Series: Mathematics Research Developments

Pub. Date: 2014 - 1st Quarter

Pages (Approximate):233 Pages (7x10) - (NBC-C)

Binding: Hardcover

ISBN: 978-1-62948-831-8

This book explores a number of new applications of invariant quasi-finite diffused Borel measures in Polish groups for a solution of various problems stated by famous mathematicians (for example, Carmichael, Erdos, Fremlin, Darji and so on). By using natural Borel embeddings of an infinite-dimensional function space into the standard topological vector space of all real-valued sequences, (endowed with the Tychonoff topology) a new approach for the construction of different translation-invariant quasi-finite diffused Borel measures with suitable properties and for their applications in a solution of various partial differential equations in an entire vector space is proposed.

Introduction

Chapter 1. On Ordinary and Standard Lebesgue Measures in R

Chapter 2. On Uniformly Distributed Sequences of an Increasing Family of Finite Sets in Infinite-Dimensional Rectangles

Chapter 3. Change of variable formula for the ¿-ordinary Lebesgue measure in RN

Chapter 4. On Existence and Uniqueness of Generators of Shy Sets in Polish Groups

Chapter 5. On a Certain Criterion of Shyness for Subsets in the Product of Unimodular Polish Groups that are not Compact

Chapter 6. On Ordinary and Standard hLebesgue Measuresh in Separable Banach Spaces

Chapter 7. On a Standard Product of an Arbitrary Family of Ð-Finite Borel Measures with Domain in Polish Spaces

For Complete Table of Contents, please visit our website at

This book presents and discusses new developments in the study of evolution equations. Topics discussed include parabolic equations; generalized gradient in weak maximum principle with non-differentiable drift; bifluid systems; Yamabe-type flows; stochastic evolution equations; heat equations; Navier-Stokes equations; Cahn-Hilliard equations; and more.(Imprint: Nova)

Preface

Regularity of IBPV for Parabolic Equations in Polyhedral Domains

Vu Trong Luong and Do Van Loi

Generalized Gradient in Weak Maximum Principle with Non-differentiable Drift

Mokhtar Hafayed and Syed Abbas

Renormalized Solutions of Nonlinear Degenerated Parabolic Equations without Sign Condition and L1 Data

Youssef Akdim, Jaouad Bennouna, and Mounir Mekkour

On a Bifluid System with High Density Ratio

Mohamed Dahi, Timack Ngom, and Mamadou Sy

Notes on Yamabe-Type Flows

Huan Zhu

Unbounded Perturbation of the Controllability for Stochastic Evolution Equations

Hugo Leiva, Nelson Merentes, Miguel Narvaez, and Nguyen Dinh Phu

Nonlinear Evolution Equation for a Magneto-Convective Flow in a Passive Mushy Layer

Dambaru Bhatta and Daniel N. Riahi

Abstract Fractional Integro-differential Equations with State-dependent Delay

Mouffak Benchohra, Sara Litimein, Juan J. Trujillo, and M. P. Velasco

Alternative Food Induced Predator-Prey Oscillations in an Eco-epidemiological Model

Krishna Pada Das, Kakali Ghosh, and J. Chattopadhyay

On the Gevrey Wave Front Set

Amor Kessab

Boundedness Properties of Solutions to Stochastic Set Differential Equations with Selectors

Nguyen Dinh Phu, Ngo Van Hoa, Nguyen Minh Triet, and Ho Vu

Blow Up Rate and Blow Up Sets for Degenerate Parabolic Equations Coupled via Nonlinear Boundary Flux

Si Xu

The Boundary Value Condition of a Degenerate Parabolic Equation

Huashui Zhan

Asymptotic Behavior for Retarded Non-Autonomous Quasilinear Parabolic Equations

Cung The Anh and Le Van Hieu

Asymptotic Equivalence of Abstract Evolution Equations

Dang Dinh Chau and Nguyen Manh Cuong

Periodic Solutions in the Alpha-Norm for Some Partial Functional Differential Equations with Finite Delay

Khalil Ezzinbi, Bila Adolphe Kyelem, and Stanislas Ouaro

C-Semigroups and Almost Periodic Solution of Non-Autonomous Evolution Equation

Indira Mishra

Global Unique Weak Solutions of Non-local Degenerate Reaction-Diffusion Equations in Weighted Sobolev's Spaces

Sikiru Adigun Sanni

Identification and Control of a Heat Equation

Fadhel Al-Musallam and Amin Boumenir

The Structural Role of the Basilar Membrane in the Hearing Process

Bernard Singleton and Hadi Alkahby

Observation of Signal Transmission on the Nerve Axon with Time Delay

Bernard Singleton and Hadi Alkahby

On General Axisymmetric Explicit Solutions for the Navier-Stokes Equations

Dao Quang Khai and Nguyen Minh Tri

On the Viscous Convective Cahn-Hilliard Equation

Assane Lo

Existence and No Existence of Mild Solutions for a Fractional Evolution System

Samira Rihani and Amor Kessab

Boundary Value Problems for Abstract First Order Non Homogeneous Differential Equations with Integral Conditions

M. Denche and A. Berkane

Index

Series:

Mathematics Research Developments

Binding: Hardcover

Pub. Date: 2014 - 1st Quarter

Pages: 7x10 - (NBC-M)

ISBN: 978-1-63117-025-6

A Very Short Introduction

152 pages | 15 black and white line drawings | 174x111mm

978-0-19-966254-8 | Paperback | July 2014 (estimated)

Focuses on the scientific and mathematical principles underlying the science of complexity

Shows how many processes in nature can be understood using complexity

Explores a series of examples to highlight and explain the various types of complex systems

Includes examples from the physical and biological sciences, as well as the applications in economics and management

Part of the bestselling Very Short Introductions series - over six million copies sold worldwide

The importance of complexity is well-captured by Hawking's comment: "Complexity is the science of the 21st century". From the movement of flocks of birds to the Internet, environmental sustainability, and market regulation, the study and understanding of complex non-linear systems has become highly influential over the last 30 years.

In this Very Short Introduction, one of the leading figures in the field, John Holland, introduces the key elements and conceptual framework of complexity. From complex physical systems such as fluid flow and the difficulties of predicting weather, to complex adaptive systems such as the highly diverse and interdependent ecosystems of rainforests, he combines simple, well-known examples ? Adam Smith's pin factory, Darwin's comet orchid, and Simon's 'watchmaker' ? with an account of the approaches, involving agents and urn models, taken by complexity theory.

ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Readership: Ideal for students in a number of disciplines, including the physical and biological sciences, economics, mathematics, and business and management, as well as the general reader who wishes to understand the underlying scientific principles of complexity.

1: Complex systems

2: Complex physical systems

3: Complex adaptive systems

4: Agents, networks, degree, and recirculation

5: Specialization and diversity

6: Emergence

7: Co-evolution and the formation of niches

8: Putting it all together

Further reading

Index

Paper | February 2014 | ISBN: 9780691160788

Cloth | February 2014 | ISBN: 9780691160757

208 pp. | 7 x 10 | 1 line illus

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic.

Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

Christopher D. Sogge is the J. J. Sylvester Professor of Mathematics at Johns Hopkins University. He is the author of Fourier Integrals in Classical Analysis and Lectures on Nonlinear Wave Equations.

Preface ix

1 A review: The Laplacian and the d'Alembertian 1

1.1 The Laplacian 1

1.2 Fundamental solutions of the d'Alembertian 6

2 Geodesics and the Hadamard parametrix 16

2.1 Laplace-Beltrami operators 16

2.2 Some elliptic regularity estimates 20

2.3 Geodesics and normal coordinates|a brief review 24

2.4 The Hadamard parametrix 31

3 The sharp Weyl formula 39

3.1 Eigenfunction expansions 39

3.2 Sup-norm estimates for eigenfunctions and spectral clusters 48

3.3 Spectral asymptotics: The sharp Weyl formula 53

3.4 Sharpness: Spherical harmonics 55

3.5 Improved results: The torus 58

3.6 Further improvements: Manifolds with nonpositive curvature 65

4 Stationary phase and microlocal analysis 71

4.1 The method of stationary phase 71

4.2 Pseudodi erential operators 86

4.3 Propagation of singularities and Egorov's theorem 103

4.4 The Friedrichs quantization 111

5 Improved spectral asymptotics and periodic geodesics 120

5.1 Periodic geodesics and trace regularity 120

5.2 Trace estimates 123

5.3 The Duistermaat-Guillemin theorem 132

5.4 Geodesic loops and improved sup-norm estimates 136

6 Classical and quantum ergodicity 141

6.1 Classical ergodicity 141

6.2 Quantum ergodicity 153

Appendix 165

A.1 The Fourier transform and the spaces S(?n) and S'(?n)) 165

A.2 The spaces D'(¶) and E'(¶) 169

A.3 Homogeneous distributions 173

A.4 Pullbacks of distributions 176

A.5 Convolution of distributions 179

Notes 183

Bibliography 185

Index 191

Symbol Glossary 193

Paper | February 2014 |ISBN: 9780691160511

Cloth | February 2014 | ISBN: 9780691160504

176 pp. | 6 x 9 |

In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by Voisin. The book focuses on two central objects: the diagonal of a variety--and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups--as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by Voisin looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kahler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Preface vii

1 Introduction 1

1.1 Decomposition of the diagonal and spread 3

1.2 The generalized Bloch conjecture 7

1.3 Decomposition of the small diagonal and application to the topology of families 9

1.4 Integral coefficients and birational invariants 11

1.5 Organization of the text 13

2 Review of Hodge theory and algebraic cycles 15

2.1 Chow groups 15

2.2 Hodge structures 24

3 Decomposition of the diagonal 36

3.1 A general principle 36

3.2 Varieties with small Chow groups 44

4 Chow groups of large coniveau complete intersections 55

4.1 Hodge coniveau of complete intersections 55

4.2 Coniveau 2 complete intersections 64

4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67

4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86

5 On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88

5.1 Tautological ring of a K3 surface 88

5.2 A decomposition of the small diagonal 96

5.3 Deligne's decomposition theorem for families of K3 surfaces 106

6 Integral coefficients 123

6.1 Integral Hodge classes and birational invariants 123

6.2 Rationally connected varieties and the rationality problem 127

6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139

Bibliography 155

Index 163

Paper | June 2014 | ISBN: 9780691161341

602 pp. | 6 x 9 | 10 line illus.

This book provides a comprehensive and up-to-date introduction to Hodge theory--one of the central and most vibrant areas of contemporary mathematics--from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch-Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ITCP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and doesn't require a deep background. At the same time, the book presents some topics at the forefront of current research.

The book is divided between introductory and advanced lectures. The introductory lectures address Kahler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch-Beilinson Conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's Algebraic De Rham Theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's Theorem on Absolute Hodge Cycles), and variation of mixed Hodge structures.

The contributors include Patrick Brosnan, James A. Carlson, Eduardo Cattani, Francois Charles, Mark Andrea de Cataldo, Fouad El Zein, Mark Green, Phillip A. Griffiths, Matt Kerr, Luca Migliorini, Jacob Murre, Christian Schnell, Le D?ng Trang, and Loring Tu.

Eduardo Cattani is professor of mathematics at the University of Massachusetts, Amherst. Fouad El Zein is a researcher at the Institut de Mathematiques de Jussieu, Universite de Paris VII. Phillip A. Griffiths is former director and professor emeritus of mathematics at the Institute for Advanced Study in Princeton. Le D?ng Trang is professor emeritus of mathematics at the Universite d'Aix-Marseille.