Christensen, L.W., University of Copenhagen, Denmark
Gorenstein Dimensions
2000. XIII, 204 pp. Softcover
3-540-41132-1
This book is intended as a reference for mathematicians working with homologicaldimensions in
commutative algebra and as an introduction to Gorenstein dimensions for graduate students with an interest in the same. Any admirer of classics like the Auslander-Buchsbaum-Serre characterization of regular rings, and the Bass and Auslander-Buchsbaum formulas for injective and projective dimension of f.g. modules will be intrigued by this book's content. Readers should be well-versed in commutative algebra and standard applications of homological methods.
The framework is that of complexes, but all major results are restated for modules in traditional notation, and an appendix makes the proofs accessible for even the casual user of hyperhomological methods.
Keywords: Gorenstein dimensions, Gorenstein rings, Auslander-Buchsbaum formulas, Foxby equivalence, Cohen-Macaulay rings . Mathematics Subject Classification : 13-02, 13C15, 13D02, 13D05, 13D07, 13D25, 13E05, 13H10, 18G25
Contents: Introduction.- Synopsis.- Conventions and prerequisites.- The classical Gorenstein dimension.- G-dimension and reflexive complexes.- Auslander categories.- G-projectivity. - G-injectivity.- Appendix: Hyperhomology. Basic definitions and notation. Standard functors and morphisms. Resolutions. Almost derived functors. Homological dimensions. Depth and width. Numerical and formal invariants. Dualizing complexes.
Series: Lecture Notes in Mathematics.VOL. 1747
Conrad, B., University of Michigan, Ann Arbor, MI, USA
Grothendieck Duality and Base Change
2000. X, 296 pp. Softcover
3-540-41134-8
Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curvesto the arithmetic theory of modular forms. Presented is a systematic overview ofthe entire theory, including many basic definitions and a detailed study of duality on curves, dualizing sheaves, and Grothendieck's residue symbol. Along the way proofs are given of some widely used foundational results which are not proven in existing treatments of the subject, such as the general base change compatibility of the trace map for proper Cohen-Macaulay morphisms (e.g., semistable curves). This should be of interest to mathematicians who have some familiarity with Grothendieck's work and wish to understand the details of this theory.
Keywords: Dualizing sheaves, residues, Grothendieck duality, trace map . MSC : 14A15
Contents: Introduction.- Basic compatibilities.- Duality foundations.- Proof of main theorom.- Examples: Higher direct images. Curves.- Residues and cohomology with supports.- Trace map on smooth curves.
Series: Lecture Notes in Mathematics.VOL. 1750
Wohlmuth, B.I., University of Augsburg, Germany
Discretization Methods and Iterative Solvers Based on Domain Decomposition
2001. X, 170 pp. Softcover
3-540-41083-X
Domain decomposition methods provide powerful and flexible tools for the numerical approximation of partial differential equations arising in the modeling of many interesting applications in science and engineering. This book deals with discretization techniques on non-matching triangulations and iterative solvers with particular emphasis on mortar finite elements, Schwarz methods and multigrid techniques. New results on non-standard situations as mortar methods based on dual basis functions and vector field discretizations are analyzed and illustrated by numerical results. The role of trace theorems, harmonic extensions, dual norms and weak interface conditions is emphasized. Although the original idea was used successfully more than a hundred years ago, these methods are relatively new for the numerical approximation. The possibilites of high performance computations and the interest in large- scale problems have led to an increased research activity.
Keywords: Domain decomposition techniques, linear elasticity, mortar finite elements, multigrid methods, Schwarz methods
Series: Lecture Notes in Computational Science and Engineering.VOL. 17
Boltyanskii, V.G./ Efremovich, V.A.
Intuitive Combinatorial Topology
2001. Approx. 215 pp. 162 figs. Hardcover
0-387-95114-8
Topology is a relatively young and very important branch of mathematics. It studies properties of objects that are preserved by deformations, twistings, and stretchings, but not tearing. This book deals with the topology of curves and surfaces as well as with the fundamental concepts of homotopy and homology, and does this in a lively and well-motivated way. There is hardly an area of mathematics that does not make use of topological results and concepts. The importance of topological methods for different areas of physics is also beyond doubt. They are used in field theory and general relativity, in the physics of low temperatures, and in modern quantum theory. The book is well suited not only as preparation for students who plan to take a course in algebraic topology but also for advanced undergraduates or beginning graduates interested in finding out what topology is all about. The book has more than 200 problems, many examples, and over 200 illustrations.
Contents: Topology of Curves.- Topology of Surfaces.- Homotopy and homology.- Supplement.
Topological Objects in Nematic Liquid Crystals.
Series: Universitext.
Cherry, W., University of North Texas, Denton, TX, USA
Ye, Z., Northern Illinois University, DeKalb, IL, USANevanlinna's Theory of Value Distribution
The Second Main Theorem and its Error Terms
2001. Approx. 200 pp. Hardcover
3-540-66416-5
On the one hand, this monograph serves as a self-contained introduction to Nevanlinna's theory of value distribution because the authors only assume the reader is familiar with the basics of complex analysis. On the other hand, the monograph also serves as a valuable reference for the research specialist because the authors present, for the first time in book form, the most modern and refined versions of the Second Main Theorem with precise error terms, in both the geometric and logarithmic derivative based approaches. A unique feature of the monograph is its "number theoretic digressions". These special sections assume no background in number theory and explore the exciting interconnections between Nevanlinna theory and the theory of Diophantine approximation.
Keywords: Nevanlinna ; value distribution ; error terms ; diophantine approximation 30D35 ; 11J97
Contents: I The First Main Theorem.- II The Second Main Theorem via Negative Curvature.- III Logarithmic Derivatives.- IV The Second Main Theorem via Logarithmic Derivatives.- V Some Applications Chapter.- VI A Further Digression into Number Theory: Theorems of Roth and Khinchin.- VII More on the Error Term.
Series: Springer Monographs in Mathematics.
Devroye, L., McGill University, Montreal, Que., Canada
Lugosi, G., Universitat Pompeu Fabra, Barcelona, SpainCombinatorial Methods in Density Estimation
2001. Approx. 215 pp. Hardcover
0-387-95117-2
Density estimation has evolved enormously since the days of bar plots and histograms, but researchers and users are still struggling with the problem of the selection of the bin widths. This text explores a new paradigm for the data-based or automatic selection of the free parameters of density estimates in general so that the expected error is within a given constant multiple of the best possible error. The paradigm can be used in nearly all density estimates and for most model selection problems, both parametric and nonparametric. It is the first book on this topic. The text is intended for first-year graduate students in statistics and learning theory, and offers a host of opportunities for further research and thesis topics. Each chapter corresponds roughly to one lecture, and is supplemented with many classroom exercises. A one year course in probability theory at the level of Feller's Volume 1 should be more than adequate preparation.
Gabor Lugosi is Professor at Universitat Pompeu Fabra in Barcelona, and Luc Debroye is Professor at McGill University in Montreal. In 1996, the authors, together with L疽zlo Gyfi, published the successful text, A Probabilistic Theory of Pattern Recognition with Springer-Verlag. Both authors have made many contributions in the area of nonparametric estimation.
Contents: Introduction.- Concentration Inequalities.- Uniform Deviation Inequalities.- Combinatorial Tools.- Total Variation.- Choosing a Density Estimate from a Collection.- Skeleton Estimates.- The Minimum Distance Estimate: Examples.- The Kernel Density Estimate.- Additive Estimates and Data Splitting.- Bandwidth Selection for Kernel Estimates.- Multiparameter Kernel Estimates.- Wavelet Estimates.- The Transformed Kernel Estimate.- Minimax Theory.- Choosing the Kernel Order.- Bandwidth Choice with Superkernels.
Series: Springer Series in Statistics.