John Stillwell - Böcker

From the reviews of the second edition:"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations.

Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity.Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a large extent, the story of philosophy of mathematics.

The collection of reliable and comprehensive data on the magnitude, composition and distribution of a country’s population is essential in order for governments to provide services, administer effectively and guide a country’s development. The primary source of basic demographic statistics is frequently a population census, which provides hugely important data sets for policy makers, practitioners and researchers working in a wide range of different socio-demographic contexts.The Routledge Handbook of Census Resources, Methods and Applications provides a comprehensive and authoritative guide to the collection, processing, quality assessment and delivery of the different data products that constitute the results of the population censuses conducted across the United Kingdom in 2011. It provides those interested in using census data with an introduction to the collection, processing and quality assessment of the 2011 Census, together with guidance on the various types of data resources that are available and how they can be accessed. It demonstrates how new methods and technologies, such as interactive infographics and web-based mapping, are now being used to visualise census data in new and exciting ways. Perhaps most importantly, it presents a collection of applications of census data in different social and health science research contexts that reveal key messages about the characteristics of the UK population and the ways in which society is changing. The operation of the 2011 Census and the use of its results are set in the context of census-taking around the world and its historical development in the UK over the last 200 years.The results of the UK 2011 Census are a unique and reliable source of detailed information that are immensely important for users from a wide range of public and private sector organisations, as well as those working in Population Studies, Human Geography, Migration Studies and the Social Sciences more generally.

This new textbook demonstrates that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. In this book, two chapters are devoted to each approach, the first being concrete and introductory, while the second is more abstract.Geometry, of all subjects, should be about taking different viewpoints, and geometry is unique among mathematical disciplines in its ability to look different from different angles. Some students prefer to visualize, while others prefer to reason or to calculate.Geometry has something for everyone, and students will find themselves building on their strengths at times, and working to overcome weaknesses at other times. This book will be suitable for a second course in geometry and contains more than 100 figures, and a large selection of exercises in each chapter.

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates.

This book is a concise, self-contained introduction to abstract algebra which stresses its unifying role in geometry and number theory. There is a strong emphasis on historical motivation - both to trace abstract concepts to their concrete roots, but also to show the power of new ideas to solve old problems. This approach shows algebra as an integral part of mathematics and makes this text more informative to both beginners and experts than others. Classical results of geometry and number theory (such as straightedge-and-compass construction and its relation to Fermat primes) are used to motivate and illustrate algebraic techniques, and classical algebra itself (solutions of cubic and quartic equations) is used to motivate the problem of solvability by radicals and its solution via Galois theory. Modern methods are used whenever they are clearer or more efficient, but technical machinery is introduced only when needed. The lively style and clear expositions make this book a pleasure to read and to learn from.

This book is intended to complement my Elements oi Algebra, and it is similarly motivated by the problem of solving polynomial equations. However, it is independent of the algebra book, and probably easier. In Elements oi Algebra we sought solution by radicals, and this led to the concepts of fields and groups and their fusion in the celebrated theory of Galois. In the present book we seek integer solutions, and this leads to the concepts of rings and ideals which merge in the equally celebrated theory of ideals due to Kummer and Dedekind. Solving equations in integers is the central problem of number theory, so this book is truly a number theory book, with most of the results found in standard number theory courses. However, numbers are best understood through their algebraic structure, and the necessary algebraic concepts­ rings and ideals-have no better motivation than number theory. The first nontrivial examples of rings appear in the number theory of Euler and Gauss. The concept of ideal-today as routine in ring the­ ory as the concept of normal subgroup is in group theory-also emerged from number theory, and in quite heroic fashion. Faced with failure of unique prime factorization in the arithmetic of certain generalized "inte­ gers" , Kummer created in the 1840s a new kind of number to overcome the difficulty. He called them "ideal numbers" because he did not know exactly what they were, though he knew how they behaved.

Geometry used to be the basis of a mathematical education; today it is not even a standard undergraduate topic. Much as I deplore this situation, I welcome the opportunity to make a fresh start. Classical geometry is no longer an adequate basis for mathematics or physics-both of which are becoming increasingly geometric-and geometry can no longer be divorced from algebra, topology, and analysis. Students need a geometry of greater scope, and the fact that there is no room for geometry in the curriculum un­ til the third or fourth year at least allows us to assume some mathematical background. What geometry should be taught? I believe that the geometry of surfaces of constant curvature is an ideal choice, for the following reasons: 1. It is basically simple and traditional. We are not forgetting euclidean geometry but extending it enough to be interesting and useful. The extensions offer the simplest possible introduction to fundamentals of modem geometry: curvature, group actions, and covering spaces. 2. The prerequisites are modest and standard. A little linear algebra (mostly 2 x 2 matrices), calculus as far as hyperbolic functions, ba­ sic group theory (subgroups and cosets), and basic topology (open, closed, and compact sets).

This is a well-balanced introduction to topology that stresses geometric aspects. Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group. It also present algorithms for topological problems. Most of the results and proofs are known, but some have been simplified or placed in a new perspective. Over 300 illustrations, many interesting exercises, and challenging open problems are included. New in this edition is a chapter on unsolvable problems, which includes the first textbook proof that the main problem of topology, the homeomorphism problem, is unsolvable.

A beautiful and relatively elementary account of a part of mathematics where three main fields - algebra, analysis and geometry - meet. The book provides a broad view of these subjects at the level of calculus, without being a calculus book. Its roots are in arithmetic and geometry, the two opposite poles of mathematics, and the source of historic conceptual conflict. The resolution of this conflict, and its role in the development of mathematics, is one of the main stories in the book. Stillwell has chosen an array of exciting and worthwhile topics and elegantly combines mathematical history with mathematics. He covers the main ideas of Euclid, but with 2000 years of extra insights attached. Presupposing only high school algebra, it can be read by any well prepared student entering university. Moreover, this book will be popular with graduate students and researchers in mathematics due to its attractive and unusual treatment of fundamental topics. A set of well-written exercises at the end of each section allows new ideas to be instantly tested and reinforced.

Only applications-driven book dealing with commerically-sponsored spatial analysis research. Focuses on business and public sector planning case studies, offering readers a snapshot of the use of spatial analysis across a broad range of areas.Internationally-renowned editors and contributors present a broad variety of global applications, and demonstrate GIS components and spatial methodologies in practice.

Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity.Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation.Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

The first book surveying the history and ideas behind reverse mathematicsReverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. In Reverse Mathematics, John Stillwell offers a historical and representative view, emphasizing basic analysis and giving a novel approach to logic. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

How the concept of proof has enabled the creation of mathematical knowledgeThe Story of Proof investigates the evolution of the concept of proof—one of the most significant and defining features of mathematical thought—through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved.Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field’s power and progress.

How the concept of proof has enabled the creation of mathematical knowledgeThe Story of Proof investigates the evolution of the concept of proof—one of the most significant and defining features of mathematical thought—through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved.Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field’s power and progress.

In August 1989, a Summer Institute was held at the Academie van Bouwkunst, the seventeenth century home of Amsterdam's School of Architecture, Town Planning and Landscape. The meeting brought together experts in Geographical Information Systems from throughout the world to address an international audience of planners. The contents of this book reflect many of the themes that were presented and discussed at the conference. The Summer Institute, let alone this volume, would not have been possible without the support of the International Association for the Development and Management of Existing and New Towns (INTNAIVN), the International Society of City and Regional Planners (ISoCaRP), The National Physical Planning Agency of the Netherlands (RPD) and the Berlage Studio. We wish to acknowledge the assistance provided by these organisations and by the various sponsors: The Ministry of Housing, Physical Planning and Environment, the Municipality of Amsterdam, Logisterion b.v., ESRI, UNISYS, MABON b.v., SPSS, PRIME Computer Inc., PANDATA. The provision of hardware facilities by the various computer companies allowed immensely valuable 'hands on' experience to be gained by all the participants.

This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.

Winner of a CHOICE Outstanding Academic Title Award for 2011!This book offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Gödel.

Yearning for the Impossible: The Surprising Truth of Mathematics, Second Edition explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress. The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. This new edition contains many new exercises and commentaries, clearly discussing a wide range of challenging subjects.

Yearning for the Impossible: The Surprising Truth of Mathematics, Second Edition explores the history of mathematics from the perspective of the creative tension between common sense and the "impossible" as the author follows the discovery or invention of new concepts that have marked mathematical progress. The author puts these creations into a broader context involving related "impossibilities" from art, literature, philosophy, and physics. This new edition contains many new exercises and commentaries, clearly discussing a wide range of challenging subjects.

This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.

Land use change is driven by a variety of forces, including spatial policies formulated at supra-national, national, regional and local levels. The main focus of this book is to contextualize, explain and illustrate a new methodology for simulating land use change in different parts of Europe. It considers some of the more important causal factors and identifies state-of-the-art approaches to modelling human and environmental systems, and for evaluating and visualizing alternative scenarios. The last part of the volume presents material from two case studies, one from The Netherlands and one from Portugal, of the implementation of a new simulation model called EuroScanner. The work should be of interest to researchers and practioners whose work involves geography, simulation and modelling, environmental planning, spatial decision making, the methodology of social sciences and economics. This volume contains a CD-rom with full-colour versions of many of the figures.

Models of land-use change incorporate a vast amount of knowledge from a wide range of disciplines. Geography contributes to the understanding of land-use change whilst demography and economics help explain underlying trends. This book offers a cross-sectional overview of current research progress that allows the construction of successful land-use models. The contributions range from methodology and calibration to actual applications in studies of recent policy implementation and evaluation. The contributors originate from academic and applied research institutes around the world and thus offer an interesting mix of theory and practice in different case study contexts. In summary, land-use change simulation modelling is a relatively new and dynamic field of study and this book provides a full overview of the topic, a wide range of applications (both geographically and thematically), a mix of theory and practice, a synthesis of recent research progress, and educational material for students and teachers.

Models of land-use change incorporate a vast amount of knowledge from a wide range of disciplines. Geography contributes to the understanding of land-use change whilst demography and economics help explain underlying trends. This book offers a cross-sectional overview of current research progress that allows the construction of successful land-use models. The contributions range from methodology and calibration to actual applications in studies of recent policy implementation and evaluation. The contributors originate from academic and applied research institutes around the world and thus offer an interesting mix of theory and practice in different case study contexts. In summary, land-use change simulation modelling is a relatively new and dynamic field of study and this book provides a full overview of the topic, a wide range of applications (both geographically and thematically), a mix of theory and practice, a synthesis of recent research progress, and educational material for students and teachers.

Planning Support Systems: Retrospect and Prospect It has been nearly twenty years since the term ‘planning support systems’ (PSS) first appeared in an article by Britton Harris (Harris 1989) and more than ten years since the concept was more broadly introduced in the academic literature (Harris and Batty 1993; Batty 1995; Klosterman 1997). As a result, the publication of a new book on PSS provides an excellent opportunity to assess past progress in the field and speculate on future developments. PSS have clearly become very popular in the academic world. This is the fourth edited book devoted to the topic following Brail and Klosterman (2001), Geertman and Stillwell (2003), and a third by Brail (2008). Papers devoted to PSS have been published in the leading planning journals and the topic has become a regular theme at academic conferences around the world; it has even spawned intellectual o- spring such as spatial planning and decision support systems (SPDSS) and public participationplanning support systems (PP-PSS). However, as Geertman and Stillwell point out in their introductory chapter, the experience with PSS in the world of professional practice has been disappointing. A substantial number of PSS have been developed but most of them are academic p- totypes or ‘one off’ professional applications that have not been adopted elsewhere.

Many parts of the world are experiencing rapid demographic restructuring, resulting in an ageing population with increasingly significant work and care pressures on cohorts less able or willing to provide support. This book examines some of the important trends that have underpinned reductions in fertility, including delayed child-bearing and increased childlessness. It demonstrates how relationships between partners have resulted in new living arrangements with changing attitudes from marriage to co-habitation as the social norm, and it considers the health and well-being for particular at risk groups such as the elderly and stepparents as well as aspects of mobility such as household migration and commuting to school.The book brings together a series of studies that all involve quantitative analyses of secondary data from censuses, surveys or administrative records. The trends and patterns reported provide new and interesting insights into behaviour of the household and the roles of adults and children, and point to questions of critical importance for practitioners and policy makers.

Many people think there is only one “right” way to teach geometry. For two millennia, the “right” way was Euclid’s way, and it is still good in many respects. Was this the new “right” way, or was the “right” way something else again, perhaps transformation groups?