Book Description

Fred Almgren exploited the excess method for proving regularity theorems in the calculus of variations. His techniques yielded Hölder continuous differentiability except for a small closed singular set. In the sixties and seventies Almgren refined and generalized his methods. Between 1974 and 1984 he wrote a 1,700-page proof that was his most ambitious development of his ground-breaking ideas. Originally, this monograph was available only as a three-volume work of limited circulation. The entire text is faithfully reproduced here.

This book gives a complete proof of the interior regularity of an area-minimizing rectifiable current up to Hausdorff codimension 2. The argument uses the theory of Q-valued functions, which is developed in detail. For example, this work shows how first variation estimates from squash and squeeze deformations yield a monotonicity theorem for the normalized frequency of oscillation of a Q-valued function that minimizes a generalized Dirichlet integral. The principal features of the book include an extension theorem analogous to Kirszbraun's theorem and theorems on the approximation in mass of nearly flat mass-minimizing rectifiable currents by graphs and images of Lipschitz Q-valued functions. ... Read more

Reviews (1)

Review of Almgren's big regularity paper
The publication of the Tex formatted version of Almgren's famous big regularity paper makes the original three-volume, 1700 page paper much easier to read. Especially, since Almgren developed the whole new machinery, it takes time for the beginning readers to become familiar with the terminology and to consult essential materials in the Appendix. Readers who have suffered reading the original three hand typed ``telephone books'' will appreciate this all-in-one book.

As in the title of the book, the paper provides a method to deal with the regularity problem of mass minimizing surfaces in higher codimension. This paper not only gives the optimal upper bound of the Hausdorff dimension of the singular sets, but also provides a possible approach to understand the structure of the singular sets of mass minimizing surfaces. This deep but famous paper contains several ingredients to be understood and explored. It is worthwhile for people who are interested in geometric measure theory to spend some time on reading it. ... Read more

Reviews (1)

mistake in the title of the book: replace "history" by "Theory"
I am one of the authors of this book. I am sorry to say that this book does not exist. The true title of the existing book is " Topics in Bifurcation Theory and Applications". same authors. ... Read more

Book Description

Many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebras. While function theory has often been employed to answer algebraic questions such as the existence of idempotents in a Banach algebra, concepts arising from the study of Banach algebras including the maximal ideal space, the Silov boundary, Geason parts, etc. have led to new questions and to new methods of proofs in function theory. This book is concerned with developing some of the principal applications of function theory in several complex variables to Banach algebras. The authors do not presuppose any knowledge of several complex variables on the part of the reader and all relevant material is developed within the text. Furthermore, the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. The third edition of this book contains new material on; maximum modulus algebras and subharmonicity, the hull of a smooth curve, integral kernels, perturbations of the Stone-Weierstrass Theorem, boundaries of analytic varieties, polynomial hulls of sets over the circle, areas, and the topology of hulls. The authors have also included a new chapter containing commentaries on history and recent developments and an updated and expanded reading list. ... Read more

Book Description

Vectors in 2 or 3 Dimensions provides an introduction to vectors from their very basics. The author has approached the subject from a geometrical standpoint and although applications to mechanics will be pointed out and techniques from linear algebra employed, it is the geometric view which is emphasised throughout.

Properties of vectors are initially introduced before moving on to vector algebra and transformation geometry. Vector calculus as a means of studying curves and surfaces in 3 dimensions and the concept of isometry are introduced later, providing a stepping stone to more advanced theories.

* Adopts a geometric approach
* Develops gradually, building from basics to the concept of isometry and vector calculus
* Assumes virtually no prior knowledge
* Numerous worked examples, exercises and challenge questions ... Read more

Book Description

Tensor Analysis and Nonlinear Tensor Functions embraces the basic fields of tensor calculus: tensor algebra, tensor analysis, tensor description of curves and surfaces, tensor integral calculus, the basis of tensor calculus in Riemannian spaces and affinely connected spaces, - which are used in mechanics and electrodynamics of continua, crystallophysics, quantum chemistry etc. The book suggests a new approach to definition of a tensor in space 3, which allows us to show a geometric representation of a tensor and operations on tensors. Based on this approach, the author gives a mathematically rigorous definition of a tensor as an individual object in arbitrary linear, Riemannian and other spaces for the first time. It is the first book to present a systematized theory of tensor invariants, a theory of nonlinear anisotropic tensor functions and a theory of indifferent tensors describing the physical properties of continua. The book will be useful for students and postgraduates of mathematical, mechanical engineering and physical departments of universities and also for investigators and academic scientists working in continuum mechanics, solid physics, general relativity, crystallophysics, quantum chemistry of solids and material science. ... Read more

Book Description

This text is an axiomatic treatment of the properties of continuous multivariable functions and related results from topology. Viewing multiple real variables as members of vector spaces, the author covers boundedness, extreme values, and uniform continuity of functions, along with the connections between continuity and topological concepts such as connectedness and compactness. The order of topics deliberately mimics the order of development in elementary calculus. This sequencing allows an elementary approach, with analogies to and generalizations from such familiar ideas as the Pythagorean Theorem. The reader is frequently reminded that the pictures suggested by geometry are powerful guides and tools. The definition-theorem-proof format resides within an informal exposition, containing numerous historical comments and questions within and between the proofs. The objective is to present precise proofs, but in a structure and tone that teach the student to plan and write proofs, both in general and specifically in a real analysis course. Applications are largely absent, except where they provide interesting illustrations of the principles and theorems presented. Problems, solutions, bibliography and index complete this book.Continuous Functions of Vector Variables is suitable for advanced undergraduates preparing for graduate programs in pure mathematics. Required background includes a course in the theory of one-variable calculus and the elements of linear algebra. ... Read more

Reviews (1)

though concise but thorough
I think this is one of the best introduction book to analysis and to multivariable calculus at the same time. it is more intesive than ross elementary analysis but less intensive than spivak and rudins. one reading this book will not need a multivariable calculus background to study this advance topic. though it is reserve for students who lack a some what mathematical maturity, it is still an excellent book, because it makes the students understand the subject.from there a student can advance further. the only problem with this book is that the problems are a bit routine. Guzman should have put nonclassic problems i.e. challenging problems. ... Read more

Book Description

Includes scalars and vectors, algebra of vectors, vector differentiation, gradient, divergence, curl, vector integration, and integral theorems. ... Read more

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A helpful supplement
The guide is a wonderful for aid for either help or catch-up in a Vector Calculus course. A quick look at the divergence and curl, as well as complete statements and proofs and Green's and Stoke's Theorems. helpful. ... Read more

Book Description

The collection of articles in this volume are based on lecturespresented during the Winter School on Mirror Symmetry held atHarvard University. There are many new directions suggested bymirror symmetry which could potentially have very richconnections in physics and mathematics.

This book brings together the latest research in a major area ofmathematical physics, including the recent progress in mirrormanifolds and Lagrangian submanifolds. In particular, severalarticles describing homological approach and related topics areincluded.

Other AMS titles edited by S.-T Yau published in the AMS/IPStudies in Advanced Mathematics series include, Mirror SymmetryIII, Volume 10, Mirror symmetry II, Volume 1, and MirrorSymmetry I, Volume 9. ... Read more

Book Description

The calculus of variations is a classical area of mathematical analysis-300 years old-yet its myriad applications in science and technology continue to hold great interest and keep it an active area of research. These two volumes contain the refereed proceedings of the international conference on Calculus of Variations and Related Topics held at the Technion-Israel Institute of Technology in March 1998. The conference commemorated 300 years of work in the field and brought together many of its leading experts.The papers in the first volume focus on critical point theory and differential equations. The other volume deals with variational aspects of optimal control. Together they provide a unique opportunity to review the state-of-the-art of the calculus of variations, as presented by an international panel of masters in the field. ... Read more

Book Description

Hemivariational inequalities represent an important class of problems in nonsmooth and nonconvex mechanics. By means of them, problems with nonmonotone, possibly multivalued, constitutive laws can be formulated, mathematically analyzed and finally numerically solved. The present book gives a rigorous analysis of finite element approximation for a class of hemivariational inequalities of elliptic and parabolic type. Finite element models are described and their convergence properties are established. Discretized models are numerically treated as nonconvex and nonsmooth optimization problems. The book includes a comprehensive description of typical representants of nonsmooth optimization methods. Basic knowledge of finite element mathematics, functional and nonsmooth analysis is needed. The book is self-contained, and all necessary results from these disciplines are summarized in the introductory chapter.Audience: Engineers and applied mathematicians at universities and working in industry. Also graduate-level students in advanced nonlinear computational mechanics, mathematics of finite elements and approximation theory. Chapter 1 includes the necessary prerequisite materials. ... Read more

Book Description

This book collects applications of nonstandard methods to the theory of vector lattices. Primary attention is paid to combining infinitesimal and Boolean-valued constructions of use in the classical problems of representing abstract analytical objects, such as Banach--Kantorovich spaces, vector measures, and dominated and integral operators. This book is a complement to Volume 358 of "Mathematics and Its Applications": Vector Lattices and Integral Operators, printed in 1996.Audience: The book is intended for the reader interested in the modern tools of nonstandard models of set theory as applied to problems of contemporary functional analysis. It will also be of use to mathematicians, students and postgraduates interested in measure and integration, operator theory, and mathematical logic and foundation. ... Read more

Book Description

Unmatched in its coverage of the topic, the first edition of GENERALIZED VECTOR AND DYADIC ANALYSIS helped revolutionize the treatment of boundary-value problems, establishing itself as a classic in the field. This expanded, revised edition is the most comprehensive book available on vector analysis founded upon the new method symbolic vector. GENERALIZED VECTOR AND DYADIC ANALYSIS presents a copious list of vector and dyadic identities, along with various forms of Green's theorems with derivations. In addition, this edition presents an historical study of the past mis-understandings and contradictions that have occurred in vector analysis presentations, furthering the reader's understanding of the subject.

Sponsored by:
IEEE Antennas and Propagation Society. ... Read more

Reviews (1)

long awaited book
If you ever came across the need of expressing vector operators in curvilinear coordinates, while writing numerical codes or solving PDEs in generalised coordinates, then you might have felt the need of a rigourous presentation of the subject.
Rigorous yet usable, I would say. This is not a book for pure mathematicians (the theorem you need is at pag. 435 consisting of one line saying the proof follows from the 2000 previous Lemma-theorem-corollary scattered all around...). The author has the physical problems well in his mind (he's Emeritus at Univ of Michigan, dept of Electrical Engineering) and makes frequent references to where in physics or geometry you will encounter such mathematical entities.
Chapter 8, devoted to the history of vector analysis, makes you clear why you haven't been able to find a neat rigorous presentation so far. This digression (I'm reluctant to use this expression for it's not a minor chapter in my opinion) gives you an even sharper feeling of what's the core of the matter by comparing critically different approaches.
The first introductory chapters on geometry, coordinate systems, dyadics provide a sound general background, presenting derivatives of vectors in generalised coordinate system from the beginning, resulting in a powerful and unifying exposition of the matter. All is extremely clear, a characteristic of the book in its whole.
The author always stress upon the transformation properties of the objects which have been defined, thus clarifying their meaning and mathematical character.
The appendix on vector analysis in the special theory of relativity is enjoyable and makes you clear how E and B can be treated as vectors even if they transform like the components of a tensor (OK, you already knew that, but you were using tensor analysis from the start, didn't you?).
There is much more in this book, which I would suggest to keep as a reference to any physicist, applied mathematician, biologist, engineer, numerical people and (even!) mathematician (developing communication skills...). First years students, which are always a bit confused on why one should use vectors at all and having accepted that think that everything is a vector, which is not, should definitely start with this book. Next time I will be asked to teach anything related to mechanics, I will adopt this book for the fundamental mathematics (but also check the excellent Biscari-Poggi-Virga Mechanics Notebook, 1999 Liguori ed. Napoli (IT) ).
It's also an excellent way to approach what you want next, which is likely to be tensor analysis which this book, unfortunately, cannot cover (can you hear me, Professor Tai?...). ... Read more

Reviews (1)

Vector Spaces and Matrices in Physics
I am a first year graduate student in physics. I found this book extremely useful. It is very easy to read, easy to understand, and very concise. The material is presented in a logical way and the entire book can be read in a few hours if you are short on time. It is one of my favorite linear algebra books. ... Read more