Series: Lecture Notes in Mathematics, Vol. 2034
2011, XXII, 498 p. 3 illus.
Softcover, ISBN 978-3-642-22596-3
Due: October 31, 2011
Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin Theorem in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, Numerical Analysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to:
1) fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra or not) to understand and apply the statements and numerous corollaries of the main result;
2) provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation;
3) provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincare, Pascal, Campbell, Baker, Hausdorff and Dynkin;
4) give an outlook on the applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type);
5) quickly enable the reader, through a description of the state-of-art and open problems, to understand the modern literature concerning a theorem which, though having its roots in the beginning of the 20th century, has not ceased to provide new problems and applications.
The book assumes some undergraduate-level knowledge of algebra and analysis, but apart from that is self-contained. Part II of the monograph is devoted to the proofs of the algebraic background. The monograph may therefore provide a tool for beginners in Algebra.
1 Historical Overview.- Part I Algebraic Proofs of the CBHD Theorem.- 2 Background Algebra.- 3 The Main Proof of the CBHD Theorem.- 4 Some eShortf Proofs of the CBHD Theorem.- 5 Convergence and Associativity for the CBHD Theorem.- 6 CBHD, PBW and the Free Lie Algebras.- Part II Proofs of the Algebraic Prerequisites.- 7 Proofs of the Algebraic Prerequisites.- 8 Construction of Free Lie Algebras.- 9 Formal Power Series in One Indeterminate.- 10 Symmetric Algebra.
Series: Lecture Notes in Mathematics, Vol. 2035
Subseries: Mathematical Biosciences Subseries
2012, X, 158 p. 43 illus., 38 in color.
Softcover, ISBN 978-3-642-22989-3
Due: October 31, 2011
This volume reports on recent mathematical and computational advances in optical, ultrasound, and opto-acoustic tomographies. It outlines the state-of-the-art and future directions in these fields and provides readers with the most recently developed mathematical and computational tools. It is particularly suitable for researchers and graduate students in applied mathematics and biomedical engineering.
Content Level â Graduate
Keywords â 35-XX; 65-XX; 92-XX - imaging algorithms - opto-acoustic imaging - stability and resolution analysis - ultrasound imaging
Related subjects â Applications - Biomedical Engineering - Radiology - Theoretical, Mathematical & Computational Physics
Direct Reconstruction Methods in Optical Tomography.- Direct Reconstruction Methods in Ultrasound Imaging of Small Anomalies.- Photoacoustic Imaging for Attenuating Acoustic Media.- Attenuation Models in Photoacoustics.- Quantitative Photoacoustic Tomography.
Series: Universitext
2012, XVIII, 322 p. 3 illus.
Softcover, ISBN 978-1-4471-2130-5
Due: September 30, 2011
Number theory is a branch of mathematics which draws its vitality from a rich historical background. It is also traditionally nourished through interactions with other areas of research, such as algebra, algebraic geometry, topology, complex analysis and harmonic analysis. More recently, it has made a spectacular appearance in the field of theoretical computer science and in questions of communication, cryptography and error-correcting codes.
Providing an elementary introduction to the central topics in number theory, this book spans multiple areas of research. The first part corresponds to an advanced undergraduate course. All of the statements given in this part are of course accompanied by their proofs, with perhaps the exception of some results appearing at the end of the chapters. A copious list of exercises, of varying difficulty, are also included here. The second part is of a higher level and is relevant for the first year of graduate school. It contains an introduction to elliptic curves and a chapter entitled gDevelopments and Open Problemsh, which introduces and brings together various themes oriented toward ongoing mathematical research.
Given the multifaceted nature of number theory, the primary aims of this book are to:
- provide an overview of the various forms of mathematics useful for studying numbers
- demonstrate the necessity of deep and classical themes such as Gauss sums
- highlight the role that arithmetic plays in modern applied mathematics
- include recent proofs such as the polynomial primality algorithm
- approach subjects of contemporary research such as elliptic curves
- illustrate the beauty of arithmetic
The prerequisites for this text are undergraduate level algebra and a little topology of Rn. It will be of use to undergraduates, graduates and phd students, and may also appeal to professional mathematicians as a reference text.
Finite Structures.- Applications: Algorithms, Primality and Factorization, Codes.- Algebra and Diophantine Equations.- Analytic Number Theory.- Elliptic Curves.- Developments and Open Problems.- Factorization.- Elementary Projective Geometry.- Galois Theory
Series: Applied and Numerical Harmonic Analysis
2011, XII, 248 p. 17 illus.
ISBN 978-0-8176-8255-2
Due: October 28, 2011
SAGE, a free open-source mathematics software system, is used with concrete examples in order to emphasize the computational aspects of coding theory
Introduces and explores unsolved open problems in an effort to illuminate the fundamentals of coding theory and stimulate further research
Appeals to academia and industry with real-world applications in electrical engineering and digital communication
Using an original mode of presentation and emphasizing the computational nature of the subject, this book explores a number of the unsolved problems that continue to exist in coding theory. A well-established and still highly relevant branch of mathematics, the theory of error-correcting codes is concerned with reliably transmitting data over a enoisyf channel. Despite its frequent use in a range of contexts?the first close-up pictures of the surface of Mars, taken by the NASA spacecraft Mariner 9, were transmitted back to Earth using a Reed?Muller code?the subject contains interesting problems that have to date resisted solution by some of the most prominent mathematicians of recent decades.
Employing SAGE "a free open-source mathematics software system "to illustrate their ideas, the authors begin by providing background on linear block codes and introducing some of the special families of codes explored in later chapters, such as quadratic residue and algebraic-geometric codes. Also surveyed is the theory that intersects self-dual codes, lattices, and invariant theory, which leads to an intriguing analogy between the Duursma zeta function and the zeta function attached to an algebraic curve over a finite field. The authors then examine a connection between the theory of block designs and the Assmus?Mattson theorem and scrutinize the knotty problem of finding a non-trivial estimate for the number of solutions over a finite field to a hyperelliptic polynomial equation of "small" degree, as well as the best asymptotic bounds for a binary linear block code. Finally, some of the more mysterious aspects relating modular forms and algebraic-geometric codes are discussed.
Selected Unsolved Problems in Coding Theory is intended for graduate students and researchers in algebraic coding theory, especially those who are interested in finding current unsolved problems. Familiarity with concepts in algebra, number theory, and modular forms is assumed. The work may be used as supplementary reading material in a graduate course on coding theory or for self-study.
Content Level â Graduate
Keywords â Assmus?Mattson theorem - Duursma zeta function - Reed?Muller code - SAGE - algebraic coding theory - algebraic-geometric codes - block codes - coding theory - error-correcting codes - hyperelliptic polynomial equation - invariant theory - modular forms - quadratic residue - self-dual codes - zeta function
Related subjects â Birkhauser Computer Science - Birkhauser Engineering - Birkhauser Mathematics
Preface.- Background.- Codes and Lattices.- Kittens and Blackjack.- RH and Coding Theory.- Hyperelliptic Curves and QR Codes.- Codes from Modular Curves.- Appendix.- Bibliography.- Index.
Series: Developments in Mathematics, Vol. 24
2011, XIV, 482 p.
Hardcover, ISBN 978-1-4614-0528-3
Due: October 28, 2011
A large mathematical community throughout the world actively works in functional analysis and uses profound techniques from topology. As the first monograph to approach the topic of topological vector spaces from the perspective of descriptive topology, this work provides also new insights into the connections between the topological properties of linear function spaces and their role in functional analysis.
Descriptive Topology in Selected Topics of Functional Analysis is a self-contained volume that applies recent developments and classical results in descriptive topology to study the classes of infinite-dimensional topological vector spaces that appear in functional analysis. Such spaces include Frechet spaces, LF-spaces and their duals, and the space of continuous real-valued functions C(X) on a completely regular Hausdorff space X, to name a few. These vector spaces appear in distribution theory, differential equations, complex analysis, and various other areas of functional analysis.
Written by three experts in the field, this book is a treasure trove for researchers and graduate students studying the interplay among the areas of point-set and descriptive topology, modern analysis, set theory, topological vector spaces and Banach spaces, and continuous function spaces. Moreover, it will serve as a reference for present and future work done in this area and could serve as a valuable supplement to advanced graduate courses in functional analysis, set-theoretic topology, or the theory of function spaces.
Preface.- 1. Overview.- 2. Elementary Facts about Baire and Baire-Type Spaces.- 3. K-analytic and quasi-Suslin Spaces.- 4. Web-Compact Spaces and Angelic Theorems.- 5. Strongly Web-Compact Spaces and a Closed Graph Theorem.- 6. Weakly Analytic Spaces.- 7. K-analytic Baire Spaces.- 8. A Three-Space Property for Analytic Spaces.- 9. K-analytic and Analytic Spaces Cp(X).- 10. Precompact sets in (LM)-Spaces and Dual Metric Spaces.- 11. Metrizability of Compact Sets in the Class G.- 12. Weakly Realcompact Locally Convex Spaces.- 13. Corsonfs Propery (C) and tightness.- 14. Frechet-Urysohn Spaces and Groups.- 15. Sequential Properties in the Class G.- 16. Tightness and Distinguished Frechet Spaces.- 17. Banach Spaces with Many Projections.- 18. Spaces of Continuous Functions Over Compact Lines.- 19. Compact Spaces Generated by Retractions.- 20. Complementably Universival Banach Space.- Index.