Singular Sets of Minimizers for the Mumford-Shah Functional


Book Description

The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of analysis and geometric measure theory. The main object under scrutiny is a free boundary K where the minimizer may have jumps. The book presents an extensive description of the known regularity properties of the singular sets K, and the techniques to get them. It is largely self-contained, and should be accessible to graduate students in analysis. The core of the book is composed of regularity results that were proved in the last ten years and which are presented in a more detailed and unified way.




Mathematical Analysis of Continuum Mechanics and Industrial Applications III


Book Description

This book focuses on mathematical theory and numerical simulation related to various areas of continuum mechanics, such as fracture mechanics, (visco)elasticity, optimal shape design, modelling of earthquakes and Tsunami waves, material structure, interface dynamics and complex systems. Written by leading researchers from the fields of applied mathematics, physics, seismology, engineering, and industry with an extensive knowledge of mathematical analysis, it helps readers understand how mathematical theory can be applied to various phenomena, and conversely, how to formulate actual phenomena as mathematical problems. This book is the sequel to the proceedings of the International Conference of Continuum Mechanics Focusing on Singularities (CoMFoS) 15 and CoMFoS16.




Free Discontinuity Problems


Book Description

This book presents a series of lectures on three of the best known examples of free discontinuity problems: the Mumford-Shah model for image segmentation, a variational model for the epitaxial growth of thin films, and the sharp interface limit of the Ohta-Kawasaki model for pattern formation in dyblock copolymers.




Journal of Fourier Analysis and Applications Special Issue


Book Description

The Journal of Fourier Analysis and Applications is a journal of the mathematical sciences devoted to Fourier analysis and its applications. The subject of Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientific phenomena, and on the solution of some of the most important problems in mathematics and the sciences. At the end of June 1993, a large Conference in Harmonic Analysis was held at the University of Paris-Sud at Orsay to celebrate the prominent role played by Jean-Pierre Kahane and his numerous achievements in this field. The large variety of topics discussed in this meeting, ranging from classical Harmonic Analysis to Probability Theory, reflects the intense mathematical curiosity and the broad mathematical interest of Jean-Pierre Kahane. Indeed, all of them are connected to his work. The mornings were devoted to plenary addresses while up to four parallel sessions took place in the afternoons. Altogether, there were about eighty speakers. This wide range of subjects appears in these proceedings which include thirty six articles.




Foundations of Computational Mathematics


Book Description

This book contains a collection of articles corresponding to some of the talks delivered at the Foundations of Computational Mathematics conference held at IMPA in Rio de Janeiro in January 1997. Some ofthe others are published in the December 1996 issue of the Journal of Complexity. Both of these publications were available and distributed at the meeting. Even in this aspect we hope to have achieved a synthesis of the mathematics and computer science cultures as well as of the disciplines. The reaction to the Park City meeting on Mathematics of Numerical Analy sis: Real Number Algorithms which was chaired by Steve Smale and had around 275 participants, was very enthusiastic. At the suggestion of Narendra Karmar mar a lunch time meeting of Felipe Cucker, Arieh Iserles, Narendra Karmarkar, Jim Renegar, Mike Shub and Steve Smale decided to try to hold a periodic meeting entitled "Foundations of Computational Mathematics" and to form an organization with the same name whose primary purpose will be to hold the meeting. This is then the first edition of FoCM as such. It has been organized around a small collection of workshops, namely - Systems of algebraic equations and computational algebraic geometry - Homotopy methods and real machines - Information-based complexity - Numerical linear algebra - Approximation and PDEs - Optimization - Differential equations and dynamical systems - Relations to computer science - Vision and related computational tools There were also twelve plenary speakers.




Analysis and Geometry of Metric Measure Spaces


Book Description

Contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieure in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation.




European Congress of Mathematics


Book Description

This is the first volume of the procedings of the second European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners. Together with volume II it contains a collection of contributions by the invited lecturers. Finally, volume II also presents reports on some of the Round Table discussions. This two-volume set thus gives an overview of the state of the art in many fields of mathematics and is therefore of interest to every professional mathematician. Contributors: Vol. I: N. Alon, L. Ambrosio, K. Astala, R. Benedetti, Ch. Bessenrodt, F. Bethuel, P. Bjørstad, E. Bolthausen, J. Bricmont, A. Kupiainen, D. Burago, L. Caporaso, U. Dierkes, I. Dynnikov, L.H. Eliasson, W.T. Gowers, H. Hedenmalm, A. Huber, J. Kaczorowski, J. Kollár, D.O. Kramkov, A.N. Shiryaev, C. Lescop, R. März. Vol. II: J. Matousek, D. McDuff, A.S. Merkurjev, V. Milman, St. Müller, T. Nowicki, E. Olivieri, E. Scoppola, V.P. Platonov, J. Pöschel, L. Polterovich , L. Pyber, N. Simányi, J.P. Solovej, A. Stipsicz, G. Tardos, J.-P. Tignol, A.P. Veselov, E. Zuazua




Progress in Partial Differential Equations


Book Description

This Research Note presents some recent advances in various important domains of partial differential equations and applied mathematics, in particular for calculus of variations and fluid flows. These topics are now part of various areas of science and have experienced tremendous development during the last decades.




Proceedings of the 2009 Joint Workshop of Fraunhofer IOSB and Institute for Anthropomatics, Vision and Fusion Laboratory


Book Description

The joint workshop of the Fraunhofer Institute of Optronics, System Technologies and Image Exploitation IOSB, Karlsruhe, and the Vision and Fusion Laboratory (Institute for Anthropomatics, Karlsruhe Institute of Technology (KIT)), is organized annually since 2005 with the aim to report on the latest research and development findings of the doctoral students of both institutions. This book provides a collection of 16 technical reports on the research results presented on the 2009 workshop.




Geometry of Sets and Measures in Euclidean Spaces


Book Description

Now in paperback, the main theme of this book is the study of geometric properties of general sets and measures in euclidean spaces. Applications of this theory include fractal-type objects such as strange attractors for dynamical systems and those fractals used as models in the sciences. The author provides a firm and unified foundation and develops all the necessary main tools, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. The last third of the book is devoted to the Beisovich-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of euclidean space posessing many of the properties of smooth surfaces. These sets have wide application including the higher-dimensional calculus of variations. Their relations to complex analysis and singular integrals are also studied. Essentially self-contained, this book is suitable for graduate students and researchers in mathematics.