Series: Lecture Notes in Mathematics, Vol. 2037
1st Edition., 2012, XII, 236 p.
Softcover, ISBN 978-3-642-23646-4
Due: November 30, 2011
In the study of algebraic/analytic varieties a key aspect is the description of the invariants of their singularities. This book targets the challenging non-isolated case. Let f be a complex analytic hypersurface germ in three variables whose zero set has a 1-dimensional singular locus. We develop an explicit procedure and algorithm that describe the boundary M of the Milnor fiber of f as an oriented plumbed 3-manifold. This method also provides the characteristic polynomial of the algebraic monodromy. We then determine the multiplicity system of the open book decomposition of M cut out by the argument of g for any complex analytic germ g such that the pair (f,g) is an ICIS. Moreover, the horizontal and vertical monodromies of the transversal type singularities associated with the singular locus of f and of the ICIS (f,g) are also described. The theory is supported by a substantial amount of examples, including homogeneous and composed singularities and suspensions. The peculiar properties of M are also emphasized.
1 Introduction.- 2 The topology of a hypersurface germ f in three variables Milnor fiber.- 3 The topology of a pair (f ; g).- 4 Plumbing graphs and oriented plumbed 3-manifolds.- 5 Cyclic coverings of graphs.- 6 The graph GC of a pair (f ; g). The definition.- 7 The graph GC . Properties.- 8 Examples. Homogeneous singularities.- 9 Examples. Families associated with plane curve singularities.- 10 The Main Algorithm.- 11 Proof of the Main Algorithm.- 12 The Collapsing Main Algorithm.- 13 Vertical/horizontal monodromies.- 14 The algebraic monodromy of H1(¶ F). Starting point.- 15 The ranks of H1(¶ F) and H1(¶ F nVg) via plumbing.- 16 The characteristic polynomial of ¶ F via P# and P#.- 18 The mixed Hodge structure of H1(¶ F).- 19 Homogeneous singularities.- 20 Cylinders of plane curve singularities: f = f 0(x;y).- 21 Germs f of type z f 0(x;y).- 22 The T?;?;??family.- 23 Germs f of type ? f (xayb; z). Suspensions.- 24 Peculiar structures on ¶ F. Topics for future research.- 25 List of examples.- 26 List of notations
Series: Lecture Notes in Mathematics, Vol. 2038
1st Edition., 2012, 318 p.
Softcover, ISBN 978-3-642-23668-6
Due: November 30, 2011
The purpose of these lecture notes is to provide an introduction to the theory of complex Monge?Ampere operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Kaehler manifolds (with or without boundary).
These operators are of central use in several fundamental problems of complex differential geometry (Kaehler?Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford?Taylor), Monge?Ampere foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi?Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of Kaehler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli-Kohn-Nirenberg-Spruck (after Guan, Chen and Blocki) and Bergman approximation of geodesics (after Phong-Sturm and Berndtsson).
Each chapter can be read independently and is based on a series of lectures delivered to non-experts. The book is thus addressed to any mathematician with some interest in one of the following fields, complex differential geometry, complex analysis, complex dynamics, fully non-linear PDE's and stochastic analysis.
1.Introduction.- I. The Local Homogenious Dirichlet Problem.-2. Dirichlet Problem in Domains of Cn.- 3. Geometric Maximality.- II. Stochastic Analysis for the Monge-Ampere Equation.- 4. Probabilistic Approach to Regularity.- III. Monge-Ampere Equations on Compact Manifolds.- 5.The Calabi-Yau Theorem.- IV Geodesics in the Space of Kahler Metrics.- 6. The Riemannian Space of Kahler Metrics.- 7. MA Equations on Manifolds with Boundary.- 8. Bergman Geodesics.
Series: Lecture Notes in Mathematics, Vol. 2039
1st Edition., 2012, X, 416 p. 29 illus.
Softcover, ISBN 978-3-642-23839-0
Due: November 30, 2011
Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis.
In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances.
1 Introduction.- 2 Graphs and associated Laplacians.- 3 Scales of Hilbert space and boundary triples.- 4 Two operators in different Hilbert spaces.- 5 Manifolds, tubular neighbourhoods and their perturbations.- 6 Plumber’s shop: Estimates for star graphs and related spaces.- 7 Global convergence results.
Series: Universitext
1st Edition., 2012, Approx. 470 p. 6 illus.
Softcover, ISBN 978-0-387-79427-3
Due: April 30, 2012
-Ranges in topics from an introduction to fields and linear systems to an introduction to ring theory and field extensions -Accessible to advanced undergraduates/graduate students, in a variety of subject areas, including mathematics, physics, engineering and computer science - Useful reference material for advanced mathematicians and professionals - Numerous practice problems at the end of each sub-section of each chapter -emphasis on interplay between algebra and geometry
Abstract Linear Algebra, Groups and Rings is an excellent introduction to the theory of groups, rings and fields. To emphasize the importance of a foundation of knowledge in both geometry and algebra, this text includes an introduction to Euclidean Spaces, and a brief treatment of algebraic topics such as matrix algebra, linear systems, vector spaces, linear coding theory, determinants, eigentheory, group theory, ring theory, and field extensions, even covering an introduction to cryptography.
This text has evolved from sets of notes from two different semester-long courses that the author has taught for several years at the University of British Columbia. The first portion of the book is from an undergraduate honors course on abstract linear algebra. The final chapters were developed from a course on the theory of groups and rings for math students. With content from these two courses, this book is ideal as a textbook for a year long course. Additionally, every sub-section of each of chapter ends with practice problems to aid in the understanding of the subject matter.
The readership does not need to have knowledge of calculus, but should have an understanding of mathematical induction and the notion of proof. Advanced undergraduate or graduate students in mathematics, physics, computer science, and engineering will find this book useful and enjoyable. The text can also be used as a supplementary text for various professional applications.
2012, 4930 p. In 7 volumes, not available separately.
print (book), Hardcover, ISBN 978-1-4419-7996-4
Due: April 9, 2012
This second edition features 30% new additional content, additional chapters as well as updated content
Editors-in-chief are renowned members of the operations research and mathematics communities
Subject spans much of applied mathematics, computer science and operations research as well as overlaps with many other fields such as computation complexity, computational biology, VLSI design, communications networks, and management science
The second edition of this 7-volume handbook is intended to be a basic, yet comprehensive reference work in combinatorial optimization that will benefit newcomers and researchers for years to come. The multi-volume work deals with several algorithmic approaches for discrete problems as well as with many combinatorial problems. The editors have brought together almost every aspect of this enormous field of combinatorial optimization, an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communications networks, and management science.
Ron Graham has been added as an EiC for this second edition to work directly with Ding-Zhu Du on new material by way of new applications to combinatorial optimization.
An international team of 30-40 experts in the field form the editorial board.
The Handbook of Combinatorial Optimization, second edition is addressed to all scientists who use combinatorial optimization methods to model and solve problems. Experts in the field as well as non-specialists will find the material stimulating and helpful.