Fractal Zeta Functions And Fractal Drums PDF Download
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Author | : Michel L. Lapidus |
Publisher | : Springer |
Total Pages | : 685 |
Release | : 2017-06-07 |
Genre | : Mathematics |
ISBN | : 3319447068 |
Download Fractal Zeta Functions and Fractal Drums Book in PDF, ePub and Kindle
This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension. Two new classes of fractal zeta functions are introduced, namely, the distance and tube zeta functions of bounded sets, and their key properties are investigated. The theory is developed step-by-step at a slow pace, and every step is well motivated by numerous examples, historical remarks and comments, relating the objects under investigation to other concepts. Special emphasis is placed on the study of complex dimensions of bounded sets and their connections with the notions of Minkowski content and Minkowski measurability, as well as on fractal tube formulas. It is shown for the first time that essential singularities of fractal zeta functions can naturally emerge for various classes of fractal sets and have a significant geometric effect. The theory developed in this book leads naturally to a new definition of fractality, expressed in terms of the existence of underlying geometric oscillations or, equivalently, in terms of the existence of nonreal complex dimensions. The connections to previous extensive work of the first author and his collaborators on geometric zeta functions of fractal strings are clearly explained. Many concepts are discussed for the first time, making the book a rich source of new thoughts and ideas to be developed further. The book contains a large number of open problems and describes many possible directions for further research. The beginning chapters may be used as a part of a course on fractal geometry. The primary readership is aimed at graduate students and researchers working in Fractal Geometry and other related fields, such as Complex Analysis, Dynamical Systems, Geometric Measure Theory, Harmonic Analysis, Mathematical Physics, Analytic Number Theory and the Spectral Theory of Elliptic Differential Operators. The book should be accessible to nonexperts and newcomers to the field.
Author | : Michel L. Lapidus |
Publisher | : |
Total Pages | : |
Release | : 2017 |
Genre | : |
ISBN | : |
Download Zeta Functions and Complex Dimensions of Relative Fractal Drums Book in PDF, ePub and Kindle
Author | : Christina Q. He |
Publisher | : American Mathematical Soc. |
Total Pages | : 114 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 0821805975 |
Download Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions Book in PDF, ePub and Kindle
This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of "fractal drums" (and especially of "fractal strings"). In this work, the authors extend previous results in this area by using the notionof generalized Minkowski content which is defined through some suitable "gauge functions" other than power functions. (This content is used to measure the irregularity (or "fractality") of the boundary of an open set in R]n by evaluating the volume of its small tubular neighborhoods). In the situation when the power function is not the natural "gauge function", this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators. This text will also be of interest to those working in mathematical physics.
Author | : Michel L. Lapidus |
Publisher | : Springer Science & Business Media |
Total Pages | : 277 |
Release | : 2013-12-01 |
Genre | : Mathematics |
ISBN | : 1461253144 |
Download Fractal Geometry and Number Theory Book in PDF, ePub and Kindle
A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap pendix B.) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.
Author | : Michel L. Lapidus |
Publisher | : Springer Science & Business Media |
Total Pages | : 583 |
Release | : 2012-09-20 |
Genre | : Mathematics |
ISBN | : 1461421764 |
Download Fractal Geometry, Complex Dimensions and Zeta Functions Book in PDF, ePub and Kindle
Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Throughout Geometry, Complex Dimensions and Zeta Functions, Second Edition, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.
Author | : Michel L. Lapidus |
Publisher | : Springer Science & Business Media |
Total Pages | : 472 |
Release | : 2007-08-08 |
Genre | : Mathematics |
ISBN | : 0387352082 |
Download Fractal Geometry, Complex Dimensions and Zeta Functions Book in PDF, ePub and Kindle
Number theory, spectral geometry, and fractal geometry are interlinked in this study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. The Riemann hypothesis is given a natural geometric reformulation in context of vibrating fractal strings, and the book offers explicit formulas extended to apply to the geometric, spectral and dynamic zeta functions associated with a fractal.
Author | : Christoph Bandt |
Publisher | : Birkhäuser |
Total Pages | : 339 |
Release | : 2015-07-08 |
Genre | : Mathematics |
ISBN | : 3319186604 |
Download Fractal Geometry and Stochastics V Book in PDF, ePub and Kindle
This book collects significant contributions from the fifth conference on Fractal Geometry and Stochastics held in Tabarz, Germany, in March 2014. The book is divided into five topical sections: geometric measure theory, self-similar fractals and recurrent structures, analysis and algebra on fractals, multifractal theory, and random constructions. Each part starts with a state-of-the-art survey followed by papers covering a specific aspect of the topic. The authors are leading world experts and present their topics comprehensibly and attractively. Both newcomers and specialists in the field will benefit from this book.
Author | : Michel Lapidus |
Publisher | : Springer Science & Business Media |
Total Pages | : 583 |
Release | : 2012-09-20 |
Genre | : Mathematics |
ISBN | : 1461421756 |
Download Fractal Geometry, Complex Dimensions and Zeta Functions Book in PDF, ePub and Kindle
Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Throughout Geometry, Complex Dimensions and Zeta Functions, Second Edition, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.
Author | : Patricia Alonso Ruiz |
Publisher | : World Scientific |
Total Pages | : 594 |
Release | : 2020-02-26 |
Genre | : Mathematics |
ISBN | : 9811215545 |
Download Analysis, Probability And Mathematical Physics On Fractals Book in PDF, ePub and Kindle
In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Using classical and abstract analytic tools developed by Cantor, Hausdorff, and Sierpinski, they have sought to address fundamental questions: How can we measure the size of a fractal set? How do waves and heat travel on irregular structures? How are analysis, geometry and stochastic processes related in the absence of Euclidean smooth structure? What new physical phenomena arise in the fractal-like settings that are ubiquitous in nature?This book introduces background and recent progress on these problems, from both established leaders in the field and early career researchers. The book gives a broad introduction to several foundational techniques in fractal mathematics, while also introducing some specific new and significant results of interest to experts, such as that waves have infinite propagation speed on fractals. It contains sufficient introductory material that it can be read by new researchers or researchers from other areas who want to learn about fractal methods and results.
Author | : Christina Q. He |
Publisher | : Oxford University Press, USA |
Total Pages | : 114 |
Release | : 2014-09-11 |
Genre | : Differential equations, Partial |
ISBN | : 9781470401931 |
Download Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings Book in PDF, ePub and Kindle
This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of fractal drums (and especially of fractal strings). In this work, the authors extend previous results in this area by using the notion of generalized Minkowski content which is defined through some suitable gauge functions other than power functions. (This content is used to measure the irregularity (or fractality) of the boundary of an open set in R ]n by evaluating the volume of its small tubular neighbourhoods). In the situation when the power function is not the natural gauge function, this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators. This text will also be of interest to those working in mathematical physics.