Fractal Geometry and Number Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Fractal Geometry and Number Theory write by Michel L. Lapidus. This book was released on 2013-12-01. Fractal Geometry and Number Theory available 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.
Fractal Geometry, Complex Dimensions and Zeta Functions
Fractal Geometry, Complex Dimensions and Zeta Functions - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Fractal Geometry, Complex Dimensions and Zeta Functions write by Springer. This book was released on 2012-09-01. Fractal Geometry, Complex Dimensions and Zeta Functions available in PDF, EPUB and Kindle.
Fractal Geometry, Complex Dimensions and Zeta Functions
Fractal Geometry, Complex Dimensions and Zeta Functions - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Fractal Geometry, Complex Dimensions and Zeta Functions write by Michel L. Lapidus. This book was released on 2007-08-08. Fractal Geometry, Complex Dimensions and Zeta Functions available 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.
Fractal Geometry, Complex Dimensions and Zeta Functions
Fractal Geometry, Complex Dimensions and Zeta Functions - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Fractal Geometry, Complex Dimensions and Zeta Functions write by Michel L. Lapidus. This book was released on 2012-09-20. Fractal Geometry, Complex Dimensions and Zeta Functions available 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.
Fractal Zeta Functions and Fractal Drums
Fractal Zeta Functions and Fractal Drums - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Fractal Zeta Functions and Fractal Drums write by Michel L. Lapidus. This book was released on 2017-06-07. Fractal Zeta Functions and Fractal Drums available 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.
