2010, XVIII, 630 p. 61 illus., Hardcover
ISBN: 978-1-4419-1720-1
Due: January 2010
This book provides an exposition of the algebraic aspects of the theory of lattice-ordered rings and lattice-ordered modules. All of the background material on rings, modules, and lattice-ordered groups necessary to make the work self-contained and accessible to a variety of readers is included.
Steinberg includes in his presentation of the material 800+ extensive exercises of varying levels of difficulty at the end of each of the sections. The first two chapters of the book provide a thorough introduction to the material, while the following four chapters delve into more specific topics.
*lattice-ordered groups, rings, and fields;
*archimedean l-groups;
*f-rings and larger varieties of l-rings;
*the category of f-modules;
*various commutativity results.
Filling a gap in the literature, Lattice-Ordered Rings and Modules may be used as a textbook or for self-study by graduate students and researchers studying lattice-ordered rings and lattice-ordered modules.
Preface.- List of Symbols.- 1 Partially Ordered Sets and Lattices.- 1.1 Partially Ordered Sets.- 1.2 Lattices.- 1.3 Completion.- 1.4 Universal Algebra.- 2 Lattice-ordered Groups.- 2.1 Basic Identities and Examples.- 2.2 Subobjects and Homomorphisms.- 2.3 Archimedean `-groups.- 2.4 Prime Subgroups, Representability, and Operator Sets.- 2.5 Values.- 2.6 Hahn Products and the Embedding Theorem.- 3 Lattice-ordered Rings.- 3.1 Basics, Examples, and Nonexamples.- 3.2 Radical Theory.- 3.3 f -Rings.- 3.4 Embedding in a Unital f -Algebra.- 3.5 Generalized Power Series Rings.- 3.6 Archimedean f -Rings.- 3.7 Squares Positive.- 3.8 Polynomial Constraints.- 4 The Category of f -Modules.- 4.1 Rings of Quotients and Essential Extensions.- 4.2 Torsion Theories and Rings of Quotients.- 4.3 Lattice-ordered Rings and Modules of Quotients.- 4.4 Injective f -Modules.- 4.5 Free f -Modules.- 5 Lattice-ordered Fields.- 5.1 Totally Ordered Extensions of Ordered Fields.- 5.2 Valuations and the Hahn Embedding Theorem.- 5.3 Lattice-ordered Fields.- 6 Additional Topics.- 6.1 Lattice-ordered Semigroup Rings.- 6.2 Algebraic f -Elements Are Central.- 6.3 More Polynomial Constraints on Totally Ordered Domains.- 6.4 Lattice-ordered Matrix Algebras.- Open Problems.- References.- Index.-
Series: Lecture Notes of the Unione Matematica Italiana , Vol. 7
2009, XVIII, 434 p., Softcover
ISBN: 978-3-642-05194-4
Due: December 2009
Homogenization is not about periodicity, or Gamma-convergence, but about understanding which effective equations to use at macroscopic level, knowing which partial differential equations govern mesoscopic levels, without using probabilities (which destroy physical reality); instead, one uses various topologies of weak type, the G-convergence of Sergio Spagnolo, the H-convergence of Francois Murat and the author, and some responsible for the appearance of nonlocal effects, which many theories in continuum mechanics or physics guessed wrongly.
For a better understanding of 20th century science, new mathematical tools must be introduced, like the author’s H-measures, variants by Patrick Gerard, and others yet to be discovered.
Researchers and graduate students in mathematics, continuum mechanics and physics
1 Why Do I Write?- 2 A Personalized Overview of Homogenization I.- 3 A Personalized Overview of Homogenization II.- 4 An Academic Question of Jacques-Louis Lions.- 5 A Useful Generalization by Francois Murat .- 6 Homogenization of an Elliptic Equation .- 7 The Div-Curl Lemma.- 8 Physical Implications of Homogenization.- 9 A Framework with Differential Forms.- 10 Properties of H-convergence.- 11 Homogenization of Monotone Operators.- 12 Homogenization of Laminated Materials.- 13 Correctors in Linear Homogenization.- 14 Correctors in Nonlinear Homogenization.- 15 Holes with Dirichlet Conditions.- 16 Holes with Neumann Conditions.- 17 Compensated Compactness.- 18 A Lemma for Studying Boundary Layers.- 19 A Model in Hydrodynamics.- 20 Problems in Dimension N = 2.- 21 Bounds on Effective Coefficients.- 22 Functions Attached to Geometries .- 23 Memory Effects.- 24 Other Nonlocal Effects.- 25 The Hashin-Shtrikman Construction.- 26 Confocal Ellipsoids and Spheres.- 27 Laminations Again, and Again.- 28 Wave Front Sets, H-Measures.- 29 Small-Amplitude Homogenization.- 30 H-Measures and Bounds on Effective Coeffcients.- 31 H-Measures and Propagation Effects.- 32 Variants of H-Measures.- 33 Relations Between Young Measures and H-Measures.- 34 Conclusion.- 35 Biographical Information.- 36 Abbreviations and Mathematical Notation.
Series: Lecture Notes of the Unione Matematica Italiana , Vol. 8
2010, XVII, 163 p., Softcover
ISBN: 978-3-642-05202-6
Due: December 2009
The famous zeros of the Riemann zeta function and its generalizations (L-functions, Dedekind and Selberg zeta functions) are analyzed through several zeta functions built over those zeros. These ‘second-generation’ zeta functions have surprisingly many explicit, yet largely unnoticed properties, which are surveyed here in an accessible and synthetic manner, and then compiled in numerous tables. No previous book has addressed this neglected topic in analytic number theory. Concretely, this handbook will help anyone faced with symmetric sums over zeros like Riemann’s. More generally, it aims at reviving the interest of number theorists and complex analysts toward those unfamiliar functions, on the 150th anniversary of Riemann’s work.
Researchers and graduate students in analytic number theory, complex analysis in one variable, special functions and integral transforms
11Mxx, 11-02, 11M26, 11M41, 11M36, 30B40, 30B50, 30E15, 41A60
L-functions
Mellin transforms
Riemann Zeros
Zeta Functions
Zeta-regularization
Series: Algebra and Applications , Vol. 10
2009, VII, 128 p., Hardcover
ISBN: 978-1-84882-940-4
Due: December 2009
Polycyclic groups are built from cyclic groups in a specific way. They arise in many contexts within group theory itself but also more generally in algebra, for example in the theory of Noetherian rings. They also touch on some aspects of topology, geometry and number theory. The first half of this book develops the standard group theoretic techniques for studying polycyclic groups and the basic properties of these groups. The second half then focuses specifically on the ring theoretic properties of polycyclic groups and their applications, often to purely group theoretic situations.
The book is not intended to be encyclopedic. Instead, it is a study manual for graduate students and researchers coming into contact with polycyclic groups, where the main lines of the subject can be learned from scratch by any reader who has been exposed to some undergraduate algebra, especially groups, rings and vector spaces. Thus the book has been kept short and readable with a view that it can be read and worked through from cover to cover. At the end of each topic covered there is a description without proofs, but with full references, of further developments in the area. The book then concludes with an extensive bibliography of items relating to polycyclic groups.
Graduate students; researchers
Foreword.- Some basic group theory.- Some ring theory.- Soluble linear groups.- Further group-theoretic properties of polycyclic groups.- Groups acting on finitely generated commutative rings.- Prime ideals in polycyclic-group rings.- The structure of modules over polycyclic groups.- Semilinear and skew linear groups.
Series: Lecture Notes in Mathematics , Vol. 1985
2010, XII, 326 p., Softcover
ISBN: 978-3-642-05133-3
Due: December 2009
This book on numerical methods for singular perturbation problems - in particular, stationary reaction-convection-diffusion problems exhibiting layer behaviour is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. A classification and a survey of layer-adapted meshes for reaction-convection-diffusion problems are included.
This structured and comprehensive account of current ideas in the numerical analysis for various methods on layer-adapted meshes is addressed to researchers in finite element theory and perturbation problems. Finite differences, finite elements and finite volumes are all covered.
Researchers and graduate students in mathematics and computer science
1 Introduction.- 2 Layer-adapted meshes.- Part I One dimensional problems.- 3 The analytical behaviour of solutions.- 4 Finite difference schemes for convection-diffusion problems.- 5 Finite element and finite volume methods.- 6 Discretisations of reaction-convection-diffusion problems.- Part II Two dimensional problems.- 7 The analytical behaviour of solutions.- 8 Reaction-diffusion problems.- 9 Convection-diffusion problems
Series: Lecture Notes in Mathematics , Vol. 1987
2009, Approx. 250 p., Softcover
ISBN: 978-3-642-05204-0
Due: December 2009
Vector fields on manifolds play a major role in mathematics and other sciences. In particular, the Poincare-Hopf index theorem gives rise to the theory of Chern classes, key manifold-invariants in geometry and topology.
It is natural to ask what is the ‘good’ notion of the index of a vector field, and of Chern classes, if the underlying space becomes singular. The question has been explored by several authors resulting in various answers, starting with the pioneering work of M.-H. Schwartz and R. MacPherson.
We present these notions in the framework of the obstruction theory and the Chern-Weil theory. The interplay between these two methods is one of the main features of the monograph.
Researchers and graduate students in algebraic geometry, complex analytic geometry, algebraic topology, and differential geometry
32S65, 37F75, 57R20, 57R25, 58K45
Chern classes
Indices of vector fields
Poincare-Hopf Theorem
Singular foliations
Singular varieties