Series: Lecture Notes in Mathematics, Vol. 1973
Subseries: Fondazione C.I.M.E., Firenze ,
2009, XII, 223 p. 101 illus., Softcover
ISBN: 978-3-642-00836-8
Due: May 21, 2009
Helmholtz's seminal paper on vortex motion (1858) marks the beginning of what is now called topological fluid mechanics.After 150 years of work, the field has grown considerably. In the last several decades unexpected developments have given topological fluid mechanics new impetus, benefiting from the impressive progress in knot theory and geometric topology on the one hand, and in mathematical and computational fluid dynamics on the other.
This volume contains a wide-ranging collection of up-to-date, valuable research papers written by some of the most eminent experts in the field. Topics range from fundamental aspects of mathematical fluid mechanics, including topological vortex dynamics and magnetohydrodynamics, integrability issues, Hamiltonian structures and singularity formation, to DNA tangles and knotted DNAs in sedimentation. A substantial introductory chapter on knots and links, covering elements of modern braid theory and knot polynomials, as well as more advanced topics in knot classification, provides an invaluable addition to this material.
Braids and Knots.- Topological quantities: calculating winding, writhing, linking, and higher order invariants.- Tangles, Rational Knots and DNA.- The group and Hamiltonian descriptions of hydrodynamical systems.- Singularities in fluid dynamics and their resolution.- Structural complexity and dynamical systems.- Random Knotting: Theorems, Simulations and Applications
Series: Nonlinear Physical Science
2009, Approx. 700 p. 14 illus., Hardcover
ISBN: 978-3-642-00250-2
Due: May 2009
"Partial Differential Equations and Solitary Waves Theory" is a self-contained book divided into two parts: Part I is a coherent survey bringing together newly developed methods for solving PDEs. While some traditional techniques are presented, this part does not require thorough understanding of abstract theories or compact concepts. Well-selected worked examples and exercises shall guide the reader through the text. Part II provides an extensive exposition of the solitary waves theory. This part handles nonlinear evolution equations by methods such as Hirotafs bilinear method or the tanh-coth method. A self-contained treatment is presented to discuss complete integrability of a wide class of nonlinear equations. This part presents in an accessible manner a systematic presentation of solitons, multi-soliton solutions, kinks, peakons, cuspons, and compactons.
While the whole book can be used as a text for advanced undergraduate and graduate students in applied mathematics, physics and engineering, Part II will be most useful for graduate students and researchers in mathematics, engineering, and other related fields.
1. Basic Concepts.- 2. First Order PDE.- 3. One-Dimensional Heat Flow.- 4. Higher Dimensional Heat Flow.- 5. One Dimensional Wave Equation.- 6. Higher Dimensional Wave Equation.- 7. Laplace's Equation.- 8. Nonlinear Equations.- 9. Physical Models.- 10. Numerical Applications.- 11. Solitons and Compactons.- 12. Solitary Wave Theory.- 13. The Family of the KdV Equations.- 14. KdV and mKdV Equations of Higher-Orders.- 15. Family of KdV-Type Equations.- 16. Boussinesq, Klein-Gordon and Liouville Equations.- 17. Burgers, Fisher and Related Equations.- 18. Camassa-Holm and Schrodinger Equations.
Series: Abel Symposia , Vol. 4
2009, Approx. 390 p., Hardcover
ISBN: 978-3-642-01199-3
Due: June 24, 2009
The 2007 Abel Symposium took place at the University of Oslo in August 2007. The goal of the symposium was to bring together mathematicians whose research efforts have led to recent advances in algebraic geometry, algebraic K-theory, algebraic topology, and mathematical physics. A common theme of this symposium was the development of new perspectives and new constructions with a categorical flavor. As the lectures at the symposium and the papers of this volume demonstrate, these perspectives and constructions have enabled a broadening of vistas, a synergy between once-differentiated subjects, and solutions to mathematical problems both old and new.
C.Baez, D.Stevenson: The classifying space of a topological 2-group.- M.Chas, D.Sullivan: String topology in dimensions two and three.- R.Cohen: Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners.- G.Cortinas, C.Weibel: Relative Chern characters for nilpotent ideals.- H.Esnault: Algebraic differential characters of flat connections with nilpotent residues.- C.Haesemeyer, C.Weibel: Norm varieties and the Chain Lemma.- L.Hesselholt: On the Whitehead spectrum of the circle.- J.F.Jardine: Cocycle categories.- J.Lurie: A survey of elliptic cohomology.- I.Panin, K.Pimenov, O.Rondigs: On Voevodsky's algebraic K-theory spectrum.- B.Toen, G.Vezzosi: Chern character, loop spaces and derived algebraic geometry.- V.Voevodsky: Lectures on motivic cohomology 2000/2001
Series: Lecture Notes in Mathematics , Vol. 1976
2009, Approx. 150 p., Softcover
ISBN: 978-3-642-01569-4
Due: June 25, 2009
This volume surveys the spectral properties of the spin Dirac operator. After a brief introduction to spin geometry, we present the main known estimates for Dirac eigenvalues on compact manifolds with or without boundaries. We give examples where the spectrum can be made explicit and present a chapter dealing with the non-compact setting. The methods mostly involve elementary analytical techniques and are therefore accessible for Master students entering the subject. A complete and updated list of references is also included.
1 Basics of spin geometry.- 2 Explicit computations of spectra.- 3 Lower eigenvalue estimates on closed manifolds.- 4 Lower eigenvalue estimates on compact manifolds with boundary.- 5 Upper eigenvalue bounds on closed manifolds.- 6 Prescription of eigenvalues on closed manifolds.- 7 The Dirac spectrum on non-compact manifolds.- 8 Other topics related with the Dirac spectrum.- A The twistor and Killing spinor equations
Series: Lecture Notes in Mathematics, Vol. 1977
Subseries: Fondazione C.I.M.E., Firenze
2009, Approx. 270 p., Softcover
ISBN: 978-3-642-01673-8
Due: June 26, 2009
This volume contains lecture notes on some topics in geometric analysis, a growing mathematical subject which uses analytical techniques, mostly of partial differential equations, to treat problems in differential geometry and mathematical physics. The presentation of the material should be rather accessible to non-experts in the field, since the presentation is didactic in nature. The reader will be provided with a survey containing some of the most exciting topics in the field, with a series of techniques used to treat such problems.
PDEs in Conformal Geometry.- Heat kernels in sub-Riemannian settings.- Concentration of solutions for some singularly perturbed Neumann problems.- On some elliptic problems in the study of selfdualChern-Simons vortices.- The k-Hessian equation.- Minimal Surfaces in CR Geometry