Series: Springer Monographs in Mathematics
1st Edition., 2011, X, 480 p., Hardcover
ISBN: 978-3-642-16151-3
Due: December 2010
This monograph deals with products of Dedekind's eta function, with Hecke theta series on quadratic number fields, and with Eisenstein series. The author brings to the public the large number of identities that have been discovered over the past 20 years, the majority of which have not been published elsewhere. The book will be of interest to graduate students and scholars in the field of number theory and, in particular, modular forms. It is not an introductory text in this field. Nevertheless, some theoretical background material is presented that is important for understanding the examples in Part II. In Part I relevant definitions and essential theorems -- such as a complete proof of the structure theorems for coprime residue class groups in quadratic number fields that are not easily accessible in the literature -- are provided. Another example is a thorough description of an algorithm for listing all eta products of given weight and level, together with proofs of some results on the bijection between these eta products and lattice simplices.
Introduction.- Part I: Theoretical background.- 1. Dedekind’s eta function and modular forms.- 2. Eta products.- 3. Eta products and lattice points in simplices.- 4. An algorithm for listing lattice points in a simplex.- 5. Theta series with Hecke character.- 6. Groups of coprime residues in quadratic fields.- Part II: Examples.-7. Ideal numbers for quadratic fields.- 8 Eta products of weight .- 9. Level 1: The full modular group.- 10. The prime level N = 2.- 11. The prime level N = 3.- 12. Prime levels N = p ? 5.- 13. Level N = 4.- 14. Levels N = p2 with primes p ? 3.- 15 Levels N = p3 and p4 for primes p.- 16. Levels N = pq with primes 3 ? p < q.- 17. Weight 1 for levels N = 2p with primes p ? 5.- 18. Level N = 6.- 19. Weight 1 for prime power levels p5 and p6.- 20. Levels p2q for distinct primes p = 2 and q.- 21. Levels 4p for the primes p = 23 and 19.- 22. Levels 4p for p = 17 and 13.- 23. Levels 4p for p = 11 and 7.- 24. Weight 1 for level N = 20.- 25. Cuspidal eta products of weight 1 for level 12.- 26. Non-cuspidal eta products of weight 1 for level 12.- 27. Weight 1 for Fricke groups Γ?(q3p).- 28. Weight 1 for Fricke groups Γ?(2pq).- 29. Weight 1 for Fricke groups Γ?(p2q2).- 30. Weight 1 for the Fricke groups Γ?(60) and Γ?(84).- 31. Some more levels 4pq with odd primes p _= q.- References.- Directory of Characters.- Index of Notations.- Index.
Series: Progress in Probability, Vol. 63
1st Edition., 2011, VIII, 492 p., Hardcover
ISBN: 978-3-0348-0020-4
Due: December 2010
This volume contains refereed research or review papers presented at the 6th Seminar on Stochastic Processes, Random Fields and Applications, which took place at the Centro Stefano Franscini (Monte Verita) in Ascona, Switzerland, in May 2008. The seminar focused mainly on stochastic partial differential equations, especially large deviations and control problems, on infinite dimensional analysis, particle systems and financial engineering, especially energy markets and climate models.
Preface.- List of participants.- I Stochastic Analysis and Random Fields.- The trace formula for the heat semigroup with polynomial potential.- Existence results for Fokker?Planck equations in Hilbert spaces.- Uniqueness in law of the Ito integral with respect to Levy noise.- Statistical inference and Malliavin calculus.- Hydrodynamics, probability and the geometry of the diffeomorphisms group.- On stochastic ergodic control in infinite dimensions.- Yet another look at Harris’ ergodic theorem for Markov chains.- Old and new examples of scale functions for spectrally negative Levy processes.- A visual criterion for identifying Ito diffusions as martingales or strict local martingales.- Are fractional Brownian motions predictable?.- Control of exit time for Lagrangian systems with weak noise.- A probabilistic deformation of calculus of variations with constraints.- Exponential integrability and DLR consistence of some rough functional.- A family of series representations of the multiparameter fractional Brownian motion.- The martingale problem for Markov solutions to the Navier-Stokes equations.- Functional inequalities for the Wasserstein Dirichlet form.- Entropic measure on multidimensional spaces.- Properties of strong local nondeterminism and local times of stable random fields.- II Stochastic Methods in Financial Models.- Hedging with residual risk: a BSDE approach.- Auto-tail dependence coefficients for stationary solutions of linear stochastic recurrence equations and for GARCH(1, 1).- The clean development mechanism and joint price formation for allowances and CERs.- Optimal investment problems with marked point processes.- Doubly stochastic CDO term structures.- A framework for dynamic hedging under convex risk measures.- On the stability of prices of contingent claims in incomplete models under statistical estimations.- Analyzing the fine structure of continous time stochastic processes.
Series: Lecture Notes in Mathematics, Number 2011
1st Edition., 2011, X, 276 p. 13 illus., 2 in color., Softcover
ISBN: 978-3-642-16285-5
Due: December 2010
This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Bohm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.
1 Introduction.- 2 Background Material.- 3 Harmonic Mappings.- 4 Evolution of the Curvature.- 5 Short-Time Existence.- 6 Uhlenbeck’s Trick.- 7 The Weak Maximum Principle.- 8 Regularity and Long-Time Existence.- 9 The Compactness Theorem for Riemannian Manifolds.- 10 The F-Functional and Gradient Flows.- 11 The W-Functional and Local Noncollapsing.- 12 An Algebraic Identity for Curvature Operators.- 13 The Cone Construction of Bohm and Wilking.- 14 Preserving Positive Isotropic Curvature.- 15 The Final Argument
1st Edition., 2011, Approx. 340 p. 1000 illus. in color., Hardcover
ISBN: 978-3-642-14647-3
Due: January 10, 2011
How can one visualize a curve that fills the entire plane or all of space? Can a polyhedron be smoothly turned inside out? What is the projective plane? What does four-dimensional space look like? Can soap bubbles exist that are not spherical? How can one better understand the structure of vortices and currents? In this book you will experience mathematics from the visual point of view, discovering fascinating and never previously published images that offer illustrative examples to the above questions. Every picture is accompanied by a brief explanatory text, references to further reading, and a number of web links where you can obtain further information. This book is intended for all friends of mathematics?students, teachers, amateurs, and professionals?who want to see something beyond dry text and endless formulas. It will provide inspiration for pursuing further one or another topic that may previously have seemed inaccessible. You will get to know mathematics from a totally new and colorful viewpoint.
1. Polyhedral Models: Platonic Solids.- Duality and Symmetry.- Archimedean Solids.- Johnson and Catalan Solids.- The Geometry of the Soccer Ball.- Special Tetrahedra.- The Altitude Regulator.- The Art of Unfolding.- 2. Geometry in the Plane: The Pythagorean Theorem.- The Nine-Point Circle.- Concentric Circles.- Metric and Projective Scales.- The Fermat Point.- Morley’s Theorem.- The Theorem of Fukuta and ?erin.- Maclaurin?Braikenridge Problems.- Derivation of the Addition Theorems.- Inscribed Squares and Equilateral Triangles.- Halving the Surface of a Triangle.- Every Angle Is a Right Angle? 3. Problems New and Old: Trisecting an Angle.- The Delian Cube Duplication Problem.- The Collatz Conjecture.- Dominoes on a Chessboard.- The Ham Sandwich Theorem.- Pick’s Theorem.- Goldbach’s Conjecture.- The Riemann Zeta Function.- 4. Formulas and the Integers: The Gauss Summation Formula.- Sums of Squares.- Sums of Fractions.- Pascal’s Triangle.- Pascal and Fibonacci.- Pascal’s Pyramids.- Estimating the Distribution of Prime Numbers.- Ulam’s Prime Number Spiral.- How Many Integers Are There?.- Mad Formulas Involving π.- 5. Functions and Limits: Nondifferentiable Functions.- Taylor Series.- Fourier Series and Periodic Waves.- Total versus Partial Differentiability.- The Weierstrass A-Function.- Solitons.- The Volume of the Sphere.- The Brouwer Fixed-Point Theorem.- 6. Curves and Knots: Conic Sections?Defined Planimetrically and Spatially.- Spherical Conic Sections and Confocal Conic Sections.- Dandelin Spheres.- Apollonian Circles.- Cubic Curves.- The Cassini Oval.- The Astroid.- Conchoids.- Geodesic Curves and Straightest Lines.- Zoll Surfaces.- Geodesics on Polyhedra.- Knots.- Celtic Knots.- Borromean Rings.- 7. Geometry and Topology of Surfaces: Hyperboloids and Paraboloids.- Quadrics and Circular Sections.- The Clebsch Surface and Singular Cubics.- Dupin Cyclides.- Supercyclides.- Plucker’s Conoid.- Helices and Spirals.- Rotoid Helicoids.- Collar Surfaces and Developable Strips.- The Pseudosphere.- The Kuen Surface.- The Csaszar Torus.- The Mobius Strip.- The Klein Bottle.- Models of the Projective Plane.- Seifert Surfaces.- Alexander’s Horned Sphere.- Turning the Sphere Inside Out.- 8. Minimal Surfaces and Soap Bubbles: Minimal Surfaces and Soap Films.- Classical Minimal Surfaces.- The Gergonne Problem.- From Catenoid to Helicoid.- The Catenoid and Its Variations.- Periodic Minimal Surfaces.- Costa’s Minimal Surface.- Discrete Minimal Surfaces.- Surfaces from Circle Patterns.- The Wente Surface.- Closed Soap Bubbles.- The Penta Surface.- 9. Tilings and Packings: Frieze Ornaments.- Ornamentation.- Aperiodic Tilings.- Kissing Number.- Space Tilings.- The Weaire?Phelan Foam and Optimal Space Packings.- Planar and Spatial Voronoi Diagrams.- 10. Space Forms and Dimension.- The Hyperbolic Plane.- Escher’s Hyperbolic Plane.- Ideal Polyhedra in Hyperbolic Space.- The Shape of Space.- The Four-Dimensional Cube and Its Unfolding.- The Hyperdodecahedron.- 120 Cells and More!.- 11. Graphs and Incidence Geometry: Pascal’s Theorem and Its Dual.- Desargues’s Theorem.- Tangent Circles.- Escape into Space.- Systems of Curves Define Regions.- The Petersen Graph.- Hamiltonian and Eulerian Circuits.- Venn Diagrams.- Schlegel Diagrams.- Minimal Spanning Trees.- Counting Triangulations.- 12. Movable Forms: Elliptic Motion.- Movable Polyhedra.- Trajectories and Envelopes.- Constrained Spatial Motion.- Degrees of Freedom.- The Rolling Reuleaux Triangle.- The Gomboc.- 13. Fractals: The Pythagoras Tree.- Filling Space and the Plane with a Closed Curve.- Hilbert Curves on the Sphere.- Fractal Dimension.- The Menger Sponge.- Julia Sets and the Mandelbrot Set.- The Feigenbaum Diagram.- The Lorenz Attractor.- Curlicue Fractals.- Random Walks.- Percolation.- 14. Maps and Mappings: Isometric Maps.- Gnomonic Projection.- Inversion and Projection.- The Silhouette of a Sphere.- Mobius Transformations from Motions of the Sphere.- The Riemann Mapping Theorem.- The Schwarz?Christoffel Mapping.- Parameterization of Surfaces.- Space Collineation.- Zeros of Complex Functions.- The Riemann Sphere.- Domain Coloring.- The Szeg? Curve.- Polynomiography.- Zeros of Polynomials.- 15. Forms and Processes in Nature and Technology: Numbers in Motion.- The Karman Vortex Street.- The Topology of Currents.- Streamlines.- Electric Field Lines.- Smoothing of Three-Dimensional Scanned Data.- Vibrations.- The Traveling Salesman Problem.- Sorting Algorithms.- The DNA Double Helix.- Virtual Maxillary Surgery.- Radiolarians.- Epipolar Geometry.- From Photograph to Spatial Location.- Reflections.- Picture Credits.- Index
Series: Operator Theory: Advances and Applications, Vol. 214
1st Edition., 2011, 350 p., Hardcover
ISBN: 978-3-0348-0048-8
Due: January 31, 2011
This volume consists of eighteen peer-reviewed papers related to lectures on pseudo-differential operators presented at the meeting of the ISAAC Group in Pseudo-Differential Operators (IGPDO) held at Imperial College London on July 13-18, 2009. Featured in this volume are the analysis, applications and computations of pseudo-differential operators in mathematics, physics and signal analysis. This volume is a useful complement to the volumes “Advances in Pseudo-Differential Operators”, “Pseudo-Differential Operators and Related Topics”, “Modern Trends in Pseudo-Differential Operators”, “New Developments in Pseudo-Differential Operators” and “Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations” published in the same series in, respectively, 2004, 2006, 2007, 2009 and 2010.
Preface.- Adaptive Wavelet Computations for Inverses of Pseudo-Differential
Operators.- Spectral Theory of Pseudo-Differential Operators on S1.- A
Characterization of Compact Pseudo-Differential Operators on S1.- Melin
Operator with Asymptotics on Manifolds with Corners.- The Iterative Structure
of the Corner Calculus.- Elliptic Equations and Boundary Value Problems
in Non-Smooth Domains.- Calculus of Pseudo-Differential Operators and a
Local Index of Dirac Operators.- Lp Bounds for a Class of Fractional Powers
of Subelliptic Operators.- The Heat Kernel and Green Function of the Generalized
Hermite Operator, and the Abstract Cauchy Problem for the Abstract Hermite
Operator.- Local Exponential Estimates for h-Pseudo-Differential Operators
with Operator-Valued Symbols.- Global Solvability in Functional Spaces
for Smooth Nonsingular Vector Fields in the Plane.- Fuchsian Mild Microfunctions
with Fractional Order and their Applications to Hyperbolic Equations.-
The Continuity of Solutions with respect to a Parameter to Symmetric Hyperbolic
Systems.- Generalized Gevrey Ultradistributions and their Microlocal Analysis.-
Weyl Rule and Pseudo-Differential Operators for Arbitrary Operators.- Time-Frequency
Characterization of Stochastic Differential Equations.- Wigner Representation
Associated with Linear Transformations of the Time-Frequency Plane.- Some
Remarks on Localization Operators