A Tale of Two Fractals

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April 2013



Since Benoit Mandelbrot's pioneering work in the late 1970s, scores of research articles and books have been published on the topic of fractals. Despite the volume of literature in the field, the general level of theoretical understanding has remained low; most work is aimed either at too mainstream an audience to achieve any depth or at too specialized a community to achieve widespread use. Written by celebrated mathematician and educator A.A. Kirillov, A Tale of Two Fractals is intended to help bridge this gap, providing an original treatment of fractals that is at once accessible to beginners and sufficiently rigorous for serious mathematicians. The work is designed to give young, non-specialist mathematicians a solid foundation in the theory of fractals, and, in the process, to equip them with exposure to a variety of geometric, analytical, and algebraic tools with applications across other areas.


Introduction.- Part 1. The Sierpinski Gasket.- Definition and General Properties.- The Laplace Operator on the Sierpinski Gasket.- Harmonic Functions on the Sierpinski Gasket.- Part 2. The Apollonian Gasket.- Introduction.- Circles and Disks on Spheres.- Definition of the Apollonian Gasket.- Arithmetic Properties of Apollonian Gaskets.- Geometric and Group-Theoretic Approach.- Many-Dimensional Apollonian Gaskets.- Bibliography.


From the reviews: "A Tale of Two Fractals was intended to bridge the gap between popular expositions on fractals and peer-reviewed research papers. ... Kirillov (Univ. of Pennsylvania) includes open questions and conjectures along with references to papers and books that expand upon particular topics beyond the scope of the present volume. ... the book is a useful addition to libraries supporting graduate programs in mathematics ... . Summing Up: Recommended. Graduate students and above." (C. Bauer, Choice, Vol. 51 (4), December, 2013) "This volume deals with several mathematical problems concerning the qualitative analysis of some models on fractal domains. The author is mainly concerned with the Sierpinski and Apollonian gaskets. ... This volume contains important advances in the theory of fractal sets or hyperbolic geometry. The book is a valuable resource for graduate students and researchers in applied nonlinear analysis." (Vicentiu D. Radulescu, zbMATH, Vol. 1273, 2013)
EAN: 9780817683818
ISBN: 081768381X
Untertitel: 2013. Auflage. Book. Sprache: Englisch.
Verlag: Birkhäuser
Erscheinungsdatum: April 2013
Seitenanzahl: 152 Seiten
Format: gebunden
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