Series: Lecture Notes in Mathematics , Vol. 1955
2008, Approx. 215 p., Softcover
ISBN: 978-3-540-69314-7
Due: August 6, 2008
The transportation problem can be formalized as the problem of finding the optimal way to transport a given measure into another with the same mass. In contrast to the Monge-Kantorovitch problem, recent approaches model the branched structure of such supply networks as minima of an energy functional whose essential feature is to favour wide roads. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electrical power supply systems and in natural counterparts such as blood vessels or the branches of trees.
These lectures provide mathematical proof of several existence, structure and regularity properties empirically observed in transportation networks. The link with previous discrete physical models of irrigation and erosion models in geomorphology and with discrete telecommunication and transportation models is discussed. It will be mathematically proven that the majority fit in the simple model sketched in this volume.
1 Introduction: the models.- 2 The mathematical models.- 3 Traffic plans.- 4 The structure of optimal traffic plans.- 5 Operations on traffic plans.- 6 Traffic plans and distances between measures.- 7 The tree structure of optimal traffic plans and their approximation.- 8 Interior and boundary regularity.- 9 The equivalence of various models.- 10 Irrigability and dimension.- 11 The landscape of an optimal pattern.- 12 The Gilbert-Steiner problem.- 13 Dirac to Lebesgue segment: a case study.- 14 Application: embedded irrigation networks .- 15 Open problems.- A Skorokhod Theorem.- B Flows in tubes.- C Notations
Series: Lecture Notes in Mathematics , Vol. 1956
2008, Approx. 130 p., Softcover
ISBN: 978-3-540-69797-8
Due: August 27, 2008
In the last decade, convolution operators of matrix functions have received unusual attention due to their diverse applications. This monograph presents some new developments in the spectral theory of these operators. The setting is the Lp spaces of matrix-valued functions on locally compact groups. The focus is on the spectra and eigenspaces of convolution operators on these spaces, defined by matrix-valued measures. Among various spectral results, the L2-spectrum of such an operator is completely determined and as an application, the spectrum of a discrete Laplacian on a homogeneous graph is computed using this result. The contractivity properties of matrix convolution semigroups are studied and applications to harmonic functions on Lie groups and Riemannian symmetric spaces are discussed. An interesting feature is the presence of Jordan algebraic structures in matrix-harmonic functions.
1. Introduction.- 2. Lebesgue spaces of matrix functions.- 3. Matrix convolution operators.- 4. Convolution semigroups
Series: Lecture Notes in Mathematics , Vol. 1958
2008, Approx. 290 p., Softcover
ISBN: 978-3-540-70518-5
Due: August 20, 2008
This volume presents the construction of canonical modular compactifications of moduli spaces for polarized Abelian varieties (possibly with level structure), building on the earlier work of Alexeev, Nakamura, and Namikawa. This provides a different approach to compactifying these spaces than the more classical approach using toroical embeddings, which are not canonical. There are two main new contributions in this monograph: (1) The introduction of logarithmic geometry as understood by Fontaine, Illusie, and Kato to the study of degenerating Abelian varieties; and (2) the construction of canonical compactifications for moduli spaces with higher degree polarizations based on stack-theoretic techniques and a study of the theta group.
1. A brief primer on algebraic stacks.- 2. Preliminaries.- 3. Moduli of broken toric varieties.- 4. Moduli of principally polarized abelian varieties.- 5. Moduli of abelian varieties with higher degree polarizations.- 6. Level Structure
2008, Approx. 400 p. 30 illus., Softcover
ISBN: 978-0-387-79145-6
Due: September 2008
Preface.- Solvable Differential Equations.- Second Order Differential Equations.- Preliminaries to Series Solutions.- Solution at an Ordinary Point.- Solution at a Singular Point.- Solution at a Singular Point (Continued).- Legendre Polynomials and Functions.- Chebyshev, Hermite and Laguerre Polynomials.- Bessel Functions.- Hypergeometric Functions.- Piecewise Continuous and Periodic Functions.- Orthogonal Functions and Polynomials.- Orthogonal Functions and Polynomials (Continued).- Boundary Value Problems.- Boundary Value Problems (Continued).- Greenfs Functions.- Regular Perturbations.- Singular Perturbations.- Sturm-Liouville Problems.- Eigenfunction Expansions.- Eigenfunction Expansions (Continued).- Convergence of the Fourier Series.- Convergence of the Fourier Series (Continued).- Fourier Series Solutions of Ordinary Differential Equations.- Partial Differential Equations.- First Order Partial Differential Equations.- Solvable Partial Differential Equations.- The Canonical Forms.- The Method of Separation of Variables.- The One-Dimensional Heat Equation.- The One-Dimensional Heat Equation (Continued).- The One-Dimensional Wave Equation.- The One-Dimensional Wave Equation (Continued).- Laplace Equation in Two Dimensions.- Laplace Equation in Polar Coordinates.- Two-Dimensional Heat Equation.- Two-Dimensional Wave Equation.- Laplace Equation in Three Dimensions.- Laplace Equation in Three Dimensions (Continued).- Nonhomogeneous Equations.- Fourier Integral and Transforms.- Fourier Integral and Transforms (Continued).- Fourier Transform Method for PDEs.- Fourier Transform Method for PDEs (Continued).- Laplace Transforms.- Laplace Transforms (Continued).- Laplace Transform Method for ODEs.- Laplace Transform Method for PDEs.- Well-Posed Problems.- Verification of Solutions.- References for Further Reading.- Index.
Series: Universitext
2008, Approx. 385 p., Softcover
ISBN: 978-3-540-69318-5
Due: September 10, 2008
1 Historical Background.- 2 Primitive Recursive Functions and Relations.- 2.1 Primitive Recursive Functions.- 2.2 Primitive Recursive Relations.- 3 Ordinals.- 3.1 Heuristic.- 3.2 Some Basic Facts on Ordinals.- 3.3 Fundamentals of Ordinal Arithmetic.- 3.3.1 A Notation System for the Ordinals below epsilon nought.- 3.4 The Veblen Hierarchy.- 3.4.1 Preliminaries.- 3.4.2 The Veblen Hierarchy.- 3.4.3 A Notation System for the Ordinals below Gamma nought.- 4 Pure Logic.- 4.1 Heiristics.- 4.2 First and Second Order Logic.- 4.3 The Tait calculus.- 4.4 Trees and the Completeness Theorem.- 4.5 Gentzens Hauptsatz for Pure First Order Logic.- 4.6 Second Order Logic.- 5 Truth Complexities for Pi 1-1-Sentences.- 5.1 The language of Arithmetic.- 5.2 The Tait language for Second Order Arithmetic.- 5.3 Truth Complexities for Arithmetical Sentences.- 5.4 Truth Complexities for Pi 1-1-Sentences.- 6 Inductive Definitions.- 6.1 Motivation.- 6.2 Inductive Definitions as Monotone Operators.- 6.3 The Stages of an Inductive Definition.- 6.4 Arithmetically Definable Inductive Definitions.- 6.5 Inductive Definitions, Well-Orderings and Well-Founded Trees.- 6.6 Inductive Definitions and Truth Complexities.- 6.7 The Pi-1-1- Ordinal of a Theory.- 7 The Ordinal Analysis for Pean Arithmetic.- 7.1 The Theory PA.- 7.2 The Theory NT.- 7.3 The Upper Bound.- 7.4 The Lower Bound.- 7.5 The Use of Gentzen's Consistency Proof for Hilbert's Programme.- 7.5.1 On the Consistency of Formal and Semi-Formal Systems.- 7.5.2 The Consistency of NT.- 7.5.3 Kreisel's Counterexample.- 7.5.4 Gentzen's Consistency Proof in the Light of Hilbert's Programme.- 8 Autonomous Ordinals and the Limits of Predicativity.- 8.1 The Language L-kappa.- 8.2 Semantics for L-kappa.- 8.3 Autonomous Ordinals.- 8.4 The Upper Bound for Autonomous Ordinals.- 8.5 The Lower Bound for Autonomous Ordinals.- 9 Ordinal Analysis of the Theory for Inductive Definitions.- 9.1 The Theory ID1.- 9.2 The Language L infinity (NT).- 9.3 The Semi-Formal System for L infinity (NT).- 9.3.1 Semantical Cut-Elimination.- 9.3.2 Operator Controlled Derivations.- 9.4 The Collapsing Theorem for ID1.- 9.5 The Upper Bound.- 9.6 The Lower Bound.- 9.6.1 Coding Ordinals in L(NT).- 9.6.2 The Well-Ordering Proof.- 9.7 Alternative Interpretations for Omega.- 10 Provably Recursive Functions of NT.- 10.1 Provably Recursive Functions of a Theory.- 10.2 Operator Controlled Derivations.- 10.3 Iterating Operators.- 10.4 Cut Elimination for Operator Controlled Derivations.- 10.5 The Embedding of NT.- 10.6 Discussion.- 11 Ordinal Analysis for Kripke Platek Set Theory with infinity.- 11.1 Naive Set Theory.- 11.2 The Language of Set Theory.- 11.3 Constructible Sets.- 11.4 Kripke Platek Set Theory.- 11.5 ID1 as a Subtheory of Kp-omega.- 11.6 Variations of KP-omega and Axiom beta.- 11.7 The Sigma Ordinal of KP-omega.- 11.8 The Theory of Pi-2 Reflection.- 11.9 An Infinite Verification Calculus for the Constructible Hierarchy.- 11.10 A Semi-Formal System for Ramified Set Theory.- 11.11 The Collapsing Theorem for Ramified Set Theory.- 11.12 Ordinal Analysis for Kripke Platek Set Theory.- 12 Predicativity Revisited.- 12.1 Admissible Extensions.- 12.2 M-Logic.- 12.3 Extending Semi-Formal Systems.- 12.4 Asymmetric Interpretations.- 12.5 Reduction of T+ to T.- 12.6 The Theories KP n and KP 0-n.- 12.7 The Theories KPl 0 and KP i 0.- 13 Non-Monotone Inductive Definitions.- 13.1 Non-Monotone Inductive Definitions.- 13.2 Prewellorderings.- 13.3 The Theory for Pi 0-1 definable Fixed-Points.- 13.4 ID1 as a Sub-Theory of the Theory for Pi 0-1 definable Fixed-Points.- 13.5 The Upper Bound for the Proof theoretical Ordinal of Pi 0-1-FXP.- 14 Epilogue.