Gauss on Interpolation.- 4.13. Gauss on Rounding Errors.- 5. Other Nineteenth Century Figures.- 5.1. Introduction.- 5.2. Jacobi on Numerical Integration.- 5.3. Jacobi on the Euler-Maclaurin Formula.- 5.4. Jacobi on Linear Equations.- 5.5. Cauchy on Interpolation.- 5.6. Cauchy on the Newton-Raphson Method.- 5.7. Cauchy on Operational Methods.- 5.8. Other Nineteenth Century Results.- 5.9. Integration of Differential Equations.- 5.10. Successive Approximation Methods.- 5.11. Hermite.- 5.12. Sums.

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Book SynopsisDeveloped from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes.Trade Review“Bott and Tu give us an introduction to algebraic topology via differential forms, imbued with the spirit of a master who knew differential forms way back when, yet written from a mature point of view which draws together the separate paths traversed by de Rham theory and homotopy theory. Indeed they assume "an audience with prior exposure to algebraic or differential topology". It would be interesting to use Bott and Tu as the text for a first graduate course in algebraic topology; it would certainly be a wonderful supplement to a standard text. “Bott and Tu write with a consistent point of view and a style which is very readable, flowing smoothly from topic to topic. Moreover, the differential forms and the general homotopy theory are well integrated so that the whole is more than the sum of its parts. "Not intended to be foundational", the book presents most key ideas, at least in sketch form, from scratch, but does not hesitate to quote as needed, without proof, major results of a technical nature, e.g., Sard's Theorem, Whitney's Embedding Theorem and the Morse Lemma on the form of a nondegenerate critical point.” —James D. Stasheff (Bulletin of the American Mathematical Society) “This book is an excellent presentation of algebraic topology via differential forms. The first chapter contains the de Rham theory, with stress on computability. Thus, the Mayer-Vietoris technique plays an important role in the exposition. The force of this technique is demonstrated by the fact that the authors at the end of this chapter arrive at a really comprehensive exposition of Poincaré duality, the Euler and Thom classes and the Thom isomorphism. “The second chapter develops and generalizes the Mayer-Vietoris technique to obtain in a very natural way the Čech-de Rham complex and the Čech cohomology for presheaves. The third chapter on spectral sequences is the most difficult one, but also the richest one by the various applications and digressions into other topics of algebraic topology: singular homology and cohomology with integer coefficients and an important part of homotopy theory, including the Hopf invariant, the Postnikov approximation, the Whitehead tower and Serre’s theorem on the homotopy of spheres. The last chapter is devoted to a brief and comprehensive description of the Chern and Pontryagin classes. “A book which covers such an interesting and important subject deserves some remarks on the style: On the back cover one can read “With its stress on concreteness, motivation, and readability, Differential forms in algebraic topology should be suitable for self-study.” This must not be misunderstood in the ense that it is always easy to read the book. The authors invite the reader to understand algebraic topology by completing himself proofs and examples in the exercises. The reader who seriously follows this invitation really learns a lot of algebraic topology and mathematics in general.” —Hansklaus Rummler (American Mathematical Society)Table of ContentsI De Rham Theory.- II The ?ech-de Rham Complex.- III Spectral Sequences and Applications.- IV Characteristic Classes.- References.- List of Notations.

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Book Synopsis1 Introduction to Lie Groups.- 1.1. Manifolds.- 1.2. Lie Groups.- 1.3. Vector Fields.- 1.4. Lie Algebras.- 1.5. Differential Forms.- Notes.- Exercises.- 2 Symmetry Groups of Differential Equations.- 2.1. Symmetries of Algebraic Equations.- 2.2. Groups and Differential Equations.- 2.3. Prolongation.- 2.4. Calculation of Symmetry Groups.- 2.5. Integration of Ordinary Differential Equations.- 2.6. Nondegeneracy Conditions for Differential Equations.- Notes.- Exercises.- 3 Group-Invariant Solutions.- 3.1. Construction of Group-Invariant Solutions.- 3.2. Examples of Group-Invariant Solutions.- 3.3. Classification of Group-Invariant Solutions.- 3.4. Quotient Manifolds.- 3.5. Group-Invariant Prolongations and Reduction.- Notes.- Exercises.- 4 Symmetry Groups and Conservation Laws.- 4.1. The Calculus of Variations.- 4.2. Variational Symmetries.- 4.3. Conservation Laws.- 4.4. Noether's Theorem.- Notes.- Exercises.- 5 Generalized Symmetries.- 5.1. Generalized Symmetries of Differential Equations.- 5.2. Récursion Operators, Master Symmetries and Formal Symmetries.- 5.3. Generalized Symmetries and Conservation Laws.- 5.4. The Variational Complex.- Notes.- Exercises.- 6 Finite-Dimensional Hamiltonian Systems.- 6.1. Poisson Brackets.- 6.2. Symplectic Structures and Foliations.- 6.3. Symmetries, First Integrals and Reduction of Order.- Notes.- Exercises.- 7 Hamiltonian Methods for Evolution Equations.- 7.1. Poisson Brackets.- 7.2. Symmetries and Conservation Laws.- 7.3. Bi-Hamiltonian Systems.- Notes.- Exercises.- References.- Symbol Index.- Author Index.Table of Contents1 Introduction to Lie Groups.- 1.1. Manifolds.- Change of Coordinates.- Maps Between Manifolds.- The Maximal Rank Condition.- Submanifolds.- Regular Submanifolds.- Implicit Submanifolds.- Curves and Connectedness.- 1.2. Lie Groups.- Lie Subgroups.- Local Lie Groups.- Local Transformation Groups.- Orbits.- 1.3. Vector Fields.- Flows.- Action on Functions.- Differentials.- Lie Brackets.- Tangent Spaces and Vectors Fields on Submanifolds.- Frobenius’ Theorem.- 1.4. Lie Algebras.- One-Parameter Subgroups.- Subalgebras.- The Exponential Map.- Lie Algebras of Local Lie Groups.- Structure Constants.- Commutator Tables.- Infinitesimal Group Actions.- 1.5. Differential Forms.- Pull-Back and Change of Coordinates.- Interior Products.- The Differential.- The de Rham Complex.- Lie Derivatives.- Homotopy Operators.- Integration and Stokes’ Theorem.- Notes.- Exercises.- 2 Symmetry Groups of Differential Equations.- 2.1. Symmetries of Algebraic Equations.- Invariant Subsets.- Invariant Functions.- Infinitesimal Invariance.- Local Invariance.- Invariants and Functional Dependence.- Methods for Constructing Invariants.- 2.2. Groups and Differential Equations.- 2.3. Prolongation.- Systems of Differential Equations.- Prolongation of Group Actions.- Invariance of Differential Equations.- Prolongation of Vector Fields.- Infinitesimal Invariance.- The Prolongation Formula.- Total Derivatives.- The General Prolongation Formula.- Properties of Prolonged Vector Fields.- Characteristics of Symmetries.- 2.4. Calculation of Symmetry Groups.- 2.5. Integration of Ordinary Differential Equations.- First Order Equations.- Higher Order Equations.- Differential Invariants.- Multi-parameter Symmetry Groups.- Solvable Groups.- Systems of Ordinary Differential Equations.- 2.6. Nondegeneracy Conditions for Differential Equations.- Local Solvability.- In variance Criteria.- The Cauchy—Kovalevskaya Theorem.- Characteristics.- Normal Systems.- Prolongation of Differential Equations.- Notes.- Exercises.- 3 Group-Invariant Solutions.- 3.1. Construction of Group-Invariant Solutions.- 3.2. Examples of Group-Invariant Solutions.- 3.3. Classification of Group-Invariant Solutions.- The Adjoint Representation.- Classification of Subgroups and Subalgebras.- Classification of Group-Invariant Solutions.- 3.4. Quotient Manifolds.- Dimensional Analysis.- 3.5. Group-Invariant Prolongations and Reduction.- Extended Jet Bundles.- Differential Equations.- Group Actions.- The Invariant Jet Space.- Connection with the Quotient Manifold.- The Reduced Equation.- Local Coordinates.- Notes.- Exercises.- 4 Symmetry Groups and Conservation Laws.- 4.1. The Calculus of Variations.- The Variational Derivative.- Null Lagrangians and Divergences.- Invariance of the Euler Operator.- 4.2. Variational Symmetries.- Infinitesimal Criterion of Invariance.- Symmetries of the Euler—Lagrange Equations.- Reduction of Order.- 4.3. Conservation Laws.- Trivial Conservation Laws.- Characteristics of Conservation Laws.- 4.4. Noether’s Theorem.- Divergence Symmetries.- Notes.- Exercises.- 5 Generalized Symmetries.- 5.1. Generalized Symmetries of Differential Equations.- Differential Functions.- Generalized Vector Fields.- Evolutionary Vector Fields.- Equivalence and Trivial Symmetries.- Computation of Generalized Symmetries.- Group Transformations.- Symmetries and Prolongations.- The Lie Bracket.- Evolution Equations.- 5.2. Récursion Operators, Master Symmetries and Formal Symmetries.- Frechet Derivatives.- Lie Derivatives of Differential Operators.- Criteria for Recursion Operators.- The Korteweg—de Vries Equation.- Master Symmetries.- Pseudo-differential Operators.- Formal Symmetries.- 5.3. Generalized Symmetries and Conservation Laws.- Adjoints of Differential Operators.- Characteristics of Conservation Laws.- Variational Symmetries.- Group Transformations.- Noether’s Theorem.- Self-adjoint Linear Systems.- Action of Symmetries on Conservation Laws.- Abnormal Systems and Noether’s Second Theorem.- Formal Symmetries and Conservation Laws.- 5.4. The Variational Complex.- The D-Complex.- Vertical Forms.- Total Derivatives of Vertical Forms.- Functionals and Functional Forms.- The Variational Differential.- Higher Euler Operators.- The Total Homotopy Operator.- Notes.- Exercises.- 6 Finite-Dimensional Hamiltonian Systems.- 6.1. Poisson Brackets.- Hamiltonian Vector Fields.- The Structure Functions.- The Lie-Poisson Structure.- 6.2. Symplectic Structures and Foliations.- The Correspondence Between One-Forms and Vector Fields.- Rank of a Poisson Structure.- Symplectic Manifolds.- Maps Between Poisson Manifolds.- Poisson Submanifolds.- Darboux’ Theorem.- The Co-adjoint Representation.- 6.3. Symmetries, First Integrals and Reduction of Order.- First Integrals.- Hamiltonian Symmetry Groups.- Reduction of Order in Hamiltonian Systems.- Reduction Using Multi-parameter Groups.- Hamiltonian Transformation Groups.- The Momentum Map.- Notes.- Exercises.- 7 Hamiltonian Methods for Evolution Equations.- 7.1. Poisson Brackets.- The Jacobi Identity.- Functional Multi-vectors.- 7.2. Symmetries and Conservation Laws.- Distinguished Functionals.- Lie Brackets.- Conservation Laws.- 7.3. Bi-Hamiltonian Systems.- Recursion Operators.- Notes.- Exercises.- References.- Symbol Index.- Author Index.

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Book SynopsisA unique and comprehensive text on the philosophy of model-based data analysis and strategy for the analysis of empirical data. The book introduces information theoretic approaches and focuses critical attention on a priori modeling and the selection of a good approximating model that best represents the inference supported by the data.Table of ContentsIntroduction * Information and Likelihood Theory: A Basis for Model Selection and Inference * Basic Use of the Information-Theoretic Approach * Formal Inference From More Than One Model: Multi-Model Inference (MMI) * Monte Carlo Insights and Extended Examples * Statistical Theory and Numerical Results * Summary

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Book SynopsisThe first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level.Trade ReviewFrom the reviews of the second edition:“In the second edition of this text, Koblitz presents a wide-ranging introduction to the theory of p-adic numbers and functions. … there are some really nice exercises that allow the reader to explore the material. … And with the exercises, the book would make a good textbook for a graduate course, provided the students have a decent background in analysis and number theory.” (Donald L. Vestal, The Mathematical Association of America, April, 2011)Table of ContentsI p-adic numbers.- 1. Basic concepts.- 2. Metrics on the rational numbers.- Exercises.- 3. Review of building up the complex numbers.- 4. The field of p-adic numbers.- 5. Arithmetic in ?p.- Exercises.- II p-adic interpolation of the Riemann zeta-function.- 1. A formula for ?(2k).- 2. p-adic interpolation of the function f(s) = as.- Exercises.- 3. p-adic distributions.- Exercises.- 4. Bernoulli distributions.- 5. Measures and integration.- Exercises.- 6. The p-adic ?-function as a Mellin-Mazur transform.- 7. A brief survey (no proofs).- Exercises.- III Building up ?.- 1. Finite fields.- Exercises.- 2. Extension of norms.- Exercises.- 3. The algebraic closure of ?p.- 4. ?.- Exercises.- IV p-adic power series.- 1. Elementary functions.- Exercises.- 2. The logarithm, gamma and Artin-Hasse exponential functions.- Exercises.- 3. Newton polygons for polynomials.- 4. Newton polygons for power series.- Exercises.- V Rationality of the zeta-function of a set of equations over a finite field.- 1. Hypersurfaces and their zeta-functions.- Exercises.- 2. Characters and their lifting.- 3. A linear map on the vector space of power series.- 4. p-adic analytic expression for the zeta-function.- Exercises.- 5. The end of the proof.- Answers and Hints for the Exercises.

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Book SynopsisThis is a short text in linear algebra, intended for a one-term course. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues.Trade ReviewSecond Edition S. Lang Introduction to Linear Algebra "Excellent! Rigorous yet straightforward, all answers included!"—Dr. J. Adam, Old Dominion UniversityTable of ContentsI Vectors.- II Matrices and Linear Equations.- III Vector Spaces.- IV Linear Mappings.- V Composition and Inverse Mappings.- VI Scalar Products and Orthogonality.- VII Determinants.- VIII Eigenvectors and Eigenvalues.- Answers to Exercises.

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Book SynopsisOne Basic Material.- I Vectors.- II Differentiation of Vectors.- III Functions of Several Variables.- IV The Chain Rule and the Gradient.- Two Maxima, Minima, and Taylor's Formula.- V Maximum and Minimum.- VI Higher Derivatives.- Three Curve Integrals and Double Integrals.- VII Potential Functions.- VIII Curve Integrals.- IX Double Integrals.- X Green's Theorem.- Four Triple and Surface Integrals.- XI Triple Integrals.- XII Surface Integrals.- Five Mappings, Inverse Mappings, and Change of Variables Formula..- XIII Matrices.- XIV Linear Mappings.- XV Determinants.- XVI Applications to Functions of Several Variables.- XVII The Change of Variables Formula.- Appendix Fourier Series.- 1. General Scalar Products.- 2. Computation of Fourier Series.- Answers to Exercises.Table of ContentsI: Basic Material. 1: Vectors. 2: Differentiation of Vectors. 3: Functions of Several Variables. 4: The Chain Rule and the Gradient. II: Maxima, Minima, and Taylor's Formula. 5: Maximum and Minimum. 6: Higher Derivatives. III: Curve Integrals and Double Integrals. 7: Potential Functions. 8: Curve Integrals. 9: Double Integrals. 10: Green's Theorem. IV: Triple and Surface Integrals. 12: Triple Integrals. V: Mappings, Inverse Mappings, and Change of Variables Formula. 13: Matrices. 14: Linear Mappings. 15: Determinants. 16: Applications to Functions of Several Variables. 17: The Change of Variables Formula. Appendix: Fourier Series.

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Book SynopsisI Vector Spaces.- II Matrices.- III Linear Mappings.- IV Linear Maps and Matrices.- V Scalar Products and Orthogonality.- VI Determinants.- VII Symmetric, Hermitian, and Unitary Operators.- VIII Eigenvectors and Eigenvalues.- IX Polynomials and Matrices.- X Triangulation of Matrices and Linear Maps.- XI Polynomials and Primary Decomposition.- XII Convex Sets.- Appendix I Complex Numbers.- Appendix II Iwasawa Decomposition and Others.Trade Review"The present textbook is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of finite-dimensional vector spaces and proceeds to explain the structure theorems for linear maps, including eigenvectors and eigenvalues, quadratic and Hermitian forms, diagonalization of symmetric, Hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. It also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants, and linear maps. In this third edition, many parts of the book have been rewritten and reorganized, and new exercises have been added." (S. Lajos, Mathematical Reviews) Table of Contents1. Vector Spaces; 2. Matrices; 3. Linear Mappings; 4. Linear Maps and Matrices; 5. Scalar Products and Orthogonality; 6. Determinants; 7. Symmetric, Hermitian, and Unitary Operators; 8. Eigenvectors and Eigenvalues; 9. Polynomials and Matrices; 10. Triangulation of Matrices and Linear Maps; 11. Polynomials and Primary Decomposition; 12. Convex Sets

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Book SynopsisIntuition.- I1: Geometry is about shapes.- I2: and more shapes.- I3: Polygons in the plane.- I4: Angles in the plane.- I5: Walking north, east, south, and west in the plane.- I6: Areas of rectangles.- I7: What is the area of the shaded triangle?.- I8: Adding the angles of a triangle.- I9: Pythagorean theorem.- I10: Side Side Side (SSS).- I11: Parallel lines.- I12: Rectangles between parallels and the Z-principle.- I13: Areas: The principle of parallel slices.- I14: If two lines in the plane do not intersect, they are parallel.- I15: The first magnification principle: preliminary form.- I16: The first magnification principle: final form.- I17: Area inside a circle of radius one.- I18: When are triangles congruent?.- I19: Magnifications preserve parallelism and angles.- I20: The principle of similarity.- I21: Proportionality of segments cut by parallels.- I22: Finding the center of a triangle.- I23: Concurrence theorem for altitudes of a triangle.- I24: Inscribing angles in circles.- I25: Fun facts about circles, and limiting cases.- I26: Degrees and radians.- I27: Trigonometry.- I28: Tangent a =(rise)/(run).- I29: Everything you always wanted to know about trigonometry but were afraid to ask.- I30: The law of sines and the law of cosines.- I31: Figuring areas.- I32: The second magnification principle.- I33: Volume of a pyramid.- I34: Of cones and collars.- I35: Sphereworld.- I36: Segments and angles in sphereworld.- I37: Of boxes, cylinders, and spheres.- I38: If it takes one can of paint to paint a square one widget on a side, how many cans does it take to paint a sphere with radius r widgets?.- I39: Excess angle formula for spherical triangles.- I40: Hyperbolic-land.- Construction.- C1: Copying triangles.- C2: Copying angles.- C3: Constructing perpendiculars.- C4:Constructing parallels.- C5: Constructing numbers as lengths.- C6 Given a number, construct its square root.- C7: Constructing parallelograms.- C8: Constructing a regular 3-gon and 4-gon.- C9: Constructing a regular 5-gon.- C10: Constructing a regular 6-gon.- C11: Constructing a regular 7-gon (almost).- C12: Constructing a regular tetrahedron.- C13: Constructing a cube and an octohedron.- C14: Constructing a dodecahedron and an icosahedron.- C15: Constructing the baricenter of a triangle.- C16: Constructing the altitudes of a triangle.- C17: Constructing a circle through three points.- C18: Bisecting a given angle.- C19: Putting circles inside angles.- C20: Inscribing circles in polygons.- C21: Circumscribing circles about polygons.- C22: Drawing triangles on the sphere.- C23: Constructing hyperbolic lines.- Proof.- P1: Distance on the line, motions of the line.- P2: Distance in the plane.- P3: Motions of the plane.- P4: A list of motions of the line.- P5: A complete list of motions of the line.- P6: Motions of the plane: Translations.- P7: Motions of the plane: Rotations.- P8: Motions of the plane: Vertical flip.- P9: Motions of the plane fixing (0,0) and (a,0).- P10: A complete list of motions of the plane.- P11: Distance in space.- P12: Motions of space.- P13: The triangle inequality.- P14: Co-ordinate geometry is about shapes and more shapes.- P15: The shortest path between two points.- P16: The unique line through two given points.- P17: Proving SSS.- Computer Programs.- CP1: Information you'll need about the CP-pages.- CP2: Given two points, construct the segment, ray, and line that pass through them.- CP3: Given a line and a point, construct the perpendicular to the line through the point, or the parallel to the line through the point.- CP4: Given asegment, construct its perpendicular bisector.- CP5: Given an angle, construct the bisector.- CP6: Given three vertices, construct the triangle and its medians.- CP7: Given three vertices, construct the triangle and its angle bisectors.- CP8: Given three vertices, construct the triangle and its altitudes.- CP9: Given a figure in the plane and a positive number R, magnify the figure by a factor of R.- CP10: Given a figure in the plane and two positive numbers R and S, magnify the figure by a factor of R in the horizontal direction and by a factor of S in the vertical direction.- CP11: Given the center and radius of a circle, and two positive numbers R and S, magnify the circle by a factor of R in the horizontal direction and by a factor of S in the vertical direction.- CP12: TRANSLATIONS: Given a figure in the plane and two numbers a and b, show the motion m(x,y) = (x + a, y + b).- CP13: ROTATIONS: Given a figure in the plane and two numbers c and s, so that c2 + s2 = 1, show the motion m(x,y) = (cx - sy, sx + cy).- CP14: FLIPS: Given a figure in the plane, show the motion m(x,y) = (x, -y).- CP15: Composing a set of two motions.- CP16: Composing a series of motions.- CP17: Given a point and a positive number R, construct the circle of radius R about the point.- CP18: Given three points in the plane, construct the unique circle that passes through all three points.- CP19: Given the center of a circle and a point on the circle, construct the tangent to the circle through the point.- CP20: Given a circle and a point outside the circle, construct the two lines tangent to the circle that pass through the point.- CP21: Given a point X inside or outside the circle of radius one and center O, construct the reciprocal point X'.- CP22: Given two points inside the circle ofradius one about (0,0), construct the hyperbolic line containing the two points.Table of ContentsIntuition.- I1: Geometry is about shapes.- I2:… and more shapes.- I3: Polygons in the plane.- I4: Angles in the plane.- I5: Walking north, east, south, and west in the plane.- I6: Areas of rectangles.- I7: What is the area of the shaded triangle?.- I8: Adding the angles of a triangle.- I9: Pythagorean theorem.- I10: Side Side Side (SSS).- I11: Parallel lines.- I12: Rectangles between parallels and the Z-principle.- I13: Areas: The principle of parallel slices.- I14: If two lines in the plane do not intersect, they are parallel.- I15: The first magnification principle: preliminary form.- I16: The first magnification principle: final form.- I17: Area inside a circle of radius one.- I18: When are triangles congruent?.- I19: Magnifications preserve parallelism and angles.- I20: The principle of similarity.- I21: Proportionality of segments cut by parallels.- I22: Finding the center of a triangle.- I23: Concurrence theorem for altitudes of a triangle.- I24: Inscribing angles in circles.- I25: Fun facts about circles, and limiting cases.- I26: Degrees and radians.- I27: Trigonometry.- I28: Tangent a =(rise)/(run).- I29: Everything you always wanted to know about trigonometry but were afraid to ask.- I30: The law of sines and the law of cosines.- I31: Figuring areas.- I32: The second magnification principle.- I33: Volume of a pyramid.- I34: Of cones and collars.- I35: Sphereworld.- I36: Segments and angles in sphereworld.- I37: Of boxes, cylinders, and spheres.- I38: If it takes one can of paint to paint a square one widget on a side, how many cans does it take to paint a sphere with radius r widgets?.- I39: Excess angle formula for spherical triangles.- I40: Hyperbolic-land.- Construction.- C1: Copying triangles.- C2: Copying angles.- C3: Constructing perpendiculars.- C4: Constructing parallels.- C5: Constructing numbers as lengths.- C6 Given a number, construct its square root.- C7: Constructing parallelograms.- C8: Constructing a regular 3-gon and 4-gon.- C9: Constructing a regular 5-gon.- C10: Constructing a regular 6-gon.- C11: Constructing a regular 7-gon (almost).- C12: Constructing a regular tetrahedron.- C13: Constructing a cube and an octohedron.- C14: Constructing a dodecahedron and an icosahedron.- C15: Constructing the baricenter of a triangle.- C16: Constructing the altitudes of a triangle.- C17: Constructing a circle through three points.- C18: Bisecting a given angle.- C19: Putting circles inside angles.- C20: Inscribing circles in polygons.- C21: Circumscribing circles about polygons.- C22: Drawing triangles on the sphere.- C23: Constructing hyperbolic lines.- Proof.- P1: Distance on the line, motions of the line.- P2: Distance in the plane.- P3: Motions of the plane.- P4: A list of motions of the line.- P5: A complete list of motions of the line.- P6: Motions of the plane: Translations.- P7: Motions of the plane: Rotations.- P8: Motions of the plane: Vertical flip.- P9: Motions of the plane fixing (0,0) and (a,0).- P10: A complete list of motions of the plane.- P11: Distance in space.- P12: Motions of space.- P13: The triangle inequality.- P14: Co-ordinate geometry is about shapes and more shapes.- P15: The shortest path between two points….- P16: The unique line through two given points.- P17: Proving SSS.- Computer Programs.- CP1: Information you’ll need about the CP-pages.- CP2: Given two points, construct the segment, ray, and line that pass through them.- CP3: Given a line and a point, construct the perpendicular to the line through the point, or the parallel to the line through the point.- CP4: Given a segment, construct its perpendicular bisector.- CP5: Given an angle, construct the bisector.- CP6: Given three vertices, construct the triangle and its medians.- CP7: Given three vertices, construct the triangle and its angle bisectors.- CP8: Given three vertices, construct the triangle and its altitudes.- CP9: Given a figure in the plane and a positive number R, magnify the figure by a factor of R.- CP10: Given a figure in the plane and two positive numbers R and S, magnify the figure by a factor of R in the horizontal direction and by a factor of S in the vertical direction.- CP11: Given the center and radius of a circle, and two positive numbers R and S, magnify the circle by a factor of R in the horizontal direction and by a factor of S in the vertical direction.- CP12: TRANSLATIONS: Given a figure in the plane and two numbers a and b, show the motion m(x,y) = (x + a, y + b).- CP13: ROTATIONS: Given a figure in the plane and two numbers c and s, so that c2 + s2 = 1, show the motion m(x,y) = (cx - sy, sx + cy).- CP14: FLIPS: Given a figure in the plane, show the motion m(x,y) = (x, -y).- CP15: Composing a set of two motions.- CP16: Composing a series of motions.- CP17: Given a point and a positive number R, construct the circle of radius R about the point.- CP18: Given three points in the plane, construct the unique circle that passes through all three points.- CP19: Given the center of a circle and a point on the circle, construct the tangent to the circle through the point.- CP20: Given a circle and a point outside the circle, construct the two lines tangent to the circle that pass through the point.- CP21: Given a point X inside or outside the circle of radius one and center O, construct the reciprocal point X’.- CP22: Given two points inside the circle of radius one about (0,0), construct the hyperbolic line containing the two points.

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Book SynopsisOne Basic Theory.- I Complex Numbers and Functions.- II Power Series.- III Cauchy's Theorem, First Part.- IV Winding Numbers and Cauchy's Theorem.- V Applications of Cauchy's integral Formula.- VI Calculus of Residues.- VII Conformal Mappings.- VIII Harmonic Functions.- Two Geometric Function Theory.- IX Schwarz Reflection.- X The Riemann Mapping Theorem.- XI Analytic Continuation Along Curves.- Three Various Analytic Topics.- XII Applications of the Maximum Modulus Principle and Jensen's Formula.- XIII Entire and Meromorphic Functions.- XIV Elliptic Functions.- XV The Gamma and Zeta Functions.- XVI The Prime Number Theorem.- 1. Summation by Parts and Non-Absolute Convergence.- 2. Difference Equations.- 3. Analytic Differential Equations.- 4. Fixed Points of a Fractional Linear Transformation.- 6. Cauchy's Theorem for Locally Integrable Vector Fields.- 7. More on Cauchy-Riemann.Trade Review"The very understandable style of explanation, which is typical for this author, makes the book valuable for both students and teachers."EMS Newsletter, Vol. 37, Sept. 2000 Fourth Edition S. Lang Complex Analysis "A highly recommendable book for a two semester course on complex analysis." —ZENTRALBLATTMATHTable of ContentsI: BASIC THEORY. 1: Complex Numbers and Functions. 2: Power Series. 3: Cauchy's Theorem, First Part. 4: Winding Numbers and Cauchy's Theorem. 5: Applications of Cauchy's Integral Formula. 6: Calculus of Residues. 7: Conformal Mappings. 8: Harmonic Functions. II: GEOMETRIC FUNCTION THEORY. 9: Schwarz Reflection. 10: The Riemann Mapping Theorem. 11: Analytic Continuation Along Curves. III: VARIOUS ANALYTIC TOPICS. 12: Applications of the Maximum Modulus Principle and Jensen's Formula. 13: Entire and Meromorphic Functions. 14: Elliptic Functions. 15: The Gamma and Zeta Functions. 16: The Prime Number Theorem.

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Book SynopsisI The Methodology of Statistical Consulting.- 1 Introduction to Statistical Consulting.- 2 Communication.- 3 Methodological Aspects.- 4 A Consulting Project from A to Z.- II Case Studies.- 5 Introduction to the Case Studies.- 6 Case Studies from Group I.- 7 Case Studies from Group II.- 8 Case Studies from Group III.- 9 Additional Case Studies.- A Resources.- A.1 References.- A.2 Datasets for Case Studies in Part II.- A.3 Statistical Consulting Course.- A.3.1 Course Description.- A.3.2 List of Topics by Week.- A.3.3 Reference List.- B Statistical Software.- B.1 SAS.- B.1.1 The SAS Setup.- B.1.2 Details on the DATA Step.- B.1.3 SAS Procedures.- B.1.4 Further Details of SAS.- B.2 S-PLUS.- B.2.1 S-PLUS Preliminaries.- B.2.2 The S-PLUS Setup.- B.2.3 Basic S-PLUS Commands.- B.2.4 Efficient Use of S-PLUS.- B.2.5 S-PLUS Statistical Procedures.- B.2.6 S-PLUS Glossary.- C Statistical Addendum.- C.1 Univariate Distributions.- C.2 Multivariate Distributions.- C.3 Statistical Tests.- C.4 Sample SizTrade ReviewFrom the reviews: THE AMERICAN STATISTICIAN "Although there are other books that effectively tackle the individual aspects described above, this book seems to be the most ideally suited to teaching a well-rounded statistics course at the undergraduate of graduate level…[It] gives informative and self-contained discussions for the many aspects of consulting in balanced proportions that would make using the book for a textbook delightfully straightforward. The collection of case studies is diverse in disciplines considered and level of difficulty, and seems to focus on interesting problems that students will find highly motivating…a valuable resource for statistical consultants, both beginning and established…a prime candidate for use as a stand-alone textbook…since it contains a desirable balance of materials with statistical methodology, oral and written communication skills, and rich case studies…It will make a solid long-term reference for students. Also, for instructors of more traditional senior undergraduate and junior graduate courses, it provides useful case studies to illustrate standard methods in realistic settings that can easily be implemented."Table of ContentsIntroduction to Statistical Consulting * Communication * Methodological Aspects * A Consulting Project from A to Z * Introduction to Case Studies * Case Studies from Group I * Case Studies from Group II * Case Studies from Group III * Additional Case Studies

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Book SynopsisThis classic textbook is suitable for a first course in the theory of statistics for students with a background in calculus, multivariate calculus, and the elements of matrix algebra.Table of ContentsPreface, 1 Preliminaries, 2 Probability, 3 Random Variables, 4 Expectations, 5 Limit Theorems, 6 Some Parametric Families, 7 Sampling and Reduction of Data, 8 Estimation, 9 Testing Hypotheses, 10 Analysis of Categorical Data, 11 Sequential Analysis, 12 Multivariate Distributions, 13 Nonpararnetric Tests, 14 Linear Models and Analysis of Variance, 15 Decision Theory, Tables, References and Further Reading, Answers to Problems, Index

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Book SynopsisDesigned for undergraduate students of set theory, this book presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. It aims to give students a grounding to the results of set theory as well as to tackle significant problems that arise from the theory.Table of ContentsINTRODUCTIONOutline of the bookAssumed knowledgeTHE REAL NUMBERSIntroductionDedekind's constructionAlternative constructionsThe rational numbersTHE NATURAL NUMBERSIntroductionThe construction of the natural numbersArithmeticFinite setsTHE ZERMELO-FRAENKEL AXIOMSIntroductionA formal languageAxioms 1 to 3Axioms 4 to 6Axioms 7 to 9CARDINAL (Without the Axiom of Choice)IntroductionComparing SizesBasic properties of ˜ and =Infinite sets without AC-countable setsUncountable sets and cardinal arithmetic without ACORDERED SETSIntroductionLinearly ordered setsOrder arithmeticWell-ordered setsORDINAL NUMBERSIntroductionOrdinal numbersBeginning ordinal arithmeticOrdinal arithmeticThe ÀsSET THEORY WITH THE AXIOM OF CHOICEIntroductionThe well-ordering principleCardinal arithmetic and the axiom of choiceThe continuum hypothesisBIBLIOGRAPHYINDEX

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Book SynopsisFundamental Constructs in Mathematics Education is a unique sourcebook crafted from classic texts, research papers and books in mathematics education. Linked together by the editors'' narrative, the book provides a fascinating examination of, and insight into, key constructs in mathematics education and how they link together. The choice of constructs is based on (some of) the many constructs which have proved fruitful in research and which have informed choices made by teachers.The book is divided into two parts: learning and teaching. The first part includes views about how people learn - from Plato to Dewey, as well as constructivism, activity theory and French didactiques. The second part includes extracts concerned with initiating, sustaining and bringing to a conclusion learners'' work on mathematical tasks.Fundamental Constructs in Mathematics Education provides access to a wide range of constructs in mathematics education and orients the Table of ContentsIntroduction Section 1: Activating and analysing learning 1. Probing Thinking 2. Conditions for Learning 3. Analysis of Learning for Informal Teaching 4. Affect in Learning Mathematics 5. Learners' Powers 6. Learning as Action 7. Learning What? Section 2: Guiding and directing learning 8. Teachers' Roles 9. Initiating Mathematical Activity 10. Sustaining Mathematical Activity 11. Concluding Mathematical Activity 12. Having Learned ?

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Book SynopsisDon't let your mathematical skills fail you! In Engineering, Construction, and Science examinations, marks are often lost through carelessness or from not properly understanding the mathematics involved. When there are only a few marks on offer for a part of a question, there may be full marks for a right answer and none for a wrong one, regardless of the thought that went into the answer. If you want to avoid losing these marks by improving the clarity both of your mathematical work and your mathematical understanding, then Essential Maths for Engineering and Construction is the book for you.We all make mistakes; who doesn't? But mistakes can be avoided when we understand why we make them. Taking mistakes commonly made by undergraduate students as its entry point, this book not only looks at how you can prevent mistakes, but also provides a primer for the fundamental mathematical skills required for your degree discipline.Whether you strugglTrade ReviewAs a principal lecturer who teaches maths to undergraduates in civil engineering and architecture, Mark Breach is well aware of the mistakes and insecurities that beset his students. ... Throughout the book, there are a lot of both worked examples, and exercises for the student to use as practice, with the solutions printed upside-down in blocks of text. ... a very useful resource for students entering tertiary education in a field that requires maths.—Peter R. Smith, University of Sydney, Australia, in Architectural Science Review, Vol. 55, No. 2, May 2012Table of ContentsFinding the Pitfalls: Introduction. Find an Independent Check. Misleading with Mathematics: When Ink Meets Paper. Juggling with Numbers. False Assumptions and False Logic. Lies, Damned Lies and Statistics. Some Mistakes that we Make: Errors in Arithmetic. Errors in Algebra. Errors in Trigonometry. Calculator Errors. Bad Notation. Errors in Calculus. Test Yourself.

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Book SynopsisStatistics for Sport and Exercise Studies guides the student through the full research process, from selecting the most appropriate statistical procedure, to analysing data, to the presentation of results, illustrating every key step in the process with clear examples, case-studies and data taken from real sport and exercise settings.Every chapter includes a range of features designed to help the student grasp the underlying concepts and relate each statistical procedure to their own research project, including definitions of key terms, practical exercises, worked examples and clear summaries. The book also offers an in-depth and practical guide to using SPSS in sport and exercise research, the most commonly used data analysis software in sport and exercise departments. In addition, a companion website includes more than 100 downloadable data sets and work sheets for use in or out of the classroom, full solutions to exercises contained in the book, plus over 1,300 PoTable of Contents1. Data, Information and Statistics 2. Using this book 3. Descriptive Statistics 4. Standardized scores 5. Probability 6. Data distributions 7. Hypothesis testing 8. Correlation 9. Linear Regression 10. t Tests 11. Analysis of Variances 12. Factorial ANOVA 13. Multivariate ANOVA 14. Nonparametric tests 15. Chi Square 16. Statistical Classification 17. Cluster Analysis 18. Data Reduction using Principal Components Analysis 19. Reliability 20. Statistical Power

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Book SynopsisMechanics 2 was written to provide thorough preparation for the revised 2004 specification. Based on the first editions, this series helps you to prepare for the new exams.Table of Contents1. Moments and equilibrium2. Centres of mass.3. Energy.4. Kinematics and variable acceleration.5. Circular motion.6. Circular motion with variable speed.7. Application of differential equations in mechanics.Exam style practice papers.Answers.Index.

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Book SynopsisTable of Contents1. Linguistics: Core Concepts and Principles 2. Grammars 3. Open-Source Libraries, Application Frameworks, Workflow Systems, and Other Resources 4. Mathematical Essentials 5. Probability 6. Inference and Prediction Methods 7. Random Processes 8. Bayesian Methods 9. Machine Learning 10. Artificial Neural Networks for Natural Language Processing 11. Information Retrieval 12. Language Core Tasks 1 13. Language Core Tasks 2 14. Language Understanding Applications 1 15. Language Understanding Applications 2 16. Deep Learning for Natural Language Processing 17. Text Mining for Modeling Cyberattacks 18. World Languages and Crosslinguistics 19. Linguistic Elegance of the Languages of South India 20. Current Trends and Open Problems

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Book SynopsisDifferential Equations Workbook For Dummies is a course supplement and practice guide for students taking a course that involves the use of differential equations. This book takes readers step-by-step through this intimidating subject and features numerous practice exercises and clear, concise examples to improve problem-solving skills.Table of ContentsIntroduction 1 Part I: Tackling First Order Differential Equations 5 Chapter 1: Looking Closely at Linear First Order Differential Equations 7 Chapter 2: Surveying Separable First Order Differential Equations 29 Chapter 3: Examining Exact First Order Differential Equations 59 Part II: Finding Solutions to Second and Higher Order Differential Equations 79 Chapter 4: Working with Linear Second Order Differential Equations 81 Chapter 5: Tackling Nonhomogeneous Linear Second Order Differential Equations 105 Chapter 6: Handling Homogeneous Linear Higher Order Differential Equations 129 Chapter 7: Taking On Nonhomogeneous Linear Higher Order Differential Equations 153 Part III: The Power Stuff: Advanced Techniques 175 Chapter 8: Using Power Series to Solve Ordinary Differential Equations 177 Chapter 9: Solving Differential Equations with Series Solutions Near Singular Points 199 Chapter 10: Using Laplace Transforms to Solve Differential Equations 225 Chapter 11: Solving Systems of Linear First Order Differential Equations 249 Part IV: The Part of Tens 273 Chapter 12: Ten Common Ways of Solving Differential Equations 275 Chapter 13: Ten Real-World Applications of Differential Equations 279 Index 283

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Book SynopsisA new edition of the definitive guide to logistic regression modeling for health science and other applications This thoroughly expanded Third Edition provides an easily accessible introduction to the logistic regression (LR) model and highlights the power of this model by examining the relationship between a dichotomous outcome and a set of covariables. Applied Logistic Regression, Third Edition emphasizes applications in the health sciences and handpicks topics that best suit the use of modern statistical software. The book provides readers with state-of-the-art techniques for building, interpreting, and assessing the performance of LR models. New and updated features include: A chapter on the analysis of correlated outcome data A wealth of additional material for topics ranging from Bayesian methods to assessing model fit Rich data sets from real-world studies that demonstrate each method under discussion Trade Review“In conclusion, the index was mercifully complete, and all items searched for were found (nice cross-referencing too) In summary: Highly recommended.” (Scientific Computing, 1 May 2013) Table of ContentsPreface to the Third Edition xiii 1 Introduction to the Logistic Regression Model 1 1.1 Introduction 1 1.2 Fitting the Logistic Regression Model 8 1.3 Testing for the Significance of the Coefficients 10 1.4 Confidence Interval Estimation 15 1.5 Other Estimation Methods 20 1.6 Data Sets Used in Examples and Exercises 22 1.6.1 The ICU Study 22 1.6.2 The Low Birth Weight Study 24 1.6.3 The Global Longitudinal Study of Osteoporosis in Women 24 1.6.4 The Adolescent Placement Study 26 1.6.5 The Burn Injury Study 27 1.6.6 The Myopia Study 29 1.6.7 The NHANES Study 31 1.6.8 The Polypharmacy Study 31 Exercises 32 2 The Multiple Logistic Regression Model 35 2.1 Introduction 35 2.2 The Multiple Logistic Regression Model 35 2.3 Fitting the Multiple Logistic Regression Model 37 2.4 Testing for the Significance of the Model 39 2.5 Confidence Interval Estimation 42 2.6 Other Estimation Methods 45 Exercises 46 3 Interpretation of the Fitted Logistic Regression Model 49 3.1 Introduction 49 3.2 Dichotomous Independent Variable 50 3.3 Polychotomous Independent Variable 56 3.4 Continuous Independent Variable 62 3.5 Multivariable Models 64 3.6 Presentation and Interpretation of the Fitted Values 77 3.7 A Comparison of Logistic Regression and Stratified Analysis for 2 × 2 Tables 82 Exercises 87 4 Model-Building Strategies and Methods for Logistic Regression 89 4.1 Introduction 89 4.2 Purposeful Selection of Covariates 89 4.2.1 Methods to Examine the Scale of a Continuous Covariate in the Logit 94 4.2.2 Examples of Purposeful Selection 107 4.3 Other Methods for Selecting Covariates 124 4.3.1 Stepwise Selection of Covariates 125 4.3.2 Best Subsets Logistic Regression 133 4.3.3 Selecting Covariates and Checking their Scale Using Multivariable Fractional Polynomials 139 4.4 Numerical Problems 145 Exercises 150 5 Assessing the Fit of the Model 153 5.1 Introduction 153 5.2 Summary Measures of Goodness of Fit 154 5.2.1 Pearson Chi-Square Statistic, Deviance, and Sum-of-Squares 155 5.2.2 The Hosmer–Lemeshow Tests 157 5.2.3 Classification Tables 169 5.2.4 Area Under the Receiver Operating Characteristic Curve 173 5.2.5 Other Summary Measures 182 5.3 Logistic Regression Diagnostics 186 5.4 Assessment of Fit via External Validation 202 5.5 Interpretation and Presentation of the Results from a Fitted Logistic Regression Model 212 Exercises 223 6 Application of Logistic Regression with Different Sampling Models 227 6.1 Introduction 227 6.2 Cohort Studies 227 6.3 Case-Control Studies 229 6.4 Fitting Logistic Regression Models to Data from Complex Sample Surveys 233 Exercises 242 7 Logistic Regression for Matched Case-Control Studies 243 7.1 Introduction 243 7.2 Methods For Assessment of Fit in a 1–M Matched Study 248 7.3 An Example Using the Logistic Regression Model in a 1–1 Matched Study 251 7.4 An Example Using the Logistic Regression Model in a 1–M Matched Study 260 Exercises 267 8 Logistic Regression Models for Multinomial and Ordinal Outcomes 269 8.1 The Multinomial Logistic Regression Model 269 8.1.1 Introduction to the Model and Estimation of Model Parameters 269 8.1.2 Interpreting and Assessing the Significance of the Estimated Coefficients 272 8.1.3 Model-Building Strategies for Multinomial Logistic Regression 278 8.1.4 Assessment of Fit and Diagnostic Statistics for the Multinomial Logistic Regression Model 283 8.2 Ordinal Logistic Regression Models 289 8.2.1 Introduction to the Models, Methods for Fitting, and Interpretation of Model Parameters 289 8.2.2 Model Building Strategies for Ordinal Logistic Regression Models 305 Exercises 310 9 Logistic Regression Models for the Analysis of Correlated Data 313 9.1 Introduction 313 9.2 Logistic Regression Models for the Analysis of Correlated Data 315 9.3 Estimation Methods for Correlated Data Logistic Regression Models 318 9.4 Interpretation of Coefficients from Logistic Regression Models for the Analysis of Correlated Data 323 9.4.1 Population Average Model 324 9.4.2 Cluster-Specific Model 326 9.4.3 Alternative Estimation Methods for the Cluster-Specific Model 333 9.4.4 Comparison of Population Average and Cluster-Specific Model 334 9.5 An Example of Logistic Regression Modeling with Correlated Data 337 9.5.1 Choice of Model for Correlated Data Analysis 338 9.5.2 Population Average Model 339 9.5.3 Cluster-Specific Model 344 9.5.4 Additional Points to Consider when Fitting Logistic Regression Models to Correlated Data 351 9.6 Assessment of Model Fit 354 9.6.1 Assessment of Population Average Model Fit 354 9.6.2 Assessment of Cluster-Specific Model Fit 365 9.6.3 Conclusions 374 Exercises 375 10 Special Topics 377 10.1 Introduction 377 10.2 Application of Propensity Score Methods in Logistic Regression Modeling 377 10.3 Exact Methods for Logistic Regression Models 387 10.4 Missing Data 395 10.5 Sample Size Issues when Fitting Logistic Regression Models 401 10.6 Bayesian Methods for Logistic Regression 408 10.6.1 The Bayesian Logistic Regression Model 410 10.6.2 MCMC Simulation 411 10.6.3 An Example of a Bayesian Analysis and Its Interpretation 419 10.7 Other Link Functions for Binary Regression Models 434 10.8 Mediation 441 10.8.1 Distinguishing Mediators from Confounders 441 10.8.2 Implications for the Interpretation of an Adjusted Logistic Regression Coefficient 443 10.8.3 Why Adjust for a Mediator? 444 10.8.4 Using Logistic Regression to Assess Mediation: Assumptions 445 10.9 More About Statistical Interaction 448 10.9.1 Additive versus Multiplicative Scale–Risk Difference versus Odds Ratios 448 10.9.2 Estimating and Testing Additive Interaction 451 Exercises 456 References 459 Index 479

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Book SynopsisBasic Statistics for Social Research offers an introduction to core general statistical concepts and methods. It covers procedural aspects of the application of statistical methods for data-description; and hypothesis-testing; distributions, tabulations, central tendency, variability, independence, correlation and regression.Table of ContentsTables and Figures ix Preface xv About the Authors xix Part I Univariate Description 1 Chapter 1 Using Statistics 3 Why Study Statistics? 4 Tasks for Statistics: Describing, Inferring, Testing, Predicting 4 Statistics in the Research Process 9 Basic Elements of Research: Units of Analysis and Variables 14 Chapter 2 Displaying One Distribution 25 Summarizing Variation in One Variable 26 Frequency Distributions for Nominal Variables 26 Frequency Distributions for Ordinal Variables 32 Frequency Distributions for Interval/Ratio Variables 38 Summarizing Data Using Excel 43 Chapter 3 Central Tendency 81 The Basic Idea of Central Tendency 82 The Mode 83 The Median 88 The Mean 95 Chapter 4 Dispersion 113 The Basic Idea of Dispersion 114 Dispersion of Categorical Data 115 Dispersion of Interval/Ratio Data 121 Chapter 5 Describing the Shape of a Distribution 149 The Basic Ideas of Distributional Shape 150 The Shape of Nominal and Ordinal Distributions 152 Unimodality 158 Skewness 163 Kurtosis 169 Some Common Distributional Shapes 175 Chapter 6 The Normal Distribution 187 Introduction to the Normal Distribution 188 Properties of Normal Distributions 189 The Standard Normal, or Z, Distribution 192 Working with Standard Normal (Z) Scores 194 Finding Areas “Under the Curve” 197 Part II Inference and Hypothesis Testing 209 Chapter 7 Basic Ideas of Statistical Inference 211 Introduction to Statistical Inference 212 Sampling Concepts 214 Central Tendency Estimates 219 Assessing Confidence in Point Estimates 229 Chapter 8 Hypothesis Testing for One Sample 247 Hypothesis Testing 248 The Testing Process 250 Tests about One Mean 258 Tests about One Proportion 267 Chapter 9 Hypothesis Testing for Two Samples 279 Comparing Two Groups 280 Comparing Two Groups’ Means 280 Comparing Two Groups’ Proportions 289 Non independent Samples 296 Using Excel for Two-Sample Tests 301 Interpreting Group Differences 302 Chapter 10 Multiple Sample Tests of Proportions: Chi-Squared 313 Comparing Proportions across Several Groups 314 Testing for Multiple Group Differences 315 Describing Group Differences 327 Chapter 11 Multiple Sample Tests for Means: One-Way ANOVA 337 Comparing Several Group Means with Analysis of Variance 338 Analyzing Variance and the F-Test 339 Analyzing Variance 342 The F-Test 350 Comparing Means 356 Part III Association and Prediction 369 Chapter 12 Association with Categorical Variables 371 The Concept of Statistical Association 372 Association with Nominal Variables 375 Association with Ordinal Variables 391 Chapter 13 Association of Interval/Ratio Variables 425 Visualizing Interval/Ratio Association 426 Significance Testing for Interval/Ratio Association 434 Chapter 14 Regression Analysis 453 Predicting Outcomes with Regression 454 Simple Linear Regression 454 Applying Simple Regression Analysis 465 Multiple Regression 469 Applying Multiple Regression 474 Chapter 15 Logistic Regression Analysis 489 Predicting with Nonlinear Relationships 490 Logistic Regression 492 The Logistic Regression Model 492 Interpreting Effects in Logistic Regression 493 Estimating Logistic Regression Models with Maximum Likelihood 495 Applying Logistic Regression 496 Assessing Partial Effects 498 Extending Logistic Regression 499 Appendix Chi-Squared Distribution: Critical Values for Commonly Used Alpha=0.05 and Alpha=0.01 505 F-Distribution: Critical Values for Commonly Used Alpha=0.05 and Alpha=0.01 507 Standard Normal Scores (Z-Scores), and Cumulative Probabilities (Proportion of Cases Having Scores below Z) 511 Student’s t-Distribution: Critical Values for Commonly Used Alpha Levels 517 Index 519

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Book SynopsisA state of the art volume on statistical causality Causality: Statistical Perspectives and Applications presents a wide-ranging collection of seminal contributions by renowned experts in the field, providing a thorough treatment of all aspects of statistical causality. It covers the various formalisms in current use, methods for applying them to specific problems, and the special requirements of a range of examples from medicine, biology and economics to political science. This book: Provides a clear account and comparison of formal languages, concepts and models for statistical causality. Addresses examples from medicine, biology, economics and political science to aid the reader''s understanding. Is authored by leading experts in their field. Is written in an accessible style. Postgraduates, professional statisticians and researchers in academia and industry will benefit from this book.Table of ContentsList of contributors xv An overview of statistical causality xvii Carlo Berzuini, Philip Dawid and Luisa Bernardinelli 1 Statistical causality: Some historical remarks 1 D.R. Cox 1.1 Introduction 1 1.2 Key issues 2 1.3 Rothamsted view 2 1.4 An earlier controversy and its implications 3 1.5 Three versions of causality 4 1.6 Conclusion 4 References 4 2 The language of potential outcomes 6 Arvid Sjölander 2.1 Introduction 6 2.2 Definition of causal effects through potential outcomes 7 2.2.1 Subject-specific causal effects 7 2.2.2 Population causal effects 8 2.2.3 Association versus causation 9 2.3 Identification of population causal effects 9 2.3.1 Randomized experiments 9 2.3.2 Observational studies 11 2.4 Discussion 11 References 13 3 Structural equations, graphs and interventions 15 Ilya Shpitser 3.1 Introduction 15 3.2 Structural equations, graphs, and interventions 16 3.2.1 Graph terminology 16 3.2.2 Markovian models 17 3.2.3 Latent projections and semi-Markovian models 19 3.2.4 Interventions in semi-Markovian models 19 3.2.5 Counterfactual distributions in NPSEMs 20 3.2.6 Causal diagrams and counterfactual independence 22 3.2.7 Relation to potential outcomes 22 References 23 4 The decision-theoretic approach to causal inference 25 Philip Dawid 4.1 Introduction 25 4.2 Decision theory and causality 26 4.2.1 A simple decision problem 26 4.2.2 Causal inference 27 4.3 No confounding 28 4.4 Confounding 29 4.4.1 Unconfounding 29 4.4.2 Nonconfounding 30 4.4.3 Back-door formula 31 4.5 Propensity analysis 33 4.6 Instrumental variable 34 4.6.1 Linear model 36 4.6.2 Binary variables 36 4.7 Effect of treatment of the treated 37 4.8 Connections and contrasts 37 4.8.1 Potential responses 37 4.8.2 Causal graphs 39 4.9 Postscript 40 Acknowledgements 40 References 40 5 Causal inference as a prediction problem: Assumptions, identification and evidence synthesis 43 Sander Greenland 5.1 Introduction 43 5.2 A brief commentary on developments since 1970 44 5.2.1 Potential outcomes and missing data 45 5.2.2 The prognostic view 45 5.3 Ambiguities of observational extensions 46 5.4 Causal diagrams and structural equations 47 5.5 Compelling versus plausible assumptions, models and inferences 47 5.6 Nonidentification and the curse of dimensionality 50 5.7 Identification in practice 51 5.8 Identification and bounded rationality 53 5.9 Conclusion 54 Acknowledgments 55 References 55 6 Graph-based criteria of identifiability of causal questions 59 Ilya Shpitser 6.1 Introduction 59 6.2 Interventions from observations 59 6.3 The back-door criterion, conditional ignorability, and covariate adjustment 61 6.4 The front-door criterion 63 6.5 Do-calculus 64 6.6 General identification 65 6.7 Dormant independences and post-truncation constraints 68 References 69 7 Causal inference from observational data: A Bayesian predictive approach 71 Elja Arjas 7.1 Background 71 7.2 A model prototype 72 7.3 Extension to sequential regimes 76 7.4 Providing a causal interpretation: Predictive inference from data 80 7.5 Discussion 82 Acknowledgement 83 References 83 8 Assessing dynamic treatment strategies 85 Carlo Berzuini, Philip Dawid, and Vanessa Didelez 8.1 Introduction 85 8.2 Motivating example 86 8.3 Descriptive versus causal inference 87 8.4 Notation and problem definition 88 8.5 HIV example continued 89 8.6 Latent variables 89 8.7 Conditions for sequential plan identifiability 90 8.7.1 Stability 90 8.7.2 Positivity 91 8.8 Graphical representations of dynamic plans 92 8.9 Abdominal aortic aneurysm surveillance 94 8.10 Statistical inference and computation 95 8.11 Transparent actions 97 8.12 Refinements 98 8.13 Discussion 99 Acknowledgements 99 References 99 9 Causal effects and natural laws: Towards a conceptualization of causal counterfactuals for nonmanipulable exposures, with application to the effects of race and sex 101 Tyler J. VanderWeele and Miguel A. Hernán 9.1 Introduction 101 9.2 Laws of nature and contrary to fact statements 102 9.3 Association and causation in the social and biomedical sciences 103 9.4 Manipulation and counterfactuals 103 9.5 Natural laws and causal effects 104 9.6 Consequences of randomization 107 9.7 On the causal effects of sex and race 108 9.8 Discussion 111 Acknowledgements 112 References 112 10 Cross-classifications by joint potential outcomes 114 Arvid Sjölander 10.1 Introduction 114 10.2 Bounds for the causal treatment effect in randomized trials with imperfect compliance 115 10.3 Identifying the complier causal effect in randomized trials with imperfect compliance 119 10.4 Defining the appropriate causal effect in studies suffering from truncation by death 121 10.5 Discussion 123 References 124 11 Estimation of direct and indirect effects 126 Stijn Vansteelandt 11.1 Introduction 126 11.2 Identification of the direct and indirect effect 127 11.2.1 Definitions 127 11.2.2 Identification 129 11.3 Estimation of controlled direct effects 132 11.3.1 G-computation 132 11.3.2 Inverse probability of treatment weighting 133 11.3.3 G-estimation for additive and multiplicative models 137 11.3.4 G-estimation for logistic models 141 11.3.5 Case-control studies 142 11.3.6 G-estimation for additive hazard models 143 11.4 Estimation of natural direct and indirect effects 146 11.5 Discussion 147 Acknowledgements 147 References 148 12 The mediation formula: A guide to the assessment of causal pathways in nonlinear models 151 Judea Pearl 12.1 Mediation: Direct and indirect effects 151 12.1.1 Direct versus total effects 151 12.1.2 Controlled direct effects 152 12.1.3 Natural direct effects 154 12.1.4 Indirect effects 156 12.1.5 Effect decomposition 157 12.2 The mediation formula: A simple solution to a thorny problem 157 12.2.1 Mediation in nonparametric models 157 12.2.2 Mediation effects in linear, logistic, and probit models 159 12.2.3 Special cases of mediation models 164 12.2.4 Numerical example 169 12.3 Relation to other methods 170 12.3.1 Methods based on differences and products 170 12.3.2 Relation to the principal-strata direct effect 171 12.4 Conclusions 173 Acknowledgments 174 References 175 13 The sufficient cause framework in statistics, philosophy and the biomedical and social sciences 180 Tyler J. VanderWeele 13.1 Introduction 180 13.2 The sufficient cause framework in philosophy 181 13.3 The sufficient cause framework in epidemiology and biomedicine 181 13.4 The sufficient cause framework in statistics 185 13.5 The sufficient cause framework in the social sciences 185 13.6 Other notions of sufficiency and necessity in causal inference 187 13.7 Conclusion 188 Acknowledgements 189 References 189 14 Analysis of interaction for identifying causal mechanisms 192 Carlo Berzuini, Philip Dawid, Hu Zhang and Miles Parkes 14.1 Introduction 192 14.2 What is a mechanism? 193 14.3 Statistical versus mechanistic interaction 193 14.4 Illustrative example 194 14.5 Mechanistic interaction defined 196 14.6 Epistasis 197 14.7 Excess risk and superadditivity 197 14.8 Conditions under which excess risk and superadditivity indicate the presence of mechanistic interaction 200 14.9 Collapsibility 201 14.10 Back to the illustrative study 202 14.11 Alternative approaches 204 14.12 Discussion 204 Ethics statement 205 Financial disclosure 205 References 206 15 Ion channels as a possible mechanism of neurodegeneration in multiple sclerosis 208 Luisa Bernardinelli, Carlo Berzuini, Luisa Foco, and Roberta Pastorino 15.1 Introduction 208 15.2 Background 209 15.3 The scientific hypothesis 209 15.4 Data 210 15.5 A simple preliminary analysis 211 15.6 Testing for qualitative interaction 213 15.7 Discussion 214 Acknowledgments 216 References 216 16 Supplementary variables for causal estimation 218 Roland R. Ramsahai 16.1 Introduction 218 16.2 Multiple expressions for causal effect 220 16.3 Asymptotic variance of causal estimators 222 16.4 Comparison of causal estimators 222 16.4.1 Supplement C with L or not 223 16.4.2 Supplement L with C or not 224 16.4.3 Replace C with L or not 225 16.5 Discussion 226 Acknowledgements 226 Appendices 227 16.A Estimator given all X’s recorded 227 16.B Derivations of asymptotic variances 227 16.C Expressions with correlation coefficients 229 16.D Derivation of I’s 230 16.E Relation between ρ2 rl|t and ρ2 rl|c 231 References 232 17 Time-varying confounding: Some practical considerations in a likelihood framework 234 Rhian Daniel, Bianca De Stavola and Simon Cousens 17.1 Introduction 234 17.2 General setting 235 17.2.1 Notation 235 17.2.2 Observed data structure 235 17.2.3 Intervention strategies 236 17.2.4 Potential outcomes 237 17.2.5 Time-to-event outcomes 237 17.2.6 Causal estimands 238 17.3 Identifying assumptions 238 17.4 G-computation formula 239 17.4.1 The formula 239 17.4.2 Plug-in regression estimation 240 17.5 Implementation by Monte Carlo simulation 242 17.5.1 Simulating an end-of-study outcome 242 17.5.2 Simulating a time-to-event outcome 242 17.5.3 Inference 242 17.5.4 Losses to follow-up 243 17.5.5 Software 243 17.6 Analyses of simulated data 243 17.6.1 The data 243 17.6.2 Regimes to be compared 244 17.6.3 Parametric modelling choices 245 17.6.4 Results 246 17.7 Further considerations 249 17.7.1 Parametric model misspecification 249 17.7.2 Competing events 249 17.7.3 Unbalanced measurement times 250 17.8 Summary 251 References 251 18 ‘Natural experiments’ as a means of testing causal inferences 253 Michael Rutter 18.1 Introduction 253 18.2 Noncausal interpretations of an association 253 18.3 Dealing with confounders 255 18.4 ‘Natural experiments’ 256 18.4.1 Genetically sensitive designs 257 18.4.2 Children of twins (CoT) design 259 18.4.3 Strategies to identify the key environmental risk feature 261 18.4.4 Designs for dealing with selection bias 263 18.4.5 Instrumental variables to rule out reverse causation 264 18.4.6 Regression discontinuity (RD) designs to deal with unmeasured confounders 265 18.5 Overall conclusion on ‘natural experiments’ 266 18.5.1 Supported causes 266 18.5.2 Disconfirmed causes 267 Acknowledgement 267 References 268 19 Nonreactive and purely reactive doses in observational studies 273 Paul R. Rosenbaum 19.1 Introduction: Background, example 273 19.1.1 Does a dose–response relationship provide information that distinguishes treatment effects from biases due to unmeasured covariates? 273 19.1.2 Is more chemotherapy for ovarian cancer more effective or more toxic? 274 19.2 Various concepts of dose 277 19.2.1 Some notation: Covariates, outcomes, and treatment assignment in matched pairs 277 19.2.2 Reactive and nonreactive doses of treatment 278 19.2.3 Three test statistics that use doses in different ways 279 19.2.4 Randomization inference in randomized experiments 280 19.2.5 Sensitivity analysis 281 19.2.6 Sensitivity analysis in the example 283 19.3 Design sensitivity 284 19.3.1 What is design sensitivity? 284 19.3.2 Comparison of design sensitivity with purely reactive doses 286 19.4 Summary 287 References 287 20 Evaluation of potential mediators in randomised trials of complex interventions (psychotherapies) 290 Richard Emsley and Graham Dunn 20.1 Introduction 290 20.2 Potential mediators in psychological treatment trials 291 20.3 Methods for mediation in psychological treatment trials 293 20.4 Causal mediation analysis using instrumental variables estimation 297 20.5 Causal mediation analysis using principal stratification 301 20.6 Our motivating example: The SoCRATES trial 302 20.6.1 What are the joint effects of sessions attended and therapeutic alliance on the PANSS score at 18 months? 303 20.6.2 What is the direct effect of random allocation on the PANSS score at 18 months and how is this influenced by the therapeutic alliance? 304 20.6.3 Is the direct effect of the number of sessions attended on the PANSS score at 18 months influenced by therapeutic alliance? 305 20.7 Conclusions 305 Acknowledgements 306 References 307 21 Causal inference in clinical trials 310 Krista Fischer and Ian R. White 21.1 Introduction 310 21.2 Causal effect of treatment in randomized trials 312 21.2.1 Observed data and notation 312 21.2.2 Defining the effects of interest via potential outcomes 312 21.2.3 Adherence-adjusted ITT analysis 314 21.3 Estimation for a linear structural mean model 316 21.3.1 A general estimation procedure 316 21.3.2 Identifiability and closed-form estimation of the parameters in a linear SMM 317 21.3.3 Analysis of the EPHT trial 319 21.4 Alternative approaches for causal inference in randomized trials comparing experimental treatment with a control 321 21.4.1 Principal stratification 321 21.4.2 SMM for the average treatment effect on the treated (ATT) 322 21.5 Discussion 324 References 325 22 Causal inference in time series analysis 327 Michael Eichler 22.1 Introduction 327 22.2 Causality for time series 328 22.2.1 Intervention causality 328 22.2.2 Structural causality 331 22.2.3 Granger causality 332 22.2.4 Sims causality 334 22.3 Graphical representations for time series 335 22.3.1 Conditional distributions and chain graphs 336 22.3.2 Path diagrams and Granger causality graphs 337 22.3.3 Markov properties for Granger causality graphs 338 22.4 Representation of systems with latent variables 339 22.4.1 Marginalization 341 22.4.2 Ancestral graphs 342 22.5 Identification of causal effects 343 22.6 Learning causal structures 346 22.7 A new parametric model 349 22.8 Concluding remarks 351 References 352 23 Dynamic molecular networks and mechanisms in the biosciences: A statistical framework 355 Clive G. Bowsher 23.1 Introduction 355 23.2 SKMs and biochemical reaction networks 356 23.3 Local independence properties of SKMs 358 23.3.1 Local independence and kinetic independence graphs 358 23.3.2 Local independence and causal influence 361 23.4 Modularisation of SKMs 362 23.4.1 Modularisations and dynamic independence 362 23.4.2 MIDIA Algorithm 363 23.5 Illustrative example – MAPK cell signalling 365 23.6 Conclusion 369 23.7 Appendix: SKM regularity conditions 369 Acknowledgements 370 References 370 Index 371

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Book Synopsis* Novel approach to the mathematics of problem solving, in particular how to do logical calculations. * Many of the problems are well-known from (mathematical) puzzle books. * The solution method in the book is new and more relevant to the true nature of problem solving in the modern IT-dominated world.Table of ContentsPreface xi PART I Algorithmic Problem Solving 1 CHAPTER 1 – Introduction 3 1.1 Algorithms 3 1.2 Algorithmic Problem Solving 4 1.3 Overview 5 1.4 Bibliographic Remarks 6 CHAPTER 2 – Invariants 7 2.1 Chocolate Bars 10 2.1.1 The Solution 10 2.1.2 The Mathematical Solution 11 2.2 Empty Boxes 16 2.2.1 Review 19 2.3 The Tumbler Problem 22 2.3.1 Non-deterministic Choice 23 2.4 Tetrominoes 24 2.5 Summary 30 2.6 Bibliographic Remarks 34 CHAPTER 3 – Crossing a River 35 3.1 Problems 36 3.2 Brute Force 37 3.2.1 Goat, Cabbage and Wolf 37 3.2.2 State-Space Explosion 39 3.2.3 Abstraction 41 3.3 Nervous Couples 42 3.3.1 What Is the Problem? 42 3.3.2 Problem Structure 43 3.3.3 Denoting States and Transitions 44 3.3.4 Problem Decomposition 45 3.3.5 A Review 48 3.4 Rule of Sequential Composition 50 3.5 The Bridge Problem 54 3.6 Conditional Statements 63 3.7 Summary 65 3.8 Bibliographic Remarks 65 CHAPTER 4 – Games 67 4.1 Matchstick Games 67 4.2 Winning Strategies 69 4.2.1 Assumptions 69 4.2.2 Labelling Positions 70 4.2.3 Formulating Requirements 72 4.3 Subtraction-Set Games 74 4.4 Sums of Games 78 4.4.1 A Simple Sum Game 79 4.4.2 Maintain Symmetry! 81 4.4.3 More Simple Sums 82 4.4.4 Evaluating Positions 83 4.4.5 Using the Mex Function 87 4.5 Summary 91 4.6 Bibliographic Remarks 92 CHAPTER 5 – Knights and Knaves 95 5.1 Logic Puzzles 95 5.2 Calculational Logic 96 5.2.1 Propositions 96 5.2.2 Knights and Knaves 97 5.2.3 Boolean Equality 98 5.2.4 Hidden Treasures 100 5.2.5 Equals for Equals 101 5.3 Equivalence and Continued Equalities 102 5.3.1 Examples of the Associativity of Equivalence 104 5.3.2 On Natural Language 105 5.4 Negation 106 5.4.1 Contraposition 109 5.4.2 Handshake Problems 112 5.4.3 Inequivalence 113 5.5 Summary 117 5.6 Bibliographic Remarks 117 CHAPTER 6 – Induction 119 6.1 Example Problems 120 6.2 Cutting the Plane 123 6.3 Triominoes 126 6.4 Looking for Patterns 128 6.5 The Need for Proof 129 6.6 From Verification to Construction 130 6.7 Summary 134 6.8 Bibliographic Remarks 134 CHAPTER 7 – Fake-Coin Detection 137 7.1 Problem Formulation 137 7.2 Problem Solution 139 7.2.1 The Basis 139 7.2.2 Induction Step 139 7.2.3 The Marked-Coin Problem 140 7.2.4 The Complete Solution 141 7.3 Summary 146 7.4 Bibliographic Remarks 146 CHAPTER 8 – The Tower of Hanoi 147 8.1 Specification and Solution 147 8.1.1 The End of the World! 147 8.1.2 Iterative Solution 148 8.1.3 Why? 149 8.2 Inductive Solution 149 8.3 The Iterative Solution 153 8.4 Summary 156 8.5 Bibliographic Remarks 156 CHAPTER 9 – Principles of Algorithm Design 157 9.1 Iteration, Invariants and Making Progress 158 9.2 A Simple Sorting Problem 160 9.3 Binary Search 163 9.4 Sam Loyd’s Chicken-Chasing Problem 166 9.4.1 Cornering the Prey 170 9.4.2 Catching the Prey 174 9.4.3 Optimality 176 9.5 Projects 177 9.6 Summary 178 9.7 Bibliographic Remarks 180 CHAPTER 10 – The Bridge Problem 183 10.1 Lower and Upper Bounds 183 10.2 Outline Strategy 185 10.3 Regular Sequences 187 10.4 Sequencing Forward Trips 189 10.5 Choosing Settlers and Nomads 193 10.6 The Algorithm 196 10.7 Summary 199 10.8 Bibliographic Remarks 200 CHAPTER 11 – Knight’s Circuit 201 11.1 Straight-Move Circuits 202 11.2 Supersquares 206 11.3 Partitioning the Board 209 11.4 Summary 216 11.5 Bibliographic Remarks 218 PART II Mathematical Techniques 219 CHAPTER 12 – The Language of Mathematics 221 12.1 Variables, Expressions and Laws 222 12.2 Sets 224 12.2.1 The Membership Relation 224 12.2.2 The Empty Set 224 12.2.3 Types/Universes 224 12.2.4 Union and Intersection 225 12.2.5 Set Comprehension 225 12.2.6 Bags 227 12.3 Functions 227 12.3.1 Function Application 228 12.3.2 Binary Operators 230 12.3.3 Operator Precedence 230 12.4 Types and Type Checking 232 12.4.1 Cartesian Product and Disjoint Sum 233 12.4.2 Function Types 235 12.5 Algebraic Properties 236 12.5.1 Symmetry 237 12.5.2 Zero and Unit 238 12.5.3 Idempotence 239 12.5.4 Associativity 240 12.5.5 Distributivity/Factorisation 241 12.5.6 Algebras 243 12.6 Boolean Operators 244 12.7 Binary Relations 246 12.7.1 Reflexivity 247 12.7.2 Symmetry 248 12.7.3 Converse 249 12.7.4 Transitivity 249 12.7.5 Anti-symmetry 251 12.7.6 Orderings 252 12.7.7 Equality 255 12.7.8 Equivalence Relations 256 12.8 Calculations 257 12.8.1 Steps in a Calculation 259 12.8.2 Relations between Steps 260 12.8.3 ‘‘If’’ and ‘‘Only If’’ 262 12.9 Exercises 264 CHAPTER 13 – Boolean Algebra 267 13.1 Boolean Equality 267 13.2 Negation 269 13.3 Disjunction 270 13.4 Conjunction 271 13.5 Implication 274 13.5.1 Definitions and Basic Properties 275 13.5.2 Replacement Rules 276 13.6 Set Calculus 279 13.7 Exercises 281 CHAPTER 14 – Quantifiers 285 14.1 DotDotDot and Sigmas 285 14.2 Introducing Quantifier Notation 286 14.2.1 Summation 287 14.2.2 Free and Bound Variables 289 14.2.3 Properties of Summation 291 14.2.4 Warning 297 14.3 Universal and Existential Quantification 297 14.3.1 Universal Quantification 298 14.3.2 Existential Quantification 300 14.4 Quantifier Rules 301 14.4.1 The Notation 302 14.4.2 Free and Bound Variables 303 14.4.3 Dummies 303 14.4.4 Range Part 303 14.4.5 Trading 304 14.4.6 Term Part 304 14.4.7 Distributivity Properties 304 14.5 Exercises 306 CHAPTER 15 – Elements of Number Theory 309 15.1 Inequalities 309 15.2 Minimum and Maximum 312 15.3 The Divides Relation 315 15.4 Modular Arithmetic 316 15.4.1 Integer Division 316 15.4.2 Remainders and Modulo Arithmetic 320 15.5 Exercises 322 CHAPTER 16 – Relations, Graphs and Path Algebras 325 16.1 Paths in a Directed Graph 325 16.2 Graphs and Relations 328 16.2.1 Relation Composition 330 16.2.2 Union of Relations 332 16.2.3 Transitive Closure 334 16.2.4 Reflexive Transitive Closure 338 16.3 Functional and Total Relations 339 16.4 Path-Finding Problems 341 16.4.1 Counting Paths 341 16.4.2 Frequencies 343 16.4.3 Shortest Distances 344 16.4.4 All Paths 345 16.4.5 Semirings and Operations on Graphs 347 16.5 Matrices 351 16.6 Closure Operators 353 16.7 Acyclic Graphs 354 16.7.1 Topological Ordering 355 16.8 Combinatorics 357 16.8.1 Basic Laws 358 16.8.2 Counting Choices 359 16.8.3 Counting Paths 361 16.9 Exercises 366 Solutions to Exercises 369 References 405 Index 407

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Book SynopsisThis book is about some of the basic ideas in mathematics that people are likely to encounter in the normal course of their lives. It is about money-statistics, relations, probabilit graphs, decision making, codes, logic, languages, and much more. It was written to help people develop their mathematical problem solving skills and to help them see how mathematics is a part of modern society.Table of ContentsMathematics of the Marketplace. Mathematics of Description. Mathematics of Collections. Mathematics of Prediction. Mathematics of Relationships. Mathematics of Optimization. Mathematics of Space. Mathematics of Numbers. Mathematics of Games. Mathematics of Other Cultures and Other Times. Mathematics of Reasoning. Mathematics of Computers. Answers to Selected Exercises. Appendices. Index.

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Book SynopsisLooking for a textbook to help you motivate your students? Sullivan/Mizrahi''s Mathematics: An Applied Approach 8/e continues its rich tradition of engaging students and demonstrating how mathematics applies to various fields of study. The text is packed with real data and real-life applications to business, economics, social and life sciences. The new Eighth Edition also features a new full color design and improved goal-oriented pedagogy to further help student understanding.Table of ContentsChapter 1. Linear Equations. Chapter 2. Systems of Linear Equations; Matrices. Chapter 3. Linear Programming: Geometric Approach. Chapter 4. Linear Programming: Simplex Method. Chapter 5. Finance. Chapter 6. Sets; Counting Techniques. Chapter 7. Probability. Chapter 8. Additional Probability Topics. Chapter 9. Statistics. Chapter 10. Functions and Their Graphs. Chapter 11. Classes of Functions. Chapter 12.The Limit of a Function. Chapter 13. The Derivative of a Function. Chapter 14. Applications: Graphing Functions; Optimization. Chapter 15. The Integral of a Function and Applications. Chapter 16. Other Applications and Extensions of the Integral. Chapter 17. Calculus of Functions of Two or More Variables. Appendix A: Review. Appendix B: Using LINDO to Solve Linear Programming Problems. Appendix C: Graphing Utilities. Answers to Odd-Numbered Problems. Photo Credits. Index.

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Book SynopsisOwing to great demand, this standard mathematics textbook has been reprinted in a paperback, Wiley Classics edition.Table of ContentsThe Basic Theorems of Fourier Analysis. The Structure of Locally Compact Abelian Groups. Idempotent Measures. Homomorphisms of Group Algebras. Measures and Fourier Transforms on Thin Sets. Functions of Fourier Transforms. Closed Ideals in L?1(G). Fourier Analysis on Ordered Groups. Closed Subalgebras of L?1(G). Appendices: Topology, Topological Groups, Banach Spaces, Banach Algebras, Measure Theory. Bibliography. List of Special Symbols. Index.

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