Author: Guy David
Publisher: American Mathematical Soc.
ISBN: 0821815377
Category : Mathematics
Languages : en
Pages : 370
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Book Description
* The only available reference on uniform rectifiabilityThe text covers the understanding of uniform rectifiability of a given set in terms of the approximate behaviour of the set at most locations and scales.
Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 1009288083
Category : Mathematics
Languages : en
Pages : 181
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Book Description
A broad survey of the theory of rectifiability and its deep connections to numerous different areas of mathematics.
Author: Xavier Tolsa
Publisher: American Mathematical Soc.
ISBN: 1470422522
Category : Mathematics
Languages : en
Pages : 142
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Book Description
This monograph is devoted to the proof of two related results. The first one asserts that if is a Radon measure in satisfyingfor -a.e. , then is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set with finite -dimensional Hausdorff measure is rectifiable if and only ifH^1x2EThe second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform . Assuming that has linear growth, it is proved that is bounded in if and only iffor every square .
Author: Hervé M. Pajot
Publisher: Springer
ISBN: 3540360743
Category : Mathematics
Languages : en
Pages : 133
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Book Description
Based on a graduate course given by the author at Yale University this book deals with complex analysis (analytic capacity), geometric measure theory (rectifiable and uniformly rectifiable sets) and harmonic analysis (boundedness of singular integral operators on Ahlfors-regular sets). In particular, these notes contain a description of Peter Jones' geometric traveling salesman theorem, the proof of the equivalence between uniform rectifiability and boundedness of the Cauchy operator on Ahlfors-regular sets, the complete proofs of the Denjoy conjecture and the Vitushkin conjecture (for the latter, only the Ahlfors-regular case) and a discussion of X. Tolsa's solution of the Painlevé problem.
Author: Carlos E. Kenig
Publisher: American Mathematical Soc.
ISBN: 1470461277
Category : Education
Languages : en
Pages : 345
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Book Description
The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.
Author: Benjamin Jaye
Publisher: American Mathematical Soc.
ISBN: 1470442132
Category : Mathematics
Languages : en
Pages : 110
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Book Description
Fix $dgeq 2$, and $sin (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $mu $ in $mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Delta )^alpha /2$, $alpha in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
Author: Dorina Mitrea
Publisher: Springer Nature
ISBN: 3031059506
Category : Mathematics
Languages : en
Pages : 940
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Book Description
This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Volume I establishes a sharp version of the Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense.
Author: S.D. Chatterji
Publisher: Birkhäuser
ISBN: 3034890788
Category : Mathematics
Languages : en
Pages : 1669
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Book Description
Since the first ICM was held in Zürich in 1897, it has become the pinnacle of mathematical gatherings. It aims at giving an overview of the current state of different branches of mathematics and its applications as well as an insight into the treatment of special problems of exceptional importance. The proceedings of the ICMs have provided a rich chronology of mathematical development in all its branches and a unique documentation of contemporary research. They form an indispensable part of every mathematical library. The Proceedings of the International Congress of Mathematicians 1994, held in Zürich from August 3rd to 11th, 1994, are published in two volumes. Volume I contains an account of the organization of the Congress, the list of ordinary members, the reports on the work of the Fields Medalists and the Nevanlinna Prize Winner, the plenary one-hour addresses, and the invited addresses presented at Section Meetings 1 - 6. Volume II contains the invited address for Section Meetings 7 - 19. A complete author index is included in both volumes. '...the content of these impressive two volumes sheds a certain light on the present state of mathematical sciences and anybody doing research in mathematics should look carefully at these Proceedings. For young people beginning research, this is even more important, so these are a must for any serious mathematics library. The graphical presentation is, as always with Birkhäuser, excellent....' (Revue Roumaine de Mathematiques pures et Appliquées)
Author: Julien Barral
Publisher: Springer Science & Business Media
ISBN: 081768400X
Category : Mathematics
Languages : en
Pages : 296
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Book Description
This volume, following in the tradition of a similar 2010 publication by the same editors, is an outgrowth of an international conference, “Fractals and Related Fields II,” held in June 2011. The book provides readers with an overview of developments in the mathematical fields related to fractals, including original research contributions as well as surveys from many of the leading experts on modern fractal theory and applications. The chapters cover fields related to fractals such as: *geometric measure theory *ergodic theory *dynamical systems *harmonic and functional analysis *number theory *probability theory Further Developments in Fractals and Related Fields is aimed at pure and applied mathematicians working in the above-mentioned areas as well as other researchers interested in discovering the fractal domain. Throughout the volume, readers will find interesting and motivating results as well as new avenues for further research.
Author: Christoph Bandt
Publisher: Birkhäuser
ISBN: 3034877552
Category : Mathematics
Languages : en
Pages : 250
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Book Description
Fractal geometry is a new and promising field for researchers from different disciplines such as mathematics, physics, chemistry, biology and medicine. It is used to model complicated natural and technical phenomena. The most convincing models contain an element of randomness so that the combination of fractal geometry and stochastics arises in between these two fields. It contains contributions by outstanding mathematicians and is meant to highlight the principal directions of research in the area. The contributors were the main speakers attending the conference "Fractal Geometry and Stochastics" held at Finsterbergen, Germany, in June 1994. This was the first international conference ever to be held on the topic. The book is addressed to mathematicians and other scientists who are interested in the mathematical theory concerning: • Fractal sets and measures • Iterated function systems • Random fractals • Fractals and dynamical systems, and • Harmonic analysis on fractals. The reader will be introduced to the most recent results in these subjects. Researchers and graduate students alike will benefit from the clear expositions.
