Series: Graduate Texts in Mathematics , Vol. 53
2010, XII, 384 p. 12 illus., Hardcover
ISBN: 978-1-4419-0614-4
Due: November 2009
A Course in Mathematical Logic for Mathematicians, Second Edition offers a straightforward introduction to modern mathematical logic that will appeal to the intuition of working mathematicians. The book begins with an elementary introduction to formal languages and proceeds to a discussion of proof theory. It then presents several highlights of 20th century mathematical logic, including theorems of Godel and Tarski, and Cohen's theorem on the independence of the continuum hypothesis. A unique feature of the text is a discussion of quantum logic.
The exposition then moves to a discussion of computability theory that
is based on the notion of recursive functions and stresses number-theoretic
connections. The text present a complete proof of the theorem of Davis-Putnam-Robinson-Matiyasevich
as well as a proof of Higman's theorem on recursive groups. Kolmogorov
complexity is also treated.
Part III establishes the essential equivalence of proof theory and computation theory and gives applications such as Godel's theorem on the length of proofs. A new Chapter IX, written by Yuri Manin, treats, among other things, a categorical approach to the theory of computation, quantum computation, and the P/NP problem. A new Chapter X, written by Boris Zilber, contains basic results of model theory and its applications to mainstream mathematics. This theory has found deep applications in algebraic and diophantine geometry.
Yuri Ivanovich Manin is Professor Emeritus at Max-Planck-Institute for Mathematics in Bonn, Germany, Board of Trustees Professor at the Northwestern University, Evanston, IL, USA, and Principal Researcher at the Steklov Institute of Mathematics, Moscow, Russia. Boris Zilber, Professor of Mathematical Logic at the University of Oxford, has contributed the Model Theory Chapter for the second edition.
Preface to the Second Edition.- Preface to the First Edition.- Introduction to Formal Languages.- Truth and Deducibility. The Continuum Problem and Forcing.- The Continuum Problem and Constructible Sets.- Recursive Functions and Church's Thesis.- Diophantine Sets and Algorithmic Undecidability.- Godel's Incompleteness Theorem.- Recursive Groups.- Constructive Universe and Computation.- Model Theory.- Suggestions for Further Reading.- Index.-
Series: Problem Books in Mathematics
2010, X, 254 p. 17 illus., 8 in color., Hardcover
ISBN: 978-1-4419-1295-4
Due: November 2009
Today, nearly every undergraduate mathematics program requires at least one semester of real analysis. Often, students consider this course to be the most challenging or even intimidating of all their mathematics major requirements. The primary goal of A Problem Book in Real Analysis is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, the authors hope that learning analysis becomes less taxing and more satisfying.
The wide variety of exercises presented in this book range from the computational to the more conceptual and varies in difficulty. They cover the following subjects: set theory; real numbers; sequences; limits of the functions; continuity; differentiability; integration; series; metric spaces; sequences; and series of functions and fundamentals of topology. Furthermore, the authors define the concepts and cite the theorems used at the beginning of each chapter. A Problem Book in Real Analysis is not simply a collection of problems; it will stimulate its readers to independent thinking in discovering analysis.
Prerequisites for the reader are a robust understanding of calculus and linear algebra.
Preface.- Elementary Logic and Set Theory.- Real Numbers.- Sequences.- Limits of Functions.- Continuity.- Differentiability.- Integration.- Series.- Metric Spaces.- Fundamentals of Topology.- Sequences and Series of Functions.- Index.- References.-
2010, Approx. 640 p. Hardcover
ISBN: 978-1-4419-1162-9
Due: November 2009
The book provides a comprehensive introduction to the theory of ordinary differential equations at the graduate level and includes applications to Newtonian and Hamiltonian mechanics. It not only has a large number of examples and computer graphics, but also has a complete collection of proofs for the major theorems, ranging from the usual existence and uniqueness results to the Hartman-Grobman linearization theorem and the Jordan canonical form theorem.
The book can be used almost exclusively in the traditional way for graduate math courses, or it can be used in an applied way for interdisciplinary courses involving physics, engineering, and other science majors. For this reason an extensive computer component using Maple is provided on Springerfs website.
This new edition has been extensively revised throughout, particularly the chapters on linear systems, stability theory and Hamiltonian systems.
The computer component is an in-depth supplement and complement to the material in the text and contains an introduction to discrete dynamical systems and iterated maps, special-purpose Maple code for animating phase portraits, stair diagrams, N-body motions, and rigid-body motions, and numerous tutorial Maple worksheets pertaining to all aspects of using Maple to study the topics in the text.
Introduction.- Techniques, Concepts and Examples.- Existence and Uniqueness:The Flow Map.-Linear Systems.- Linearization & Transformation.- Stability Theory.- Integrable Systems.- Newtonian Mechanics.-Hamiltonian Systems.- Appendix A: Elementary Analysis.- Appendix B: Lipschitz Maps and Linearization.- Appendix C: Linear Algebra.-Appendix D: Electronic Contents.
Series: International Mathematical Series , Vol. 11-13
2010, Approx. 1200 p. 3-volume-set.
ISBN: 978-1-4419-1346-3
Due: February 2010
Research articles and surveys from world-recognized mathematicians cover large areas in Analysis where the contributions of Prof. Maz'ya are fundamental, influential, and/or pioneering. Recent advantages in the study of Sobolev type spaces, PDEs and important boundary value problems in mathematical physics, spectral problems, asymptotic expansions, various actual problems in Analysis and applications are presented. Archive photos and List of references to Maz'ya's works companion the collection.
Around the Research of Vladimir Maz'ya I
Function Spaces
Hardy Inequalities for Nonconvex Domains, F. Avkhadiev, A. Laptev.- Distributions with Slow Tails and Ergodicity of Markov Semigroups in Infinite Dimensions, S. Bobkov, B. Zegarlinski.- On Some Aspects of the Theory of Orlicz-Sobolev Spaces, A. Cianchi.- Mellin Analysis of Weighted Sobolev Spaces with Nonhomogeneous Norms on Cones, M. Costabel et al.- Optimal Hardy-Sobolev-Maz'ya Inequalities with Multiple Interior Singularities, S. Filippas et al.- Sharp Fractional Hardy Inequalities in Half-Spaces, R.L. Frank, R. Seiringer.- Collapsing Riemannian Metrics to Sub-Riemannian and the Geometry of Hypersurfaces in Carnot Groups, N. Garofalo, C. Selby.- Sobolev Homeomorphisms and Composition Operators, V. Gol'dshtein, A. Ukhlov.- Extended Lp Dirichlet Spaces, N. Jacob, R.L. Schilling.- Characterizations for the Hardy Inequality, J. Kinnunen, R. Korte.- Geometric Properties of Planar BV-Extension Domains, P. Koskela et al.- On a New Characterization of Besov Spaces with Negative Exponents, M. Marcus, L. Veron.- Isoperimetric Hardy Type and Poincare Inequalities on Metric Spaces, J. Martin, M. Milman.- Gauge Functions and Sobolev Inequalities on Fluctuating Domains, E. Mbakop, U. Mosco.- A Converse to Maz'ya's Inequality for Capacities under Curvature Lower Bound, E. Milman.- Pseudo-Poincare Inequalities and Applications to Sobolev Inequalities, L. Saloff-Coste.- The p-Faber-Krahn Inequality Noted, J. Xiao.
Around the Research of Vladimir Maz'ya II
Partial Differential Equations
Large Solutions to Semilinear Elliptic Equations with Hardy Potential and Exponential Nonlinearity, C. Bandle et al.- Stability Estimates for Resolvents, Eigenvalues, and Eigenfunctions of Elliptic Operators on Variable Domains, G. Barbatis et al.- Operator Pencil in a Domain with Concentrated Masses. A Scalar Analog of Linear Hydrodynamics, G. Chechkin.- Selfsimilar Perturbation near a Corner: Matching Versus Multiscale Expansions for a Model Problem, M. Dauge et al.- Stationary Navier-Stokes Equation on Lipschitz Domains in Riemannian Manifolds with Nonvanishing Boundary Conditions, M. Dindo?.- On the Regularity of Nonlinear Subelliptic Equations, A. Domokos, J.J. Manfredi.- Rigorous and Heuristic Treatment of Sensitive Singular Perturbations Arising in Elliptic Shells, Y.V. Egorov et al.- On the Existence of Positive Solutions of Semilinear Elliptic Inequalities on Riemannian Manifolds, A. Grigor'yan, V.A. Kondratiev.- Recurrence Relations for Orthogonal Polynomials and Algebraicity of Solutions of the Dirichlet Problem, D. Khavinson, N. Stylianopoulos.- On First Neumann Eigenvalue Bounds for Conformal Metrics, G. Kokarev, N. Nadirashvili.- Necessary Condition for the Regularity of a Boundary Point for Porous Medium Equations with Coefficients of Kato Class, V. Liskevich, I.I. Skrypnik.- The Problem of Steady Flow over a Two-Dimensional Bottom Obstacle, O. Motygin, N. Kuznetsov.- Well Posedness and Asymptotic Expansion of Solution of Stokes Equation Set in a Thin Cylindrical Elastic Tube, G.P. Panasenko, R. Stavre.- On Solvability of Integral Equations for Harmonic Single Layer Potential on the Boundary of a Domain with Cusp, S.V. Poborchi.- Holder Estimates for Green's Matrix of the Stokes System in Convex Polyhedra, J. Rossmann.- Boundary Integral Methods for Periodic Scattering Problems, G. Schmidt.- Boundary Coerciveness and the Neumann Problem for 4th Order Linear Partial Differential Operators, G.C. Verchota.
Around the Research of Vladimir Maz'ya III
Analysis and Applications
Optimal Control of a Biharmonic Obstacle Problem, D. Adams et al.- Minimal Thinness and Beurling's Minimum Principle, H. Aikawa.- Progress in the Problem of Lp-Contractivity of Semigroups for Partial Differential Operators, A. Cialdea.- Uniqueness and Nonuniqueness in Inverse Hyperbolic Problems and the Black Hole Phenomenon, G. Eskin.- Global Green's Function Estimates, M.W. Frazier, I.E. Verbitsky.- On Spectral Minimal Partitions: the Case of the Sphere, B. Helffer et al.- Weighted Sobolev Space Estimates for a Class of Singular Integral Operators, D. Mitrea et al.- On general Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities, S. Molchanov, B. Vainberg.- Estimates for the Counting Function of the Laplace Operator on Domains with Rough Boundaries, Y. Netrusov, Y. Safarov.- W2,p-Theory of the Poincare Problem, D.K. Palagachev.- Weighted Inequalities for Integral and Supremum Operators, L. Pick.- Finite Rank Toeplitz Operators in the Bergman Space, G. Rozenblum.- Resolvent Estimates for Non-Selfadjoint Operators via Semigroups, J. Sjostrand.
Series: Springer Undergraduate Mathematics Series
Originally published by Oxford University Press
2010, Approx. 260 p. 42 illus., Softcover
ISBN: 978-1-84882-815-5
Due: November 2009
A new edition of a classic text, extensively revised and updated in order to simplify the presentation and offer a more modern outlook, showing how ideas from classical mechanics link with contemporary research.
Aims to give readers a confident grasp of the material by confronting, rather than evading, common notational and pedagogical difficulties encountered on the journey from Newton to Lagrange and Hamilton.
Analytical dynamics forms an important part of any undergraduate programme in applied mathematics and physics: it develops intuition about three-dimensional space and provides invaluable practice in problem solving.
First published in 1987, this text is an introduction to the core ideas. It offers concise but clear explanations and derivations to give readers a confident grasp of the chain of argument that leads from Newtonfs laws through Lagrangefs equations and Hamiltonfs principle, to Hamiltonfs equations and canonical transformations.
This new edition has been extensively revised and updated to include:
A chapter on symplectic geometry and the geometric interpretation of some of the coordinate calculations.
A more systematic treatment of the conections with the phase-plane analysis of ODEs; and an improved treatment of Euler angles.
A greater emphasis on the links to special relativity and quantum theory, e.g., linking Schrodingerfs equation to Hamilton-Jacobi theory, showing how ideas from this classical subject link into contemporary areas of mathematics and theoretical physics.
Aimed at second- and third-year undergraduates, the book assumes some familiarity
with elementary linear algebra, the chain rule for partial derivatives,
and vector mechanics in three dimensions, although the latter is not essential.
A wealth of examples show the subject in action and a range of exercises-?
with solutions-? are provided to help test understanding.
Frames of Reference.- One Degree of Freedom.- Lagrangian Mechanics.- Noether's Theorem.- Rigid Bodies.- Oscillations.- Hamiltonian mechanics.- Geometry of Classical Mechanics.- Epilogue: Relativity and Quantum Theory.- Notes on Exercises.
Series: Springer Monographs in Mathematics
2010, Approx. 470 p., Hardcover
ISBN: 978-3-642-04630-8
Due: November 2009
The semigroup methods are known as a powerful tool for analyzing nonlinear diffusion equations and systems. The author has studied abstract parabolic evolution equations and their applications to nonlinear diffusion equations and systems for more than 30 years. He gives first, after reviewing the theory of analytic semigroups, an overview of the theories of linear, semilinear and quasilinear abstract parabolic evolution equations as well as general strategies for constructing dynamical systems, attractors and stable-unstable manifolds associated with those nonlinear evolution equations.
In the second half of the book, he shows how to apply the abstract results to various models in the real world focusing on various self-organization models: semiconductor model, activator-inhibitor model, B-Z reaction model, forest kinematic model, chemotaxis model, termite mound building model, phase transition model, and Lotka-Volterra competition model. The process and techniques are explained concretely in order to analyze nonlinear diffusion models by using the methods of abstract evolution equations.
Thus the present book fills the gaps of related titles that either treat only very theoretical examples of equations or introduce many interesting models from Biology and Ecology, but do not base analytical arguments upon rigorous mathematical theories.