Robert S. Strichartz
Differential Equations on Fractals: A Tutorial
Paper | 2006 | ISBN: 0-691-12731-X
Cloth | 2006 | ISBN: 0-691-12542-2
192 pp. | 6 x 9 | 43 line illus.
Differential Equations on Fractals opens the door to
understanding the recently developed area of analysis on
fractals, focusing on the construction of a Laplacian on the
Sierpinski gasket and related fractals. Written in a lively and
informal style, with lots of intriguing exercises on all levels
of difficulty, the book is accessible to advanced undergraduates,
graduate students, and mathematicians who seek an understanding
of analysis on fractals. Robert Strichartz takes the reader to
the frontiers of research, starting with carefully motivated
examples and constructions.
One of the great accomplishments of geometric analysis in the
nineteenth and twentieth centuries was the development of the
theory of Laplacians on smooth manifolds. But what happens when
the underlying space is rough? Fractals provide models of rough
spaces that nevertheless have a strong structure, specifically
self-similarity. Exploiting this structure, researchers in
probability theory in the 1980s were able to prove the existence
of Brownian motion, and therefore of a Laplacian, on certain
fractals. An explicit analytic construction was provided in 1989
by Jun Kigami. Differential Equations on Fractals explains
Kigami's construction, shows why it is natural and important, and
unfolds many of the interesting consequences that have recently
been discovered.
This book can be used as a self-study guide for students
interested in fractal analysis, or as a textbook for a special
topics course.
Robert S. Strichartz is Professor of Mathematics at Cornell
University. He is the author of The Way of Analysis and A Guide
to Distribution Theory and Fourier Transforms.
Table of Contents
Zou, Wenming, Schechter, Martin
Critical Point Theory and Its Applications
2006, XII, 318 p., Hardcover
ISBN: 0-387-32965-X
About this book
Since the birth of the calculus of variations, researchers have
discovered that variational methods, when they apply, can obtain
better results than most other methods. Moreover, they apply in a
very large number of situations. It was realized many years ago
that the solutions of a great number of problems are in effect
critical points of functionals. Critical Point Theory and Its
Applications presents some of the latest research in the area of
critical point theory. Researchers have obtained many new results
recently using this approach, and in most cases comparable
results have not been obtained with other methods. This book
describes the methods and presents the newest applications.
The topics covered in the book include extrema, even valued
functionals, weak and double linking, sign changing solutions,
Morse inequalities, and cohomology groups. The applications
described include Hamiltonian systems, Schrodinger equations and
systems, jumping nonlinearities, elliptic equations and systems,
superlinear problems and beam equations. Many minimax theorems
are established without the use of the (PS) compactness condition.
Table of contents
Preface.- 1. Preliminaries.- 2. Functionals Bounded Below.- 3.
Even Functionals.- 4. Weak Linking and Homoclinic Orbits.- 5.
Double Linking Theorems.- 6. Superlinear Problems.- 7. Systems
with Hamiltonian Potentials.- 8. Linking and Elliptic Systems.- 9.
Sign-changing Solutions.- 10. Cohomology Groups.- Bibliography.-
Index.
A.V.Bolsinov, A.T.Fomenko and A.A.Oshemkov, Moscow State University, Russia
Topological Methods in the Theory of Integrable Systems
300pp Hbk 2006
This volume comprises selected papers on the subject of the
topology of integrable systems ---- theory which studies their
qualitative properties, singularities and topological invariants.
It presents new trends in the theory of integrable systems and
also give the reader some background to this field.
In particular, the volume includes original papers devoted to the
classification of singularities with symmetries, semi-classical
integrability, construction of "exotic" integrable
systems, applications of integrability to Riemannian geometry,
and analysis of specific integrable systems in classical
mechanics. All of the contributing authors are well-known
specialists actively working in this area of mathematics.
Braides, Andrea; Chiado Piat, Valeria (Eds.)
Topics on Concentration Phenomena and Problems with Multiple Scales
Series: Lecture Notes of the Unione Matematica Italiana , Vol. 2
2006, XI, 317 p., 5 illus., Softcover
ISBN: 3-540-36241-X
About this book
The study of variational problems showing multi-scale behaviour with oscillation or concentration phenomena are a challenging topic of very active research. This volume collects lecture notes devoted to the asymptotic analysis of such problems when the multi-scale behaviour derives from scale separation in the passage from atomistic systems to continuous functionals, from competition between bulk and surface energies, from various types of homogenization processes (random, in perforated domains, on fractals), and to concentration effects in Ginzburg-Landau energies and in subcritical growth problems.
Table of contents
Tartar, Luc
An Introduction to Navier-Stokes Equation and Oceanography
Series: Lecture Notes of the Unione Matematica Italiana , Vol. 1
2006, XXVIII, 247 p., Softcover
ISBN: 3-540-35743-2
About this book
The Introduction to Navier-Stokes Equation and Oceanography corresponds to a graduate course in mathematics, taught at Carnegie Mellon University in the spring of 1999. Comments were added to the lecture notes distributed to the students, as well as short biographical information for all scientists mentioned in the text, the purpose being to show that the creation of scientific knowledge is an international enterprise, and who contributed to it, from where, and when. The goal of the course is to teach a critical point of view concerning the partial differential equations of continuum mechanics, and to show the need for developing new adapted mathematical tools.
Table of contents
Batagelj, V.; Bock, H.-H.; Ferligoj, A.; ?iberna, A. (Eds.)
Data Science and Classification
Series: Studies in Classification, Data Analysis, and Knowledge Organization
2006, XII, 358 p., 67 illus., Softcover
ISBN: 3-540-34415-2
About this book
This volume provides new methodological developments in data analysis and classification. A wide range of topics is covered that includes the measurement of similarity and dissimilarity, methods for classification and clustering, network and graph analyses, analysis of symbolic data, and web mining. Apart from structural and theoretical results the book shows how to apply the proposed to a variety of problems, for example in medicine, microarray analysis, social network structures, and music. The combination of new methodological advances with the wide range of real applications collected in this volume is of special value for researchers when choosing the appropriate among newly developed analytical tools for their research problems in classification and data analysis.
Table of contents
Similarity and Dissimilarity.- Classification and Clustering.- Network and Graph Analysis.- Analysis of Symbolic Data.- General Data Analysis Methods.- Data and Web Mining.- Analysis of Music Data.- Gene and Microarray Analysis.