Mathematics of Discrete Structures for Computer Science

How did ancient scientific and knowledge-ordering writers make their work authoritative? This book answers that question for a wide range of ancient disciplines, from mathematics, medicine, architecture and agriculture, through to law, historiography and philosophy - focusing mainly, but not exclusively, on the literature of the Roman Empire. It draws attention to habits that these different fields had in common, while also showing how individual texts and authors manipulated standard techniques of self-authorisation in distinctive ways. It stresses the importance of competitive and assertive styles of self-presentation, and also examines some of the pressures that pulled in the opposite direction by looking at authors who chose to acknowledge the limitations of their own knowledge or resisted close identification with narrow versions of expert identity. A final chapter by Sir Geoffrey Lloyd offers a comparative account of scientific authority and expertise in ancient Chinese, Indian a

Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a concise review of classical extreme value statistics and a basic introduction to mean-field spin glasses, the author then focuses on branching Brownian motion. Here, the classical results of Bramson on the asymptotics of solutions of the F-KPP equation are reviewed in detail and applied to the recent construction of the extremal process of BBM. The extension of these results to branching Brownian motion with variable speed are then explained. As a self-contained exposition that is accessible to graduate students with some background in probability theory, this book makes a good introduction for anyone interested in accessing this exciting field of mathematics.

Cooperative game theory deals with situations where objectives of participants of the game are partially cooperative and partially conflicting. It is in the interest of participants to cooperate in the sense of making binding agreements to achieve the maximum possible benefit. When it comes to distribution of benefit/payoffs, participants have conflicting interests. Such situations are usually modelled as cooperative games. While the book mainly discusses transferable utility games, there is also a brief analysis of non-transferable utility games. Alternative solution concepts to cooperative game theoretic problems are presented in chapters 1-9 and the next four chapters present issues related to computations of solutions discussed in the earlier chapters. The proofs of all results presented in the book are quite explicit. Additionally the mathematical techniques employed in demonstrating the results will be helpful to those who wish to learn application of mathematics for solving prob

This book was first published in 1978. It responds to a rapid growth of scientific and popular interest in questions of climatic change, variability and stability. The many distinguished contributors have combined to present a broad and authoritative interdisciplinary approach to the study of climatic change and its impact on mankind. There is a particular reference to data from the southern hemisphere. The topics discussed include a description of the physical basis of climate; the long- and short-term climatic record; models and mechanisms of climatic change; climate forecasting; and the economic, social and political implications of climatic change. While the book was intended as a university text suitable for introductory courses in the earth sciences, it is written with a minimum of mathematics and jargon. It is therefore intelligible to the interested lay person, and of relevance to those having responsibilities in areas such as economics, agriculture and environmental planning.

The study of permutation patterns is a thriving area of combinatorics that relates to many other areas of mathematics, including graph theory, enumerative combinatorics, model theory, the theory of automata and languages, and bioinformatics. Arising from the Fifth International Conference on Permutation Patterns, held in St Andrews in June 2007, this volume contains a mixture of survey and research articles by leading experts, and includes the two invited speakers, Martin Klazar and Mike Atkinson. Together, the collected articles cover all the significant strands of current research: structural methods and simple patterns, generalisations of patterns, various enumerative aspects, machines and networks, packing, and more. Specialists in this area and other researchers in combinatorics and related fields will find much of interest in this book. In addition, the volume provides plenty of material accessible to advanced undergraduates and is a suitable reference for projects and dissertati

This short book sets out the principles of the methods commonly employed in obtaining numerical solutions to mathematical equations and shows how they are applied in solving particular types of equations. Now that computing facilities are available to most universities, scientific and engineering laboratories and design shops, an introduction to numerical method is an essential part of the training of scientists and engineers. A course on the lines of Professor Wilkes's book is given to graduate or undergraduate students of mathematics, the physical sciences and engineering at many universities and the number will increase. By concentrating on the essentials of his subject and giving it a modern slant, Professor Wilkes has written a book that is both concise and that covers the needs of a great many users of digital computers; it will serve also as a sound introduction for those who need to consult more detailed works.

Economics is probably the most subtle, precise and powerful of the social sciences and its theories have deep philosophical import. Yet the dominant alliance between economics and philosophy has long been cheerfully simple. This is the textbook alliance of neo-Classicism and Positivism, so crucial to the defence of orthodox economics against by now familiar objections. This is an unusual book and a deliberately controversial one. The authors cast doubt on assumptions which neo-Classicists often find too obvious to defend or, indeed, to mention. They set out to disturb an influential concensus and to champion an unpopular cause. Although they go deeper into both philosophy and economics than is usual in interdisciplinary works, they start from first principles and the text is provokingly clear. This will be a stimulating book for all economic theorists and philosophers interested in the philosophy of science and social science.

Revolutionary advances in experimental techniques and spectacular increases in computer power over recent years have enabled researchers to develop a much more profound understanding of the atomic few-body problem. One area of intense focus has been the study of fragmentation processes. Covering the latest research in the field, this edited text is the first to provide a focussed and systematic treatment of fragmentation processes, bringing together contributions from a range of leading experts. As well as tackling the more established electron-impact ionization processes, (e,2e), this book also guides the reader through topics such as molecular fragmentation, ion-atom collisions and multi-photon processes. Combining a broad range of topics with an equal mix of theoretical and experimental discussion, this is an invaluable text for graduate students and researchers in atomic collisions, laser physics and chemistry.

Master the basic concepts and methodologies of digital signal processing with this systematic introduction, without the need for an extensive mathematical background. The authors lead the reader through the fundamental mathematical principles underlying the operation of key signal processing techniques, providing simple arguments and cases rather than detailed general proofs. Coverage of practical implementation, discussion of the limitations of particular methods and plentiful MATLAB illustrations allow readers to better connect theory and practice. A focus on algorithms that are of theoretical importance or useful in real-world applications ensures that students cover material relevant to engineering practice, and equips students and practitioners alike with the basic principles necessary to apply DSP techniques to a variety of applications. Chapters include worked examples, problems and computer experiments, helping students to absorb the material they have just read. Lecture slides

The Book: The Ultimate Guide to Rebuilding Civilization is a hand-illustrated journey into the art of starting over. With whimsy and insight, it explores how to rebuild society from scratch if the modern world were to disappear tomorrow. Blending captivating illustrations with thought-provoking ideas from science, art, engineering, and more, it’s both an imaginative guide to reestablishing the world we take for granted and a heartfelt celebration of human ingenuity.

Originally published during the early part of the twentieth century, the Cambridge Manuals of Science and Literature were designed to provide concise introductions to a broad range of topics. They were written by experts for the general reader and combined a comprehensive approach to knowledge with an emphasis on accessibility. Wireless Telegraphy by C. L. Fortescue was first published in 1913. The volume contains a technical account of the electrical phenomena involved in wireless telegraphy, followed by descriptions of its use in various practical contexts.

Originally published during the early part of the twentieth century, the Cambridge Manuals of Science and Literature were designed to provide concise introductions to a broad range of topics. They were written by experts for the general reader and combined a comprehensive approach to knowledge with an emphasis on accessibility. Australia by J. W. Gregory was first published in 1916. The volume presents a discussion of Australia's social and political character, together with information on botany, geography and the historical development of the area.

Originally published during the early part of the twentieth century, the Cambridge Manuals of Science and Literature were designed to provide concise introductions to a broad range of topics. They were written by experts for the general reader and combined a comprehensive approach to knowledge with an emphasis on accessibility. H. J. Chaytor's volume The Troubadours was published in 1912. It introduces the reader to the concept of courtly love and to the role of the troubadour poets, tracing their influence across medieval Europe.

Originally published during the early part of the twentieth century, the Cambridge Manuals of Science and Literature were designed to provide concise introductions to a broad range of topics. They were written by experts for the general reader and combined a comprehensive approach to knowledge with an emphasis on accessibility. The English Puritans, written by John Brown and first published in 1910, presents an historical overview of the rise, growth and decline of the Puritan movement in England in the sixteenth and seventeenth centuries.

Originally published during the early part of the twentieth century, the Cambridge Manuals of Science and Literature were designed to provide concise introductions to a broad range of topics. They were written by experts for the general reader and combined a comprehensive approach to knowledge with an emphasis on accessibility. Beyond the Atom by John Cox was first published in 1913. It offers a contemporaneous look at the rapid advances occurring in the study of radioactivity during the first decade of the twentieth century.

Linear algebra is something all mathematics undergraduates and many other students, in subjects ranging from engineering to economics, have to learn. The fifth edition of this hugely successful textbook retains the quality of earlier editions while at the same time seeing numerous minor improvements and major additions. The latter include: a new chapter on singular values and singular vectors, including ways to analyze a matrix of data; a revised chapter on computing in linear algebra, with professional-level algorithms and code that can be downloaded for a variety of languages; a new section on linear algebra and cryptography; and a new chapter on linear algebra in probability and statistics. A dedicated and active website also offers solutions to exercises as well as new exercises from many different sources (e.g. practice problems, exams, development of textbook examples), plus codes in MATLAB, Julia, and Python.

A concise yet elementary introduction to measure and integration theory, which are vital in many areas of mathematics, including analysis, probability, mathematical physics and finance. In this highly successful textbook, core ideas of measure and integration are explored, and martingales are used to develop the theory further. Other topics are also covered such as Jacobi's transformation theorem, the Radon–Nikodym theorem, differentiation of measures and Hardy–Littlewood maximal functions. In this second edition, readers will find newly added chapters on Hausdorff measures, Fourier analysis, vague convergence and classical proofs of Radon–Nikodym and Riesz representation theorems. All proofs are carefully worked out to ensure full understanding of the material and its background. Requiring few prerequisites, this book is suitable for undergraduate lecture courses or self-study. Numerous illustrations and over 400 exercises help to consolidate and broaden knowledge. Full solutions to a

This undergraduate textbook is a rigorous mathematical introduction to dynamical systems and an accessible guide for students transitioning from calculus to advanced mathematics. It has many student-friendly features, such as graded exercises that range from straightforward to more difficult with hints, and includes concrete applications of real analysis and metric space theory to dynamical problems. Proofs are complete and carefully explained, and there is opportunity to practice manipulating algebraic expressions in an applied context of dynamical problems. After presenting a foundation in one-dimensional dynamical systems, the text introduces students to advanced subjects in the latter chapters, such as topological and symbolic dynamics. It includes two-dimensional dynamics, Sharkovsky's theorem, and the theory of substitutions, and takes special care in covering Newton's method. Mathematica code is available online, so that students can see implementation of many of the dynamical a

Christiane L. Joost-Gaugier offers the first systematic study of Pythagoras and his influence on mathematics, astronomy, philosophy, religion, medicine, music, the occult, and social life - as well as on architecture and art - in the late medieval and early modern eras. Following the threads of admiration for this ancient Greek sage from the fourteenth century to Kepler and Galileo in the seventeenth, this book demonstrates that Pythagoras's influence in intellectual circles - Christian, Jewish, and Arab - was more widespread than has previously been acknowledged. Joost-Gaugier shows how this admiration was reflected in ideas that were applied to the visual arts by a number of well known architects and artists who sought, through the use of a visual language inspired by the memory of Pythagoras, to obtain perfect harmony in their creations. Among these were Alberti, Bramante, Leonardo da Vinci, Michelangelo, and Raphael.

Like differentiability, convexity is a natural and powerful property of functions that plays a significant role in many areas of mathematics, both pure and applied. It ties together notions from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. This book, which is the product of a collaboration of over 15 years, is unique in that it focuses on convex functions themselves, rather than on convex analysis. The authors explore the various classes and their characteristics and applications, treating convex functions in both Euclidean and Banach spaces. The book can either be read sequentially for a graduate course, or dipped into by researchers and practitioners. Each chapter contains a variety of specific examples, and over 600 exercises are included, ranging in difficulty from early graduate to research level.

Temperature is probably the most influential of all climatic variables. Our only direct, quantitative knowledge of global temperatures comes from instruments operated over the last 150 or so years. Yet as crucial and as central as these data are to our understanding of the climate, they are largely taken for granted, even by many of those using them. Measuring Global Temperatures will fill this gap by explaining how global temperatures are measured, how the data are analysed, what the potential errors are, and what needs to be done to improve temperature measurement in the future. The book is of great interest to all meteorologists, climatologists, and hydrologists, and especially those concerned with climate change and global warming. It is written in accessible language with little mathematics, and so will appeal to students and amateur meteorologists with a strong interest in weather and climate.

Christiane L. Joost-Gaugier offers the first systematic study of Pythagoras and his influence on mathematics, astronomy, philosophy, religion, medicine, music, the occult, and social life - as well as on architecture and art - in the late medieval and early modern eras. Following the threads of admiration for this ancient Greek sage from the fourteenth century to Kepler and Galileo in the seventeenth, this book demonstrates that Pythagoras's influence in intellectual circles - Christian, Jewish, and Arab - was more widespread than has previously been acknowledged. Joost-Gaugier shows how this admiration was reflected in ideas that were applied to the visual arts by a number of well known architects and artists who sought, through the use of a visual language inspired by the memory of Pythagoras, to obtain perfect harmony in their creations. Among these were Alberti, Bramante, Leonardo da Vinci, Michelangelo, and Raphael.

Functional Magnetic Resonance Imaging (fMRI) has become a standard tool for mapping the working brain's activation patterns, both in health and in disease. It is an interdisciplinary field and crosses the borders of neuroscience, psychology, psychiatry, radiology, mathematics, physics and engineering. Developments in techniques, procedures and our understanding of this field are expanding rapidly. In this second edition of Introduction to Functional Magnetic Resonance Imaging, Richard Buxton – a leading authority on fMRI – provides an invaluable guide to how fMRI works, from introducing the basic ideas and principles to the underlying physics and physiology. He covers the relationship between fMRI and other imaging techniques and includes a guide to the statistical analysis of fMRI data. This book will be useful both to the experienced radiographer, and the clinician or researcher with no previous knowledge of the technology.

The most basic algebraic varieties are the projective spaces, and rational varieties are their closest relatives. In many applications where algebraic varieties appear in mathematics and the sciences, we see rational ones emerging as the most interesting examples. The authors have given an elementary treatment of rationality questions using a mix of classical and modern methods. Arising from a summer school course taught by János Kollár, this book develops the modern theory of rational and nearly rational varieties at a level that will particularly suit graduate students. There are numerous examples and exercises, all of which are accompanied by fully worked out solutions, that will make this book ideal as the basis of a graduate course. It will act as a valuable reference for researchers whilst helping graduate students to reach the point where they can begin to tackle contemporary research problems.

It is not uncommon to be frustrated by the outcome of an election or a decision in voting, law, economics, engineering, and other fields. Does this 'bad' result reflect poor data or poorly informed voters? Or does the disturbing conclusion reflect the choice of the decision/election procedure? Nobel Laureate Kenneth Arrow's famed theorem has been interpreted to mean 'no decision procedure is without flaws'. Similarly, Nobel Laureate Amartya Sen dashes hope for individual liberties by showing their incompatibility with societal needs. This highly accessible book offers a new, different interpretation and resolution of Arrow's and Sen's theorems. Using simple mathematics, it shows that these negative conclusions arise because, in each case, some of their assumptions negate other crucial assumptions. Once this is understood, not only do the conclusions become expected, but a wide class of other phenomena can also be anticipated.

Climate change is one of the great challenges of modern politics. In this volume, leading political theorists and historians investigate how the history of political ideas can help us make sense of it. The contributors add a historical perspective to contemporary debates in political theory. They also show that the history of political thought offers new directions for thinking about the environment today. By situating the relationship between humans and nature within a wider history of ideas, the essays provide alternative ways of thinking about the most intractable problems of environmental politics - the status of science in modern democracies, problems of collective action, and the challenges of fatalism. This volume will create new avenues of research for scholars and students in the history of political thought. It is essential reading for undergraduate students interested in environmental challenges: both those in politics seeking a historical perspective, and those in history w

This 1996 volume reviewed advances in the field of human tumour immunology, particularly in relation to the potential for immune intervention in preventing or treating tumours. The editors and contributors, all prominent workers in the field, survey advances in the understanding of the relationship between the cancer cell and the immune response at the time of publication. Chapters review in depth the function of immune surveillance and mechanisms of tumour immunity, the role of T lymphocytes and oncogenes in the immune response to cancer, and the potential for immunotherapy of cancer. Areas of tumour immunology that were new at the time of this book's publication are presented, including progress in the development of tumour vaccines with particular reference to melanoma. The volumes in this series are intended for a wide audience of clinicians and researchers with an interest in the applications of biomedical science to the understanding and management of cancer.

Life is a compelling addition to the Darwin College Lecture Series, in which eight distinguished authors each present an essay from their area of expertise devoted to the theme of 'life'. The book forges connections between art, science and the humanities in a vibrant and thought-provoking collection that exposes both conventional and unconventional views on the meaning of life, the enigmatic boundaries between the living and the dead, and what may or may not follow afterwards. This collection arises from the Darwin College Lecture Series of 2012 and includes contributions from eight distinguished scholars, all of whom are held in esteem not only for their research, but also for their ability to communicate their subject to popular audiences.

The scope of the detection techniques in particle detectors is very wide, depending on the aim of the measurement. Detectors cover the measurement of energies from the very low to the highest of energies observed in cosmic rays. Describing the instrumentation for experiments in high energy physics and astroparticle physics, this edition describes track detectors, calorimeters, particle identification, neutrino detectors, momentum measurement, electronics, and data analysis. It also discusses applications of these detectors in other fields such as nuclear medicine, radiation protection and environmental science. Problem sets have been added to each chapter and additional instructive material has been provided, making this an excellent reference for graduate students and researchers in particle physics.

The scope of the detection techniques in particle detectors is very wide, depending on the aim of the measurement. Detectors cover the measurement of energies from the very low to the highest of energies observed in cosmic rays. Describing the instrumentation for experiments in high energy physics and astroparticle physics, this edition describes track detectors, calorimeters, particle identification, neutrino detectors, momentum measurement, electronics, and data analysis. It also discusses applications of these detectors in other fields such as nuclear medicine, radiation protection and environmental science. Problem sets have been added to each chapter and additional instructive material has been provided, making this an excellent reference for graduate students and researchers in particle physics.

Thomas Hobbes (1588–1679) is recognized as one of the fathers of modern philosophy and political theory. In his own time he was as famous for his work in physics, geometry, and religion. He associated with some of the greatest writers, scientists, and politicians of his age. Martinich has written a complete and accessible biography of Hobbes. The book takes full account of the historical and cultural context in which Hobbes lived, drawing on both published and unpublished sources. It will be a great resource for philosophers, political theorists and historians of ideas. The clear, crisp prose style will also ensure that the book appeals to general readers with an interest in the history of philosophy, the rise of modern science and the English Civil War.

Thomas Hobbes (1588–1679) is recognized as one of the fathers of modern philosophy and political theory. In his own time he was as famous for his work in physics, geometry, and religion. He associated with some of the greatest writers, scientists, and politicians of his age. Martinich has written a complete and accessible biography of Hobbes. The book takes full account of the historical and cultural context in which Hobbes lived, drawing on both published and unpublished sources. It will be a great resource for philosophers, political theorists and historians of ideas. The clear, crisp prose style will also ensure that the book appeals to general readers with an interest in the history of philosophy, the rise of modern science and the English Civil War.

Writing and the Ancient State explores the early development of writing and its relationship to the growth of political structures. The first part of the book focuses on the contribution of writing to the state's legitimating project. The second part deals with the state's use of writing in administration, analyzing both textual and archaeological evidence to reconstruct how the state used bookkeeping to allocate land, police its people, and extract taxes from them. The third part focuses on education, the state's system for replenishing its staff of scribe-officials. The first half of each part surveys evidence from Mesopotamia, Egypt, the Maya lowlands, Central Mexico, and the Andes; against this background the second half examines the evidence from China. The chief aim of this book is to shed new light on early China (from the second millennium BC through the end of the Han period, ca. 220 AD) while bringing to bear the lens of cross-cultural analysis on each of the civilizations un

How genomics reveals deep histories of inequality, going back many thousands of years. Inequality is an urgent global concern, with pundits, politicians, academics, and best-selling books all taking up its causes and consequences. In Inequality, Carles Lalueza-Fox offers an entirely new perspective on the subject, examining the genetic marks left by inequality on humans throughout history. Lalueza-Fox describes genetic studies, made possible by novel DNA sequencing technologies, that reveal layers of inequality in past societies, manifested in patterns of migration, social structures, and funerary practices. Through their DNA, ancient skeletons have much to tell us, yielding anonymous stories of inequality, bias, and suffering. Lalueza-Fox, a leader in paleogenomics, offers the deep history of inequality. He explores the ancestral shifts associated with migration and describes the gender bias unearthed in these migrations―the brutal sexual asymmetries, for example, between male Euro

This book deals with a key feature of globalization: the rise of regulation beyond the state. It examines the emergence of transnational regulatory cooperation between public and private actors and pursues an inquiry that is at once legal, empirical and theoretical. It asks why a private actor and an international organization would regulate cooperatively and what this tells us about the material meaning of concepts such as 'expertise', 'authority' and 'legitimacy' in specific domains of global governance. Additionally, the book addresses the structures and patterns in which cooperation evolves and how this affects the broader global order. It does so through an investigation of two public-private cooperative agreements: one between the International Standards Organization, the Organisation for Economic Co-operation and Development, the Global Compact and the International Labor Organization and one between the International Olympic Committee and the United Nations Environment Programm

This book deals with a key feature of globalization: the rise of regulation beyond the state. It examines the emergence of transnational regulatory cooperation between public and private actors and pursues an inquiry that is at once legal, empirical and theoretical. It asks why a private actor and an international organization would regulate cooperatively and what this tells us about the material meaning of concepts such as 'expertise', 'authority' and 'legitimacy' in specific domains of global governance. Additionally, the book addresses the structures and patterns in which cooperation evolves and how this affects the broader global order. It does so through an investigation of two public-private cooperative agreements: one between the International Standards Organization, the Organisation for Economic Co-operation and Development, the Global Compact and the International Labor Organization and one between the International Olympic Committee and the United Nations Environment Programm

The book is a comprehensive study of India's relationship with the non-proliferation regime, and its transformative evolution from a perennial outlier to one seeking greater integration with the regime and its normative structures. The highlight of this study is its incisive conceptual analysis of the regime as a functional system and its structural complexities, which brings forth new insights on the regime's core ideas like non-proliferation and counter-proliferation. The book also provides an extensive non-Western narrative on the concept of counter-proliferation and its conceivable role and influence in the regime. It breaks new ground in explaining India's quest for an anti-proliferation strategy, which could determine its status and future in the emerging global nuclear order. It will be a substantial contribution to the literature on India's approach towards non-proliferation, counter-proliferation and disarmament, and will enhance the understanding of the impact of the regime's

Who marched in religious processions and why? How were blood sacrifice and communal feasting related to identities in the ancient Greek city? With questions such as these, current scholarship aims to demonstrate the ways in which religion maps on to the socio-political structures of the Greek polis ('polis religion'). In this book Dr Kindt explores a more comprehensive conception of ancient Greek religion beyond this traditional paradigm. Comparative in method and outlook, the book invites its readers to embark on an interdisciplinary journey touching upon such diverse topics as religious belief, personal religion, magic and theology. Specific examples include the transformation of tyrant property into ritual objects, the cultural practice of setting up dedications at Olympia, and a man attempting to make love to Praxiteles' famous statue of Aphrodite. The book will be valuable for all students and scholars seeking to understand the complex phenomenon of ancient Greek religion.

Who marched in religious processions and why? How were blood sacrifice and communal feasting related to identities in the ancient Greek city? With questions such as these, current scholarship aims to demonstrate the ways in which religion maps on to the socio-political structures of the Greek polis ('polis religion'). In this book Dr Kindt explores a more comprehensive conception of ancient Greek religion beyond this traditional paradigm. Comparative in method and outlook, the book invites its readers to embark on an interdisciplinary journey touching upon such diverse topics as religious belief, personal religion, magic and theology. Specific examples include the transformation of tyrant property into ritual objects, the cultural practice of setting up dedications at Olympia, and a man attempting to make love to Praxiteles' famous statue of Aphrodite. The book will be valuable for all students and scholars seeking to understand the complex phenomenon of ancient Greek religion.

While much has been written on asymmetric aspects of sentence structure, symmetric aspects have been largely ignored, or claimed to be non-existent. Does symmetry in syntax exist, and if it does, how do we account for it? In this book, Barbara Citko sets out to tackle these questions and offers a unified approach to a number of phenomena that have so far been studied only in isolation. Focusing on three core minimalist mechanisms: merge, move and labeling, she advances a new theory of these mechanisms, by showing that under certain well-defined circumstances merge can create symmetric structures, move can target either of two potentially moveable objects, and labels can be constructed symmetrically from the features of two objects. This book is aimed at researchers and graduate students interested in minimalist syntax, the structure of questions, relative clauses, coordination, double object constructions and copular sentences.