This volume is a systematic treatment of the additive number theory of polynomials over a finite field, an area possessing deep and fascinating parallels with classical number theory. In providing asymptomatic proofs of both the Polynomial Three Primes Problem (an analog of Vinogradov's theorem) and the Polynomial Waring Problem, the book develops the various tools necessary to apply an adelic "circle method" to a wide variety of additive problems in both the polynomial and classical settings. A key to the methods employed here is that the generalized Riemann hypothesis is valid in this polynomial setting. The authors presuppose a familiarity with algebra and number theory as might be gained from the first two years of graduate course, but otherwise the book is self-contained. Starting with analysis on local fields, the main technical results are all proved in detail so that there are extensive discussions of the theory of characters in a non-Archimidean field, adele class groups, the global singular series and Radon-Nikodyn derivatives, L-functions of Dirichlet type, and K-ideles.

Multivariate Analysis deals with observations on more than one variable where there is some inherent interdependence between the variables. With several texts already available in this area, one may very well enquire of the authors as to the need for yet another book. Most of the available books fall into two categories, either theoretical or data analytic. The present book not only combines the two approaches but it also has been guided by the need to give suitable matter for the beginner as well as illustrating some deeper aspects of the subject for the research worker. Practical examples are kept to the forefront and, wherever feasible, each technique is motivated by such an example.

In recent years there has been a resurgence of interest in the study of delay differential equations motivated largely by new applications in physics, biology, ecology, and physiology. The aim of this monograph is to present a reasonably self-contained account of the advances in the oscillation theory of this class of equations. Throughout, the main topics of study are shown in action, with applications to such diverse problems as insect population estimations, logistic equations in ecology, the survival of red blood cells in animals, integro-differential equations, and the motion of the tips of growing plants. The authors begin by reviewing the basic theory of delay differential equations, including the fundamental results of existence and uniqueness of solutions and the theory of the Laplace and z-transforms. Little prior knowledge of the subject is required other than a firm grounding in the main techniques of differential equation theory. As a result, this book provides an invaluable reference to the recent work both for mathematicians and for all those whose research includes the study of this fascinating class of differential equations.

The UK's most trusted A level Mathematics resources With over 900,000 copies sold (plus 1.3 million copies sold of the previous edition), Pearson's own resources for Pearson Edexcel are the market-leading and most trusted for AS and A level Mathematics. This book can be used alongside the Year 1 book to cover all the content needed for the compulsory Edexcel A level Core Pure Mathematics exams Enhanced focus on problem-solving and modelling, as well as supporting the large data set and calculators Packed with worked examples with guidance, lots of exam-style questions, practice papers, and plenty of mixed and review exercises Full worked solutions to every question available free and online for quick and easy access. Plus free additional online content with GeoGebra interactives and Casio calculator tutorials Practice books also available offering the most comprehensive and flexible AS/A level Maths practice with over 2000 extra questions Includes access to an online digital edition (valid for 3 years once activated) Pearson Edexcel A level Further Mathematics Core Pure Mathematics Book 2 Textbook + e-book matches the Pearson Edexcel exam structure and is fully integrated with Pearson Edexcel's interactive scheme of work. All of the books in this series focus on problem-solving and modelling, as well as supporting the large data set and calculators. They are packed with worked examples with guidance, lots of exam-style questions, practice papers, and plenty of mixed and review exercises. There are full worked solutions to every question available free and online for quick and easy access. You will also have access to lots of free additional online content with GeoGebra interactives and Casio calculator tutorials. There are separate Pure and Applied textbooks for AS and A level Maths, and a textbook per option for AS and A level Further Maths. Practice books are also available offering the most comprehensive and flexible AS/A level Maths practice with over 2000 extra questions. Pearson's revision resources are the smart choice for those revising for Pearson Edexcel AS and A level Mathematics - there is a Revision Workbook for exam practice and a Revision Guide for classroom and independent study. Practice Papers Plus+ books contain additional full length practice papers, so you can practice answering questions by writing straight into the book and perfect your responses with targeted hints, guidance and support for every question, including fully worked solutions.

Written by one of the subject's foremost experts, this is the first book on division space integration theory. It is intended to present a unified account of many classes of integrals including the Lebesgue-Bochner, Denjoy-Perron gauge, Denjoy-Hincin, Cesaro-Perron, and Marcinkiewicz-Zygmund integrals. Professor Henstock develops here the general axiomatic theory of Riemann-type integration from first principles in such a way that familiar classes of integrals (such as Lebesgue and Wiener integrals) are subsumed into the general theory in a systematic fashion. In particular, the theory seeks to place Feynman integration on a secure analytical footing. By adopting an axiomatic approach, proofs are, in general, simpler and more transparent than have previously appeared. The author also shows how one proof can prove corresponding results for a wide variety of integrals. As a result, this book will be the central reference work in this subject for many years to come.

The biological and social sciences often generalize causal conclusions from one context or location to others that may differ in some relevant respects, as is illustrated by inferences from animal models to humans or from a pilot study to a broader population. Inferences like these are known as extrapolations. The question of how and when extrapolation can be legitimate is a fundamental issue for the biological and social sciences that has not received the attention it deserves. In Across the Boundaries, Steel argues that previous accounts of extrapolation are inadequate and proposes a better approach that is able to answer methodological critiques of extrapolation from animal models to humans.
Across the Boundaries develops the thought that knowledge of mechanisms linking cause to effect can serve as a basis for extrapolation. Despite its intuitive appeal, this idea faces several obstacles. Extrapolation is worthwhile only when there are stringent practical or ethical limitations on what can be learned about the target (say, human) population by studying it directly. Meanwhile, the mechanisms approach rests on the idea that extrapolation is justified when mechanisms are the same or similar enough. Yet since mechanisms may differ significantly between model and target, it needs to be explained how the suitability of the model could be established given only very limited information about the target. Moreover, since model and target are rarely alike in all relevant respects, an adequate account of extrapolation must also explain how extrapolation can be legitimate even when some causally relevant differences are present.
Steel explains how his proposal can answer thesechallenges, illustrates his account with a detailed biological case study, and explores its implications for such traditional philosophy of science topics ceteris paribus laws and reductionism. Finally, he considers whether mechanisms-based extrapolation can work in social science.

This textbook offers an accessible, modern introduction at undergraduate level to an area known variously as general topology, point-set topology or analytic topology with a particular focus on helping students to build theory for themselves. It is the result of several years of the authors' combined university teaching experience stimulated by sustained interest in advanced mathematical thinking and learning, alongside established research careers in analytic topology. Point-set topology is a discipline that needs relatively little background knowledge, but sufficient determination to grasp ideas precisely and to argue with straight and careful logic. Research and long experience in undergraduate mathematics education suggests that an optimal way to learn such a subject is to teach it to yourself, pro-actively, by guided reading of brief skeleton notes and by doing your own spadework to fill in the details and to flesh out the examples. This text will facilitate such an approach for those learners who opt to do it this way and for those instructors who would like to encourage this so-called 'Moore approach', even for a modest segment of the teaching term or for part of the class. In reality, most students simply do not have the combination of time, background and motivation needed to implement such a plan fully. The accessibility, flexibility and completeness of this text enable it to be used equally effectively for more conventional instructor-led courses. Critically, it furnishes a rich variety of exercises and examples, many of which have specimen solutions, through which to gain in confidence and competence.

This book is a unique introduction to graph theory, written by one of its founding fathers. It is not intended as a comprehensive treatise, but rather as an account of those parts of the theory that have been of special interest to the author. Professor Tutte details his experiences in the area, and provides a fascinating insight into the processes leading to his proofs.

This series contains the most focused resources for the 'mainstream' modules of the AS and A-Level courses. Providing thorough exam preparation through detailed worked examples, practise questions and further practise on the CD-ROM, these are the most supportive, AQA revision resources available. Checked by examiners, students can have complete confidence in covering all the topics for their specification Maximises students' marks by honing their exam technique with examiner tips, a sample module paper and step-by-step solutions on CD-ROM Includes theory and strategies for answering each type of question with progressive questions and Powerpoint model answers to strengthen understanding and application. A perfect match to each module of AQA

The burgeoning field of data analysis is expanding at an incredible pace due to the proliferation of data collection in almost every area of science. The enormous data sets now routinely encountered in the sciences provide an incentive to develop mathematical techniques and computational algorithms that help synthesize, interpret and give meaning to the data in the context of its scientific setting. A specific aim of this book is to integrate standard scientific computing methods with data analysis. By doing so, it brings together, in a self-consistent fashion, the key ideas from: * statistics, * time-frequency analysis, and * low-dimensional reductions The blend of these ideas provides meaningful insight into the data sets one is faced with in every scientific subject today, including those generated from complex dynamical systems. This is a particularly exciting field and much of the final part of the book is driven by intuitive examples from it, showing how the three areas can be used in combination to give critical insight into the fundamental workings of various problems. Data-Driven Modeling and Scientific Computation is a survey of practical numerical solution techniques for ordinary and partial differential equations as well as algorithms for data manipulation and analysis. Emphasis is on the implementation of numerical schemes to practical problems in the engineering, biological and physical sciences. An accessible introductory-to-advanced text, this book fully integrates MATLAB and its versatile and high-level programming functionality, while bringing together computational and data skills for both undergraduate and graduate students in scientific computing.

While the significance of networks in various human behavior and activities has a history as long as human's existence, network awareness is a recent scientific phenomenon. The neologism network science is just one or two decades old. Nevertheless, with this limited time, network thinking has substantially reshaped the recent development in economics, and almost all solutions to real-world problems involve the network element. This book integrates agent-based modeling and network science. It is divided into three parts, namely, foundations, primary dynamics on and of social networks, and applications. The authors begin with the network origin of agent-based models, known as cellular automata, and introduce a number of classic models, such as Schelling's segregation model and Axelrod's spatial game. The essence of the foundation part is the network-based agent-based models in which agents follow network-based decision rules. Under the influence of the substantial progress in network science in late 1990s, these models have been extended from using lattices into using small-world networks, scale-free networks, etc. The text also shows that the modern network science mainly driven by game-theorists and sociophysicists has inspired agent-based social scientists to develop alternative formation algorithms, known as agent-based social networks. It reviews a number of pioneering and representative models in this family. Upon the given foundation, the second part reviews three primary forms of network dynamics, such as diffusions, cascades, and influences. These primary dynamics are further extended and enriched by practical networks in goods-and-service markets, labor markets, and international trade. At the end, the book considers two challenging issues using agent-based models of networks: network risks and economic growth.

The Oxford Handbook of the Economics of Networks represents the frontier of research into how and why networks form, how they influence behavior, how they help govern outcomes in an interactive world, and how they shape collective decision making, opinion formation, and diffusion dynamics. From a methodological perspective, the contributors to this volume devote attention to theory, field experiments, laboratory experiments, and econometrics. Theoretical work in network formation, games played on networks, repeated games, and the interaction between linking and behavior is synthesized. A number of chapters are devoted to studying social process mediated by networks. Topics here include opinion formation, diffusion of information and disease, and learning. There are also chapters devoted to financial contagion and systemic risk, motivated in part by the recent financial crises. Another section discusses communities, with applications including social trust, favor exchange, and social collateral; the importance of communities for migration patterns; and the role that networks and communities play in the labor market. A prominent role of networks, from an economic perspective, is that they mediate trade. Several chapters cover bilateral trade in networks, strategic intermediation, and the role of networks in international trade. Contributions discuss as well the role of networks for organizations. On the one hand, one chapter discusses the role of networks for the performance of organizations, while two other chapters discuss managing networks of consumers and pricing in the presence of network-based spillovers. Finally, the authors discuss the internet as a network with attention to the issue of net neutrality.

Developed for the new International A Level specification, these new resources are specifically designed for international students, with a strong focus on progression, recognition and transferable skills, allowing learning in a local context to a global standard. Recognised by universities worldwide and fully comparable to UK reformed GCE A levels. Supports a modular approach, in line with the specification. Appropriate international content puts learning in a real-world context, to a global standard, making it engaging and relevant for all learners. Reviewed by a language specialist to ensure materials are written in a clear and accessible style. The embedded transferable skills, needed for progression to higher education and employment, are signposted so students understand what skills they are developing and therefore go on to use these skills more effectively in the future. Exam practice provides opportunities to assess understanding and progress, so students can make the best progress they can.

This hands-on guide is primarily intended to be used in undergraduate laboratories in the physical sciences and engineering. It assumes no prior knowledge of statistics. It introduces the necessary concepts where needed, with key points illustrated with worked examples and graphic illustrations. In contrast to traditional mathematical treatments it uses a combination of spreadsheet and calculus-based approaches, suitable as a quick and easy on-the-spot reference. The emphasis throughout is on practical strategies to be adopted in the laboratory.
Error analysis is introduced at a level accessible to school leavers, and carried through to research level. Error calculation and propagation is presented though a series of rules-of-thumb, look-up tables and approaches amenable to computer analysis. The general approach uses the chi-square statistic extensively. Particular attention is given to hypothesis testing and extraction of parameters and their uncertainties by fitting mathematical models to experimental data. Routines implemented by most contemporary data analysis packages are analysed and explained. The book finishes with a discussion of advanced fitting strategies and an introduction to Bayesian analysis.

The UK's most trusted A level Mathematics resources With over 900,000 copies sold (plus 1.3 million copies sold of the previous edition), Pearson's own resources for Pearson Edexcel are the market-leading and most trusted for AS and A level Mathematics. Our A level Mathematics Pure Year 2 Practice Book helps you get exam-ready with confidence and practice at the right pace. Coverage: the practice workbooks cover all Pure, Statistics and Mechanics topics Quantity: the most A level question practice available, with over 2,000 extra questions per book Practice at the right pace: start with the essentials, build your skills with various practice questions to make connections between topics, then apply this to exam-style questions at the end of each chapter Get exam-ready with confidence: differentiated questions including 'Bronze, Silver, Gold' in each chapter, and a mixed problem-solving section for each book, will guide and help you to develop the skills you need for your exams Designed to be used flexibly, the practice books are fully mapped to the scheme of work and textbooks so you can use them seamlessly in and out of the classroom and all year round. Use them lesson by lesson, topic by topic, for homework, revision and more - the choice is yours Great value practice materials that are cheaper than photocopying, saves more time than independently sourcing questions and answers, and are all in one place Pearson Edexcel AS and A level Mathematics Pure Mathematics Year 2 Practice Book matches the Pearson Edexcel exam structure and is fully integrated with Pearson Edexcel's interactive scheme of work. Practice books are also available offering the most comprehensive and flexible AS/A level Maths practice with over 2000 extra questions. Pearson's revision resources are the smart choice for those revising for Pearson Edexcel AS and A level Mathematics - there is a Revision Workbook for exam practice and a Revision Guide for classroom and independent study. Practice Papers Plus+ books contain additional full length practice papers, so you can practice answering questions by writing straight into the book and perfect your responses with targeted hints, guidance and support for every question, including fully worked solutions.

The UK's most trusted A level Mathematics resources With over 900,000 copies sold (plus 1.3 million copies sold of the previous edition), Pearson's own resources for Pearson Edexcel are the market-leading and most trusted for AS and A level Mathematics. This book covers all the content needed for the optional Edexcel AS and A level Further Statistics 1 exams Enhanced focus on problem-solving and modelling, as well as supporting the large data set and calculators Packed with worked examples with guidance, lots of exam-style questions, practice papers, and plenty of mixed and review exercises Full worked solutions to every question available free and online for quick and easy access. Plus free additional online content with GeoGebra interactives and Casio calculator tutorials Practice books also available offering the most comprehensive and flexible AS/A level Maths practice with over 2000 extra questions Includes access to an online digital edition (valid for 3 years once activated) Pearson Edexcel AS and A level Further Mathematics Further Statistics 1 Textbook + e-book matches the Pearson Edexcel exam structure and is fully integrated with Pearson Edexcel's interactive scheme of work. All of the books in this series focus on problem-solving and modelling, as well as supporting the large data set and calculators. They are packed with worked examples with guidance, lots of exam-style questions, practice papers, and plenty of mixed and review exercises. There are full worked solutions to every question available free and online for quick and easy access. You will also have access to lots of free additional online content with GeoGebra interactives and Casio calculator tutorials. There are separate Pure and Applied textbooks for AS and A level Maths, and a textbook per option for AS and A level Further Maths. Practice books are also available offering the most comprehensive and flexible AS/A level Maths practice with over 2000 extra questions. Pearson's revision resources are the smart choice for those revising for Pearson Edexcel AS and A level Mathematics - there is a Revision Workbook for exam practice and a Revision Guide for classroom and independent study. Practice Papers Plus+ books contain additional full length practice papers, so you can practice answering questions by writing straight into the book and perfect your responses with targeted hints, guidance and support for every question, including fully worked solutions.

The UK's most trusted A level Mathematics resources With over 900,000 copies sold (plus 1.3 million copies sold of the previous edition), Pearson's own resources for Pearson Edexcel are the market-leading and most trusted for AS and A level Mathematics. Our A level Mathematics Statistics and Mechanics Year 2 Practice Book helps you get exam-ready with confidence and practice at the right pace. Coverage: the practice workbooks cover all Pure, Statistics and Mechanics topics Quantity: the most A level question practice available, with over 2,000 extra questions per book Practice at the right pace: start with the essentials, build your skills with various practice questions to make connections between topics, then apply this to exam-style questions at the end of each chapter Get exam-ready with confidence: differentiated questions including 'Bronze, Silver, Gold' in each chapter, and a mixed problem-solving section for each book, will guide and help you to develop the skills you need for your exams Designed to be used flexibly, the practice books are fully mapped to the scheme of work and textbooks so you can use them seamlessly in and out of the classroom and all year round. Use them lesson by lesson, topic by topic, for homework, revision and more - the choice is yours Great value practice materials that are cheaper than photocopying, saves more time than independently sourcing questions and answers, and are all in one place Pearson Edexcel A level Mathematics Statistics & Mechanics Year 2 Practice Book matches the Pearson Edexcel exam structure and is fully integrated with Pearson Edexcel's interactive scheme of work. Practice books are also available offering the most comprehensive and flexible AS/A level Maths practice with over 2000 extra questions. Pearson's revision resources are the smart choice for those revising for Pearson Edexcel AS and A level Mathematics - there is a Revision Workbook for exam practice and a Revision Guide for classroom and independent study. Practice Papers Plus+ books contain additional full length practice papers, so you can practice answering questions by writing straight into the book and perfect your responses with targeted hints, guidance and support for every question, including fully worked solutions.

At first glance, the concept of equality in maths seems unambiguous. When we see the equality sign, we think of 'solving for x' or balancing two sides of an equation or maybe even the many famous equations that make use of this elegant, innocuous symbol.

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But between those parallel lines lies a mathematical playground of choice and abstraction, leading to far greater insight than you could have dreamed. As it turns out, sameness and difference, equality and inequality, are not nearly as straightforward as they seem.

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Unequal explores the rich and rewarding interplay between sameness and difference, from numbers to manifolds to category theory and beyond in a glorious celebration of mathematics that will change the way you look at maths - and the world around you - forever.