Maths for scientists Books

Book SynopsisProbability is an area of mathematics of tremendous contemporary importance across all aspects of human endeavour. This book is a compact account of the basic features of probability and random processes at the level of first and second year mathematics undergraduates and Masters'' students in cognate fields. It is suitable for a first course in probability, plus a follow-up course in random processes including Markov chains.A special feature is the authors'' attention to rigorous mathematics: not everything is rigorous, but the need for rigour is explained at difficult junctures. The text is enriched by simple exercises, together with problems (with very brief hints) many of which are taken from final examinations at Cambridge and Oxford. The first eight chapters form a course in basic probability, being an account of events, random variables, and distributions - discrete and continuous random variables are treated separately - together with simple versions of the law of large numbersTable of ContentsPART A BASIC PROBABILITY; PART B FURTHER PROBABILITY

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Book SynopsisJourney through the world of abstract mathematics into category theory with popular science author Eugenia Cheng. Featuring humanizing examples and demystification of mathematical thought processes, this book is for fans of How to Bake Pi who want to dig deeper into mathematical concepts and build their mathematical background.Trade Review'This book is an educational tour de force that presents mathematical thinking as a right-brained activity. Most 'left brain/right brain' education-talk is at best a crude metaphor; but by putting the main focus on the process of (mathematical) abstraction, Eugenia Cheng supplies the reader (whatever their 'brain-type') with the mental tools to make that distinction precise and potentially useful. The book takes the reader along in small steps; but make no mistake, this is a major intellectual journey. Starting not with numbers, but everyday experiences, it develops what is regarded as a very advanced branch of abstract mathematics (category theory, though Cheng really uses this as a proxy for mathematical thinking generally). This is not watered-down math; it's the real thing. And it challenges the reader to think-deeply at times. We 'left-brainers' can learn plenty from it too.' Keith Devlin, Stanford University (Emeritus), author of The Joy of Sets'Eugenia Cheng loves mathematics—not the ordinary sort that most people encounter, but the most abstract sort that she calls 'the mathematics of mathematics.' And in this lovely excursion through her abstract world of Category Theory, she aims to give those who are willing to join her a glimpse of that world. The journey will change how they view mathematics. Cheng is a brilliant writer, with prose that feels like poetry. Her contagious enthusiasm makes her the perfect guide.' John Ewing, President, Math for America'Eugenia Cheng's singular contribution is in making abstract mathematics relevant to all through her great ingenuity in developing novel connections between logic and life. Her latest book, The Joy of Abstraction, provides a long awaited fully rigorous yet gentle introduction to the 'mathematics of mathematics,' allowing anyone to experience the joy of learning to think categorically.' Emily Riehl, Johns Hopkins University, author of Category Theory in Context'Archimedes is quoted as having said once: 'Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.' In this fascinating book, Eugenia Cheng approaches the abstract mathematical area of Category Theory with pure love, to reveal its beauty to anybody interested in learning something about contemporary mathematics.' Mario Livio, astrophysicist, author of The Golden Ratio and Brilliant Blunders'Eugenia Cheng's latest book will appeal to a remarkably broad and diverse audience, from non-mathematicians who would like to get a sense of what mathematics is really about, to experienced mathematicians who are not category theorists but would like a basic understanding of category theory. Speaking as one of the latter, I found it a real pleasure to be able to read the book without constantly having to stop and puzzle over the details. I have learnt a lot from it already, including what the famous Yoneda lemma is all about, and I look forward to learning more from it in the future.' Sir Timothy Gowers, Collège de France, Fields Medalist, main editor of The Princeton Companion to Mathematics'At last: a book that makes category theory as simple as it really is. Cheng explains the subject in a clear and friendly way, in detail, not relying on material that only mathematics majors learn. Category theory – indeed, mathematics as a whole – has been waiting for a book like this.' John Baez, University of California, Riverside'Many people speak derisively of category theory as the most abstract area of mathematics, but Eugenia Cheng succeeds in redeeming the word 'abstract'. This book is loquacious, conversational, and inviting. Reading this book convinced me I could teach category theory as an introductory course, and that is a real marvel, since it is a subject most people leave for experts.' Francis Su, Harvey Mudd College, author of Mathematics for Human Flourishing'Finally, a book about category theory that doesn't assume you already know category theory! In this inviting but rigorous introduction to what she calls 'the mathematics of mathematics', Eugenia Cheng brings the subject to us with insight, wit, and a point of view. Her story of finding joy-and advantage-in abstraction will inspire you to find it, too.' Patrick Honner, award-winning high school math teacher, columnist for Quanta Magazine, author of Painless Statistics'This higher category theory is the mathematics of the twenty-first century (at least my corner of it). If you'd like a taste of it, I recommend Dr. Cheng's book. The first half is an accessible and thought-provoking insight into categorical thinking. The second half climbs into the rarified air of theoretic math, but it is worth a read to get a feel for what some parts of modern mathematics look like.' Jonathan Kujawa, 3 Quarks Daily'… a successful addition to the literature that I am sure students will use in the future and I would be happy to recommend.' Constanze Roitzheim, Mathematische SemesterberichteTable of ContentsPrologue; Part I. Building Up to Categories: 1. Categories: the idea; 2. Abstraction; 3. Patterns; 4. Context; 5. Relationships; 6. Formalism; 7. Equivalence relations; 8. Categories: the definition; Interlude: A Tour of Math: 9. Examples we've already seen, secretly; 10. Ordered sets; 11. Small mathematical structures; 12. Sets and functions; 13. Large worlds of mathematical structures; Part II. Doing Category Theory: 14. Isomorphisms; 15. Monics and epics; 16. Universal properties; 17. Duality; 18. Products and coproducts; 19. Pullbacks and pushouts; 20. Functors; 21. Categories of categories; 22. Natural transformations; 23. Yoneda; 24. Higher dimensions; 25. Epilogue: thinking categorically; Appendices: A. Background on alphabets; B. Background on basic logic; C. Background on set theory; D. Background on topological spaces; Glossary; Further reading; Acknowledgements; Index.

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Book SynopsisStatistics for Engineers and Scientists stands out for its clear presentation of applied statistics. The book takes a practical approach to methods of statistical modeling and data analysis that are most often used in scientific work. This edition features a unique approach highlighted by an engaging writing style that explains difficult concepts clearly, along with the use of contemporary real world data sets, to help motivate students and show direct connections to industry and research. While focusing on practical applications of statistics, the text makes extensive use of examples to motivate fundamental concepts and to develop intuition.The new edition of Statistics for Engineers and Scientists is also available in McGraw Hill Connect, featuring SmartBook 2.0, Adaptive Learning Assignments, and more!Table of ContentsChapter 1: Sampling and Descriptive StatisticsChapter 2: ProbabilityChapter 3: Propagation of ErrorChapter 4: Commonly Used DistributionsChapter 5: Confidence IntervalsChapter 6: Hypothesis TestingChapter 7: Correlation and Simple Linear RegressionChapter 8: Multiple RegressionChapter 9: Factorial ExperimentsChapter 10: Statistical Quality Control

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Book SynopsisCHEMISTRY STUDENT GUIDES. GUIDED BY STUDENTS For any student who has ever struggled with a mathematical understanding of chemistry, this book is for you. Mathematics is the essential tool for physical scientists. We know that confidence in using mathematics early on in a chemistry degree builds a solid foundation for further study. However, applying the abstract mathematics taught in schools to chemical phenomena is one of the biggest challenges that chemistry students face. In this book, we take a ‘chemistry-first’ approach. We link the mathematics to recognisable chemical concepts, building on high school chemistry, to facilitate deeper understanding. We cover the practical mathematical skills, including representation of data as tables and graphs, and give an overview of error handling in the physical sciences. More advanced mathematical concepts are introduced, using calculus to determine kinetic rate laws, intermolecular forces and in quantifying energetic change in thermodynamics. We also introduce the concept of the complex number and its role in considering quantum wave functions, widely used in computational chemistry. There are worked examples and problem sets to provide plenty of practise material to build proficiency. We also include insights from real students, which identify common problem areas and provide the prompts that helped them to overcome these. Chemistry Student Guides are written with current students involved at every stage, guiding the books towards the most challenging aspects of the topic.Trade ReviewThe teaching of mathematics to students of chemistry has always been a difficult undertaking. Many students do not appreciate the value of mathematics in their subject and their confidence is often further undermined by textbooks which rely heavily on examples drawn from physics. Dickinson & McKinley bring a great deal of experience of chemistry teaching at degree level and their textbook is explicitly organised so that mathematical concepts are mapped directly to topics a chemistry student will find familiar. The book also places emphasis on techniques of dimensional analysis and the statistical analysis of data, which are often poorly taught to chemistry students. Overall the book fills a gap in chemical education and is highly recommended to anyone teaching physical chemistry or laboratory skills to chemistry students in higher education. -- Dr. Ben Horrocks * University of Newcastle *The book covers a wide range of topics, giving rounded support for the maths required for a chemistry degree. It is broken down well into manageable chunks with context laced throughout which help to add depth to the concepts being learned or revised. This is useful for people, such as myself, who learn better by applying concepts to ‘the bigger picture’. The breakdown of the vocabulary is also really useful, as lecturers often don’t realise that they’re introducing a new concept, or explain it very briefly, so having a source to consolidate that and ensure I’ve understood their explanation is very helpful. The increased complexity of the equations I was dealing with compared to A -Level drove consideration of units from my mind, but the way this book breaks down the section on units and gives examples would probably save me from losing easy marks in an exam. -- Cerys Day-Williams * Student - University of Bath *I found that the book teaches Maths in a way that is easily digestible to a student. There are no "big words" or fancy Maths terminology meaning that if you haven’t seen a certain concept before, it’s easy to understand what is being taught. I would definitely recommend this book to any student, whether you have studied maths previously or not, as it teaches the fundamental concepts which are used within a Chemistry degree. -- Isabel Manuel * Student - University of Bath *Table of ContentsHow to use this book; The basics: Mathematical functions, rearranging equations and handling units; Probability and Statistics in Chemistry; Experimental uncertainty and significant figures: What are the bounds of experimental error?; Tables and Graphs - Presenting and Analysing Experimental Data; Trigonometry and coordinate systems: Crystal structures and molecular separations; Vectors, directions and crystal structures; Calculus 1, Differentiation: Mean speeds and equilibrium separations; Calculus 2, Integration: Reaction kinetics and rate laws; Complex numbers: Quantum mechanics and solving the Schrödinger equation; The Mathematical Toolkit

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Book SynopsisThis substantially revised and expanded new edition of the bestselling textbook, addresses the difficulties that can arise with the mathematics that underpins the study of symmetry, and acknowledges that group theory can be a complex concept for students to grasp.Trade Review"the best introduction to the subject, especially for those whose mathematics is weak." (Chemistry and Industry, 2nd April 2001) ".I recommend this book..." (Education in Chemistry, September 2002)Table of ContentsPreface to the Second Edition vii How to use the Programmes ix Programme 1: Symmetry Elements and Operations 1 Programme 2: Point Groups 22 Programme 3: Non-degenerate Representations 46 Programme 4: Matrices 65 Programme 5: Degenerate Representations 85 Programme 6: Applications to Chemical Bonding 102 Programme 7: Applications to Molecular Vibration 122 Programme 8: Linear Combinations 139 Bibliography 173 Mathematical Data for use with Character Tables 174 Character Tables for Chemically Important Symmetry Groups 176 Index 186

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Book SynopsisThis highly acclaimed undergraduate textbook teaches all the mathematics for undergraduate courses in the physical sciences. Containing over 800 exercises, half come with hints and answers and, in a separate manual, complete worked solutions. The remaining exercises are intended for unaided homework; full solutions are available to instructors.Trade ReviewFrom reviews of previous editions: '…a great scientific textbook. It is a tour de force … to write mathematical sections that are both complete and at an appropriate academic level. The authors have clearly succeeded in this challenge, making this a remarkable pedagogical book … The choice of exercises is excellent and possibly the best feature of the book. In summary, this textbook is a great reference at undergraduate levels, particularly for those who like to teach or learn using lots of examples and exercises.' R. Botet, European Journal of Physics'… the book provides scientists who need to use the tool of mathematics for practical purposes with a single, comprehensive book. I recommend this book not only to students in physics and engineering sciences, but also to students in other fields of natural sciences.' P. Steward, Optik'… suitable as a textbook for undergraduate use … this is a book that in view of its content and its modest softcover price, will find its way on to many bookshelves.' Nigel Steele, The Times Higher Education Supplement'Riley et al. has clear, thorough and straightforward explanations of the subjects treated. It rigorously adopts a three-stage approach throughout the book: first a heuristic, intuitive introduction, then a formal treatment, and finally one or two examples. This consistent presentation, the layout, and the print quality make the book most attractive … and value for money. It contains a thousand pages, there are plenty of exercises with each chapter.' J. M. Thijssen, European Journal of PhysicsThis is a valuable book with great potential use in present-day university physics courses. Furthermore, the book will be useful for graduate too, and researchers will find it useful for looking up material which they have forgotten since their undergraduate days.' J. M. Thijssen, European Journal of Physics'This textbook is a well-written, modern, comprehensive, and complete collection of topics in mathematical methods ranging from a review of differential and integral calculus to group and representation theory, probability, the calculus of variations, and tensors.' Science Books and Films'This is a very comprehensive textbook suitable for most students enrolling on undergraduate degree courses in engineering. It contains 31 stand-alone chapters of mathematical methods which enable the students to understand the principles of the basic mathematical techniques and the authors have produced a clear, thorough and straightforward explanation of each subject. … finding a single textbook which covers the engineering student's need throughout their entire course is by no means an easy task. I believe the authors have achieved it … complete fully worked solutions … which I think is a useful asset for both students and lecturers.' Civil Engineering' ... this highly acclaimed undergraduate textbook is suitable for teaching all the mathematics ever likely to be needed for an undergraduate course in any of the physical sciences. As well as lucid descriptions of all the topics covered and many worked examples, it contains more than 800 exercises.' L'enseignement mathematiqueTable of ContentsPrefaces; 1. Preliminary algebra; 2. Preliminary calculus; 3. Complex numbers and hyperbolic functions; 4. Series and limits; 5. Partial differentiation; 6. Multiple integrals; 7. Vector algebra; 8. Matrices and vector spaces; 9. Normal modes; 10. Vector calculus; 11. Line, surface and volume integrals; 12. Fourier series; 13. Integral transforms; 14. First-order ordinary differential equations; 15. Higher-order ordinary differential equations; 16. Series solutions of ordinary differential equations; 17. Eigenfunction methods for differential equations; 18. Special functions; 19. Quantum operators; 20. Partial differential equations: general and particular; 21. Partial differential equations: separation of variables; 22. Calculus of variations; 23. Integral equations; 24. Complex variables; 25. Application of complex variables; 26. Tensors; 27. Numerical methods; 28. Group theory; 29. Representation theory; 30. Probability; 31. Statistics; Index.

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Book SynopsisLinear algebra is a fundamental area of mathematics, and is arguably the most powerful mathematical tool ever developed. It is a core topic of study within fields as diverse as: business, economics, engineering, physics, computer science, ecology, sociology, demography and genetics. For an example of linear algebra at work, one needs to look no further than the Google search engine, which relies upon linear algebra to rank the results of a search with respect to relevance. The strength of the text is in the large number of examples and the step-by-step explanation of each topic as it is introduced. It is compiled in a way that allows distance learning, with explicit solutions to set problems freely available online. The miscellaneous exercises at the end of each chapter comprise questions from past exam papers from various universities, helping to reinforce the reader''s confidence. Also included, generally at the beginning of sections, are short historical biographies of the leading pTrade ReviewThis book gives an introduction to linear algebra for students with limited mathematical preparation. ... The steady pace of the book is so gentle that no student need be left behind. * Peter Macgregor, Mathematical Gazette *Table of Contents1. Linear Equations and Matrices ; 2. Euclidean Space ; 3. General Vector Spaces ; 4. Inner Product Spaces ; 5. Linear Transformation ; 6. Determinants ; 7. Eigenvalues and Eigenvectors

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Book SynopsisThe maths needed to succeed in A Level Science is harder now than ever before. Suitable for all awarding bodies, this practical handbook addresses all of the maths skills needed for A Level Physics specifications. Worked examples, practice questions, ''remember points'' and ''stretch yourself'' questions give students the key knowledge and then the opportunity to practise and build confidence.

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Book SynopsisThis solutions booklet is a supplement to the text book 'Group Theory in Physics' by Wu-Ki Tung. It will be useful to lecturers and students taking the subject as detailed solutions are given.

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Book SynopsisThis text provides a critique of the subjective Bayesian view of statistical inference, and proposes the author's own error-statistical approach as an alternative framework for the epistemology of experiment. It seeks to address the needs of researchers who work with statistical analysis.

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Book SynopsisExam Board: Edexcel, AQA, OCR & WJEC EduqasLevel: GCSE 9-1Subject: Maths FoundationSuitable for the 2025 examsComplete revision and practice to fully prepare for the GCSE grade 9-1 examsRevision that Sticks! Collins GCSE 9-1 Maths Foundation Complete All-in-One Revision and Practice uses a revision method that really works: repeated practice throughout.A revision guide, workbook and practice paper in one book!With clear and concise revision for every topic, plus seven practice opportunities, Collins offers the best revision at the best price.Includes:quick tests as you goend-of-topic practice questionstopic review questions later in the bookmixed practice questions at the end of the bookmore topic-by-topic practice in the workbooka complete exam-style paperfree Q&A flashcards to download onlinefree ebook version

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Book SynopsisConfusing Textbooks? Missed Lectures? Not Enough Time?Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!Schaum's Outlines-Problem Solved.Table of ContentsThe Laplace Transform.The Inverse Laplace Transform.Applications to Differential Equations.Applications to Integral and Difference Equations.Complex Variable Theory.Fourier Series and Integrals.The Complex Inversion Formula.Applications to Boundary-Value Problems.Appendix A: Table of General Properties of Laplace Transforms.Appendix B: Table of Special Laplace Transforms.Appendix C: Table of Special Functions.

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Book SynopsisConfusing Textbooks? Missed Lectures? Not Enough Time?Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!Schaum's Outlines-Problem Solved.Table of ContentsSchaum's Outline of Basic Mathematics with Applications to Science and Technology, 2ed 1. Decimal Fractions 2. Measurement and Scientific Notation 3. Common Fractions 4. Percentage 5. Essentials of Algebra 6. Ratio and Proportion 7. Linear Equations 8. Exponents and Radicals 9. Logarithms 10. Quadratic Equations and Square Roots 11. Essentials of Plane Geometry 12. Solid Figures 13. Trigonometric Figures 14. Solution of Triangles 15. Vectors 16. Radian Measure 17. Conic Sections 18. Numbering Systems 19.Arithmetic Operations in a Computer 20.Counting Methods 21.Probability and Odds 22.Statistics

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Book SynopsisTough Test Questions? Missed Lectures? Not Enough Time?Fortunately for you, there's Schaum's.More than 40 million students have trusted Schaum's Outlines to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you: Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!Schaum's Outlines-Problem Solved.Table of Contents Schaum's Outline of Advanced Mathematics for Engineers and Scientists 1.Review of Fundamental Concepts 2.Ordinary Differential Equations 3.Linear Differential Equations 4.LaPlace Transforms 5.Vector Analysis 6.Multiple Line and Surface Integrals and Integral Theorems 7.Fourier Series 8.Fourier Integrals 9.Partial Differential Equations 10. Complex Variables and Conformal Mapping 11. Complex Inversion Formula for Laplace Transforms 12. Matrices 13. Calculus of Variations

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Book SynopsisAimed at a broad audience of graduate students and researchers in ecology and evolution, this novel creates a persuasive argument that an explicit consideration of history will often lead to a deeper, more nuanced understanding of almost every eco-evolutionary system.

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Book SynopsisThis slim volume provides a very approachable guide to the techniques and basic ideas of probability and statistics and more advanced techniques such as generalised linear models, classification using logistic regression, and support-vector machines.Table of Contents1: First steps 2: Summarising data 3: Probablity 4: Probability distributions 5: Estimation and confidence 6: Models, p-values, and hypotheses 7: Comparing proportions 8: Relations between two continous variable 9: Several explanatory variables 10: Classification 11: Last Words

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Book SynopsisFor more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, the outcome of which leads to satisfactory formal ability but does not lead to real understanding or to greater intellectual independence. This new edition of Richard Courant''s and Herbert Robbins''s classic work seeks to address this problem. Its goal is to put the meaning back into mathematics. Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions and projective geometry, from topology and calculus to matters of principle and the Continuum Hypothesis, this fascinating survey allows readers to delve into mathematics as an organic whole rather than an empty drill in problem solving. With chapters largely independent of one another and sections that lead upward from basic to more advanced discussions, readers can easily pick and choose areas of particular interest without impairing their understanding of subsequent parts. Brought up to date with a new chapter by Ian Stewart, What is Mathematics?, Second Edition offers new insights into recent mathematical developments and describes proofs of the Four-Color Theorem and Fermat''s Last Theorem, problems that were still open when Courant and Robbins wrote this masterpiece, but ones that have since been solved. Formal mathematics is like spelling and grammar--a matter of the correct application of local rules. Meaningful mathematics is like journalism--it tells an interesting story. But unlike some journalism, the story has to be true. The best mathematics is like literature--it brings a story to life before your eyes and involves you in it, intellectually and emotionally. What is Mathematics is like a fine piece of literature--it opens a window onto the world of mathematics for anyone interested to view.Trade ReviewCan...be read with great profit by anyone desiring general mathematical literacy. * Mathematics Abstracts *A great book. * Ludwig Otto, Paul Quinn College *A lucid representation of the fundamental concepts and methods of the whole field of mathematics. It is an easily understandable introduction for the layman and helps to give the mathematical student a general view of the basic principles and methods. * Albert Einstein *Without doubt, the work will have great influence. It should be in the hands of everyone, professional or otherwise, who is interested in scientific thinking. * The New York Times *A work of extraordinary perfection. * Mathematical Reviews *It contains an excellent selection of material for students who have no desire to develop mathematical skills but who may be willing to look briefly into this field of intellectual activity....For the inquiring student who wishes to know what real mathematics is about, or for the trained engineer or physicist who has some interest in the justification of procedures he uses, it should prove a source of great pleasure and satisfaction. * Journal of Applied Physics *This book is a work of art. * Marston Morse *This is not a book in philosophy; but there are probably few philosophers who can not gain instruction and clarification from it. It succeeds brilliantly in conveying the intellectual excitement of mathematical inquiry and in communicating the essential ideas and methods."Journal of PhilosophyIt is a work of high perfection, whether judged by aesthetic, pedagogical or scientific standards. It is astonishing to what extent What is Mathematics? has succeeded in making clear by means of the simplest examples all the fundamental ideas and methods which we mathematicians consider the life blood of our science. * Herman Weyl *Still a book that all prospective mathematics teachers should read and experience. A rare book that has retained its "freshness" and readability for more than 50 years....Very readable. * Stephen Krulik, Temple University *

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Book SynopsisThis book presents the statistical methods that are useful in the study of molecular evolution and illustrates how to use them in actual data analysis. Molecular evolution has been developing at a great pace over the past decade or so, driven by the huge increase in genetic sequence data from many organisms, the improvement of high-speed microcomputers, and the development of several new methods for phylogenetic analysis. This book for graduate students and researchers, assuming a basic knowledge of evolution, molecular biology, and elementary statistics, should make it possible for many investigators to incorporate refined statistical analysis of large-scale data in their own work. Nei is one of the leading workers in this area. He and Kumar have developed a computer program called MEGA, which has been sold for about $20 to over 1900 users. For the book, the authors are thoroughly revising MEGA and will make it available via FTP. The book also included analysis using the other most poTrade ReviewIt is worth its price * Plant Systematics and Evolution *Table of Contents1. Molecular basis of evolution ; 2. Evolutionary changes of amino acid sequences ; 3. Evolutionary changes of DNA sequences ; 4. Synonymous and nonsynonymous nucleotide substitutions ; 5. Phylogenetic trees ; 6. Phylogenetic inference: Distance methods ; 7. Phylogenetic inference: Maximum parsimony methods ; 8. Phylogenetic inference: Maximum likelihood methods ; 9. Accuracies and statistical tests of phylogenetic trees ; 10. Molecular clocks and linearized trees ; 11. Ancestral nucleotide and amino acid sequences ; 12. Genetic polymorphism and evolution ; 13. Population trees from genetic markers ; 14. Perspectives ; Appendices ; A. Mathematical sumbols and notations ; B. Geological timescale ; C. Geological events in the Cenozoic and Meszoic eras ; D. Evolution of organisms based on the fossil record

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Book SynopsisThis title is an introduction to the mathematical analysis of p- and hp-finite elements applied to elliptic problems in solid and fluid mechanics, and is suitable for graduate students and researchers who have had some prior exposure to finite element methods (FEM).Trade Review'Summarizing the book is the first theoretical book addressing the hp-version of the finite element method which is used today in practical computations. It is very well written and gives a very good review of the techniques and results in this relatively new direction in the FEM. It is highly recommended to anybody with mathematical interest for both learning and reference' ZAMMTable of ContentsVariational formulation of boundary value problems ; The Finite Element Method (FEM): definition, basic properties ; hp- Finite Elements in one dimension ; hp- Finite Elements in two dimensions ; Finite Element analysis of saddle point problems, mixed hp-FEM in incompressible fluid flow ; hp-FEM in the theory of elasticity

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Book SynopsisThis book gives a broad range of worked mathematical examples which are appropriate for scientists and engineers, ranging from basic algebra to calculus and Fourier transforms. Together with its companion volume Foundations of Science Mathematics (Oxford Chemistry Primer 77), it summarizes the basic concepts and results that should be familiar from high school, and then extends the ideas to cover the material needed by the majority of scienceundergraduates.Table of Contents1. Basic algebra and arithmetic ; 2. Curves and graphs ; 3. Trigonometry ; 4. Differentiation ; 5. Integration ; 6. Taylor series ; 7. Complex numbers ; 8. Vectors ; 9. Matrices ; 10. Partial differentiation ; 11. Line integrals ; 12. Multiple integrals ; 13. Ordinary differential equations ; 14. Partial differential equations ; 15. Fourier series and transforms

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Book SynopsisThis is an introduction to the mathematical foundations of quantum field theory, using operator algebraic methods and emphasizing the link between the mathematical formulations and related physical concepts. It starts with a general probabilistic description of physics, which encompasses both classical and quantum physics. The basic key physical notions are clarified at this point. It then introduces operator algebraic methods for quantum theory, and goes on to discuss the theory of special relativity, scattering theory, and sector theory in this context.Trade Review'the self-contained monograph provides an introduction suitable for mathematics graduates to the basic properties of quantum fields' AslibTable of ContentsStates and observables ; Quantum theory ; The relativistic symmetry ; Local observables ; Scattering theory ; Sector theory ; Appendix A: Hilbert space and operators ; Appendix B: Operator algebras ; Appendix C: Free fields

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Book SynopsisThis monograph provides a treatment of the theory of algebraic Riccati equations, an area of increasing interest in the mathematics and engineering communities. A range of applications are covered, demonstrating the use of these equations for providing solutions to complex problems.Table of Contents1. Preliminaries from the theory of matrices ; 2. Indefinite scalar products ; 3. Skew-symmetric scalar products ; 4. Matrix theory and control ; 5. Linear matrix equations ; 6. Rational matrix functions ; 7. Geometric theory: the complex case ; 8. Geometric theory: the real case ; 9. Constructive existence and comparison theorems ; 10. Hermitian solutions and factorizations of rational matrix functions ; 11. Perturbation theory ; 12. Geometric theory for the discrete algebraic Riccati equation ; 13. Constructive existence and comparison theorems ; 14. Perturbation theory for discrete algebraic Riccati equations ; 15. Discrete algebraic Riccati equations and matrix pencils ; 16. Linear-quadratic regulator problems ; 17. The discrete Kalman filter ; 18. The total least squares technique ; 19. Canonical factorization ; 20. Hoo control problems ; 21. Contractive rational matrix functions ; 22. The matrix sign function ; 23. Structured stability radius ; Bibliography ; List of notations ; Index

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Book SynopsisMethods in theoretical quantum optics is aimed at those readers who already have some knowledge of mathematical methods and have also been introduced to the basic ideas of quantum optics. This book is ideal for students who have already explored the basics of the quantum theory of light and are seeking to acquire the mathematical skills used in real problems. This book is not primarily about the physics of quantum optics, but rather presents the mathematical methods widely used by workers in this field. There is no comparable book which covers either the range or the depth of mathematical techniques.Trade Review... many valuable insights ... Even old hands at the quantum optics game will benefit from these ... The authors cover a very wide range of material appropriate to quantum optics. By bringing it together in the way they have, they have made an important contribution to the teaching and understanding of quantum optics. * Zentralblatt MATH *Table of Contents1. Foundations ; 2. Coherent interactions ; 3. Operators and states ; 4. Quantum statistics of fields ; 5. Dissipative processes ; 6. Dressed states ; Appendices ; Selected bibliography ; Index ; 1. Foundations ; 2. Coherent interactions ; 3. Operators and states ; 4. Quantum statistics of fields ; 5. Dissipative processes ; 6. Dressed states ; Appendices ; Selected bibliography ; Index

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Book SynopsisDrawing on the author's extensive experience of supporting students undertaking projects, Scientific Data Analysis is a guide for any science undergraduate or beginning graduate who needs to analyse their own data, and wants a clear, step-by-step description of how to carry out their analysis in a robust, error-free way.Trade ReviewThis is an appealing introduction that would be accessible to a variety of students at the college level. Its strengths are clarity and directness with an abundance of good examples and case studies. * MAA Review *Table of ContentsPART I - UNDERSTANDING THE STATISTICS; PART II - ANALYSING EXPERIMENTAL DATA

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Book SynopsisThe perfect introduction to the essential mathematical concepts which all chemistry students need to master. Working from foundational principles, the book builds the student's confidence by leading them through the subject in a steady, progressive way from basic algebra to the mathematics of quantum chemistry. mathematics.Trade ReviewA very useful text to gradually guide students through both the fundamental and more advanced aspects of mathematics specifically relevant for a chemistry undergraduate degree. It is particularly useful in allowing students to test their knowledge of mathematical concepts and processes via self-test exercise and additional problems that are directly relevant to chemistry. * Dr Jon Tandy, Senior Lecturer in Physical Chemistry, London Metropolitan University *This is an outstanding and carefully thought-out introduction to the mathematical toolkit required for students embarking on a chemistry degree programme. * Dr Robert Johnson, Lecturer, School of Chemistry, University College Dublin *Table of ContentsSection A: Core mathematics: algebra, logarithms and trigonometry 1: The display of numbers 2: Algebra I 3: Algebra II 4: Algebra III 5: Algebra IV 6: Algebra V 7: Algebra VI 8: Algebra VII 9: Powers I 10: Powers II 11: Trigonometry 12: Advanced BODMAS Section B: Calculus 13: Differentiation I 14: Differentiation II 15: Differentiation III 16: Differentiation IV 17: Differentiation V 18: Differentiation VI 19: Integration I 20: Integration II 21: Integration III 22: Integration IV Section C: Matrices, vectors and complex numbers 23: Matrices I 24: Matrices II 25: Complex numbers 26: Vectors Section D: Laboratory mathematics 27: Graphs I 28: Graphs II 29: Graphs III 30: Probability I 31: Probability II 32: Statistics I 33: Statistics II 34: Statistics III 35: Statistics IV 36: Dimensional analysis

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Book SynopsisThe Chemistry Maths Book provides a complete course companion suitable for students at all levels. All the most useful and important topics are covered, with numerous examples of applications in chemistry and the physical sciences. Taking a clear, straightforward approach, the book develops ideas in a logical, coherent way, allowing students progressively to build a thorough working understanding of the subject.Topics are organized into three parts: algebra, calculus, differential equations, and expansions in series; vectors, determinants and matrices; and numerical analysis and statistics. The extensive use of examples illustrates every important concept and method in the text, and are used to demonstrate applications of the mathematics in chemistry and several basic concepts in physics. The exercises at the end of each chapter, are an essential element of the development of the subject, and have been designed to give students a working understanding of the material in the text.Online Resources:The online resources feature the following: - Figures from the book in electronic format, ready to download- Full worked solutions to all end of chapter exercisesTrade ReviewReview from previous edition It seems well suited both for its stated purpose and as a "brush-up" book for undergraduates, graduate students, and others. The mathematics are carried out briskly and with very little dressing ... there is much material to cover here and it works well through Steiner's particularly lucid presentation. The notation is standard and clear ... I am impressed with this book, I am sure that it will remain open on my desk and will become well worn in short order. * C. Michael McCallum, University of the Pacific, Journal of Chemical Education, Vol. 74 No. 12 December 1997 *Table of Contents1. Numbers, variables and units ; 2. Algebraic functions ; 3. Transcendental functions ; 4. Differentiation ; 5. Integration ; 6. Methods of integration ; 7. Sequences and series ; 8. Complex numbers ; 9. Functions of several variables ; 10. Functions in 3 dimensions ; 11. First-order differential equations ; 12. Second-order differential equations. Constant coefficients ; 13. Second-order differential equations. Some special functions ; 14. Partial differential equations ; 15. Orthogonal expansions. Fourier analysis ; 16. Vectors ; 17. Determinants ; 18. Matrices and linear transformations ; 19. The matrix eigenvalue problem ; 20. Numerical methods ; 21. Probability and statistics

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Book SynopsisCore Maths for the Biosciences introduces the range of mathematical concepts that bioscience students need to master during thier studies. Starting from fundamental concepts, it blends clear explanations and biological examples throughout as it equips the reader with the full range of mathematical tools required by biologists today.Trade ReviewExactly the sort of thing that will be helpful in showing those with biological problems how mathematics can be very useful - and that what is really important is maintaining an intuitive understanding between the mathematics - which is essentially no more, but no less, than a way of thinking very precisely - and the actual phenomena they are dealing with...Very fine indeed. * Professor Lord May of Oxford, Department of Zoology, University of Oxford *Fantastic. Easy to understand, interactive, biologically relevant and dictated in a way that seemed as though you are almost having a conversation with the author. * James Sleigh, Student, University of Oxford *Coherent and clear. The best I have seen this kind of material treated. * Stephen Hubbard, University of Dundee *This book is by far the best of its kind, a spectacular diamond in the rough. * Helen Smith, student, University of Salford *The interactive spreadsheets are a work of genius. * Stuart Fisk, student, University of Essex *Table of ContentsPART 1: ARITHMETIC, ALGEBRA & FUNCTIONS; PART 2: CALCULUS AND DIFFERENTIAL EQUATIONS

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Book SynopsisMathematical Techniques provides a complete course in mathematics, covering all the essential topics with which a physical sciences or engineering student should be familiar. It introduces and builds on concepts in a progressive, carefully-layered way, and features over 2000 end of chapter problems, plus additional self-check questions.Trade ReviewReview from previous edition This textbook offers an accessible and comprehensive grounding in many of the mathematical techniques required in the early stages of an engineering or science degree and also for the routine methods needed by first and second year mathematics students. * Engineering Designer March/April 2003 *There are also significant changes in content in the opening chapter, where the foundation material has been expanded usefully. The authors do not attempt to dodge theoretical hurdles. They are careful to explain many of the less intuitive properties of functions and to highlight generalisations without becoming over abstract. * Times Higher Education Supplement, November 2002 *Thoroughly recommended. * Zentralblatt MATH, 993:2002 *Table of ContentsPART 1. ELEMENTARY METHODS, DIFFERENTIATION, COMPLEX NUMBERS; PART 2. MATRIX AND VECTOR ALGEBRA; PART 3. INTEGRATION AND DIFFERENTIAL EQUATIONS; PART 4. TRANSFORMS AND FOURIER SERIES; PART 5. MULTIVARIABLE CALCULUS; PART 6. DISCRETE MATHEMATICS; PART 7. PROBABILITY AND STATISTICS; PART 8. PROJECTS; SELF-TESTS: SELECTED ANSWERS; ANSWERS TO SELECTED PROBLEMS; APPENDICES; FURTHER READING; INDEX

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Book SynopsisThis new edition updates Durbin & Koopman''s important text on the state space approach to time series analysis. The distinguishing feature of state space time series models is that observations are regarded as made up of distinct components such as trend, seasonal, regression elements and disturbance terms, each of which is modelled separately. The techniques that emerge from this approach are very flexible and are capable of handling a much wider range of problems than the main analytical system currently in use for time series analysis, the Box-Jenkins ARIMA system. Additions to this second edition include the filtering of nonlinear and non-Gaussian series.Part I of the book obtains the mean and variance of the state, of a variable intended to measure the effect of an interaction and of regression coefficients, in terms of the observations.Part II extends the treatment to nonlinear and non-normal models. For these, analytical solutions are not available so methods are based on simulation.Trade ReviewReview from previous edition ...provides an up-to-date exposition and comprehensive treatment of state space models in time series analysis...This book will be helpful to graduate students and applied statisticians working in the area of econometric modelling as well as researchers in the areas of engineering, medicine and biology where state space models are used. * Journal of the Royal Statistical Society *Table of ContentsPART I: THE LINEAR STATE SPACE MODEL; PART II: NON-GAUSSIAN AND NONLINEAR STATE SPACE MODELS

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Book SynopsisMathematical Physics is an introduction to such basic mathematical structures as groups, vector spaces, topological spaces, measure spaces, and Hilbert space. Geroch uses category theory to emphasize both the interrelationships among different structures and the unity of mathematics.

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Book SynopsisThis richly illustrated third edition provides a thorough training in practical mathematical biology and shows how exciting mathematical challenges can arise from a genuinely interdisciplinary involvement with the biosciences.Trade ReviewFrom the reviews: "The 2nd volume of the authors elucidating work highlights a surprisingly broad spectrum of applications in the field of mathematical biology. The sense given to the mathematical texture of thoughts broadens the reader’s insight … . The growing number of specialists in sub-disciplines of mathematical biology will be enjoying the truly concise approach … . It can so be said that the foremost results … might be essential for new interpretations of data … . It is a recommended text for mathematicians … ." (Daniel Gertsch, Bioworld, Issue 2, 2004) From the reviews of the third edition: "This is the second volume of the third edition of Murray’s ‘Mathematical Biology’. … covers a wide variety of problems in pattern formation, each discussed in its biological context. … This volume alone is a large book, with more than 800 pages and a similar number of references. … it is a valuable collection of results from different areas of mathematical biology." (Carlo Laing, New Zealand Mathematical Society Newsletter, Issue 90, April, 2004) "This book, a classical text in mathematical biology, cleverly combines mathematical tools with subject area sciences. The multi-layer way of material presentation makes the book useful for different types of reader including graduate-level students, bioscientists … . it is an enjoyable reading and I recommend it to anyone with serious interest in mathematical modelling." (V.V. Fedorov, Short Book Reviews, Vol. 23 (3), 2003) "This second volume of the third edition of Murray’s Mathematical biology focuses on partial differential equations (spatial models) and their application to the biomedical sciences. … Each chapter deals with its particular topic in great detail, usually focusing on one biological example and the associated mathematical model and results. This volume is not an introductory text … making it extremely useful in graduate courses and for reference." (Trachette L. Jackson, Mathematical Reviews, 2004b) "In this second volume … the development towards specific biological configurations and towards a mechanism for understanding morphogenesis represents an important portion of the work. … chapters deal with attractive topics … . There is an extensive index at the end. … very interesting and strongly recommended." (A. Akutowicz, Zentralblatt MATH, Vol. 1006, 2003) "In this volume it becomes clear that compiling the third edition was a ‘labor of love’. The book has a significantly different feel from the original first edition. … my reaction to the third edition was positive. … The historical and biological overviews have much interesting information. … Certainly, the spicy writing will keep students alert … . In summary, I recommend the new and expanded third edition to any serious young student interested in mathematical biology … ." (Leah Edelstein-Keshet, SIAM Review, Vol. 46 (1), 2004) "Mathematical Biology would be eminently suitable as a text for a final year undergraduate or postgraduate course in mathematical biology … . It is also a good source of examples for courses in mathematical methods … . Mathematical Biology provides a good way into the field and a useful reference for those of us already there. It may attract more mathematicians to work in biology by showing them that there is real work to be done." (Peter Saunders, The Mathematical Gazette, Vol. 90 (518), 2006)Table of ContentsMulti-Species Waves and Practical Applications * Spatial Pattern Formation with Reaction Diffusion Systems * Animal Coat Patterns and Other Practical Applications of Reaction Diffusion Mechanisms * Pattern Formation on Growing Domains: Alligators and Snakes * Bacterial Patterns and Chemotaxis * Mechanical Theory for Generating Pattern and Form in Development * Evolution, Morphogenetic Laws, Developmental Constraints and Teratologies * A Mechanical Theory of Vascular Network Formation * Epidermal Wound Healing * Dermal Wound Healing * Growth and Control of Brain Tumours * Neural Models of Pattern Formation * Geographic Spread and Control of Epidemics * Wolf Territoriality, Wolf-Deer Interaction and Survival

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Book SynopsisThe book introduces most of the basic tools of chemometrics including experimental design, signal analysis, statistical methods for analytical chemistry and multivariate methods.Trade Review"…useful for introducing chemometrics in undergraduate classes…a valuable encyclopedia for researchers…" (Journal of Chemical Education, December 2007)Table of ContentsPreface. 1 Introduction. 1.1 Development of Chemometrics. 1.2 Application Areas. 1.3 How to Use this Book. 1.4 Literature and Other Sources of Information. References. 2 Experimental Design. 2.1 Why Design Experiments in Chemistry? 2.2 Degrees of Freedom and Sources of Error. 2.3 Analysis of Variance and Interpretation of Errors. 2.4 Matrices, Vectors and the Pseudoinverse. 2.5 Design Matrices. 2.6 Factorial Designs. 2.7 An Example of a Factorial Design. 2.8 Fractional Factorial Designs. 2.9 Plackett–Burman and Taguchi Designs. 2.10 The Application of a Plackett–Burman Design to the Screening of Factors Influencing a Chemical Reaction. 2.11 Central Composite Designs. 2.12 Mixture Designs. 2.13 A Four Component Mixture Design Used to Study Blending of Olive Oils. 2.14 Simplex Optimization. 2.15 Leverage and Confidence in Models. 2.16 Designs for Multivariate Calibration. References. 3 Statistical Concepts. 3.1 Statistics for Chemists. 3.2 Errors. 3.3 Describing Data. 3.4 The Normal Distribution. 3.5 Is a Distribution Normal? 3.6 Hypothesis Tests. 3.7 Comparison of Means: the t-Test. 3.8 F-Test for Comparison of Variances. 3.9 Confidence in Linear Regression. 3.10 More about Confidence. 3.11 Consequences of Outliers and How to Deal with Them. 3.12 Detection of Outliers. 3.13 Shewhart Charts. 3.14 More about Control Charts. References. 4 Sequential Methods. 4.1 Sequential Data. 4.2 Correlograms. 4.3 Linear Smoothing Functions and Filters. 4.4 Fourier Transforms. 4.5 Maximum Entropy and Bayesian Methods. 4.6 Fourier Filters. 4.7 Peakshapes in Chromatography and Spectroscopy. 4.8 Derivatives in Spectroscopy and Chromatography. 4.9 Wavelets. References. 5 Pattern Recognition. 5.1 Introduction. 5.2 Principal Components Analysis. 5.3 Graphical Representation of Scores and Loadings. 5.4 Comparing Multivariate Patterns. 5.5 Preprocessing. 5.6 Unsupervised Pattern Recognition: Cluster Analysis. 5.7 Supervised Pattern Recognition. 5.8 Statistical Classification Techniques. 5.9 K Nearest Neighbour Method. 5.10 How Many Components Characterize a Dataset? 5.11 Multiway Pattern Recognition. References. 6 Calibration. 6.1 Introduction. 6.2 Univariate Calibration. 6.3 Multivariate Calibration and the Spectroscopy of Mixtures. 6.4 Multiple Linear Regression. 6.5 Principal Components Regression. 6.6 Partial Least Squares. 6.7 How Good is the Calibration and What is the Most Appropriate Model? 6.8 Multiway Calibration. References. 7 Coupled Chromatography. 7.1 Introduction. 7.2 Preparing the Data. 7.3 Chemical Composition of Sequential Data. 7.4 Univariate Purity Curves. 7.5 Similarity Based Methods. 7.6 Evolving and Window Factor Analysis. 7.7 Derivative Based Methods. 7.8 Deconvolution of Evolutionary Signals. 7.9 Noniterative Methods for Resolution. 7.10 Iterative Methods for Resolution. 8 Equilibria, Reactions and Process Analytics. 8.1 The Study of Equilibria using Spectroscopy. 8.2 Spectroscopic Monitoring of Reactions. 8.3 Kinetics and Multivariate Models for the Quantitative Study of Reactions 8.4 Developments in the Analysis of Reactions using On-line Spectroscopy. 8.5 The Process Analytical Technology Initiative. References. 9 Improving Yields and Processes Using Experimental Designs. 9.1 Introduction. 9.2 Use of Statistical Designs for Improving the Performance of Synthetic Reactions. 9.3 Screening for Factors that Influence the Performance of a Reaction. 9.4 Optimizing the Process Variables. 9.5 Handling Mixture Variables using Simplex Designs. 9.6 More about Mixture Variables. 10 Biological and Medical Applications of Chemometrics. 10.1 Introduction. 10.2 Taxonomy. 10.3 Discrimination. 10.4 Mahalanobis Distance. 10.5 Bayesian Methods and Contingency Tables. 10.6 Support Vector Machines. 10.7 Discriminant Partial Least Squares. 10.8 Micro-organisms. 10.9 Medical Diagnosis using Spectroscopy. 10.10 Metabolomics using Coupled Chromatography and Nuclear Magnetic Resonance. References. 11 Biological Macromolecules. 11.1 Introduction. 11.2 Sequence Alignment and Scoring Matches. 11.3 Sequence Similarity. 11.4 Tree Diagrams. 11.5 Phylogenetic Trees. References. 12 Multivariate Image Analysis. 12.1 Introduction. 12.2 Scaling Images. 12.3 Filtering and Smoothing the Image. 12.4 Principal Components for the Enhancement of Images. 12.5 Regression of Images. 12.6 Alternating Least Squares as Employed in Image Analysis. 12.7 Multiway Methods In Image Analysis. References. 13 Food. 13.1 Introduction. 13.2 How to Determine the Origin of a Food Product using Chromatography. 13.3 Near Infrared Spectroscopy. 13.4 Other Information. 13.5 Sensory Analysis: Linking Composition to Properties. 13.6 Varimax Rotation. 13.7 Calibrating Sensory Descriptors to Composition. References. Index.

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Book SynopsisThere are not many calculus books that are very accessible to students without a strong mathematical background and the large majority of financial derivatives students do not have a strong quantitative background. This book provides a short introduction to the subject with examples of its use in mathematical finance e. g pricing of derivatives.Table of ContentsPreface xiii 1 Brownian Motion 1 1.1 Origins 1 1.2 Brownian Motion Specification 2 1.3 Use of Brownian Motion in Stock Price Dynamics 4 1.4 Construction of Brownian Motion from a Symmetric Random Walk 6 1.5 Covariance of Brownian Motion 12 1.6 Correlated Brownian Motions 14 1.7 Successive Brownian Motion Increments 16 1.7.1 Numerical Illustration 17 1.8 Features of a Brownian Motion Path 19 1.8.1 Simulation of Brownian Motion Paths 19 1.8.2 Slope of Path 20 1.8.3 Non-Differentiability of Brownian Motion Path 21 1.8.4 Measuring Variability 24 1.9 Exercises 26 1.10 Summary 29 2 Martingales 31 2.1 Simple Example 31 2.2 Filtration 32 2.3 Conditional Expectation 33 2.3.1 General Properties 34 2.4 Martingale Description 36 2.4.1 Martingale Construction by Conditioning 36 2.5 Martingale Analysis Steps 37 2.6 Examples of Martingale Analysis 37 2.6.1 Sum of Independent Trials 37 2.6.2 Square of Sum of Independent Trials 38 2.6.3 Product of Independent Identical Trials 39 2.6.4 Random Process B(t) 39 2.6.5 Random Process exp[B(t) – t] 40 2.6.6 Frequently Used Expressions 40 2.7 Process of Independent Increments 41 2.8 Exercises 42 2.9 Summary 42 3 Itō Stochastic Integral 45 3.1 How a Stochastic Integral Arises 45 3.2 Stochastic Integral for Non-Random Step-Functions 47 3.3 Stochastic Integral for Non-Anticipating Random Step-Functions 49 3.4 Extension to Non-Anticipating General Random Integrands 52 3.5 Properties of an Itō Stochastic Integral 57 3.6 Significance of Integrand Position 59 3.7 Itō integral of Non-Random Integrand 61 3.8 Area under a Brownian Motion Path 62 3.9 Exercises 64 3.10 Summary 67 3.11 A Tribute to Kiyosi Itō 68 Acknowledgment 72 4 Itō Calculus 73 4.1 Stochastic Differential Notation 73 4.2 Taylor Expansion in Ordinary Calculus 74 4.3 Itō’s Formula as a Set of Rules 75 4.4 Illustrations of Itō’s Formula 78 4.4.1 Frequent Expressions for Functions of Two Processes 78 4.4.2 Function of Brownian Motion f [B(t)] 80 4.4.3 Function of Time and Brownian Motion f [t, B(t)]82 4.4.4 Finding an Expression for 83 4.4.5 Change of Numeraire 84 4.4.6 Deriving an Expectation via an ODE 85 4.5 Lévy Characterization of Brownian Motion 87 4.6 Combinations of Brownian Motions 89 4.7 Multiple Correlated Brownian Motions 92 4.8 Area under a Brownian Motion Path – Revisited 95 4.9 Justification of Itō’s Formula 96 4.10 Exercises 100 4.11 Summary 101 5 Stochastic Differential Equations 103 5.1 Structure of a Stochastic Differential Equation 103 5.2 Arithmetic Brownian Motion SDE 104 5.3 Geometric Brownian Motion SDE 105 5.4 Ornstein–Uhlenbeck SDE 108 5.5 Mean-Reversion SDE 110 5.6 Mean-Reversion with Square-Root Diffusion SDE 112 5.7 Expected Value of Square-Root Diffusion Process 112 5.8 Coupled SDEs 114 5.9 Checking the Solution of a SDE 115 5.10 General Solution Methods for Linear SDEs 115 5.11 Martingale Representation 120 5.12 Exercises 123 5.13 Summary 124 6 Option Valuation 127 6.1 Partial Differential Equation Method 128 6.2 Martingale Method in One-Period Binomial Framework 130 6.3 Martingale Method in Continuous-Time Framework 135 6.4 Overview of Risk-Neutral Method 138 6.5 Martingale Method Valuation of Some European Options 139 6.5.1 Digital Call 139 6.5.2 Asset-or-Nothing Call 141 6.5.3 Standard European Call 142 6.6 Links between Methods 144 6.6.1 Feynman-Kač Link between PDE Method and Martingale Method 144 6.6.2 Multi-Period Binomial Link to Continuous 146 6.7 Exercise 147 6.8 Summary 148 7 Change of Probability 151 7.1 Change of Discrete Probability Mass 151 7.2 Change of Normal Density 153 7.3 Change of Brownian Motion 154 7.4 Girsanov Transformation 155 7.5 Use in Stock Price Dynamics – Revisited 160 7.6 General Drift Change 162 7.7 Use in Importance Sampling 163 7.8 Use in Deriving Conditional Expectations 167 7.9 Concept of Change of Probability 172 7.9.1 Relationship between Expected Values under Equivalent Probabilities 174 7.10 Exercises 174 7.11 Summary 176 8 Numeraire 179 8.1 Change of Numeraire 179 8.1.1 In Discrete Time 179 8.1.2 In Continuous Time 182 8.2 Forward Price Dynamics 184 8.2.1 Dynamics of Forward Price of a Bond 184 8.2.2 Dynamics of Forward Price of any Traded Asset 185 8.3 Option Valuation under most Suitable Numeraire 187 8.3.1 Exchange Option 187 8.3.2 Option on Bond 188 8.3.3 European Call under Stochastic Interest Rate 188 8.4 Relating Change of Numeraire to Change of Probability 190 8.5 Change of Numeraire for Geometric Brownian Motion 192 8.6 Change of Numeraire in LIBOR Market Model 194 8.7 Application in Credit Risk Modelling 198 8.8 Exercises 200 8.9 Summary 201 Annexes A Annex A: Computations with Brownian Motion 205 A. 1 Moment Generating Function and Moments of Brownian Motion 205 A. 2 Probability of Brownian Motion Position 208 A. 3 Brownian Motion Reflected at the Origin 208 A. 4 First Passage of a Barrier 214 A. 5 Alternative Brownian Motion Specification 216 B Annex B: Ordinary Integration 221 B. 1 Riemann Integral 221 B. 2 Riemann–Stieltjes Integral 226 B. 3 Other Useful Properties 231 B. 4 References 234 C Annex C: Brownian Motion Variability 235 C. 1 Quadratic Variation 235 C. 2 First Variation 238 D Annex D: Norms 239 D. 1 Distance between Points 239 D. 2 Norm of a Function 242 D. 3 Norm of a Random Variable 244 D. 4 Norm of a Random Process 244 D. 5 Reference 246 E Annex E: Convergence Concepts 247 E. 1 Central Limit Theorem 247 E. 2 Mean-Square Convergence 248 E. 3 Almost Sure Convergence 249 E. 4 Convergence in Probability 250 E. 5 Summary 250 Answers to Exercises 253 References 299 Index 303

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Book SynopsisThe objective of this book is to motivate an appreciation of contemporary statistical techniques within the context of engineering. The author presents an optimum blend between statistical thinking and statistical methodology through emphasis of a broad sweep of tools rather than endless streams of seemingly unrelated methods and formulae.Trade Review"Overall this is an excellent book, which defines a broader mandate than many of its competing texts. By providing, clear, understandable discussion of the basics of statistics through to more advanced methods commonly used by engineers, this book is an essential reference for practitioners, and an ideal text for a two semester course introducing engineers to the power and utility of statistics." (The American Statistician, August 2008) "In this book on modern engineering statistics, Ryan does an excellent job of providing the appropriate statistical concepts and tools using engineering resources.... Highly recommended. Lower- and upper-division undergraduates" (CHOICE, April 2008) "This self-contained volume motivates an appreciation of statistical techniques within the context of engineering; many datasets that are used in the chapters and exercises are from engineering sources. This book is ideal for either a one- or two-semester course in engineering statistics." (Computing Reviews, April 2008)Table of ContentsPreface xvii 1. Methods of Collecting and Presenting Data 1 1.1 Observational Data and Data from Designed Experiments 3 1.2 Populations and Samples 5 1.3 Variables 6 1.4 Methods of Displaying Small Data Sets 7 1.5 Methods of Displaying Large Data Sets 16 1.6 Outliers 22 1.7 Other Methods 22 1.8 Extremely Large Data Sets: Data Mining 23 1.9 Graphical Methods: Recommendations 23 1.10 Summary 24 References 24 Exercises 25 2. Measures of Location and Dispersion 45 2.1 Estimating Location Parameters 46 2.2 Estimating Dispersion Parameters 50 2.3 Estimating Parameters from Grouped Data 55 2.4 Estimates from a Boxplot 57 2.5 Computing Sample Statistics with MINITAB 58 2.6 Summary 58 Reference 58 Exercises 58 3. Probability and Common Probability Distributions 68 3.1 Probability: From the Ethereal to the Concrete 68 3.3 Common Discrete Distributions 76 3.4 Common Continuous Distributions 92 3.5 General Distribution Fitting 106 3.6 How to Select a Distribution 107 3.7 Summary 108 References 109 Exercises 109 4. Point Estimation 121 4.1 Point Estimators and Point Estimates 121 4.2 Desirable Properties of Point Estimators 121 4.3 Distributions of Sampling Statistics 125 4.4 Methods of Obtaining Estimators 128 4.5 Estimating σθ 132 4.6 Estimating Parameters Without Data 133 4.7 Summary 133 References 134 Exercises 134 5. Confidence Intervals and Hypothesis Tests—One Sample 140 5.1 Confidence Interval for μ: Normal Distribution σ Not Estimated from Sample Data 140 5.2 Confidence Interval for μ: Normal Distribution σ Estimated from Sample Data 146 5.3 Hypothesis Tests for μ: Using Z and t 147 5.4 Confidence Intervals and Hypothesis Tests for a Proportion 157 5.5 Confidence Intervals and Hypothesis Tests for σ2 and σ 161 5.6 Confidence Intervals and Hypothesis Tests for the Poisson Mean 164 5.7 Confidence Intervals and Hypothesis Tests When Standard Error Expressions are Not Available 166 5.8 Type I and Type II Errors 168 5.9 Practical Significance and Narrow Intervals: The Role of n 172 5.10 Other Types of Confidence Intervals 173 5.11 Abstract of Main Procedures 174 5.12 Summary 175 Appendix: Derivation 176 References 176 Exercises 177 6. Confidence Intervals and Hypothesis Tests—Two Samples 189 6.1 Confidence Intervals and Hypothesis Tests for Means: Independent Samples 189 6.2 Confidence Intervals and Hypothesis Tests for Means: Dependent Samples 197 6.3 Confidence Intervals and Hypothesis Tests for Two Proportions 200 6.4 Confidence Intervals and Hypothesis Tests for Two Variances 202 6.5 Abstract of Procedures 204 6.6 Summary 205 References 205 Exercises 205 7. Tolerance Intervals and Prediction Intervals 214 7.1 Tolerance Intervals: Normality Assumed 215 7.2 Tolerance Intervals and Six Sigma 219 7.3 Distribution-Free Tolerance Intervals 219 7.4 Prediction Intervals 221 7.5 Choice Between Intervals 227 7.6 Summary 227 References 228 Exercises 229 8. Simple Linear Regression Correlation and Calibration 232 8.1 Introduction 232 8.2 Simple Linear Regression 232 8.3 Correlation 254 8.4 Miscellaneous Uses of Regression 256 8.5 Summary 264 References 264 Exercises 265 9. Multiple Regression 276 9.1 How Do We Start? 277 9.2 Interpreting Regression Coefficients 278 9.3 Example with Fixed Regressors 279 9.4 Example with Random Regressors 281 9.5 Example of Section 8.2.4 Extended 291 9.6 Selecting Regression Variables 293 9.7 Transformations 299 9.8 Indicator Variables 300 9.9 Regression Graphics 300 9.10 Logistic Regression and Nonlinear Regression Models 301 9.11 Regression with Matrix Algebra 302 9.12 Summary 302 References 303 Exercises 304 10. Mechanistic Models 314 10.1 Mechanistic Models 315 10.2 Empirical–Mechanistic Models 316 10.3 Additional Examples 324 10.4 Software 325 10.5 Summary 326 References 326 Exercises 327 11. Control Charts and Quality Improvement 330 11.1 Basic Control Chart Principles 330 11.2 Stages of Control Chart Usage 331 11.3 Assumptions and Methods of Determining Control Limits 334 11.4 Control Chart Properties 335 11.5 Types of Charts 336 11.6 Shewhart Charts for Controlling a Process Mean and Variability (Without Subgrouping) 336 11.7 Shewhart Charts for Controlling a Process Mean and Variability (With Subgrouping) 344 11.8 Important Use of Control Charts for Measurement Data 349 11.9 Shewhart Control Charts for Nonconformities and Nonconforming Units 349 11.10 Alternatives to Shewhart Charts 356 11.11 Finding Assignable Causes 359 11.12 Multivariate Charts 362 11.13 Case Study 362 11.14 Engineering Process Control 364 11.15 Process Capability 365 11.16 Improving Quality with Designed Experiments 366 11.17 Six Sigma 367 11.18 Acceptance Sampling 368 11.19 Measurement Error 368 11.20 Summary 368 References 369 Exercises 370 12. Design and Analysis of Experiments 382 12.1 Processes Must be in Statistical Control 383 12.2 One-Factor Experiments 384 12.3 One Treatment Factor and at Least One Blocking Factor 392 12.4 More Than One Factor 395 12.5 Factorial Designs 396 12.6 Crossed and Nested Designs 405 12.7 Fixed and Random Factors 406 12.8 ANOM for Factorial Designs 407 12.9 Fractional Factorials 409 12.10 Split-Plot Designs 413 12.11 Response Surface Designs 414 12.12 Raw Form Analysis Versus Coded Form Analysis 415 12.13 Supersaturated Designs 416 12.14 Hard-to-Change Factors 416 12.15 One-Factor-at-a-Time Designs 417 12.16 Multiple Responses 418 12.17 Taguchi Methods of Design 419 12.18 Multi-Vari Chart 420 12.19 Design of Experiments for Binary Data 420 12.20 Evolutionary Operation (EVOP) 421 12.21 Measurement Error 422 12.22 Analysis of Covariance 422 12.23 Summary of MINITAB and Design-Expert® Capabilities for Design of Experiments 422 12.24 Training for Experimental Design Use 423 12.25 Summary 423 Appendix A Computing Formulas 424 Appendix B Relationship Between Effect Estimates and Regression Coefficients 426 References 426 Exercises 428 13. Measurement System Appraisal 441 13.1 Terminology 442 13.2 Components of Measurement Variability 443 13.3 Graphical Methods 449 13.4 Bias and Calibration 449 13.5 Propagation of Error 454 13.6 Software 455 13.7 Summary 456 References 456 Exercises 457 14. Reliability Analysis and Life Testing 460 14.1 Basic Reliability Concepts 461 14.2 Nonrepairable and Repairable Populations 463 14.3 Accelerated Testing 463 14.4 Types of Reliability Data 466 14.5 Statistical Terms and Reliability Models 467 14.6 Reliability Engineering 473 14.7 Example 474 14.8 Improving Reliability with Designed Experiments 474 14.9 Confidence Intervals 477 14.10 Sample Size Determination 478 14.11 Reliability Growth and Demonstration Testing 479 14.12 Early Determination of Product Reliability 480 14.13 Software 480 14.14 Summary 481 References 481 Exercises 482 15. Analysis of Categorical Data 487 15.1 Contingency Tables 487 15.2 Design of Experiments: Categorical Response Variable 497 15.3 Goodness-of-Fit Tests 498 15.4 Summary 500 References 500 Exercises 501 16. Distribution-Free Procedures 507 16.1 Introduction 507 16.2 One-Sample Procedures 508 16.3 Two-Sample Procedures 512 16.4 Nonparametric Analysis of Variance 514 16.5 Exact Versus Approximate Tests 519 16.6 Nonparametric Regression 519 16.7 Nonparametric Prediction Intervals and Tolerance Intervals 521 16.8 Summary 521 References 521 Exercises 522 17. Tying It All Together 525 17.1 Review of Book 525 17.2 The Future 527 17.3 Engineering Applications of Statistical Methods 528 Reference 528 Exercises 528 Answers to Selected Excercises 533 Appendix: Statistical Tables 562 Table A Random Numbers 562 Table B Normal Distribution 564 Table C t-Distribution 566 Table D F-Distribution 567 Table E Factors for Calculating Two-Sided 99% Statistical Intervals for a Normal Population to Contain at Least 100p% of the Population 570 Table F Control Chart Constants 571 Author Index 573 Subject Index 579

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Book SynopsisMathematical Methods in Biology uniquely covers both deterministic and probabilistic models, including algorithms in the MATLAB platform. The book focuses mostly in one area of the life sciences, focusing mainly on theoretical ecology.Trade Review"Admirably, the volume is written with bits of MATLAB code inserted at appropriate places and has exercises interspersed throughout the text (as well as hints for solutions to the exercises at the end of the book)." The Quarterly Review of Biology, June 2010) "The mathematical and reasoning sophistication increases as the chapters proceed." (Book News, December 2009)Table of ContentsPreface. 1. Introduction To Ecological Modeling. 1.1 Mathematical Models. 1.2 Rates of Change. 1.3 Balance Laws. 1.4 Temperature in the Environment. 1.5 Dimensionless Variables. 1.6 Descriptive Statistics. 1.7 Regression and Curve Fitting. 1.8 Reference Notes. 2. Population Dynamics for Single Species. 2.1 Laws of Population Dynamics. 2.2 Continuous Time Models. 2.3 Qualitative Analysis of Population Models. 2.4 Dynamics of Predation. 2.5 Discrete Time Models. 2.6 Equilibria, Stability, and Chaos. 2.7 Reference Notes. 3. Structure and Interacting Populations. 3.1 Structure--Juveniles and Adults. 3.2 Structured Linear Models. 3.3 Nonlinear Interactions. 3.4 Appendix--Matrices. 3.5 Reference Notes. 4. Interactions in Continuous Time. 4.1 Interacting Populations. 4.2 Phase Plane Analysis. 4.3 Linear Systems. 4.4 Nonlinear Systems. 4.5 Bifurcation. 4.6 Reference Notes. 5. Concepts of Probability. 5.1 Introductory Examples and Definitions. 5.2 The Hardy-Weinberg Law. 5.3 Continuous Random Variables. 5.4 Discrete Random Variables. 5.5 Joint Probability Distributions. 5.6 Covariance and Correlation. 5.7 Reference Notes. 6. Statistical Inference. 6.1 Introduction. 6.2 Interval Analysis. 6.3 Estimating Proportions. 6.4 The Chi-Squared Test. 6.5 Hypothesis Testing. 6.6 Bootstrap Methods. 6.7 Reference Notes. 7. Stochastic Processes. 7.1 Introduction. 7.2 Randomizing Discrete Dynamics. 7.3 Random Walk. 7.4 Birth Processes. 7.5 Stochastic Differential Equations. 7.6 SDEs from Markov Models. 7.7 Solving SDEs. 7.8 The Fokker-Planck Equation. 7.9 Reference Notes. A. Hints and Solutions to Exercises

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