Originally published by the Soifer, Alexander, 1987
1st ed. 1987. , 2008, Approx. 130 p., Softcover
ISBN: 978-0-387-74646-3
Due: April 2008
About this textbook
Aims to develop young mathematiciansf problem-solving abilities
Inspires readers to explore a range of methods of solving algebraic, geometric, and combinatorial problems
Offers examples and practice problems
Engages a general audience
The first edition was endorsed by the NCTM
This second edition of Alexander Sofierfs Mathematics as Problem-Solving explores various elementary solving techniques for algebraic, geometric, and combinatoric problems. Each chapter builds on the previous one, enhancing the readerfs problem-solving logic and level of understanding.
The author presents each mathematical topic in self-contained chapters by outlining classical solutions as well as his own secret discoveries found through problem-solving experience. With roughly 200 problems and multiple proofs for each, Sofier challenges the reader to approach problems from different angles and experience the mystery and beauty of mathematics.
Written by a distinguished mathematician and renowned author, this book is a wonderful resource for students at various levels.
Table of contents
Preface.- Language and a Few Celebrated Ideas.- Numbers.- Algebra.- Geometry.- Combinatorial Problems.- Literature.
Series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics , Vol. 34
2008, Approx. 320 p. 20 illus., Hardcover
ISBN: 978-3-540-74012-4
Due: February 2008
About this book
Hilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Rado. The book gives a concise introduction to variational methods and presents an overview of areas of current research in the field.
The fourth edition gives a survey on new developments in the field. In particular it includes the proof for the convergence of the Yamabe flow and a detailed treatment of the phenomenon of blow-up. Also the recently discovered results for backward bubbling in the heat flow for harmonic maps or surfaces are discussed. Aside from these more significant additions, a number of smaller changes throughout the text have been made and the references have been updated.
Table of contents
The Direct Methods in the Calculus of Variations.- Lower Semi-Continuity.- Constraints.- Compensated Compactness.- The Concentration-Compactness Principle.- Ekeland's Variational Principle.- Duality.- Minimization Problems Depending on Parameters.- Minimax Methods.- The Finite Dimensional Case.- The Palais-Smale Condition.- A General Deformation Lemma.- The Minimax Principle.- Index Theory.- The Mountain Pass Lemma and its Variants.- Perturbation Theory.- Linking.- Parameter Dependence.- Critical Points of Mountain Pass Type.- Non-Differentiable Functionals.- Ljusternik-Schnirelman Theory on Convex Sets.- Limit Cases of the Palais-Smale Condition.- Pohozaev's Non-Existence Result.- The Brezis-Nierenberg Result.- The Effect of Topology.- The Yamabe Problem.- The Dirichlet Problem for the Equation of Constant Mean Curvature.- Harmonic Maps of Riemannian Surfaces.- Appendix A.- Appendix B.- Appendix C.- References.- Index.
Series: Oberwolfach Seminars , Vol. 38
2008, Approx. 350 p., Softcover
ISBN: 978-3-7643-8620-7
Due: February 2008
About this book
First book on a newly emerging field of discrete differential geometry providing an excellent way to access this new exciting area
Carefully edited collection of essays by key researchers in the field
Surveys fascinating connections between discrete geometry and differential geometry: for instance, circle patterns and triangulated surfaces on the discrete side connect to minimal surfaces and curvature line parametrizations on the differential side
Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and mathematical physics.
This collection of essays, which documents the main lectures of the 2004 Oberwolfach Seminar on the topic, as well as a number of additional contributions by key participants, gives a lively, multi-facetted introduction to this emerging field.
Written for:
Students and researchers in mathematics, in particular, in discrete geometry, differential geometry, mathematical physics, as well as in computer science
Keywords:
computer grapics
discrete geometry
minimal surface
polyhedral surface
Series: Trends in Mathematics
2008, Approx. 320 p., Hardcover
ISBN: 978-3-7643-8603-0
Due: March 2008
About this book
This book consists of a collection of original, refereed research and expository articles on elliptic aspects of geometric analysis on manifolds, including singular, foliated and non-commutative spaces. The topics covered include the index of operators, torsion invariants, K-theory of operator algebras and L2-invariants.
The results presented in this book, which is largely inspired and stimulated by the Atiyah-Singer index theorem, should be of interest to graduates and researchers in mathematical physics, differential topology and differential analysis.
Table of contents
Preface.- Lefschetz Distribution of Lie Foliations.- Torsion as Function on the Space of Representations.- K-theory of Twisted Group Algebras.- Twisted Burnside Theorem for Two-Step Torsion-free Nilpotent Groups.- Ihara Zeta Functions for Periodic Simple Graphs.- Adiabatic Limits and the Spectrum of the Laplacian on Foliated Manifolds.- Non-standard Podle's Spheres.- Elliptic Theory on Manifolds with Corners.- Dixmier Traceability for General Pseudo-differential Operators.- Topological Invariants of Bifurcation.- L2-Invariants and Rank Metric.- Group Bundle Duality.- A New Topology on the Space of Unbounded Selfadjoint Operators.- Boundaries, Eta Invariant and the Determinant Bundle.- Modified Hochschild and Periodic Cyclic Homology.
2008, Approx. 400 p., Hardcover
ISBN: 978-3-7643-8605-4
Due: March 2008
About this book
After general properties of quadratic mappings over rings, the authors more intensely study quadratic forms, and especially their Clifford algebras. To this purpose they review the required part of commutative algebra, and they present a significant part of the theory of graded Azumaya algebras.
Interior multiplications and deformations of Clifford algebras are treated with the most efficient methods. The connection between orthogonal transformations and Clifford algebras is established in a new way, by means of Lipschitz monoids. Lipschitz monoids also allow a more efficient study of hyperbolic spaces.
Table of contents
Introduction.- 1. Algebraic Preliminaries.- 2. Quadratic Mappings.- 3. Clifford Algebras.- 4. Comultiplications. Exponentials. Deformations.- 5. Orthogonal Groups and Lipschitz Groups.- 6. Further Algebraic Developments.- 7. Hyperbolic Spaces.- 8. Other Topics.- Bibliography.- Index.