Classical Recursion Theory

This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.

Providing a pedagogical introduction to the essential principles of path integrals and Hamiltonians, this book describes cutting-edge quantum mathematical techniques applicable to a vast range of fields, from quantum mechanics, solid state physics, statistical mechanics, quantum field theory, and superstring theory to financial modeling, polymers, biology, chemistry, and quantum finance. Eschewing use of the Schrödinger equation, the powerful and flexible combination of Hamiltonian operators and path integrals is used to study a range of different quantum and classical random systems, succinctly demonstrating the interplay between a system's path integral, state space, and Hamiltonian. With a practical emphasis on the methodological and mathematical aspects of each derivation, this is a perfect introduction to these versatile mathematical methods, suitable for researchers and graduate students in physics and engineering.

This monograph is concerned with the qualitative theory of best L1-approximation from finite-dimensional subspaces. It presents a survey of recent research that extends 'classical' results concerned with best uniform approximation to the L1 case. The work is organized in such a way as to be useful for self-study or as a text for advanced courses. It begins with a basic introduction to the concepts of approximation theory before addressing one- or two-sided best approximation from finite-dimensional subspaces and approaches to the computation of these. At the end of each chapter is a series of exercises; these give the reader an opportunity to test understanding and also contain some theoretical digressions and extensions of the text.

This book describes advances in the application of chaos theory to classical scattering and nonequilibrium statistical mechanics generally, and to transport by deterministic diffusion in particular. The author presents the basic tools of dynamical systems theory, such as dynamical instability, topological analysis, periodic-orbit methods, Liouvillian dynamics, dynamical randomness and large-deviation formalism. These tools are applied to chaotic scattering and to transport in systems near equilibrium and maintained out of equilibrium. Chaotic Scattering is illustrated with disk scatterers and with examples of unimolecular chemical reactions and then generalized to transport in spatially extended systems. This book will be bought by researchers interested in chaos, dynamical systems, chaotic scattering, and statistical mechanics in theoretical, computational and mathematical physics and also in theoretical chemistry.

Paul Adrien Maurice Dirac was undoubtedly one of the most brilliant and influential physicists of the twentieth century. Between 1925 and 1934, this Nobel Laureate revolutionized physics with his contributions to quantum theory. This book, the first full length biography of Dirac, offers a comprehensive account of his life and presents his physics in its historical context, including known areas such as cosmology and classical electron theory. The author examines Dirac's successes and failures, and pays particular attention to Dirac's opposition to modern quantum electrodynamics - an opposition based on aesthetic objections. This book, which draws extensively from unpublished sources, including Dirac's correspondence with Bohr, Heisenberg, Pauli, Schrödinger, Gamow, and other physicists, is a history of modern physics as seen through one scientist's career.

Intermediate in level between an advanced textbook and a monograph, this book covers both the classical and representation theoretic views of automorphic forms in a style which is accessible to graduate students entering the field. The treatment is based on complete proofs, which reveal the uniqueness principles underlying the basic constructions. The book features extensive foundational material on the representation theory of GL(1) and GL(2) over local fields, the theory of automorphic representations, L-functions and advanced topics such as the Langlands conjectures, the Weil representation, the Rankin–Selberg method and the triple L-function, examining this subject matter from many different and complementary viewpoints. Researchers as well as students will find this a valuable guide to a notoriously difficult subject.

This 1994 book introduces the tools of modern differential geometry, exterior calculus, manifolds, vector bundles and connections, to advanced undergraduate and beginning graduate students in mathematics, physics and engineering. The book covers both classical surface theory and the modern theory of connections and curvature, and includes a chapter on applications to theoretical physics. The only prerequisites are multivariate calculus and linear algebra; no knowledge of topology is assumed. The powerful and concise calculus of differential forms is used throughout. Through the use of numerous concrete examples, the author develops computational skills in the familiar Euclidean context before exposing the reader to the more abstract setting of manifolds. There are nearly 200 exercises, making the book ideal for both classroom use and self-study.

Paul Adrien Maurice Dirac was undoubtedly one of the most brilliant and influential physicists of the twentieth century. Between 1925 and 1934, this Nobel Laureate revolutionized physics with his contributions to quantum theory. This book, the first full length biography of Dirac, offers a comprehensive account of his life and presents his physics in its historical context, including known areas such as cosmology and classical electron theory. The author examines Dirac's successes and failures, and pays particular attention to Dirac's opposition to modern quantum electrodynamics - an opposition based on aesthetic objections. This book, which draws extensively from unpublished sources, including Dirac's correspondence with Bohr, Heisenberg, Pauli, Schrödinger, Gamow, and other physicists, is a history of modern physics as seen through one scientist's career.

This monograph is concerned with the qualitative theory of best L1-approximation from finite-dimensional subspaces. It presents a survey of recent research that extends 'classical' results concerned with best uniform approximation to the L1 case. The work is organized in such a way as to be useful for self-study or as a text for advanced courses. It begins with a basic introduction to the concepts of approximation theory before addressing one- or two-sided best approximation from finite-dimensional subspaces and approaches to the computation of these. At the end of each chapter is a series of exercises; these give the reader an opportunity to test understanding and also contain some theoretical digressions and extensions of the text.

High-dimensional probability offers insight into the behavior of random vectors, random matrices, random subspaces, and objects used to quantify uncertainty in high dimensions. Drawing on ideas from probability, analysis, and geometry, it lends itself to applications in mathematics, statistics, theoretical computer science, signal processing, optimization, and more. It is the first to integrate theory, key tools, and modern applications of high-dimensional probability. Concentration inequalities form the core, and it covers both classical results such as Hoeffding's and Chernoff's inequalities and modern developments such as the matrix Bernstein's inequality. It then introduces the powerful methods based on stochastic processes, including such tools as Slepian's, Sudakov's, and Dudley's inequalities, as well as generic chaining and bounds based on VC dimension. A broad range of illustrations is embedded throughout, including classical and modern results for covariance estimation, clust

This ambitious book provides a new framework for analysing global international society (GIS). In doing so, it also links the English School's approach more closely to classical sociology, constructivism, liberal institutionalism, realism and postcolonialism. It retells the expansion of international society story to explain why the differences among states are as important as their similarities in understanding the structure and dynamics of contemporary GIS. Drawing on differentiation theory, it sets out four ideal-type models for international society. These cover the 'like units' of the classical English School, as well as differentiation by geography, hierarchy/privilege, and function. These models offer a systematic way to integrate international and world society, and to understand the relationship between the deep structure of primary institutions, and the vast array of intergovernmental and international non-governmental organisations. In this pioneering book, Buzan and Schouen

This ambitious book provides a new framework for analysing global international society (GIS). In doing so, it also links the English School's approach more closely to classical sociology, constructivism, liberal institutionalism, realism and postcolonialism. It retells the expansion of international society story to explain why the differences among states are as important as their similarities in understanding the structure and dynamics of contemporary GIS. Drawing on differentiation theory, it sets out four ideal-type models for international society. These cover the 'like units' of the classical English School, as well as differentiation by geography, hierarchy/privilege, and function. These models offer a systematic way to integrate international and world society, and to understand the relationship between the deep structure of primary institutions, and the vast array of intergovernmental and international non-governmental organisations. In this pioneering book, Buzan and Schouen

This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kać interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and conce

This book introduces the reader to the geometry of surfaces and submanifolds in the conformal n-sphere. Various models for Möbius geometry are presented: the classical projective model, the quaternionic approach, and an approach that uses the Clifford algebra of the space of homogeneous coordinates of the classical model; the use of 2-by-2 matrices in this context is elaborated. For each model in turn applications are discussed. Topics comprise conformally flat hypersurfaces, isothermic surfaces and their transformation theory, Willmore surfaces, orthogonal systems and the Ribaucour transformation, as well as analogous discrete theories for isothermic surfaces and orthogonal systems. Certain relations with curved flats, a particular type of integrable system, are revealed. Thus this book will serve both as an introduction to newcomers (with background in Riemannian geometry and elementary differential geometry) and as a reference work for researchers.

Since the time of Lagrange and Euler, it has been well known that an understanding of algebraic curves can illuminate the picture of rigid bodies provided by classical mechanics. A modern view of the role played by algebraic geometry has been established iby many mathematicians. This book presents some of these techniques, which fall within the orbit of finite dimensional integrable systems. The main body of the text presents a rich assortment of methods and ideas from algebraic geometry prompted by classical mechanics, whilst in appendices the general, abstract theory is described. The methods are given a topological application to the study of Liouville tori and their bifurcations. The book is based on courses for graduate students given by the author at Strasbourg University but the wealth of original ideas will make it also appeal to researchers.

Since the time of Lagrange and Euler, it has been well known that an understanding of algebraic curves can illuminate the picture of rigid bodies provided by classical mechanics. A modern view of the role played by algebraic geometry has been established iby many mathematicians. This book presents some of these techniques, which fall within the orbit of finite dimensional integrable systems. The main body of the text presents a rich assortment of methods and ideas from algebraic geometry prompted by classical mechanics, whilst in appendices the general, abstract theory is described. The methods are given a topological application to the study of Liouville tori and their bifurcations. The book is based on courses for graduate students given by the author at Strasbourg University but the wealth of original ideas will make it also appeal to researchers.

Supergravity, together with string theory, is one of the most significant developments in theoretical physics. Written by two of the most respected workers in the field, this is the first-ever authoritative and systematic account of supergravity. The book starts by reviewing aspects of relativistic field theory in Minkowski spacetime. After introducing the relevant ingredients of differential geometry and gravity, some basic supergravity theories (D=4 and D=11) and the main gauge theory tools are explained. In the second half of the book, complex geometry and N=1 and N=2 supergravity theories are covered. Classical solutions and a chapter on AdS/CFT complete the book. Numerous exercises and examples make it ideal for Ph.D. students, and with applications to model building, cosmology and solutions of supergravity theories, it is also invaluable to researchers. A website hosted by the authors, featuring solutions to some exercises and additional reading material, can be found at www.camb

'Mass terms' like water, rice and traffic, have proved very difficult to accommodate in any theory of meaning since, unlike count nouns such as house or dog, they cannot be treated as denoting sets of individuals. In this study, motivated by the need to design a computer program for understanding natural language utterances containing mass terms, Harry Bunt provides a thorough analysis of the problem and offers an original and detailed solution. An extension of classical set theory, Ensemble Theory, is defined. This provides the formal basis of a framework for the analysis of natural language meaning which Dr Bunt calls two-level model-theoretic semantics. The validity of the framework is convincingly demonstrated by the detailed analysis of a fragment of English including sentences with quantified and modified mass terms. This significant advance in our understanding of the formal syntactic and semantic properties of mass terms will be of interest not only to linguists and logicians,

This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on G\G and its relationship with the classical automorphic forms on X, Poincare series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unitary representations of G, and the spectral decomposition of L2 (G\G). The main prerequisites are some results in functional analysis (reviewed, with references) and some familiarity with the elementary theory of Lie groups and Lie algebras. Graduate students and researcher

This book provides an introduction to some aspects of the analytic theory of automorphic forms on G=SL2(R) or the upper-half plane X, with respect to a discrete subgroup G of G of finite covolume. The point of view is inspired by the theory of infinite dimensional unitary representations of G; this is introduced in the last sections, making this connection explicit. The topics treated include the construction of fundamental domains, the notion of automorphic form on G\G and its relationship with the classical automorphic forms on X, Poincare series, constant terms, cusp forms, finite dimensionality of the space of automorphic forms of a given type, compactness of certain convolution operators, Eisenstein series, unitary representations of G, and the spectral decomposition of L2 (G\G). The main prerequisites are some results in functional analysis (reviewed, with references) and some familiarity with the elementary theory of Lie groups and Lie algebras. Graduate students and researcher

Previous studies on the relationship between rhetorical theory and Roman poetry have generally taken the form of lists enumerating elements of style and arrangement that poets are said to have 'borrowed' from rhetorical critics. This book examines, and ultimately questions, this entrenched theoretical model and the very notion of rhetorical influence on which this paradigm is built. Tracing key moments in the poetic and the rhetorical traditions, in the context of which the problematic relationship of difference and similarity between rhetorical and poetic discourse is discussed, the book focuses on the cultural relevance of this intellectual divide in Roman literary culture. The study of rhetorical sources, such as Cicero, Seneca the Elder and Quintilian, and of select responses in Roman poetry, sheds light on long-standing scholarly assumptions about classical poetry as artless language and about the role of rhetoric in the construction of the decline of post-classical cultures.

This book presents the most comprehensive and systematic description currently available of both classical and novel theories of cloud processes, providing a much-needed link between cloud theory, observation, experimental results, and cloud modeling. This volume shows why and how modern models serve as a major tool of investigation of cloud processes responsible for atmospheric phenomena, including climate change. It systematically describes classical as well as recent advancements in cloud physics, including cloud-aerosol interaction; collisions of particles in turbulent clouds; and the formation of multiphase cloud particles. As the first of its kind to serve as a practical guide for using state-of-the-art numerical cloud models, major emphasis is placed on explaining how microphysical processes are treated in modern numerical cloud resolving models. The book will be a valuable resource for advanced students, researchers and numerical model designers in cloud physics, atmospheric sc

Focusing on the role that automorphisms and equivalence relations play in the algebraic theory of minimal sets provides an original treatment of some key aspects of abstract topological dynamics. Such an approach is presented in this lucid and self-contained book, leading to simpler proofs of classical results, as well as providing motivation for further study. Minimal flows on compact Hausdorff spaces are studied as icers on the universal minimal flow M. The group of the icer representing a minimal flow is defined as a subgroup of the automorphism group G of M, and icers are constructed explicitly as relative products using subgroups of G. Many classical results are then obtained by examining the structure of the icers on M, including a proof of the Furstenberg structure theorem for distal extensions. This book is designed as both a guide for graduate students, and a source of interesting new ideas for researchers.

'Big data' poses challenges that require both classical multivariate methods and contemporary techniques from machine learning and engineering. This modern text equips you for the new world - integrating the old and the new, fusing theory and practice and bridging the gap to statistical learning. The theoretical framework includes formal statements that set out clearly the guaranteed 'safe operating zone' for the methods and allow you to assess whether data is in the zone, or near enough. Extensive examples showcase the strengths and limitations of different methods with small classical data, data from medicine, biology, marketing and finance, high-dimensional data from bioinformatics, functional data from proteomics, and simulated data. High-dimension low-sample-size data gets special attention. Several data sets are revisited repeatedly to allow comparison of methods. Generous use of colour, algorithms, Matlab code, and problem sets complete the package. Suitable for master's/graduat

This accessible text on classical celestial mechanics, the principles governing the motions of bodies in the Solar System, provides a clear and concise treatment of virtually all of the major features of solar system dynamics. Building on advanced topics in classical mechanics such as rigid body rotation, Langrangian mechanics and orbital perturbation theory, this text has been written for advanced undergraduates and beginning graduate students in astronomy, physics, mathematics and related fields. Specific topics covered include Keplerian orbits, the perihelion precession of the planets, tidal interactions between the Earth, Moon and Sun, the Roche radius, the stability of Lagrange points in the three-body problem and lunar motion. More than 100 exercises allow students to gauge their understanding and a solutions manual is available to instructors. Suitable for a first course in celestial mechanics, this text is the ideal bridge to higher level treatments.

Hyperbolic geometry is a classical subject in pure mathematics which has exciting applications in theoretical physics. In this book leading experts introduce hyperbolic geometry and Maass waveforms and discuss applications in quantum chaos and cosmology. The book begins with an introductory chapter detailing the geometry of hyperbolic surfaces and includes numerous worked examples and exercises to give the reader a solid foundation for the rest of the book. In later chapters the classical version of Selberg's trace formula is derived in detail and transfer operators are developed as tools in the spectral theory of Laplace–Beltrami operators on modular surfaces. The computation of Maass waveforms and associated eigenvalues of the hyperbolic Laplacian on hyperbolic manifolds are also presented in a comprehensive way. This book will be valuable to graduate students and young researchers, as well as for those experienced scientists who want a detailed exposition of the subject.

Exploring important theories for understanding freezing and the liquid-glass transition, this book is useful for graduate students and researchers in soft-condensed matter physics, chemical physics and materials science. It details recent ideas and key developments, providing an up-to-date view of current understanding. The standard tools of statistical physics for the dense liquid state are covered. The freezing transition is described from the classical density functional approach. Classical nucleation theory as well as applications of density functional methods for nucleation of crystals from the melt are discussed, and compared to results from computer simulation of simple systems. Discussions of supercooled liquids form a major part of the book. Theories of slow dynamics and the dynamical heterogeneities of the glassy state are presented, as well as nonequilibrium dynamics and thermodynamic phase transitions at deep supercooling. Mathematical treatments are given in full detail so

William Emerton Heitland (1847–1935) was a Cambridge classicist, who was described as having 'a passionate desire to attain the truth'. His most distinguished work, Agricola, published in 1921, is a detailed study of agricultural labour in classical times. He makes use of a wide range of sources, from Homer in the eighth century BCE to Apollinaris Sidonius in the fifth century CE. In asking the question, by whom and under what conditions was the work done, he deals with land tenure, taxation, military service and political theory. He argues that changes in agricultural production were necessarily connected to changes in other areas of society. To a large extent, classical agriculture was based on slavery, and even those who were free tenants had limited legal rights. Roman poets such as Virgil idealised the pastoral life, but may not reflect reality. It is an important sourcebook for social and economic history.

J. P. Droop (1882–1963) was a classical field archaeologist. After graduating from Trinity College, Cambridge, in 1904 he worked as a field archaeologist for the British School at Athens, and was appointed Chair of Classical Archaeology at Liverpool University in 1914. This volume was intended as a guide to practical archaeological excavation and was first published in 1915 as part of the Cambridge Archaeological and Ethnological series. At the time of publication, archaeology was starting to become a more scientific and academic discipline, as can be seen in Droop's arguments on the importance of archaeological context and knowledge of stratification on site. The development of excavation as a scientifically based practice is shown by the emphasis on planning of the site, in contrast to earlier guides to excavation. This volume provides insights into the development of the theory as well as the practice of archaeology.

In 1920–1924 Jacob Wackernagel (1853–1938), Professor of Comparative Philology at the University of Basel, published two volumes of lectures on Greek, Latin and German syntax based on his 1918–1919 courses for beginning undergraduates (who would already have studied all three languages at school). Wackernagel introduces the principles of grammatical functions and cross-linguistic work, and provides a foundation for the understanding of comparative syntax. The book remained in print for many decades thanks to its lively style and well-chosen examples from classical texts. The Sanskritist Louis Renou (1896–1966), writing on Wackernagel in 1938, asserted that 'no linguist could possibly afford to ignore this book', and it remains of interest to historical linguists and scholars of classical languages today. In this, the first volume, Wackernagel outlines the history of syntactic theory from ancient times to the present, and provides a detailed treatment of the verb.

In considering the practice and theory of translating Classical Greek plays into English from a theatrical perspective, Found in Translation, first published in 2006, also addresses the wider issues of transferring any piece of theatre from a source into a target language. The history of translating classical tragedy and comedy, here fully investigated, demonstrates how through the ages translators have, wittingly or unwittingly, appropriated Greek plays and made them reflect socio-political concerns of their own era. Chapters are devoted to topics including verse and prose, mask and non-verbal language, stage directions and subtext and translating the comic. Among the plays discussed as 'case studies' are Aeschylus' Agamemnon, Sophocles' Oedipus Tyrannus and Euripides' Medea and Alcestis. The book concludes with a consideration of the boundaries between 'translation' and 'adaptation', followed by an appendix of every translation of Greek tragedy and comedy into English from the 1550s

Topological solitons occur in many nonlinear classical field theories. They are stable, particle-like objects, with finite mass and a smooth structure. Examples are monopoles and Skyrmions, Ginzburg-Landau vortices and sigma-model lumps, and Yang-Mills instantons. This book is a comprehensive survey of static topological solitons and their dynamical interactions. Particular emphasis is placed on the solitons which satisfy first-order Bogomolny equations. For these, the soliton dynamics can be investigated by finding the geodesics on the moduli space of static multi-soliton solutions. Remarkable scattering processes can be understood this way. The book starts with an introduction to classical field theory, and a survey of several mathematical techniques useful for understanding many types of topological soliton. Subsequent chapters explore key examples of solitons in one, two, three and four dimensions. The final chapter discusses the unstable sphaleron solutions which exist in several

In considering the practice and theory of translating Classical Greek plays into English from a theatrical perspective, Found in Translation, first published in 2006, also addresses the wider issues of transferring any piece of theatre from a source into a target language. The history of translating classical tragedy and comedy, here fully investigated, demonstrates how through the ages translators have, wittingly or unwittingly, appropriated Greek plays and made them reflect socio-political concerns of their own era. Chapters are devoted to topics including verse and prose, mask and non-verbal language, stage directions and subtext and translating the comic. Among the plays discussed as 'case studies' are Aeschylus' Agamemnon, Sophocles' Oedipus Tyrannus and Euripides' Medea and Alcestis. The book concludes with a consideration of the boundaries between 'translation' and 'adaptation', followed by an appendix of every translation of Greek tragedy and comedy into English from the 1550s

Topological solitons occur in many nonlinear classical field theories. They are stable, particle-like objects, with finite mass and a smooth structure. Examples are monopoles and Skyrmions, Ginzburg-Landau vortices and sigma-model lumps, and Yang-Mills instantons. This book is a comprehensive survey of static topological solitons and their dynamical interactions. Particular emphasis is placed on the solitons which satisfy first-order Bogomolny equations. For these, the soliton dynamics can be investigated by finding the geodesics on the moduli space of static multi-soliton solutions. Remarkable scattering processes can be understood this way. The book starts with an introduction to classical field theory, and a survey of several mathematical techniques useful for understanding many types of topological soliton. Subsequent chapters explore key examples of solitons in one, two, three and four dimensions. The final chapter discusses the unstable sphaleron solutions which exist in several

An introductory 2002 textbook, Process Control covers the most essential aspects of process control suitable for a two-semester course. While classical techniques are discussed, also included is a discussion of state space modeling and control, a modern control topic lacking in most introductory texts. MATLAB, a popular engineering software package, is employed as a powerful yet approachable computational tool. Text examples demonstrate how root locus, Bode plots, and time domain simulations can be integrated to tackle a control problem. Classical control and state space designs are compared. Despite the reliance on MATLAB, theory and analysis of process control are well-presented, creating a well-rounded pedagogical text. Each chapter concludes with problem sets, to which hints or solutions are provided. A web site provides excellent support in the way of MATLAB outputs of text examples and MATLAB sessions, references, and supplementary notes. Students and professionals will find it a

This thorough work presents the fundamental results of modular function theory as developed during the nineteenth and early-twentieth centuries. It features beautiful formulas and derives them using skillful and ingenious manipulations, especially classical methods often overlooked today. Starting with the work of Gauss, Abel, and Jacobi, the book then discusses the attempt by Dedekind to construct a theory of modular functions independent of elliptic functions. The latter part of the book explains how Hurwitz completed this task and includes one of Hurwitz's landmark papers, translated by the author, and delves into the work of Ramanujan, Mordell, and Hecke. For graduate students and experts in modular forms, this book demonstrates the relevance of these original sources and thereby provides the reader with new insights into contemporary work in this area.

There are many interactions between noncommutative algebra and representation theory on the one hand and classical algebraic geometry on the other, with important applications in both directions. The aim of this book is to provide a comprehensive introduction to some of the most significant topics in this area, including noncommutative projective algebraic geometry, deformation theory, symplectic reflection algebras, and noncommutative resolutions of singularities. The book is based on lecture courses in noncommutative algebraic geometry given by the authors at a Summer Graduate School at the Mathematical Sciences Research Institute, California in 2012 and, as such, is suitable for advanced graduate students and those undertaking early post-doctorate research. In keeping with the lectures on which the book is based, a large number of exercises are provided, for which partial solutions are included.

There are many interactions between noncommutative algebra and representation theory on the one hand and classical algebraic geometry on the other, with important applications in both directions. The aim of this book is to provide a comprehensive introduction to some of the most significant topics in this area, including noncommutative projective algebraic geometry, deformation theory, symplectic reflection algebras, and noncommutative resolutions of singularities. The book is based on lecture courses in noncommutative algebraic geometry given by the authors at a Summer Graduate School at the Mathematical Sciences Research Institute, California in 2012 and, as such, is suitable for advanced graduate students and those undertaking early post-doctorate research. In keeping with the lectures on which the book is based, a large number of exercises are provided, for which partial solutions are included.

Continuum mechanics and thermodynamics are foundational theories of many fields of science and engineering. This book presents a fresh perspective on these fundamental topics, connecting micro- and nanoscopic theories and emphasizing topics relevant to understanding solid-state thermo-mechanical behavior. Providing clear, in-depth coverage, the book gives a self-contained treatment of topics directly related to nonlinear materials modeling. It starts with vectors and tensors, finite deformation kinematics, the fundamental balance and conservation laws, and classical thermodynamics. It then discusses the principles of constitutive theory and examples of constitutive models, presents a foundational treatment of energy principles and stability theory, and concludes with example closed-form solutions and the essentials of finite elements. Together with its companion book, Modeling Materials, (Cambridge University Press, 2011), this work presents the fundamentals of multiscale materials mod

This book provides a comprehensive presentation of geometric results, primarily from the theory of convex sets, that have been proved by the use of Fourier series or spherical harmonics. An important feature of the book is that all necessary tools from the classical theory of spherical harmonics are presented with full proofs. These tools are used to prove geometric inequalities, stability results, uniqueness results for projections and intersections by hyperplanes or half-spaces and characterisations of rotors in convex polytopes. Again, full proofs are given. To make the treatment as self-contained as possible the book begins with background material in analysis and the geometry of convex sets. This treatise will be welcomed both as an introduction to the subject and as a reference book for pure and applied mathematics.