Series; :Springer Monographs in Mathematics
1st Edition, 2010, XX, 443 p., Hardcover
ISBN: 978-3-642-14033-4
Due: August 20, 2010
Cellular automata were introduced by John von Neumann who used them as models for self-reproducing machines. The authors present a self-contained exposition of the theory of cellular automata on groups and explore its deep connections with recent developments in geometric group theory and other branches of mathematics and theoretical computer science. The topics treated include in particular the Garden of Eden theorem for amenable groups and the Gromov-Weiss surjunctivity theorem as well as the solution of the Kaplansky conjecture on the stable finiteness of group rings for sofic groups. The volume is entirely self-contained, includes more than 300 exercises, and appeals to a large audience including specialists as well as newcomers in the field. Based on the interplay between amenability, geometric and combinatorial group theory, and symbolic dynamics it considers linear cellular automata: this gives applications to the theory of group rings (Kaplansky conjectures on the structure theory of group rings) that have no counterpart in other books on the same topics.
Preface.- 1.Cellular automata.- 2.Residually finite groups.- 3.Surjunctive groups.- 4.Amenable groups.- 5.The Garden of Eden theorem.- 6.Finitely generated amenable groups.- 7.Local embeddability and sofic groups.- 8.Linear cellular automata.- Appendices: A.Nets and the Tychonoff product theorem.- B.Uniform structures.- C.Symmetric groups.- D.Free groups.- E.Inductive limits and projective limits of groups.- F.The Banach-Alaoglu theorem.- G.The Markov-Kakutani fixed point theorem.- H.The Hall harem Theorem .- I.Complements of functional analysis.- J.Ultrafilters.- Open problems.- References.- List of symbols.- Index.
Series: Springer Undergraduate Texts in Mathematics and Technology
1st Edition., 2010, XVIII, 402 p. 10 illus., 5 in color., Hardcover
ISBN: 978-0-387-78976-7
Due: July 29, 2010
Optimization is used in almost all branches of applied sciences today
Text appeals to a wide readership because of its real life applications
Self-contained text with a rich collection of detailed examples and two-color graphics to assist the reader in full comprehension
Book includes many exercises, often supplemented by helpful hints or Matlab/Maple supplements
Optimization is an important field in its own right but also plays a central role in numerous applied sciences, including operations research, management science, economics, finance, and engineering.
Optimization - Theory and Practice offers a modern and well-balanced presentation of various optimization techniques and their applications. The book's clear structure, sound theoretical basics complemented by insightful illustrations and instructive examples, makes it an ideal introductory textbook and provides the reader with a comprehensive foundation in one of the most fascinating and useful branches of mathematics.
1. Introduction: Examples of Optimization Problems, Historical Overview.- 2. Optimality Conditions: Convex Sets, Inequalities, Local First- and Second-Order Optimality Conditions, Duality.- 3. Unconstrained Optimization Problems: Elementary Search and Localization Methods, Descent Methods with Line Search, Trust Region Methods, Conjugate Gradient Methods, Quasi-Newton Methods.- 4. Linearly Constrained Optimization Problems: Linear and Quadratic Optimization, Projection Methods.- 5. Nonlinearly Constrained Optimization Methods: Penalty Methods, SQP Methods.- 6. Interior-Point Methods for Linear Optimization: The Central Path, Newton's Method for the Primal-Dual System, Path-Following Algorithms, Predictor-Corrector Methods.- 7. Semidefinite Optimization: Selected Special Cases, The S-Procedure, The Function log°det, Path-Following Methods, How to Solve SDO Problems?, Icing on the Cake: Pattern Separation via Ellipsoids.- 8. Global Optimization: Branch and Bound Methods, Cutting Plane Methods.- Appendices: A Second Look at the Constraint Qualifications, The Fritz John Condition, Optimization Software Tools for Teaching and Learning.- Bibliography.- Index of Symbols.- Subject Index.
Series: Sources and Studies in the History of Mathematics and Physical Sciences
1st Edition., 2010, XII, 308 p. 2 illus., 1 in color., Hardcover
ISBN: 978-1-4419-5936-2
Due: September 11, 2010
The Hill-Brown theory of lunar motion was, from its completion in 1908 to its retirement in 1984, the most accurate model of the moon’s orbit. The mathematical, philosophical, and historical interest in the analytic solution of the lunar problem using the Hill?Brown method still engages celestial mechanicians, and is the primary focus of this work. This book, in three parts, describes three phases in the development of the modern theory and calculation of the Moon's motion. Part I explains the crisis in lunar theory in the 1870s that led G.W. Hill to lay a new foundation for an analytic solution, a preliminary orbit he called the "variational curve." Part II is devoted to E.W. Brown's completion of the new theory as a series of successive perturbations of Hill's variational curve. Part III describes the revolutionary developments in time-measurement and the determination of Earth-Moon and Earth-planet distances that led to the replacement of the Hill?Brown theory in 1984. Although some calculus and differential equations are included, the text is largely accessible without advanced knowledge in these areas. Amateurs of astronomy, as well as instructors and scholars of the general history of science, will find this book of significant interest.
Preface.-Part I: Hill lays the foundation (1877-1878).- 1. George William Hill, mathematician.-2. Lunar theory from the 1740s to the 1870s ? a sketch.-3. Hill on the motion of the lunar perigee.- 4. Hill’s Variation Curve.- 5. Early assessments of Hill’s work on the lunar theory.- Part II: Brown completes the theory (1892-1908), and constructs tables (1908-1919).-6. E.W. Brown, celestial mechanician.-7. First papers and a book.-8. Initiatives inspired by John Couch Adams’ papers.-9. Further preliminaries to the systematic development.-10. Theory of the Motion of the Moon.-11.A solution-procedure without approximations.-12. The ‘Main Problem’ solved.-13. Correcting for the idealizations.-14. Direct Planetary Perturbations of the Moon.-15. Indirect Planetary Perturbations of the Moon.-16. The effect of the figures of the Earth and Moon .-17. Perturbations of order (delta R)2.-18.The Tables.- 19. Determining the Values of the Arbitrary Constants.-20. Ernest W. Brown as theorist and computer.- Part III: Revolutionary developments in time measurement, computing, and data collection.- 21. Introduction .- 22. Tidal acceleration, fluctuations, and the variability of the Earth’s Rotation, from the 1690s to 1939.- 23. The quest for a uniform time: from Ephemeris Time to Atomic Time.- 24. 1984: The Hill-Brown theory is replaced as the basis of lunar ephemerides.- 25. The mathematical and philosophical interest in an analytical solution of the lunar problem.- Appendix: "Observations on the Desirability of New Tables of the Moon" (file of George William Hill, Naval Observatory Library).-Index.
Series: Universitext
1st Edition., 2010, XII, 272 p., Softcover
ISBN: 978-1-4419-6708-4
Due: August 29, 2010
Presents a complete proof of a major recent accomplishment of modern complex analysis, the affirmative resolution of Vitushkin's conjecture
Includes Melnikov and Verdera's proof of Denjoy's conjecture
Reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkin's conjecture
Contains important background material on removability, analytic capacity, Hausdorff measure, arclength measure, and Garabedian duality
Vitushkin's conjecture, a special case of Painleve's problem, states that a compact subset of the complex plane with finite linear Hausdorff measure is removable for bounded analytic functions if and only if it intersects every rectifiable curve in a set of zero arclength measure. Chapters 6-8 of this carefully written text present a major recent accomplishment of modern complex analysis, the affirmative resolution of this conjecture. Four of the five mathematicians whose work solved Vitushkin's conjecture have won the prestigious Salem Prize in analysis. Chapters 1-5 of this book provide important background material on removability, analytic capacity, Hausdorff measure, arclength measure, and Garabedian duality that will appeal to many analysts with interests independent of Vitushkin's conjecture. The fourth chapter contains a proof of Denjoy's conjecture that employs Melnikov curvature. A brief postscript reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkin's conjecture. Although standard notation is used throughout, there is a symbol glossary at the back of the book for the reader's convenience. This text can be used for a topics course or seminar in complex analysis. To understand it, the reader should have a firm grasp of basic real and complex analysis.
fContentsPreface.- 1 Removable Sets and Analytic Capacity.- 1.1 Removable Sets.- 1.2 Analytic Capacity.- 2 Removable Sets and Hausdor Measure.- 2.1 Hausdor
Measure and Dimension.- 2.2 Painleve's Theorem.- 2.3 Frostman's Lemma.- 2.4 Conjecture & Refutation: The Planar Cantor Quarter Set.- 3 Garabedian Duality for Hole-Punch Domains.- 3.1 Statement of the Result and an Initial Reduction.- 3.2 Interlude: Boundary Correspondence for H1(U).- 3.3 Interlude: Some F. & M. Riesz Theorems.- 3.4 Construction of the Boundary Garabedian Function.- 3.5 Construction of the Interior Garabedian Function.- 3.6 A Further Reduction.- 3.7 Interlude: Some Extension and Join Propositions.- 3.8 Analytically Extending the Ahlfors and Garabedian Functions.- 3.9 Interlude: Consequences of the Argument Principle.- 3.10 An Analytic Logarithm of the Garabedian Function.- 4 Melnikov and Verdera's Solution to the Denjoy Conjecture.- 4.1 Menger Curvature of Point Triples.- 4.2 Melnikov's Lower Capacity Estimate.- 4.3 Interlude: A Fourier Transform Review.- 4.4 Melnikov Curvature of Some Measures on Lipschitz Graphs.- 4.5 Arclength & Arclength Measure: Enough to Do the Job.- 4.6 The Denjoy Conjecture Resolved Affirmatively.- 4.7 Conjecture & Refutation: The Joyce-Morters Set.- 5 Some Measure Theory.- 5.1 The Caratheodory Criterion and Metric Outer Measures.- 5.2 Arclength & Arclength Measure: The Rest of the Story.- 5.3 A Vitali Covering Lemma and Planar Lebesgue Measure.- 5.4 Regularity Properties of Hausdor Measures.- 5.5 The Besicovitch Covering Lemma and Lebesgue Points.- 6 A Solution to Vitushkin's Conjecture Modulo Two Difficult Results.- 6.1 Statement of the Conjecture and a Reduction.- 6.2 Cauchy Integral Representation.- 6.3 Estimates of Truncated Cauchy Integrals.- 6.4 Estimates of Truncated Suppressed Cauchy Integrals.- 6.5 Vitushkin's Conjecture Resolved Affirmatively Modulo Two Difficult Results.- 6.6 Postlude: The Original Vitushkin Conjecture.- 7 The T(b) Theorem of Nazarov, Treil, and Volberg.- 7.1 Restatement of the Result.- 7.2 Random Dyadic Lattice Construction.- 7.3 Lip(1)-Functions Attached to Random Dyadic Lattices.- 7.4 Construction of the Lip(1)-Function of the Theorem.- 7.5 The Standard Martingale Decomposition.- 7.6 Interlude: The Dyadic Carleson Imbedding Inequality.- 7.7 The Adapted Martingale Decomposition.- 7.8 Bad Squares and Their Rarity.- 7.9 The Good/Bad-Function Decomposition.- 7.10 Reduction to the Good Function Estimate.- 7.11 A Sticky Point, More Reductions, and Course Setting.- 7.12 Interlude: The Schur Test.- 7.13 G1: The Crudely Handled Terms.- 7.14 G2: The Distantly Interacting Terms.- 7.15 Splitting Up the G3 Terms.- 7.16 Gterm 3 : The Suppressed Kernel Terms.- 7.17 Gtran 3 : The Telescoping Terms.- 8 The Curvature Theorem of David and Leger.- 8.1 Restatement of the Result and an Initial Reduction.- 8.2 Two Lemmas Concerning High Density Balls.- 8.3 The Beta Numbers of Peter Jones.- 8.4 Domination of Beta Numbers by Local Curvature.- 8.5 Domination of Local Curvature by Global Curvature.- 8.6 Selection of Parameters for the Construction.- 8.7 Construction of a Baseline L0.- 8.8 De nition of a Stopping-Time Region S0.- 8.9 De nition of a Lipschitz Set K0 over the Base Line.- 8.10 Construction of Adapted Dyadic Intervals on the Base Line.- 8.11 Assigning Linear Functions to Adapted Dyadic Intervals.- 8.12 Construction of a Lipschitz Graph G Threaded through K0.- 8.13 Veri cation that the Graph is Indeed Lipschitz.- 8.14 A Partition of K n K0 into Three Sets: K1, K2, & K3.- 8.15 The Smallness of the Set K2.- 8.16 The Smallness of a Horrible Set H.- 8.17 Most of K Lies in the Vicinity of the Lipschitz Graph.- 8.18 The Smallness of the Set K1.- 8.19 Gamma Functions of the Lipschitz Graph.- 8.20 A Point Estimate on One of the Gamma Functions.- 8.21 A Global Estimate on the Other Gamma Function.- 8.22 Interlude: Calderon's Formula.- 8.23 A Decomposition of the Lipschitz Function.- 8.24 The Smallness of the Set K3.- Postscript.- Bibliography.- Symbol Glossary & List.- Index
Series: Nonlinear Physical Science
1st Edition., 2010, 275 p. 110 illus., 10 in color., Hardcover
ISBN: 978-3-642-12717-5
Due: August 2010
“Hamiltonian Chaos Beyond the KAM Theory: Dedicated to George M. Zaslavsky (1935?2008)” covers the recent developments and advances in the theory and application of Hamiltonian chaos in nonlinear Hamiltonian systems. The book is dedicated to Dr. George Zaslavsky, who was one of three founders of the theory of Hamiltonian chaos. Each chapter in this book was written by well-established scientists in the field of nonlinear Hamiltonian systems. The development presented in this book goes beyond the KAM theory, and the onset and disappearance of chaos in the stochastic and resonant layers of nonlinear Hamiltonian systems are predicted analytically, instead of qualitatively. The book is intended for researchers in the field of nonlinear dynamics in mathematics, physics and engineering. Dr. Albert C.J. Luo is a Professor at Southern Illinois University Edwardsville, USA. Dr. Valentin Afraimovich is a Professor at San Luis Potosi University, Mexico.
Stochastic and Resonant Layers in Nonlinear Hamiltonian Systems.- A New Approach to The Treatment of Separatrix Chaos and Its Applications.- Hamiltonian Chaos and Anomalous Transport in Two Dimensional Flows.- Hamiltonian Chaos with A Cold Atom in An Optical Lattice.- Using Stochastic Webs to Control The Quantum Transport of Electrons in Semiconductor Superlattices.- Chaos in Ocean Acoustic Waveguide.