Surveys of Modern Mathematics, Vol. 1
Softcover. 231 pages.
ISBN: 978-1-57146-234-3
Release date: 9 March 2012
This volume is an expansion of lectures given by the author at the Park City Mathematics Institute (Utah) in 2008, and on other occasions. The purpose of this volume is to describe analytic techniques useful in the study of questions pertaining to linear series, multiplier ideals, and vanishing theorems for algebraic vector bundles. The author aims to be concise in his exposition, assuming that the reader is already somewhat acquainted with the basic concepts of sheaf theory, homological algebra, and complex differential geometry. In the final chapters, some very recent questions and open problems are addressed?such as results related to the finiteness of the canonical ring and the abundance conjecture, and results describing the geometric structure of Kahler varieties and their positive cones.
Mathematics has developed to a very high level and continues to progress rapidly. An essential characteristic of modern study and research is a strong interaction between the various areas of mathematics?an interaction fruitful and beautiful in its results. It is crucial to educate new generations of mathematicians about important existing theory together with new developments in mathematics, and in the process to give students a basis for grasping this interconnectivity of mathematics.
The book series Surveys of Modern Mathematics (SMM) has been created especially to help provide such an education to students worldwide, in volumes that are both accessible and affordable. Volumes in the SMM series will consist of lecture notes selected from introductory courses, collections of important survey papers, and expository monographs on well-known or developing topics.
With joint publication by Higher Education Press of Beijing within China and by International Press of Boston in the West and elsewhere, the Surveys of Modern Mathematics book series extends a broad reach to students and general readers worldwide.
Morningside Lectures in Mathematics, Vol. 1
Softcover. 317 pages.
ISBN: 978-1-57146-235-0
Release date: 9 March 2012
In this volume we present lectures by M. C. Lopes concerning the boundary layers of incompressible fluid flow; by C. J. Xu on the micro-local analysis and its applications to the regularities of kinetic equations; by Y. X. Zheng on the weak solutions of variational wave equation from liquid crystals; and by P. Zhang and Z. F. Zhang on the free boundary problem of Euler equations. In addition, we also included lectures by F. Nier on the hypoellipticity of Fokker-Planck operator and Witten-Laplace operator. We hope that these lectures may serve as valuable references, providing up-to-date descriptions of current developments in various related research topics, that will benefit young researchers or graduate students.
The Morningside Center of Mathematics, at the Chinese Academy of Sciences in Beijing, carefully selects research topics in basic mathematics, applied mathematics and computational mathematics fields, and invites world-class mathematicians and outstanding scholars to give lectures and to conduct collaborative research.
The Morningside Lectures in Mathematics (MLM) book series is based mainly on the lectures that these well-known experts have given at the Morningside Center of Mathematics. We hope that with the aid of these volumes more people will be enabled to broaden their vision and to understand the latest developments in various related fields.
HARDBACK 9781421404950
PAPERBACK 9781421404967
February 2012 304 pp., 9 halftones, 88 line drawings
Introduced nearly three decades ago as a variable resolution alternative to the Fourier transform, a wavelet is a short oscillatory waveform for analysis of transients. The discrete wavelet transform has remarkable multi-resolution and energy-compaction properties. Amir-Homayoon Najmi's introduction to wavelet theory explains this mathematical concept clearly and succinctly.
Wavelets are used in processing digital signals and imagery from myriad sources. They form the backbone of the JPEG2000 compression standard, and the Federal Bureau of Investigation uses biorthogonal wavelets to compress and store its vast database of fingerprints. Najmi provides the mathematics that demonstrate how wavelets work, describes how to construct them, and discusses their importance as a tool to investigate and process signals and imagery. He reviews key concepts such as frames, localizing transforms, orthogonal and biorthogonal bases, and multi-resolution. His examples include the Haar, the Shannon, and the Daubechies families of orthogonal and biorthogonal wavelets.
Our capacity and need for collecting and transmitting digital data is increasing at an astonishing rate. So too is the importance of wavelets to anyone working with and analyzing digital data. Najmi's primer will be an indispensable resource for those in computer science, the physical sciences, applied mathematics, and engineering who wish to obtain an in-depth understanding and working knowledge of this fascinating and evolving field.
Amir-Homayoon Najmi completed the mathematical tripos at Cambridge University and obtained his D.Phil. at Oxford University. He is with the Johns Hopkins University's Applied Physics Laboratory and is a faculty member of the Whiting School of Engineering CE programs in applied physics and electrical engineering.
ISBN: 978-1-1183-5706-4
Paperback
324 pages
April 2012
A unique approach to analysis that lets you apply mathematics across a range of subjects
This innovative text sets forth a thoroughly rigorous modern account of the theoretical underpinnings of calculus: continuity, differentiability, and convergence. Using a constructive approach, every proof of every result is direct and ultimately computationally verifiable. In particular, existence is never established by showing that the assumption of non-existence leads to a contradiction. The ultimate consequence of this method is that it makes sense?not just to math majors but also to students from all branches of the sciences.
The text begins with a construction of the real numbers beginning with the rationals, using interval arithmetic. This introduces readers to the reasoning and proof-writing skills necessary for doing and communicating mathematics, and it sets the foundation for the rest of the text, which includes:
*Early use of the Completeness Theorem to prove a helpful Inverse Function
Theorem
*Sequences, limits and series, and the careful derivation of formulas and
estimates for important functions
*Emphasis on uniform continuity and its consequences, such as boundedness
and the extension of uniformly continuous functions from dense subsets
*Construction of the Riemann integral for functions uniformly continuous
on an interval, and its extension to improper integrals
*Differentiation, emphasizing the derivative as a function rather than
a pointwise limit
*Properties of sequences and series of continuous and differentiable functions
*Fourier series and an introduction to more advanced ideas in functional
analysis
Examples throughout the text demonstrate the application of new concepts. Readers can test their own skills with problems and projects ranging in difficulty from basic to challenging.
This book is designed mainly for an undergraduate course, and the author understands that many readers will not go on to more advanced pure mathematics. He therefore emphasizes an approach to mathematical analysis that can be applied across a range of subjects in engineering and the sciences.
ISBN: 978-0-470-69514-2
Hardcover
488 pages
May 2012
Modelling Under Risk and Uncertainty goes beyond the eblack-boxf view that some risk analysts or statisticians develop the underlying phenomenology of the environmental or industrial processes, without valuing enough their physical properties and inner modelling potential; conversely it is also to attract environmental or engineering modellers to more elaborate statistical and risk analysis material beyond for example, elementary variance analysis, taking advantage of advanced scientific computing, to face new regulations departing from deterministic design or decision-making.
Modelling Under Risk and Uncertainty:
*Addresses a concern of growing interest for large industries or environmentalists:
risk and uncertainty in complex systems.
*Gives news insight in to the peculiar mathematical challenges generated
by recent industrial safety or environmental control analysis.
*Looks at implementing decision theory choices related to risk and uncertainty
analysis through statistical estimation and computation, in the presence
of physical modelling and risk analysis.
*Discusses key issue of differentiating or aggregating the dimensions of
risk/aleatory and epistemic uncertainty.
*Illustrated with one favourite pedagogical example developed throughout
the book to facilitate the reading and understanding.
Researchers in applied statistics, scientific computing, reliability, advanced mechanics, physics or environmental science will benefit from this book.