Classical Recursion Theory

With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms.

With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms.

Quantum information processing offers fundamental improvements over classical information processing, such as computing power, secure communication, and high-precision measurements. However, the best way to create practical devices is not yet known. This textbook describes the techniques that are likely to be used in implementing optical quantum information processors. After developing the fundamental concepts in quantum optics and quantum information theory, the book shows how optical systems can be used to build quantum computers according to the most recent ideas. It discusses implementations based on single photons and linear optics, optically controlled atoms and solid-state systems, atomic ensembles, and optical continuous variables. This book is ideal for graduate students beginning research in optical quantum information processing. It presents the most important techniques of the field using worked examples and over 120 exercises.

This elegant book is sure to become the standard introduction to synthetic differential geometry. It deals with some classical spaces in differential geometry, namely 'prolongation spaces' or neighbourhoods of the diagonal. These spaces enable a natural description of some of the basic constructions in local differential geometry and, in fact, form an inviting gateway to differential geometry, and also to some differential-geometric notions that exist in algebraic geometry. The presentation conveys the real strength of this approach to differential geometry. Concepts are clarified, proofs are streamlined, and the focus on infinitesimal spaces motivates the discussion well. Some of the specific differential-geometric theories dealt with are connection theory (notably affine connections), geometric distributions, differential forms, jet bundles, differentiable groupoids, differential operators, Riemannian metrics, and harmonic maps. Ideal for graduate students and researchers wishing to

Using classical and quantum methods with a strong emphasis on symmetry principles, this book, a reissue of the 2004 second edition, develops the theory of a variety of optical activity and related phenomena from the perspective of molecular scattering of polarised light. In addition to the traditional topic of optical rotation and circular dichroism in the visible and near-ultraviolet associated with electronic transitions, the newer topic of optical activity associated with vibrational transitions, which may be studied using both infrared and Raman techniques, is also treated. Ranging from the physics of elementary particles to the structure of viruses, the subject matter of the book reflects the importance of optical activity and chirality in much of modern science and will be of interest to a wide range of physical and life scientists.

Joseph-Louis Lagrange (1736–1813), one of the notable French mathematicians of the Revolutionary period, is remembered for his work in the fields of analysis, number theory and mechanics. Like Laplace and Legendre, Lagrange was assisted by d'Alembert, and it was on the recommendation of the latter and the urging of Frederick the Great himself that Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin. The two-volume Mécanique analytique was first published in 1788; the edition presented here is that of 1811–15, revised by the author before his death. In this work, claimed to be the most important on classical mechanics since Newton, Lagrange developed the law of virtual work, from which single principle the whole of solid and fluid mechanics can be derived.

Joseph-Louis Lagrange (1736–1813), one of the notable French mathematicians of the Revolutionary period, is remembered for his work in the fields of analysis, number theory and mechanics. Like Laplace and Legendre, Lagrange was assisted by d'Alembert, and it was on the recommendation of the latter and the urging of Frederick the Great himself that Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin. The two-volume Mécanique analytique was first published in 1788; the edition presented here is that of 1811–15, revised by the author before his death. In this work, claimed to be the most important on classical mechanics since Newton, Lagrange developed the law of virtual work, from which single principle the whole of solid and fluid mechanics can be derived.

Joseph-Louis Lagrange (1736–1813), one of the notable French mathematicians of the Revolutionary period, is remembered for his work in the fields of analysis, number theory and mechanics. Like Laplace and Legendre, Lagrange was assisted by d'Alembert, and it was on the recommendation of the latter and the urging of Frederick the Great himself that Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin. The two-volume Mécanique analytique was first published in 1788; the edition presented here is that of 1811–15, revised by the author before his death. In this work, claimed to be the most important on classical mechanics since Newton, Lagrange developed the law of virtual work, from which single principle the whole of solid and fluid mechanics can be derived.

Multiplicity diagrams can be viewed as schemes for describing the phenomenon of 'symmetry breaking' in quantum physics. The subject of this book is the multiplicity diagrams associated with the classical groups U(n), O(n), etc. It presents such topics as asymptotic distributions of multiplicities, hierarchical patterns in multiplicity diagrams, lacanae, and the multiplicity diagrams of the rank 2 and rank 3 groups. The authors take a novel approach, using the techniques of symplectic geometry. The book develops in detail some themes which were touched on in the highly successful Symplectic Techniques in Physics by V. Guillemin and S. Sternberg, including the geometry of the moment map, the Duistermaat-Heckman theorem, the interplay between coadjoint orbits and representation theory, and quantization. Students and researchers in geometry and mathematical physics will find this book fascinating.

Under what conditions is a group of scientists rational? How would rational scientists collectively agree to make their group more effective? What sorts of negotiations would occur among them and under what conditions? What effect would their final agreement have on science and society? These questions have been central to the philosophy of science for the last two decades. In this 2007 book, Husain Sarkar proposes answers to them by building on classical solutions - the skeptical view, two versions of the subjectivist view, the objectivist view, and the view of Hilary Putnam. Although he finds these solutions not completely adequate, Sarkar retrieves what is of value from them and also expropriates the arguments of John Rawls and Amartya Sen, in order to weave a richer, deeper, and more developed theory of group rationality.

Rousseau and Geneva reconstructs the main aspects of Genevan socio-economic, political and religious thought in the first half of the eighteenth century. In this way Dr Rosenblatt effectively contextualizes the development of Rousseau's thought from the First Discourse through to the Social Contract. Over time Rousseau has been adopted as a French thinker, but this adoption obscures his Genevan origin. Dr Rosenblatt points out that he is, in fact, a Genevan thinker and illustrates that Rousseau's classical republicanism, his version of natural law theory, his civil religion and his hostility to the arguments of doux commerce theorists are all responses to the political use of such arguments in Geneva. The author also points out that it was this relationship with Geneva that played an integral part in his development into an original political thinker.

Macroeconomics is evolving in an almost dialectic fashion. The latest evolution is the development of a new synthesis that combines insights of new classical, new Keynesian and real business cycle traditions into a dynamic, stochastic general equilibrium (DSGE) model that serves as a foundation for thinking about macro policy. That new synthesis has opened up the door to a new antithesis, which is being driven by advances in computing power and analytic techniques. This new synthesis is coalescing around developments in complexity theory, automated general to specific econometric modeling, agent-based models, and non-linear and statistical dynamical models. This book thus provides the reader with an introduction to what might be called a Post Walrasian research program that is developing as the antithesis of the Walrasian DSGE synthesis.

Macroeconomics is evolving in an almost dialectic fashion. The latest evolution is the development of a new synthesis that combines insights of new classical, new Keynesian and real business cycle traditions into a dynamic, stochastic general equilibrium (DSGE) model that serves as a foundation for thinking about macro policy. That new synthesis has opened up the door to a new antithesis, which is being driven by advances in computing power and analytic techniques. This new synthesis is coalescing around developments in complexity theory, automated general to specific econometric modeling, agent-based models, and non-linear and statistical dynamical models. This book thus provides the reader with an introduction to what might be called a Post Walrasian research program that is developing as the antithesis of the Walrasian DSGE synthesis.

This self-contained book is a graduate-level introduction for mathematicians and for physicists interested in the mathematical foundations of the field, and can be used as a textbook for a two-semester course on mathematical statistical mechanics. It assumes only basic knowledge of classical physics and, on the mathematics side, a good working knowledge of graduate-level probability theory. The book starts with a concise introduction to statistical mechanics, proceeds to disordered lattice spin systems, and concludes with a presentation of the latest developments in the mathematical understanding of mean-field spin glass models. In particular, progress towards a rigorous understanding of the replica symmetry-breaking solutions of the Sherrington-Kirkpatrick spin glass models, due to Guerra, Aizenman-Sims-Starr and Talagrand, is reviewed in some detail.

This is a comprehensive 1997 account of the history of literary criticism in Britain and Europe between 1660 and 1800. Unlike previous histories, it is not just a chronological survey of critical writing, but a multidisciplinary investigation of how the understanding of literature and its various genres was transformed, at the start of the modern era, by developments in philosophy, psychology, the natural sciences, linguistics, and other disciplines, as well as in society at large. In the process, modern literary theory - at first often implicit in literary texts themselves - emancipated itself from classical poetics and rhetoric, and literary criticism emerged as a full-time professional activity catering for an expanding literate public. The volume is international both in coverage and in authorship. Extensive bibliographies provide guidance for further specialised study.

This Companion volume provides a comprehensive overview of the major themes and topics pertinent to ancient Greek law. A substantial introduction establishes the recent historiography on this topic and its development over the last 30 years. Many of the 22 essays, written by an international team of experts, deal with procedural and substantive law in classical Athens, but significant attention is also paid to legal practice in the archaic and Hellenistic eras; areas that offer substantial evidence for legal practice, such as Crete and Egypt; the intersection of law with religion, philosophy, political theory, rhetoric, and drama, as well as the unity of Greek law and the role of writing in law. The volume is intended to introduce non-specialists to the field as well as to stimulate new thinking among specialists.

This Companion volume provides a comprehensive overview of the major themes and topics pertinent to ancient Greek law. A substantial introduction establishes the recent historiography on this topic and its development over the last 30 years. Many of the 22 essays, written by an international team of experts, deal with procedural and substantive law in classical Athens, but significant attention is also paid to legal practice in the archaic and Hellenistic eras; areas that offer substantial evidence for legal practice, such as Crete and Egypt; the intersection of law with religion, philosophy, political theory, rhetoric, and drama, as well as the unity of Greek law and the role of writing in law. The volume is intended to introduce non-specialists to the field as well as to stimulate new thinking among specialists.

This book describes the manifestations of chaos in atoms and molecules. The study of chaos is today one of the most active and prolific areas in atomic physics. This is the first attempt to provide a coherent introduction to this fascinating area. In line with its scope, the book is divided into two parts. The first part (chapters 1–5) deals with the theory and principles of classical chaos. The ideas developed here are then applied to actual atomic and molecular physics systems in the second part of the book (chapters 6–10) covering microwave driven surface state electrons, the hydrogen atom in a strong microwave field, the kicked hydrogen atom, chaotic scattering with CsI molecules and the helium atom. The book contains many diagrams and a detailed references list.

Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses.

At first glance the prime numbers appear to be distributed in a very irregular way amongst the integers, but it is possible to produce a simple formula that tells us (in an approximate but well defined sense) how many primes we can expect to find that are less than any integer we might choose. The prime number theorem tells us what this formula is and it is indisputably one of the great classical theorems of mathematics. This textbook gives an introduction to the prime number theorem suitable for advanced undergraduates and beginning graduate students. The author's aim is to show the reader how the tools of analysis can be used in number theory to attack a 'real' problem, and it is based on his own experiences of teaching this material.

At first glance the prime numbers appear to be distributed in a very irregular way amongst the integers, but it is possible to produce a simple formula that tells us (in an approximate but well defined sense) how many primes we can expect to find that are less than any integer we might choose. The prime number theorem tells us what this formula is and it is indisputably one of the great classical theorems of mathematics. This textbook gives an introduction to the prime number theorem suitable for advanced undergraduates and beginning graduate students. The author's aim is to show the reader how the tools of analysis can be used in number theory to attack a 'real' problem, and it is based on his own experiences of teaching this material.

Multiplicity diagrams can be viewed as schemes for describing the phenomenon of 'symmetry breaking' in quantum physics. The subject of this book is the multiplicity diagrams associated with the classical groups U(n), O(n), etc. It presents such topics as asymptotic distributions of multiplicities, hierarchical patterns in multiplicity diagrams, lacanae, and the multiplicity diagrams of the rank 2 and rank 3 groups. The authors take a novel approach, using the techniques of symplectic geometry. The book develops in detail some themes which were touched on in the highly successful Symplectic Techniques in Physics by V. Guillemin and S. Sternberg, including the geometry of the moment map, the Duistermaat-Heckman theorem, the interplay between coadjoint orbits and representation theory, and quantization. Students and researchers in geometry and mathematical physics will find this book fascinating.

Covering both theory and experiment, this text describes the behaviour of homogeneous and density-stratified fluids over and around topography. Its presentation is suitable for advanced undergraduate and graduate students in fluid mechanics, as well as for practising scientists, engineers, and researchers. Using laboratory experiments and illustrations to further understanding, the author explores topics ranging from the classical hydraulics of single-layer flow to more complex situations involving stratified flows over two- and three-dimensional topography, including complex terrain. A particular focus is placed on applications to the atmosphere and ocean, including discussions of downslope windstorms, and of oceanic flow over continental shelves and slopes. This new edition has been restructured to make it more digestible, and updated to cover significant developments in areas such as exchange flows, gravity currents, waves in stratified fluids, stability, and applications to the atm

Covering both theory and experiment, this text describes the behaviour of homogeneous and density-stratified fluids over and around topography. Its presentation is suitable for advanced undergraduate and graduate students in fluid mechanics, as well as for practising scientists, engineers, and researchers. Using laboratory experiments and illustrations to further understanding, the author explores topics ranging from the classical hydraulics of single-layer flow to more complex situations involving stratified flows over two- and three-dimensional topography, including complex terrain. A particular focus is placed on applications to the atmosphere and ocean, including discussions of downslope windstorms, and of oceanic flow over continental shelves and slopes. This new edition has been restructured to make it more digestible, and updated to cover significant developments in areas such as exchange flows, gravity currents, waves in stratified fluids, stability, and applications to the atm

Intersection homology is a version of homology theory that extends Poincaré duality and its applications to stratified spaces, such as singular varieties. This is the first comprehensive expository book-length introduction to intersection homology from the viewpoint of singular and piecewise-linear chains. Recent breakthroughs have made this approach viable by providing intersection homology and cohomology versions of all the standard tools in the homology tool box, making the subject readily accessible to graduate students and researchers in topology as well as researchers from other fields. This text includes both new research material and new proofs of previously-known results in intersection homology, as well as treatments of many classical topics in algebraic and manifold topology. Written in a detailed but expository style, this book is suitable as an introduction to intersection homology or as a thorough reference.

This clear and pedagogical text delivers a concise overview of classical and quantum statistical physics. Essential Statistical Physics shows students how to relate the macroscopic properties of physical systems to their microscopic degrees of freedom, preparing them for graduate courses in areas such as biophysics, condensed matter physics, atomic physics and statistical mechanics. Topics covered include the microcanonical, canonical, and grand canonical ensembles, Liouville's Theorem, Kinetic Theory, non-interacting Fermi and Bose systems and phase transitions, and the Ising model. Detailed steps are given in mathematical derivations, allowing students to quickly develop a deep understanding of statistical techniques. End-of-chapter problems reinforce key concepts and introduce more advanced applications, and appendices provide a detailed review of thermodynamics and related mathematical results. This succinct book offers a fresh and intuitive approach to one of the most challenging

Mathematical models of bond markets are of interest to researchers working in applied mathematics, especially in mathematical finance. This book concerns bond market models in which random elements are represented by Lévy processes. These are more flexible than classical models and are well suited to describing prices quoted in a discontinuous fashion. The book's key aims are to characterize bond markets that are free of arbitrage and to analyze their completeness. Nonlinear stochastic partial differential equations (SPDEs) are an important tool in the analysis. The authors begin with a relatively elementary analysis in discrete time, suitable for readers who are not familiar with finance or continuous time stochastic analysis. The book should be of interest to mathematicians, in particular to probabilists, who wish to learn the theory of the bond market and to be exposed to attractive open mathematical problems.

The field of economics has proliferated in complexity and importance since the Second World War. Alessandro Roncaglia recounts the history of the different approaches (marginalist, neoclassical, Keynesian, Austrian, monetarism, rational expectations, institutionalist, evolutionary, classical-Sraffian) and the different fields (micro, macro, money and finance, industrial and game theory, institutions, public finance, econometrics), illustrating the thought and personality of the most important contemporary economists (from Hayek to Sraffa, from Modigliani and Samuelson to Friedman, from Simon to Sen, and many others), focusing on the conceptual foundations of the different streams. At the same time he appraises critically the important debates and controversies in the field and concludes by discussing possible future directions for economic thought. This follow-up to The Wealth of Ideas: A History of Contemporary Economic Thought is a readable introduction to the contemporary economics

The field of economics has proliferated in complexity and importance since the Second World War. Alessandro Roncaglia recounts the history of the different approaches (marginalist, neoclassical, Keynesian, Austrian, monetarism, rational expectations, institutionalist, evolutionary, classical-Sraffian) and the different fields (micro, macro, money and finance, industrial and game theory, institutions, public finance, econometrics), illustrating the thought and personality of the most important contemporary economists (from Hayek to Sraffa, from Modigliani and Samuelson to Friedman, from Simon to Sen, and many others), focusing on the conceptual foundations of the different streams. At the same time he appraises critically the important debates and controversies in the field and concludes by discussing possible future directions for economic thought. This follow-up to The Wealth of Ideas: A History of Contemporary Economic Thought is a readable introduction to the contemporary economics

Individuality and collectivity are central concepts in sociological inquiry. Incorporating cultural history, social theory, urban and economic sociology, Borch proposes an innovative rethinking of these key terms and their interconnections via the concept of the social avalanche. Drawing on classical sociology, he argues that while individuality embodies a tension between the collective and individual autonomy, certain situations, such as crowds and other moments of group behaviour, can subsume the individual entirely within the collective. These events, or social avalanches, produce an experience of being swept away suddenly and losing one's sense of self. Cities are often on the verge of social avalanches, their urban inhabitants torn between de-individualising external pressure and autonomous self-presentation. Similarly, Borch argues that present-day financial markets, dominated by computerised trading, abound with social avalanches and the tensional interplay of mimesis and autono

Individuality and collectivity are central concepts in sociological inquiry. Incorporating cultural history, social theory, urban and economic sociology, Borch proposes an innovative rethinking of these key terms and their interconnections via the concept of the social avalanche. Drawing on classical sociology, he argues that while individuality embodies a tension between the collective and individual autonomy, certain situations, such as crowds and other moments of group behaviour, can subsume the individual entirely within the collective. These events, or social avalanches, produce an experience of being swept away suddenly and losing one's sense of self. Cities are often on the verge of social avalanches, their urban inhabitants torn between de-individualising external pressure and autonomous self-presentation. Similarly, Borch argues that present-day financial markets, dominated by computerised trading, abound with social avalanches and the tensional interplay of mimesis and autono

The poet Virgil remains the most significant and influential figure in Latin literature, and this expanded and updated Companion covers his life, work, and reception from antiquity to the present. The Aeneid, the Eclogues, the Georgics, and the Appendix Vergiliana are all discussed, as are art, history, politics, and philosophy; Virgil's literary style is carefully explored along with poetic traditions before and since, and chapters engage with his poems and their reception from perspectives including intertextuality, narratology, gender theory, philology and historicism. Leading authors cover topics from translations and commentaries to genre, authority, and characterisation, providing revised and updated recommendations for further reading. This volume is an accessible introduction to Virgil and his legacy for students and teachers, while also providing wide-ranging and in-depth investigations that will appeal to scholars of classical literature and other disciplines.

The poet Virgil remains the most significant and influential figure in Latin literature, and this expanded and updated Companion covers his life, work, and reception from antiquity to the present. The Aeneid, the Eclogues, the Georgics, and the Appendix Vergiliana are all discussed, as are art, history, politics, and philosophy; Virgil's literary style is carefully explored along with poetic traditions before and since, and chapters engage with his poems and their reception from perspectives including intertextuality, narratology, gender theory, philology and historicism. Leading authors cover topics from translations and commentaries to genre, authority, and characterisation, providing revised and updated recommendations for further reading. This volume is an accessible introduction to Virgil and his legacy for students and teachers, while also providing wide-ranging and in-depth investigations that will appeal to scholars of classical literature and other disciplines.

Modern scientific computational methods are undergoing a transformative change; big data and statistical learning methods now have the potential to outperform the classical first-principles modeling paradigm. This book bridges this transition, connecting the theory of probability, stochastic processes, functional analysis, numerical analysis, and differential geometry. It describes two classes of computational methods to leverage data for modeling dynamical systems. The first is concerned with data fitting algorithms to estimate parameters in parametric models that are postulated on the basis of physical or dynamical laws. The second is on operator estimation, which uses the data to nonparametrically approximate the operator generated by the transition function of the underlying dynamical systems. This self-contained book is suitable for graduate studies in applied mathematics, statistics, and engineering. Carefully chosen elementary examples with supplementary MATLAB® codes and append

The first systematic analysis of the arguments made against human rights from the French Revolution to the present day. Through the writings of Edmund Burke, Jeremy Bentham, Auguste Comte, Louis de Bonald, Joseph de Maistre, Karl Marx, Carl Schmitt and Hannah Arendt, the authors explore the divergences and convergences between these 'classical' arguments against human rights and the contemporary critiques made both in Anglo-American and French political philosophy. Human Rights on Trial is unique in its marriage of history of ideas with normative theory, and its integration of British/North American and continental debates on human rights. It offers a powerful rebuttal of the dominant belief in a sharp division between human rights today and the rights of man proclaimed at the end of the eighteenth century. It also offers a strong framework for a democratic defence of human rights.

The first systematic analysis of the arguments made against human rights from the French Revolution to the present day. Through the writings of Edmund Burke, Jeremy Bentham, Auguste Comte, Louis de Bonald, Joseph de Maistre, Karl Marx, Carl Schmitt and Hannah Arendt, the authors explore the divergences and convergences between these 'classical' arguments against human rights and the contemporary critiques made both in Anglo-American and French political philosophy. Human Rights on Trial is unique in its marriage of history of ideas with normative theory, and its integration of British/North American and continental debates on human rights. It offers a powerful rebuttal of the dominant belief in a sharp division between human rights today and the rights of man proclaimed at the end of the eighteenth century. It also offers a strong framework for a democratic defence of human rights.

The aim of this collection is to illustrate the pervasive influence of humanist rhetoric on early-modern literature and philosophy. The first half of the book focuses on the classical rules of judicial rhetoric. One chapter considers the place of these rules in Shakespeare's The Merchant of Venice, while two others concentrate on the technique of rhetorical redescription, pointing to its use in Machiavelli's The Prince as well as in several of Shakespeare's plays, notably Coriolanus. The second half of the book examines the humanist background to the philosophy of Thomas Hobbes. A major new essay discusses his typically humanist preoccupation with the visual presentation of his political ideas, while other chapters explore the rhetorical sources of his theory of persons and personation, thereby offering new insights into his views about citizenship, political representation, rights and obligations and the concept of the state.

The aim of this collection is to illustrate the pervasive influence of humanist rhetoric on early-modern literature and philosophy. The first half of the book focuses on the classical rules of judicial rhetoric. One chapter considers the place of these rules in Shakespeare's The Merchant of Venice, while two others concentrate on the technique of rhetorical redescription, pointing to its use in Machiavelli's The Prince as well as in several of Shakespeare's plays, notably Coriolanus. The second half of the book examines the humanist background to the philosophy of Thomas Hobbes. A major new essay discusses his typically humanist preoccupation with the visual presentation of his political ideas, while other chapters explore the rhetorical sources of his theory of persons and personation, thereby offering new insights into his views about citizenship, political representation, rights and obligations and the concept of the state.

The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appe

The Riemann hypothesis (RH) is perhaps the most important outstanding problem in mathematics. This two-volume text presents the main known equivalents to RH using analytic and computational methods. The book is gentle on the reader with definitions repeated, proofs split into logical sections, and graphical descriptions of the relations between different results. It also includes extensive tables, supplementary computational tools, and open problems suitable for research. Accompanying software is free to download. These books will interest mathematicians who wish to update their knowledge, graduate and senior undergraduate students seeking accessible research problems in number theory, and others who want to explore and extend results computationally. Each volume can be read independently. Volume 1 presents classical and modern arithmetic equivalents to RH, with some analytic methods. Volume 2 covers equivalences with a strong analytic orientation, supported by an extensive set of appe