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Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions

1997 Mathematics
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Release : 1997
Genre : Mathematics
Kind : eBook
Book Rating : 975/5 ( reviews)

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Book Synopsis Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions by : Christina Q. He

Download or read book Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions written by Christina Q. He. This book was released on 1997. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings

2014-09-11 Differential equations, Partial
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Release : 2014-09-11
Genre : Differential equations, Partial
Kind : eBook
Book Rating : 931/5 ( reviews)

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Book Synopsis Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings by : Christina Q. He

Download or read book Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings written by Christina Q. He. This book was released on 2014-09-11. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Fractal Zeta Functions and Fractal Drums

2017-06-07 Mathematics
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Release : 2017-06-07
Genre : Mathematics
Kind : eBook
Book Rating : 068/5 ( reviews)

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Book Synopsis Fractal Zeta Functions and Fractal Drums by : Michel L. Lapidus

Download or read book Fractal Zeta Functions and Fractal Drums written by Michel L. Lapidus. This book was released on 2017-06-07. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Fractal Geometry, Complex Dimensions and Zeta Functions

2012-09-20 Mathematics
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Release : 2012-09-20
Genre : Mathematics
Kind : eBook
Book Rating : 764/5 ( reviews)

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Book Synopsis Fractal Geometry, Complex Dimensions and Zeta Functions by : Michel L. Lapidus

Download or read book Fractal Geometry, Complex Dimensions and Zeta Functions written by Michel L. Lapidus. This book was released on 2012-09-20. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Advances in Non-Archimedean Analysis

2011 Mathematics
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Release : 2011
Genre : Mathematics
Kind : eBook
Book Rating : 914/5 ( reviews)

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Book Synopsis Advances in Non-Archimedean Analysis by : Jesus Araujo-Gomez

Download or read book Advances in Non-Archimedean Analysis written by Jesus Araujo-Gomez. This book was released on 2011. Available in PDF, EPUB and Kindle. Book excerpt: These collected articles feature recent developments in various areas of non-Archimedean analysis: Hilbert and Banach spaces, finite dimensional spaces, topological vector spaces and operator theory, strict topologies, spaces of continuous functions and of strictly differentiable functions, isomorphisms between Banach functions spaces, and measure and integration.

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