Book Description

The theory of generalized analytic continuation studiescontinuations of meromorphic functions in situations wheretraditional theory says there is a natural boundary. Thisbroader theory touches on a remarkable array of topics inclassical analysis, as described in the book. The authors usethe strong analogy with the summability of divergent series tomotivate the subject. In this vein, for instance, theorems canbe described as being "Abelian" or "Tauberian". The introductoryoverview carefully explains the history and context of thetheory. The book addresses the following questions: (1) When canwe say, in some reasonable way, that component functions of ameromorphic function on a disconnected domain, are"continuations" of each other? (2) What role do such"continuations" play in certain aspects of approximation theoryand operator theory?

The authors begin with a review of the works of Poincare, Borel,Wolff, Walsh, and Goncar, on continuation properties of "Borelseries" and other meromorphic functions that are limits ofrapidly convergent sequences of rational functions. They thenmove on to the work of Tumarkin, who looked at the continuationproperties of functions in the classical Hardy space of the diskin terms of the concept of "pseudocontinuation". Tumarkin's workwas seen in a different light by Douglas, Shapiro, and Shieldsin their discovery of a characterization of the cyclic vectorsfor the backward shift operator on the Hardy space. The authorscover this important concept of "pseudocontinuation" quitethoroughly since it appears in many areas of analysis. They alsoadd a new and previously unpublished method of "continuation" tothe list, based on formal multiplication of trigonometricseries, which can be used to examine the backward shift operatoron many spaces of analytic functions. The book attempts to unifythe various types of "continuations" and suggests someinteresting open questions. ... Read more

Book Description

First posed by Hermann Weyl in 1910, the limit-point/limit-circle problem has inspired, over the last century, several new developments in the asymptotic analysis of nonlinear differential equations. This self-contained monograph traces the evolution of this problem from its inception to its modern-day extensions to the study of deficiency indices and analogous properties for nonlinear equations. The book opens with a discussion of the problem in the linear case, as Weyl originally stated it, and then proceeds to a generalization for nonlinear higher-order equations. En route, the authors distill the classical theorems for second and higher-order linear equations, and carefully map the progression to nonlinear limit-point results. The relationship between the limit-point/limit-circle properties and the boundedness, oscillation, and convergence of solutions is explored, and in the final chapter, the connection between limit-point/limit-circle problems and spectral theory is examined in detail. With over 120 references, many open problems, and illustrative examples, this work will be valuable to graduate students and researchers in differential equations, functional analysis, operator theory, and related fields. ... Read more

Book Description

The authors of this volume claim that mathematics can be usefully re-conceptualized as a special form of communication. As a result, the familiar discussion of mental schemes, misconceptions, and cognitive conflict is transformed into a consideration of activity, patterns of interaction, and communication failure. By equating thinking with communicating, the discursive approach also deconstructs the problematic dichotomy between "individual" and "social" research perspectives. Although each author applies his or her own analyses to the discourse generated by students and teachers grappling with mathematical problems, their joint aim is to put discursive research into the limelight and to spur thinking about its nature and its possible advantages and pitfalls. This volume is therefore addressed both to those interested in specific questions regarding classroom communication, and to those who are looking for a general conceptual lens with which to tackle the complexity of mathematical teaching and learning. ... Read more

Book Description

The Ricci flow is currently a hot topic at the forefront of mathematics research. Ricci flow as an approach to the Geometrization Conjecture has recently received attention in the press with articles appearing in The New York Times and other popular newspapers and magazines. The NY Times article refers to recent work of Grisha Perelman on Richard Hamilton's program for Ricci flow as an approach to Thurston's Geometrization Conjecture.

In the past two decades the Ricci flow, and particularly Richard Hamilton's work, has received attention as both having a profound influence on geometric evolution equations and as a possible approach to studying Thurston's Geometrization Conjecture. This selection of papers on the Riemannian Ricci flow is intended for a variety of purposes. The graduate student or researcher unfamiliar with the Ricci flow may use it as an introduction to the Ricci flow quickly leading to current research topics and open problems. Geometers already familiar with the Ricci flow may use it as a handy reference which contains almost all of Richard Hamilton's papers on the subject to date (2002).

The recent developments of Grisha Perelman on Richard Hamilton's program for Ricci flow are exciting. And our collection is intended to make readily available, in one book, the work of Hamilton and others on Ricci flow to a wider audience. ... Read more

Book Description

This fourth volume of Research in Collegiate MathematicsEducation (RCME IV) reflects the themes of student learning andcalculus. Included are overviews of calculus reform in Franceand in the U.S. and large-scale and small-scale longitudinalcomparisons of students enrolled in first-year reform coursesand in traditional courses. The work continues with detailedstudies relating students' understanding of calculus andassociated topics. Direct focus is then placed on instructionand student comprehension of courses other than calculus, namelyabstract algebra and number theory. The volume concludes with astudy of a concept that overlaps the areas of focus,quantifiers. The book clearly reflects the trend towards agrowing community of researchers who systematically gather anddistill data regarding collegiate mathematics' teaching andlearning. ... Read more

Book Description

Since the publication of the seminal work of H. Federer which gives a rather complete and comprehensive discussion on the subject, the geometric measure theory has developed in the last three decades into an even more cohesive body of basic knowledge with an ample structure of its own, established strong ties with many other subject areas of mathematics and made numerous new striking applications. The present book is intended for the researchers in other fields of mathematics as well as graduate students for a quick overview on the subject of the geometric measure theory with emphases on various basic ideas, techniques and their applications in problems arising in the calculus of variations, geometrical analysis and nonlinear partial differential equations.

This graduate-level treatment of Geometric Measure Theory illustrates with concrete examples and emphasizes basic ideas and techniques with their applications to the calculus of variations, geometrical analysis, and non-linear PDEs. ... Read more

Book Description