Book Description

The Riemann problem is the most fundamental problem in the entire field of non-linear hyperbolic conservation laws. Since first posed and solved in 1860, great progress has been achieved in the one-dimensional case. However, the two-dimensional case is substantially different. Although research interest in it has lasted more than a century, it has yielded almost no analytical demonstration. It remains a great challenge for mathematicians.This volume presents work on the two-dimensional Riemann problem carried out over the last 20 years by a Chinese group. The authors explore four models: scalar conservation laws, compressible Euler equations, zero-pressure gas dynamics, and pressure-gradient equations. They use the method of generalized characteristic analysis plus numerical experiments to demonstrate the elementary field interaction patterns of shocks, rarefaction waves, and slip lines. They also discover a most interesting feature for zero-pressure gas dynamics: a new kind of elementary wave appearing in the interaction of slip lines-a weighted Dirac delta shock of the density function.The Two-Dimensional Riemann Problem in Gas Dynamics establishes the rigorous mathematical theory of delta-shocks and Mach reflection-like patterns for zero-pressure gas dynamics, clarifies the boundaries of interaction of elementary waves, demonstrates the interesting spatial interaction of slip lines, and proposes a series of open problems. With applications ranging from engineering to astrophysics, and as the first book to examine the two-dimensional Riemann problem, this volume will prove fascinating to mathematicians and hold great interest for physicists and engineers. ... Read more

Book Description

This independent account of modern ideas in differential geometry shows how they can be used to understand and extend classical results in integral geometry. The authors explore the influence of total curvature on the metric structure of complete, non-compact Riemannian 2-manifolds, although their work can be extended to more general spaces. Each chapter features open problems, making the volume a suitable learning aid for graduate students and non-specialists who seek an introduction to this modern area of differential geometry. ... Read more

Book Description

This book documents the recent focus on a branch of Riemannian geometry called Comparison Geometry. The simple idea of comparing the geometry of an arbitrary Riemannian manifold with the geometries of constant curvature spaces has seen a tremendous evolution recently. This volume is an up-to-date reflection of the recent development regarding spaces with lower (or two-sided) curvature bounds. The content reflects some of the most exciting activities in comparison geometry during the year and especially of the Mathematical Sciences Research Institute's workshop devoted to the subject. This volume features both survey and research articles.It also provides complete proofs:in one case, a new, unified strategy is presented and new proofs are offered. This volume will be a valuable source for advanced researchers and those who wish to learn about and contribute to this beautiful subject. ... Read more

Book Description

This book develops new unified methods which lead to results inparts of mathematical physics traditionally considered as being farapart. The emphasis is three-fold: Firstly, this volume unifies three independently developed approachesto stochastic differential equations on manifolds, namely the theoryof Itô equations in the form of Belopolskaya-Dalecky,Nelson's construction of the so-called mean derivatives of stochasticprocesses and the author's construction of stochastic line integralswith Riemannian parallel translation. Secondly, the book includes applications such as the Langevin equationof statistical mechanics. Nelson's stochastic mechanics (a version ofquantum mechanics), and the hydrodynamics of viscous incompressiblefluid treated with the modern Lagrange formalism. Considering thesetopics together has become possible following the discovery of theircommon mathematical nature. Thirdly, the work contains sufficient preliminary and backgroundmaterial from coordinate-free differential geometry and from thetheory of stochastic differential equations to make it self-containedand convenient for mathematicians and mathematical physicists notfamiliar with those branches. Audience: This volume will be of interest to mathematicalphysicists, and mathematicians whose work involves probability theory,stochastic processes, global analysis, analysis on manifolds ordifferential geometry, and is recommended for graduate level courses. ... Read more

Book Description

Based on a graduate course taught at Utrecht University, this book provides a short introduction to the theory of Foliations and Lie Groupoids to students who have already taken a first course in differential geometry. Ieke Moerdijk and Janez Mrcun include detailed references to enable students to find the requisite background material in the research literature. The text features many exercises and worked examples. ... Read more

Book Description

The orbit method influenced the development of several areas of mathematics in the second half of the 20th century and continues to be an important tool today.Among the distinguished names associated with the orbit method is that of A.A. Kirillov, whose pioneering paper on nilpotent orbits in 1962, places him as the founder of orbit theory. The origins of the orbit method lie in the search for a relationship between classical and quantum mechanics. Over the years, the orbit method has been used to link harmonic analysis (theory of unitary representations of Lie groups) with differential geometry (symplectic geometry of homogeneous spaces), and it has stimulated and invigorated many classical domains of mathematics, i.e., representation theory, integrable systems, complex algebraic geometry, to name several.It continues to be a useful and powerful tool in all of these areas of mathematics and mathematical physics. This volume, dedicated to A. A. Kirillov, covers a very broad range of key topics such as: * The orbit method in the theory of unitary representations of Lie groups * Infinite-dimensional Lie groups: their orbits and representations * Quantization and the orbit method; geometric quantization (old and new) * The Virasoro algebra; string and conformal field theories * Lie superalgebras and their representations * Combinatorial aspects of representation theory. The prominent contributors to this volume present original and expository invited papers in the areas of Lie theory, geometry, algebra, and mathematical physics.The work will be an invaluable reference for researchers in the above mentioned fields, as well as a useful text for graduate seminars and courses. Contributors include: A. Alekseev, J. Alev, R. Brylinski, J. Dixmier, D.R. Farkas, V. Ginzburg, V. Gorbounov, P. Grozman, E. Gutkin, A. Joseph, D. Kazhdan, A.A. Kirillov, B. Kostant, D. Leites, F. Malikov, A. Melnikov, Y.A. Neretin, A. Okounkov, G. Olshanski, F. Petrov, A. Polishchuk, W. Rossmann, A. Sergeev, V. Schechtman, I. Shchepochkina. ... Read more

Book Description

This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists.

One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions. The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary. ... Read more

Reviews (1)

Excellent reference book for researchers in CR geometry
The book presents an excellent discussions on some of the most active research fields in the subject of several complex variables. It includes a fairly complete list of references and an accurate account of the historical backgrounds of the materials. ... Read more

Reviews (1)

There are much better books on mathematics of tomography
I bought this book, but found it pretty much useless. In my opinion, its presentation of techniques and the history of the subject is skewed. Authors' contributions are presented in an overinflated manner, so the reader would never guess that sometimes there exist more powerful and at the same time simpler approaches. There are much better books on mathematics of tomography, for instance the ones written by F. Natterer. Analytically inclined readers would like to read other books on Radon transform, e.g. the one by S. Helgason. The old books directed towards applications by G. Herman and Kak and Slaney are still of value. There are also some good edited volumes on the subject. ... Read more

Book Description

This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum ofp on Y andp on Z, given thatp is the p form valued Laplacian and pi: Z Y is a Riemannian submersion. After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change.Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area.Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture.Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important. ... Read more

Book Description

This text features a careful treatment of flow lines and algebraic invariants in contact form geometry, a vast area of research connected to symplectic field theory, pseudo-holomorphic curves, and Gromov-Witten invariants (contact homology). In particular, it develops a novel algebraic tool in this field: rooted in the concept of critical points at infinity, the new algebraic invariants defined here are useful in the investigation of contact structures and Reeb vector fields. The book opens with a review of prior results and then proceeds through an examination of variational problems, non-Fredholm behavior, true and false critical points at infinity, and topological implications. An increasing convergence with regular and singular Yamabe-type problems is discussed, and the intersection between contact form and Riemannian geometry is emphasized. Rich in open problems and full, detailed proofs, this work lays the foundation for new avenues of study in contact form geometry and will benefit graduate students and researchers. ... Read more

Book Description

Providing a comprehensive presentation of the geometry of submanifolds, this volume expands the classical results in the theory of curves and surfaces. The geometry of submanifolds starts from the idea of the extrinsic geometry of a surface and the theory studies the position and properties of a submanifold in ambient space, in both local and global aspects. This volume also highlights the contributions made by great geometers, past and present, to the geometry of submanifolds and the developing areas of application. ... Read more

Book Description

Hardbound. This book represents a summary of the research we have been conducting since the early 1990s, and describes a conceptual framework which addresses some current shortcomings, and proposes a unified approach for a broad class of problems. While the framework is defined, our research continues, and some of the elements presented here will no doubt evolve in the coming years.It is organized in eight chapters. In the Introduction chapter, we present the definition of the problems, and give an overview of the proposed approach and its implementation. In particular, we illustrate the limitations of the 2.5D sketch, and motivate the use of a representation in terms of layers instead.In chapter 2, we review some of the relevant research in the literature. The discussion focuses on general computational approaches for early vision, and individual methods are only cited as references. Chapter 3 is the fundamental chapter, as it presents the elements ... Read more

Reviews (2)

A fusion of other papers with many errata
This book covers the essential parts of the Tensor Voting framework, describes some of its applications, and compares it with other methodologies. If you want to implement this new approach, however, you're left alone with many details that are not explained, like the construction of the voting fields (especially in the 3D case) and how to choose their parameters. Another thing that annoyed me is that the book is just a fusion of the PhD theses of the two co-authors Mi-Suen Lee and Chi-Keung Tang. Almost all of its text was simply copied from the other documents, thereby introducing several errata and inconsistencies, since some passages are hard to understand without the original accompanying figures. Some of the given equations are erroneous, which can be quite misleading, and many references to other literature are specified with an incorrect number.

Although you won't find a lot of information about the Tensor Voting framework in general and this is the only book covering the topic, I wouldn't recommend buying it. It's not worth its price. You'll be better off getting the original papers -- for free!

Amazing new Early Vision Methodology!
The book provide a brand new methodology and mature computational framework to solve early vision problems. It uses the tensor to convey the information and vote for each other based on tensor fields. After accumulating the votes on each input sites, it gives out the surface, curves and juctions all at once through feature extraction procedure. This method have found a lot of applications in motion analysis, stereo analysis, segmentation... So far, this book is the only book introducing this amazing new Tensor Voting methodology and software. ... Read more

Book Description

This monograph presents a comprehensive coverage ofthree-dimensional topology, as well as exploring some of itsfrontiers. Many important applied problems of mechanics andtheoretical physics can be reduced to algorithmic problems ofthree-dimensional topology, which can then be solved using computers.Although much progress in this field has been made in recent years,these results have not been readily accessible to a wider audience upto now. This book is based on courses the authors have given overseveral years, and summarises the most outstanding achievements ofmodern computer topology. Audience: This book will be of interest to graduate students andresearchers whose work involves such diverse disciplines as physics,mathematics, computer programmes for spline theory and itsapplications, geometrical modelling, geometry, and topology. Theillustrations by A.T. Fomenko, drawn especially for this work, addgreat value and extra appeal. ... Read more

Book Description

This book contains the proceedings of the Conference on Differential Geometry, held in Budapest, 1996. The papers presented here all give essential new results. A wide variety of topics in differential geometry is covered and applications are also studied. Beyond the traditional differential geometry subjects, several popular ones such as Einstein manifolds and symplectic geometry are also well represented.Audience: This volume will be of interest to research mathematicians whose work involves differential geometry, global analysis, analysis on manifolds, manifolds and complexes, mathematics of physics, and relativity and gravitation. ... Read more