Classical Recursion Theory

This book aims to give a self-contained presentation of a number of results, which relate the volume of convex bodies in n-dimensional Euclidean space and the geometry of the corresponding finite-dimensional normed spaces. The methods employ classical ideas from the theory of convex sets, probability theory, approximation theory and the local theory of Banach spaces. The book is in two parts. The first presents self-contained proofs of the quotient of the subspace theorem, the inverse Santalo inequality and the inverse Brunn-Minkowski inequality. The second part gives a detailed exposition of the recently introduced classes of Banach spaces of weak cotype 2 or weak type 2, and the intersection of the classes (weak Hilbert space). The book is based on courses given in Paris and in Texas.

This book generalises the classical theory of orthogonal polynomials on the complex unit circle, or on the real line to orthogonal rational functions whose poles are among a prescribed set of complex numbers. The first part treats the case where these poles are all outside the unit disk or in the lower half plane. Classical topics such as recurrence relations, numerical quadrature, interpolation properties, Favard theorems, convergence, asymptotics, and moment problems are generalised and treated in detail. The same topics are discussed for the different situation where the poles are located on the unit circle or on the extended real line. In the last chapter, several applications are mentioned including linear prediction, Pisarenko modelling, lossless inverse scattering, and network synthesis. This theory has many applications in theoretical real and complex analysis, approximation theory, numerical analysis, system theory, and in electrical engineering.

This textbook on the calculus of variations leads the reader from the basics to modern aspects of the theory. One-dimensional problems and the classical issues like Euler-Lagrange equations are treated, as are Noether's theorem, Hamilton-Jacobi theory, and in particular geodesic lines, thereby developing some important geometric and topological aspects. The basic ideas of optimal control theory are also given. The second part of the book deals with multiple integrals. After a review of Lebesgue integration, Banach and Hilbert space theory and Sobolev spaces (with complete and detailed proofs), there is a treatment of the direct methods and the fundamental lower semicontinuity theorems. Subsequent chapters introduce the basic concepts of the modern calculus of variations, namely relaxation, Gamma convergence, bifurcation theory and minimax methods based on the Palais–Smale condition. The only prerequisites are basic results from calculus of one and several variables. After having studie

Convex geometry is at once simple and amazingly rich. While the classical results go back many decades, during that previous to this book's publication in 1999, the integral geometry of convex bodies had undergone a dramatic revitalization, brought about by the introduction of methods, results and, most importantly, new viewpoints, from probability theory, harmonic analysis and the geometry of finite-dimensional normed spaces. This book is a collection of research and expository articles on convex geometry and probability, suitable for researchers and graduate students in several branches of mathematics coming under the broad heading of 'Geometric Functional Analysis'. It continues the Israel GAFA Seminar series, which is widely recognized as the most useful research source in the area. The collection reflects the work done at the program in Convex Geometry and Geometric Analysis that took place at MSRI in 1996.

Bringing together a wide range of classical and contemporary texts, Invitation to Christian Spirituality: An Ecumenical Anthology addresses the theory and practice of Christian spirituality in the bro

This book focuses on the representation theory of q-Schur algebras and connections with the representation theory of Hecke algebras and quantum general linear groups. The aim is to present, from a unified point of view, quantum analogues of certain results known already in the classical case. The approach is largely homological, based on Kempf's vanishing theorem for quantum groups and the quasi-hereditary structure of the q-Schur algebras. Beginning with an introductory chapter dealing with the relationship between the ordinary general linear groups and their quantum analogies, the text goes on to discuss the Schur Functor and the 0-Schur algebra. The next chapter considers Steinberg's tensor product and infinitesimal theory. Later sections of the book discuss tilting modules; the Ringel dual of the q-Schur algebra; Specht modules for Hecke algebras; and the global dimension of the q-Schur algebras. An appendix gives a self-contained account of the theory of quasi-hereditary algebras

Acoustics of Fluid-Structure Interactions addresses an increasingly important branch of fluid mechanics - the absorption of noise and vibration by fluid flow. This subject, which offers numerous challenges to conventional areas of acoustics, is of growing concern in places where the environment is adversely affected by sound. Howe presents useful background material on fluid mechanics and the elementary concepts of classical acoustics and structural vibrations. Using examples, many of which include complete worked solutions, he vividly illustrates the theoretical concepts involved. He provides the basis for all calculations necessary for the determination of sound generation by aircraft, ships, general ventilation and combustion systems, as well as musical instruments. Both a graduate textbook and a reference for researchers, Acoustics of Fluid-Structure Interactions is an important synthesis of information in this field. It will also aid engineers in the theory and practice of noise

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 is a genuine introduction to plane algebraic curves from a geometric viewpoint, designed as a first text for undergraduates in mathematics, or for postgraduate and research workers in the engineering and physical sciences. The book contains several hundred worked examples and exercises, making it suitable for adoption as a course text. From the lines and conics of elementary geometry the reader proceeds to general curves in the real affine plane, with excursions to more general fields to illustrate applications, such as number theory. By adding points at infinity the affine plane is extended to the projective plane, yielding a natural setting for curves and providing a flood of illumination into the underlying geometry. A minimal amount of algebra leads to the famous theorem of Bezout, whilst the ideas of linear systems are used to discuss the classical group structure on the cubic.

This is a genuine introduction to plane algebraic curves from a geometric viewpoint, designed as a first text for undergraduates in mathematics, or for postgraduate and research workers in the engineering and physical sciences. The book contains several hundred worked examples and exercises, making it suitable for adoption as a course text. From the lines and conics of elementary geometry the reader proceeds to general curves in the real affine plane, with excursions to more general fields to illustrate applications, such as number theory. By adding points at infinity the affine plane is extended to the projective plane, yielding a natural setting for curves and providing a flood of illumination into the underlying geometry. A minimal amount of algebra leads to the famous theorem of Bezout, whilst the ideas of linear systems are used to discuss the classical group structure on the cubic.

This book is devoted to the study of classical combinatorial structures such as random graphs, permutations, and systems of random linear equations in finite fields. The author shows how the application of the generalized scheme of allocation in the study of random graphs and permutations reduces the combinatorial problems to classical problems of probability theory on the summation of independent random variables. He concentrates on research by Russian mathematicians, including a discussion of equations containing an unknown permutation and a presentation of techniques for solving systems of random linear equations in finite fields. These results will interest specialists in combinatorics and probability theory and will also be useful in applied areas of probabilistic combinatorics such as communication theory, cryptology, and mathematical genetics.

This 1998 book provides a sophisticated alternative to existing accounts of the role of the intellectual in modern democracy. Arguing that society suffers from a systemic deliberation deficit, Jeffrey Goldfarb explores the potential of the intellectual as democratic agent, at once civilizing political contestation and subverting complacent consensus. The sentimental Leftist view of the intellectual as guardian of democracy and the demonising Rightist view of the intellectual as obstructor of progress, are both shown to be flawed. Instead, intellectuals are portrayed as special kinds of 'strangers' who pay careful attention to their critical faculties, equipping them uniquely to address the most pressing issues of today. Professor Goldfarb deploys classical and contemporary social theory to analyse a diverse set of intellectuals in action, from Socrates in fifth-century Athens to Malcolm X and Toni Morrison in twentieth-century America, and, drawing on personal acquaintance, the politic

This 1998 book provides a sophisticated alternative to existing accounts of the role of the intellectual in modern democracy. Arguing that society suffers from a systemic deliberation deficit, Jeffrey Goldfarb explores the potential of the intellectual as democratic agent, at once civilizing political contestation and subverting complacent consensus. The sentimental Leftist view of the intellectual as guardian of democracy and the demonising Rightist view of the intellectual as obstructor of progress, are both shown to be flawed. Instead, intellectuals are portrayed as special kinds of 'strangers' who pay careful attention to their critical faculties, equipping them uniquely to address the most pressing issues of today. Professor Goldfarb deploys classical and contemporary social theory to analyse a diverse set of intellectuals in action, from Socrates in fifth-century Athens to Malcolm X and Toni Morrison in twentieth-century America, and, drawing on personal acquaintance, the politic

The theory of natural dualities, as presented in this text, is broad enough to encompass many known dualities through a rich assortment of substantive theorems, yet concrete enough to be used to generate an array of previously undiscovered dualities. This text will serve as a user manual for algebraists, for category theorists and for those who use algebra in their work, particularly mathematicians and computer scientists interested in non-classical logics. It will also give the specialist a complete account of the foundations, leading to the research frontier of this rapidly developing field. As the first text devoted to the theory of Natural Dualities, it provides an efficient path through a large body of results, examples and applications in this subject which is otherwise available only in scattered research papers. To enable the book to be used in courses, each chapter ends with an extensive exercise set. Several fundamental unsolved problems are included.

Orthonormal Systems and Banach Space Geometry describes the interplay between orthonormal expansions and Banach space geometry. Using harmonic analysis as a starting platform, classical inequalities and special functions are used to study orthonormal systems leading to an understanding of the advantages of systems consisting of characters on compact Abelian groups. Probabilistic concepts such as random variables and martingales are employed and Ramsey's theorem is used to study the theory of super-reflexivity. The text yields a detailed insight into concepts including type and co-type of Banach spaces, B-convexity, super-reflexivity, the vector-valued Fourier transform, the vector-valued Hilbert transform and the unconditionality property for martingale differences (UMD). A long list of unsolved problems is included as a starting point for research. This book should be accessible to graduate students and researchers with some basic knowledge of Banach space theory, real analysis, proba

This book, first published in 1998, treats turbulence from the point of view of dynamical systems. The exposition centres around a number of important simplified models for turbulent behaviour in systems ranging from fluid motion (classical turbulence) to chemical reactions and interfaces in disordered systems.The modern theory of fractals and multifractals now plays a major role in turbulence research, and turbulent states are being studied as important dynamical states of matter occurring also in systems outside the realm of hydrodynamics, i.e. chemical reactions or front propagation. The presentation relies heavily on simplified models of turbulent behaviour, notably shell models, coupled map lattices, amplitude equations and interface models, and the focus is primarily on fundamental concepts such as the differences between large and small systems, the nature of correlations and the origin of fractals and of scaling behaviour. This book will be of interest to graduate students and

Real Business Cycle theory combines the remains of monetarism with the new classical macroeconomics, and has become one of the dominant approaches within contemporary macroeconomics today. This volume

Fundamental measurement problems in engineering, mechanics, manufacturing, and physics are now being solved by powerful optical methods. This book presents a lucid, up-to-date discussion of these optical methods. Beginning from a firm base in modern optics, the book proceeds through relevant theory of interference and diffraction and integrates this theory with descriptions of laboratory techniques and apparatus. Among the techniques discussed are classical interferometry, photoelasticity, geometric moire, spatial filtering, moire interferometry, holography, holographic interferometry, laser speckle interferometry, and video-based speckle methods. By providing a firm base in the physical principles and at the same time allowing the reader to perform meaningful experiments related to the topic being studied, the book offers a unique user-oriented approach that will appeal to students, researchers and practising engineers.

Every form of behaviour is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realised how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centred not on the concept of rational players but on the population dynamics of behavioural programmes. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behaviour, and of the closely related interactions between species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions which can alter the basis of their success, i.e. to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions which punctuate evolution. In short, evolutionary

In some domains of mechanics, physics and control theory boundary value problems arise for nonlinear first order PDEs. A well-known classical result states a sufficiency condition for local existence

This volume presents a comprehensive statement in defense of the doctrine known as classical, hedonistic utilitarianism. It is presented as a viable alternative in the search for a moral theory and th

Using the tools of contemporary semiotic theory to analyze classical rabbinic hermeneutics and medieval mystical exegesis, Betty Rojtman unveils a striking modernity in these early forms of textual in

This book fills the gap between the classical supply-side theory of Malthus and the modern demand-side theory of economic demography. Specifically, it investigates the dynamic macroeconomic implicatio

Classical economic theory assumes that people in risk situations follow a course of action based on a rational, consistent assessment of likely outcomes. But as Zur Shapira demonstrates inRisk Taking,

First published in 1997, Reinventing Allegory asks how and why allegory has survived as a literary mode from the late Renaissance to the postmodern present. Three chapters on Romanticism, including one on the painter J. M. W. Turner, present this era as the pivotal moment in allegory's modern survival. Other chapters describe larger historical and philosophical contexts, including classical rhetoric and Spenser, Milton and seventeenth-century rhetoric, Neoclassical distrust of allegory, and recent theory and metafiction. By using a series of key historical moments to define the special character of modern allegory, this study offers an important framework for assessing allegory's role in contemporary literary culture.

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

Classical string theory is concerned with the propagation of classical 1-dimensional curves 'strings', and the theory has connections to the calculus of variations, minimal surfaces and harmonic maps. The quantization of string theory gives rise to problems in different areas, according to the method used. The representation theory of Lie, Kac-Moody and Virasoro algebras have been used for such quantization. In this lecture note the authors give an introduction to certain global analytic and probabilistic aspects of string theory. It is their intention to bring together, and make explicit the necessary mathematical tools. Researchers with an interest in string theory, in either mathematics or theoretical physics, will find this a stimulating volume.

The James space J and the James tree space JT were constructed as counterexamples to several outstanding conjectures in Banach space theory. This book is a compendium of most of the known results about these spaces, frequently taken from the original sources, but presented in a unified and up to date fashion. Generalisations of J and JT are also discussed and other pathological Banach spaces are introduced. Specialists will welcome this book for its drawing together of classical material and recent results. Graduate students should also find that this book offers an excellent introduction to more advanced topics in Banach space theory.

A two-semester graduate textbook providing a comprehensive introduction to two topics of classical analysis. Presumes a good grounding in advanced calculus and complex-variable theory; previous knowle

This book gives an introduction to theories of computability from a mathematically sophisticated point of view. It treats not only 'the' theory of computability (created by Alan Turing and others in the 1930s), but also a variety of other theories (of Boolean functions, automata and formal languages). These are addressed from the classical perspective of their generation by grammars and from the modern perspective as rational cones. The treatment of the classical theory of computable functions and relations takes the form of a tour through basic recursive function theory, starting with an axiomatic foundation and developing the essential methods in order to survey the most memorable results of the field. This authoritative account by one of the leading lights of the subject will prove exceptionally useful reading for graduate students, and researchers in theoretical computer science and mathematics.

This book is an introductory text on the combinatorial theory of finite geometry. It assumes only a basic knowledge of set theory and analysis, but soon leads the student to results at the frontiers of research. It begins with an elementary combinatorial approach to finite geometries based on finite sets of points and lines, and moves into the classical work on affine and projective planes. The next part deals with polar spaces, partial geometries, and generalised quadrangles. The revised edition contains an entirely new chapter on blocking sets in linear spaces, which highlights some of the most important applications of blocking sets from the initial game-theoretic setting to their recent use in cryptography. Extensive exercises at the end of each chapter ensure the usefulness of this book for senior undergraduate and beginning graduate students.

This book is an introductory text on the combinatorial theory of finite geometry. It assumes only a basic knowledge of set theory and analysis, but soon leads the student to results at the frontiers of research. It begins with an elementary combinatorial approach to finite geometries based on finite sets of points and lines, and moves into the classical work on affine and projective planes. The next part deals with polar spaces, partial geometries, and generalised quadrangles. The revised edition contains an entirely new chapter on blocking sets in linear spaces, which highlights some of the most important applications of blocking sets from the initial game-theoretic setting to their recent use in cryptography. Extensive exercises at the end of each chapter ensure the usefulness of this book for senior undergraduate and beginning graduate students.

For lecture courses that cover the classical theory of nonlinear differential equations associated with Poincare and Lyapunov and introduce the student to the ideas of bifurcation theory and chaos, th

A gradual approach to learning to play the recorder, to reading music and to understanding basic music theory. As ability increases, new material is added. There are 67 familiar popular, classical and

The main purpose of this book is to give a systematic treatment of the theory of stochastic differential equations and stochastic flow of diffeomorphisms, and through the former to study the properties of stochastic flows. The classical theory was initiated by K. Itô and since then has been much developed. Professor Kunita's approach here is to regard the stochastic differential equation as a dynamical system driven by a random vector field, including thereby Itô's theory as a special case. The book can be used with advanced courses on probability theory or for self-study. The author begins with a discussion of Markov processes, martingales and Brownian motion, followed by a review of Itô's stochastic analysis. The next chapter deals with continuous semimartingales with spatial parameters, in order to study stochastic flow, and a generalisation of Ito's equation. Stochastic flows and their relation with this are generalised and considered in chapter 4. It is shown that solutions of a g

This book focuses on the role of education in relation to music and gender. Invoking a concept of musical patriarchy and a theory of the social construction musical meanings, Lucy Green shows how women's musical practices and gendered musical meanings have been reproduced, hand in hand, through history. Covering a wide range of music, including classical, jazz and popular styles, Dr Green uses ethnographic methods to convey the everyday interactions and experiences of girls, boys, and their teachers. She views the contemporary school music classroom as a microcosm of the wider society, and reveals the participation of music education in the continued production and reproduction of gendered musical practices and meanings.

For almost two decades, this has been the classical textbook on applications of operator algebra theory to quantum statistical physics. Major changes in the new edition relate to Bose-Einstein condens

In this book Christian Gourieroux and Alain Monfort provide an up-to-date and comprehensive analysis of modern time series econometrics. They have succeeded in synthesising in an organised and integrated way a broad and diverse literature. While the book does not assume a deep knowledge of economics, one of its most attractive features is the close attention it pays to economic models and phenomena throughout. The coverage represents a major reference tool for graduate students, researchers and applied economists. The book is divided into four sections. Section one gives a detailed treatment of classical seasonal adjustment or smoothing methods. Section two provides a thorough coverage of various mathematical tools. Section three is the heart of the book, and is devoted to a range of important topics including causality, exogeneity shocks, multipliers, cointegration and fractionally integrated models. The final section describes the main contribution of filtering and smoothing theory t

In this book Christian Gourieroux and Alain Monfort provide an up-to-date and comprehensive analysis of modern time series econometrics. They have succeeded in synthesising in an organised and integrated way a broad and diverse literature. While the book does not assume a deep knowledge of economics, one of its most attractive features is the close attention it pays to economic models and phenomena throughout. The coverage represents a major reference tool for graduate students, researchers and applied economists. The book is divided into four sections. Section one gives a detailed treatment of classical seasonal adjustment or smoothing methods. Section two provides a thorough coverage of various mathematical tools. Section three is the heart of the book, and is devoted to a range of important topics including causality, exogeneity shocks, multipliers, cointegration and fractionally integrated models. The final section describes the main contribution of filtering and smoothing theory t

The Hardy-Littlewood method is a means of estimating the number of integer solutions of equations and was first applied to Waring's problem on representations of integers by sums of powers. This introduction to the method deals with its classical forms and outlines some of the more recent developments. Now in its second edition, it has been fully updated; extensive revisions have been made and a new chapter added to take account of major advances by Vaughan and Wooley. The reader is expected to be familiar with elementary number theory and postgraduate students should find it of great use as an advanced textbook. It will also be indispensable to all lecturers and research workers interested in number theory and it is the standard reference on the Hardy-Littlewood method.