Part of Cambridge Mathematical Library
Not yet published - available from July 2022
FORMAT: Paperback ISBN: 9781009229821
In this volume, Olga A. Ladyzhenskaya expands on her highly successful 1991 Accademia Nazionale dei Lincei lectures. The lectures were devoted to questions of the behaviour of trajectories for semigroups of nonlinear bounded continuous operators in a locally non-compact metric space and for solutions of abstract evolution equations. The latter contain many initial boundary value problems for dissipative partial differential equations. This work, for which Ladyzhenskaya was awarded the Russian Academy of Sciences' Kovalevskaya Prize, reflects the high calibre of her lectures; it is essential reading for anyone interested in her approach to partial differential equations and dynamical systems. This edition, reissued for her centenary, includes a new technical introduction, written by Gregory A. Seregin, Varga K. Kalantarov and Sergey V. Zelik, surveying Ladyzhenskaya's works in the field and subsequent developments influenced by her results.
Ladyzhenskaya's survey of her own work on partial differential equations and dynamical systems
Contains a new technical introduction written by Gregory A. Seregin, Varga K. Kalantarov and Sergey V. Zelik
Shines a light on the often-overlooked results of Soviet mathematicians working in this area
Part I. Attractors for the Semigroups of Operators:
1. Basic notions
2. Semigroups of class K
3. Semigroups of class AK
4. On dimensions of compact invariant sets
Part II: Semigroups Generated by Evolution Equations:
5. Introduction to Part II
6. Estimates for the number of determining modes and the fractal dimension of bounded invariant sets for the Navier?Stokes equations
7. Evolution equations of hyperbolic type
References
Index.
ISBN 9780367565206
ISBN 9780367564933(Paperback)
August 23, 2022 Forthcoming
190 Pages 49 Color & 9 B/W Illustrations
Working out solutions to polynomial equations is a mathematical problem that dates from antiquity. Galois developed a theory in which the obstacle to solving a polynomial equation is an associated collection of symmetries. Obtaining a root requires "breaking" that symmetry. When the degree of an equation is at least five, Galois Theory established that there is no formula for the solutions like those found in lower degree cases. However, this negative result doesn't mean that the practice of equation-solving ends. In a recent breakthrough, Doyle and McMullen devised a solution to the fifth-degree equation that uses geometry, algebra, and dynamics to exploit icosahedral symmetry.
Polynomials, Dynamics, and Choice: The Price We Pay for Symmetry is organized in two parts, the first of which develops an account of polynomial symmetry that relies on considerations of algebra and geometry. The second explores beyond polynomials to spaces consisting of choices ranging from mundane decisions to evolutionary algorithms that search for optimal outcomes. The two algorithms in Part I provide frameworks that capture structural issues that can arise in deliberative settings. While decision-making has been approached in mathematical terms, the novelty here is in the use of equation-solving algorithms to illuminate such problems.
Treats the topic?familiar to many?of solving polynomial equations in a way that’s dramatically different from what they saw in school
Accessible to a general audience with limited mathematical background
Abundant diagrams and graphics.
Section I. Polynomials: Symmetries and Solutions. 1. Solving Equations: A Fundamental Problem. 1.1. Polynomial Primer. 1.2. What Numbers Do We Use? 1.3. Roots And Coefficients. 2. What is Symmetry? 2.1. Mirrors and Reflections. 2.2. Mathematical Symmetry. 2.3. Exploring Geometric Symmetry. 2.4. Groups in the Abstract. 2.5. Posing and Solving Problems with Symmetry. 2.6. Structure in the Abstract. 2.7. A look at Higher Dimensions. 2.8. What is Geometry? 2.9. Molecular Symmetry. 2.10. Conservation Laws. 2.11. Thermodynamic Systems. 3. Geometry of Choice: Symmetry’s Cost. 3.1. Spaces Where the Roots Live. 3.2. Shuffling Roots and Solving Equations. 4. Compute First, Then Choose. 4.1. Simplifying a Polynomial. 4.2. Solutions from a Formula and a Choice. 4.3. Reducing a Polynomial’s Symmetry. 4.4. What Goes Wrong. 5. Choose First, Then Compute. 5.1. A Line that becomes a Sphere. 5.2. Symmetrical Structures. 5.3. Fundamentals of Dynamics. 5.4. Dynamical Geometry and Symmetry. 5.5. Solving Equations by Iteration. Section II. Beyond Equation. Chapter 6. Interlude: Modeling Choice. 7. Learning to Choose. 7.1. Making Rational Decisions. 7.2. The Heart Has its Reasons. 7.3. Give Chance a Choice. 8. Choosing to Learn. 8.1. A Crowd Decides. 8.2. When in Doubt, Simulate. 8.3. Give Choice a Chance. 9. Conclusion. 9.1 Symmetry, More or Less. 9.2. Choosing As Metaphor. 9.3 Random Choice is Unavoidable.
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ISBN 9781032105710
September 15, 2022 Forthcoming
350 Pages 72 Color & 150 B/W Illustrations
The complexity of large-scale data sets ("Big Data") has stimulated the development of advanced computational methods for analyzing them. There are two different kinds of methods to aid this. The model-based method uses probability models and likelihood and Bayesian theory, while the model-free method does not require a probability model, likelihood or Bayesian theory. These two approaches are based on different philosophical principles of probability theory, espoused by the famous statisticians Ronald Fisher and Jerzy Neyman
Statistical Modelling and Inference covers simple experimental and survey designs, and probability models up to and including generalised linear (regression) models and some extensions of these, including finite mixtures. A wide range of examples from different application fields are also discussed and analyzed. No special software is used, beyond that needed for maximum likelihood analysis of generalised linear models. Students are expected to have a basic mathematical background of algebra, coordinate geometry and calculus.
Probability models are developed from the shape of the sample empirical cumulative distribution function, (cdf) or a transformation of it.
Bounds for the value of the population cumulative distribution function are obtained from the Beta distribution at each point of the empirical cdf.
Bayes's theorem is developed from the properties of the screening test for a rare condition.
The multinomial distribution provides an always-true model for any randomly sampled data.
The model-free bootstrap method for finding the precision of a sample estimate has a model-based parallel - the Bayesian bootstrap - based on the always-true multinomial distribution.
The Bayesian posterior distributions of model parameters can be obtained from the maximum likelihood analysis of the model.
This book is aimed at students in a wide range of disciplines including Data Science. The book is based on the model-based theory, used widely by scientists in many fields, and compares it, in less detail, with the model-free theory, popular in computer science, machine learning and official survey analysis. The development of the model-based theory is accelerated by recent developments in Bayesian analysis.
Preface. 1.1. What is Statistical Modelling? 1.2. What is Statistical Analysis? 1.3. What is Statistical Inference? 1.4. Why this book? 1.5. Why the focus on the Bayesian approach? 1.6. Coverage of this book. 1.7. Recent changes in technology. 1.8. Aims of the course. 2. What is (or are) Big Data? 3. Data and research studies. 3.1. Lifetimes of radio transceivers. 3.2. Clustering of V1 missile hits in South London. 3.3. Court case on vaccination risk. 3.4. Clinical trial of Depepsen for the treatment of duodenal ulcers. 3.5. Effectiveness of treatments for respiratory distress in newborn babies. 3.6. Vitamin K. 3.7. Species counts. 3.8. Toxicology in small animal experiments. 3.9. Incidence of Down’s syndrome in four regions. 3.10. Fish species in lakes. 3.11. Absence from school. 3.12. Hostility in husbands of suicide attempters. 3.13. Tolerance of racial intermarriage. 3.14. Hospital bed use. 3.15. Dugong growth. 3.16. Simulated motorcycle collision. 3.17. Global warming. 3.18. Social group membership. 4. The StatLab data base. 4.1. Types of variables. 4.2. StatLab population questions. 5. Sample surveys ? should we believe what we read? 5.1. Women and Love. 5.2. Would you have children? 5.3. Representative sampling. 5.4. Bias in the Newsday sample. 5.5. Bias in the Women and Love sample. 6. Probability. 6.1. Relative frequency. 6.2. Degree of belief. 6.3. StatLab dice sampling. 6.4. Computer sampling. 6.5. Probability for sampling. 6.6. Probability axioms. 6.7. Screening tests and Bayes’s theorem. 6.8. The misuse of probability in the Sally Clark case. 6.9. Random variables and their probability distributions. 6.10. Sums of independent random variables. 7. Statistical inference I ? discrete distributions. 7.1. Evidence-based policy. 7.2. The basis of statistical inference. 7.3. The survey sampling approach. 7.4. Model-based inference theories. 7.5. The likelihood function. 7.6. Binomial distribution. 7.7. Frequentist theory. 7.8. Bayesian theory. 7.9. Inferences from posterior sampling. 7.10. Sample design. 7.11. Parameter transformations. 7.12. The Poisson distribution. 7.13. Categorical variables.7.14. Maximum likelihood. 7.15. Bayesian analysis. 8. Comparison of binomials: the Randomised Clinical Trial. 8.1. Definition. 8.2. Example ? RCT of Depepsen for the treatment of duodenal ulcers. 8.3. Monte Carlo simulation. 8.4. RCT continued. 8.5. Bayesian hypothesis testing/model comparison. 8.6. Other measures of treatment difference. 8.7. The ECMO trials. 9. Data visualisation. 9.1. The histogram. 9.2. The empirical mass and cumulative distribution functions. 9.3. Probability models for continuous variables. 10. Statistical Inference II ? the continuous exponential, Gaussian and uniform distributions. 10.1. The exponential distribution. 10.2. The exponential likelihood. 10.3. Frequentist theory. 10.4. Bayesian theory. 10.5. The Gaussian distribution. 10.6. The Gaussian likelihood function. 10.7. Frequentist inference. 10.8. Bayesian inference. 10.9. Hypothesis testing. 10.10. Frequentist hypothesis testing. 10.11. Bayesian hypothesis testing. 10.12. Pivotal functions. 10.13. Conjugate priors. 10.14. The uniform distribution. 11. Statistical Inference III ? two-parameter continuous distributions. 11.1. The Gaussian distribution. 11.2. Frequentist analysis. 11.3. Bayesian analysis. 11.4. The lognormal distribution. 11.5. The Weibull distribution. 11.6. The gamma distribution. 11.7. The gamma likelihood. 12. Model assessment. 12.1. Gaussian model assessment. 12.2. Lognormal model assessment. 12.3. Exponential model assessment. 12.4. Weibull model assessment. 12.5. Gamma model assessment. 13. The multinomial distribution. 13.1. The multinomial likelihood. 13.2. Frequentist analysis. 13.3. Bayesian analysis. 13.4. Criticisms of the Haldane prior. 13.5. Inference for multinomial quantiles. 13.6. Dirichlet posterior weighting. 13.7. The frequentist bootstrap. 13.8. Stratified sampling and weighting. 14. Model comparison and model averaging. 14.4. The deviance. 14.5. Asymptotic distribution of the deviance. 14.6. Nested models. 14.7. Model choice and model averaging. 15. Gaussian linear regression models. 15.1. Simple linear regression. 15.2. Model assessment through residual examination. 15.3. Likelihood for the simple linear regression model. 15.4. Maximum likelihood. 15.5. Bayesian and frequentist inferences. 15.6. Model-robust analysis. 15.7. Correlation and prediction. 15.8. Probability model assessment. 15.9. "Dummy variable" regression. 15.10. Two-variable models. 15.11. Model assumptions. 15.12. The p-variable linear model. 15.13. The Gaussian multiple regression likelihood. 15.14. Interactions. 15.15. Ridge regression, the Lasso and the "elastic net". 15.16. Modelling boy birthweights. 15.17. Modelling girl intelligence at age 10 and family income 15.18. Modelling of the hostility data. 15.19. Principal component regression. 16. Incomplete data and their analysis with the EM and DA algorithms. 16.1. The general incomplete data model. 16.2. The EM algorithm. 16.3. Missingness. 16.4. Lost data. 16.5. Censoring in the exponential distribution. 16.6. Randomly missing Gaussian observations. 16.7. Missing responses and/or covariates in simple and multiple regression. 16.8. Mixture distributions. 16.9. Bayesian analysis and the Data Augmentation algorithm. 17. Generalised linear models (GLMs). 17.1. The exponential family. 17.2. Maximum likelihood 17.3 The GLM algorithm. 17.4. Bayesian package development. 17.5. Bayesian analysis from ML. 17.6. Binary response models. 17.7. The menarche data. 17.8. Poisson regression ? fish species frequency. 17.9. Gamma regression. 18. Extensions of GLMs. 18.1. Double GLMs. 18.2. Maximum likelihood. 18.3. Bayesian analysis. 18.4. Segmented or broken-stick regressions. 18.5. Heterogeneous regressions. 18.6. Highly non-linear functions. 18.7. Neural networks. 18.8. Social networks and social group membership. 18.9. The motorcycle data. 19. Appendix 1 ? length-biased sampling. 20. Appendix 2 ? Two-component Gaussian mixture. 21. Appendix 3 ? StatLab Variables. 22. Appendix 4 ? a short history of statistics from 1890.
Copyright Year 2023
ISBN 9780367554934
September 2, 2022 Forthcoming
248 Pages 68 Color Illustrations
The purpose of this unique textbook is to bridge the gap between the need for numerical solutions to modelling techniques through computer simulations to develop skill in employing sensitivity analysis to biological and life sciences applications.
The underpinning mathematics is minimalized. The focus is on the consequences, implementation, and application. Historical context motivates the models. An understanding of the earliest models provide insight into more complicated ones.
While the text avoids getting mired in the details of numerical analysis, it demonstrates how to use numerical methods and provide core codes that can be readily altered to fit a variety of situations.
Numerical scripts in both Python and MATLABTM are included. Python is compiled in Jupyter notebooks to aid classroom use. Additionally, codes are organized and available online.
One of the most important skills requiring the use of computer simulations is sensitivity analysis. Sensitivity analysis is increasingly used in biomathematics. There are numerous pitfalls to using sensitivity analysis and therefore a need for exposure to worked examples in order to successfully transfer their use from mathematicians to biologists.
The interconnections between mathematics and the life sciences have an extensive history. This book offers a new approach to using mathematics to model applications using computers, to employ numerical methods, and takes students a step further into the realm of sensitivity analysis. With some guidance and practice, the reader will have a new and incredibly powerful tool to use.
Forward
1. Introduction
1.1 What is a Model?
1.2 Projectile Motion
1.3 Problems
2. Mathematical Background
2.1 Mathematical Preliminaries
2.2 Linearization
2.3 Qualitative Analysis
2.4 Problems
2.5 Appendix: Planar Example
3. Introduction to the Numerical Methods
3.1 Introduction
3.2 Best Practices in Coding
3.3 Getting the Programs Running
3.4 Initial Programs
3.5 Problems
4. Ecology
4.1 Historical Background
4.2 Single Species Models
4.3 Competitive Exclusion
4.4 State of the Art and Caveats
4.5 Problems
5. Within-host Disease Models
5.1 Historical Background
5.2 Pathological: Tumor
5.3 Viral: Acute Infection
5.4 Chronic: Tuberculosis
5.5 Problems
5.6 Appendix
6. Between-Host Disease Models
6.1 Historical Background
6.2 Two Compartment Models
6.3 Classical SIR
6.4 Waning Antigens
6.5 Caveats and State of the Art
6.6 Problems
7. Microbiology
7.1 Historical Background
7.2 Bacterial Growth: Chemostat
7.3 Multiple State Model: Free/Attached
7.4 Cooperators, Cheaters, and Competitions
7.5 Problems
8. Circulation and Cardiac Physiology
8.1 Historical Background
8.2 Blood Circulation Models
8.3 Cardiac Physiology
8.4 Problems
9. Neuroscience
9.1 Historical Background
9.2 Action Potential
9.3 Fitzhugh-Nagumo
9.4 Problems
10. Genetics
10.1 Historical Background
10.2 Heredity
10.3 Problems
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Copyright Year 2023
ISBN 9781032101590
ISBN 9781032101583(Paperback)
September 7, 2022 Forthcoming
371 Pages 38 B/W Illustrations
Since 1973, Galois Theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fifth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today’s algebra students.
Reorganised and revised Chapters 7 and 13
New exercises and examples
Expanded, updated references
Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel
A new final chapter discussing other directions in which Galois Theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations
This bestseller continues to deliver a rigorous, yet engaging, treatment of the subject while keeping pace with current educational requirements. More than 200 exercises and a wealth of historical notes augment the proofs, formulas, and theorems.
1. Classical Algebra. 1.1. Complex Numbers. 1.2. Subfields and Subrings of the Complex Numbers. 1.3. Solving Equations. 1.4. Solution by Radicals. 2. The Fundamental Theorem of Algebra. 2.1. Polynomials. 2.2. Fundamental Theorem of Algebra. 2.3. Implications 3. Factorisation of Polynomials. 3.1. The Euclidean Algorithm. 3.2 Irreducibility. 3.3. Gauss’s Lemma. 3.4. Eisenstein’s Criterion. 3.5. Reduction Modulo p. 3.6. Zeros of Polynomials. 4. Field Extensions. 4.1. Field Extensions. 4.2. Rational Expressions. 4.3. Simple Extensions. 5. Simple Extensions. 5.1. Algebraic and Transcendental Extensions. 5.2. The Minimal Polynomial. 5.3. Simple Algebraic Extensions. 5.4. Classifying Simple Extensions. 6. The Degree of an Extension. 6.1. Definition of the Degree. 6.2. The Tower Law. 6.3. Primitive Element Theorem. 7. Ruler-and-Compass Constructions. 7.1. Approximate Constructions and More General Instruments. 7.2. Constructions in C. 7.3. Specific Constructions. 7.4. Impossibility Proofs. 7.5. Construction From a Given Set of Points. 8. The Idea Behind Galois Theory. 8.1. A First Look at Galois Theory. 8.2. Galois Groups According to Galois. 8.3. How to Use the Galois Group. 8.4. The Abstract Setting. 8.5. Polynomials and Extensions. 8.6. The Galois Correspondence. 8.7. Diet Galois. 8.8. Natural Irrationalities. 9. Normality and Separability. 9.1. Splitting Fields. 9.2. Normality. 9.3. Separability. 10. Counting Principles. 10.1. Linear Independence of Monomorphisms. 11. Field Automorphisms. 11.1. K-Monomorphisms. 11.2. Normal Closures. 12. The Galois Correspondence. 12.1. The Fundamental Theorem of Galois Theory. 13. Worked Examples. 13.1. Examples of Galois Groups. 13.2. Discussion. 14. Solubility and Simplicity. 14.1. Soluble Groups. 14.2. Simple Groups. 14.3. Cauchy’s Theorem. 15. Solution by Radicals. 15.1. Radical Extensions. 15.2. An Insoluble Quintic. 15.3. Other Methods. 16. Abstract Rings and Fields. 16.1. Rings and Fields. 16.2. General Properties of Rings and Fields. 16.3. Polynomials Over General Rings. 16.4. The Characteristic of a Field. 16.5. Integral Domains. 17. Abstract Field Extensions and Galois Groups. 17.1. Minimal Polynomials. 17.2. Simple Algebraic Extensions. 17.3. Splitting Fields. 17.4. Normality. 17.5. Separability. 17.6. Galois Theory for Abstract Fields. 17.7. Conjugates and Minimal Polynomials. 17.8. The Primitive Element Theorem. 17.9. Algebraic Closure of a Field. 18. The General Polynomial Equation. 18.1. Transcendence Degree. 18.2. Elementary Symmetric Polynomials. 18.3. The General Polynomial. 18.5. Solving Equations of Degree Four or Less. 18.6. Explicit Formulas. 19. Finite Fields. 19.1. Structure of Finite Fields. 19.2. The Multiplicative Group. 19.3. Counterexample to the Primitive Element Theorem. 19.4. Application to Solitaire. 20. Regular Polygons. 20.1. What Euclid Knew. 20.2. Which Constructions are Possible? 20.3. Regular Polygons. 20.4. Fermat Numbers. 20.5. How to Construct a Regular 17-gon. 21. Circle Division. 21.1. Genuine Radicals. 21.2. Fifth Roots Revisited. 21.3. Vandermonde Revisited. 21.4. The General Case. 21.5. Cyclotomic Polynomials. 21.6. Galois Group of Q(ζ)= Q. 21.7. Constructions Using a Trisector. 22. Calculating Galois Groups. 22.1. Transitive Subgroups. 22.2. Bare Hands on the Cubic. 22.3. The Discriminant. 22.4. General Algorithm for the Galois Group. 23. Algebraically Closed Fields. 23.1. Ordered Fields and Their Extensions. 23.2. Sylow’s Theorem. 23.3. The Algebraic Proof. 24. Transcendental Numbers. 24.1. Irrationality. 24.2. Transcendence of e. 24.3. Transcendence of π. 25. What Did Galois Do or Know? 25.1. List of the Relevant Material. 25.2. The First Memoir. 25.3. What Galois Proved. 25.4. What is Galois Up To? 25.5. Alternating Groups, Especially A5. 25.6. Simple Groups Known to Galois. 25.7. Speculations about Proofs. 25.8. A5 is Unique. 26. Further Directions. 26.1. Inverse Galois Problem. 26.2. Differential Galois Theory. 26.3. p-adic Numbers.