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Fractal Geometry, Complex Dimensions and Zeta Functions

Geometry and Spectra of Fractal Strings

$Number theory, spectral geometry, and fractal geometry are interlinked in this study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. The Riemann hypothesis is given a natural geometric reformulation in context of vibrating fractal strings, and the book offers explicit formulas extended to apply to the geometric, spectral and dynamic zeta functions associated with a fractal.$

جزئیات و درخواست

By Michel L. Lapidus, Machiel van Frankenhuijsen
Publisher: Springer Science & Business Media, 2007-08-08
Pages: 472
Language: en
ISBN_13:9780387352084
ISBN_10:0387352082
Subject: Mathematics

Quantized Number Theory, Fractal Strings And The Riemann Hypothesis: From Spectral Operators To Phase Transitions And Universality

$Studying the relationship between the geometry, arithmetic and spectra of fractals has been a subject of significant interest in contemporary mathematics. This book contributes to the literature on the subject in several different and new ways. In particular, the authors provide a rigorous and detailed study of the spectral operator, a map that sends the geometry of fractal strings onto their spectrum. To that effect, they use and develop methods from fractal geometry, functional analysis, complex analysis, operator theory, partial differential equations, analytic number theory and mathematical physics.Originally, M L Lapidus and M van Frankenhuijsen 'heuristically' introduced the spectral operator in their development of the theory of fractal strings and their complex dimensions, specifically in their reinterpretation of the earlier work of M L Lapidus and H Maier on inverse spectral problems for fractal strings and the Riemann hypothesis.One of the main themes of the book is to provide a rigorous framework within which the corresponding question 'Can one hear the shape of a fractal string?' or, equivalently, 'Can one obtain information about the geometry of a fractal string, given its spectrum?' can be further reformulated in terms of the invertibility or the quasi-invertibility of the spectral operator.The infinitesimal shift of the real line is first precisely defined as a differentiation operator on a family of suitably weighted Hilbert spaces of functions on the real line and indexed by a dimensional parameter c. Then, the spectral operator is defined via the functional calculus as a function of the infinitesimal shift. In this manner, it is viewed as a natural 'quantum' analog of the Riemann zeta function. More precisely, within this framework, the spectral operator is defined as the composite map of the Riemann zeta function with the infinitesimal shift, viewed as an unbounded normal operator acting on the above Hilbert space.It is shown that the quasi-invertibility of the spectral operator is intimately connected to the existence of critical zeros of the Riemann zeta function, leading to a new spectral and operator-theoretic reformulation of the Riemann hypothesis. Accordingly, the spectral operator is quasi-invertible for all values of the dimensional parameter c in the critical interval (0,1) (other than in the midfractal case when c =1/2) if and only if the Riemann hypothesis (RH) is true. A related, but seemingly quite different, reformulation of RH, due to the second author and referred to as an 'asymmetric criterion for RH', is also discussed in some detail: namely, the spectral operator is invertible for all values of c in the left-critical interval (0,1/2) if and only if RH is true.These spectral reformulations of RH also led to the discovery of several 'mathematical phase transitions' in this context, for the shape of the spectrum, the invertibility, the boundedness or the unboundedness of the spectral operator, and occurring either in the midfractal case or in the most fractal case when the underlying fractal dimension is equal to ½ or 1, respectively. In particular, the midfractal dimension c=1/2 is playing the role of a critical parameter in quantum statistical physics and the theory of phase transitions and critical phenomena.Furthermore, the authors provide a 'quantum analog' of Voronin's classical theorem about the universality of the Riemann zeta function. Moreover, they obtain and study quantized counterparts of the Dirichlet series and of the Euler product for the Riemann zeta function, which are shown to converge (in a suitable sense) even inside the critical strip.For pedagogical reasons, most of the book is devoted to the study of the quantized Riemann zeta function. However, the results obtained in this monograph are expected to lead to a quantization of most classic arithmetic zeta functions, hence, further 'naturally quantizing' various aspects of analytic number theory and arithmetic geometry.The book should be accessible to experts and non-experts alike, including mathematics and physics graduate students and postdoctoral researchers, interested in fractal geometry, number theory, operator theory and functional analysis, differential equations, complex analysis, spectral theory, as well as mathematical and theoretical physics. Whenever necessary, suitable background about the different subjects involved is provided and the new work is placed in its proper historical context. Several appendices supplementing the main text are also included.$

جزئیات و درخواست

By Hafedh Herichi, Michel L Lapidus
Publisher: World Scientific, 2021-07-27
Pages: 494
Language: en
ISBN_13:9789813230811
ISBN_10:9813230819
Subject: Mathematics

Fractal Zeta Functions and Fractal Drums

Higher-Dimensional Theory of Complex Dimensions

$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.$

جزئیات و درخواست

By Michel L. Lapidus, Goran Radunović, Darko Žubrinić
Publisher: Springer, 2017-06-07
Pages: 685
Language: en
ISBN_13:9783319447063
ISBN_10:3319447068
Subject: Mathematics

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics: Fractals in pure mathematics

$This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8-12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoit Mandelbrot, held January 4-7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3-4, 2012, in Honolulu, HI. Articles in this volume cover fractal geometry (and some aspects of dynamical systems) in pure mathematics. Also included are articles discussing a variety of connections of fractal geometry with other fields of mathematics, including probability theory, number theory, geometric measure theory, partial differential equations, global analysis on non-smooth spaces, harmonic analysis and spectral geometry. The companion volume (Contemporary Mathematics, Volume 601) focuses on applications of fractal geometry and dynamical systems to other sciences, including physics, engineering, computer science, economics, and finance.$

جزئیات و درخواست

By David Carfi, Michel Laurent Lapidus, Erin P. J. Pearse, Machiel Van Frankenhuysen
Publisher: American Mathematical Soc., 2013-10-22
Pages: 410
Language: en
ISBN_13:9780821891476
ISBN_10:0821891472
Subject: Mathematics

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

Fractals in Applied Mathematics

$This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8-12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoît Mandelbrot, held January 4-7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3-4, 2012, in Honolulu, HI. Articles in this volume cover fractal geometry and various aspects of dynamical systems in applied mathematics and the applications to other sciences. Also included are articles discussing a variety of connections between these subjects and various areas of physics, engineering, computer science, technology, economics and finance, as well as of mathematics (including probability theory in relation with statistical physics and heat kernel estimates, geometric measure theory, partial differential equations in relation with condensed matter physics, global analysis on non-smooth spaces, the theory of billiards, harmonic analysis and spectral geometry). The companion volume (Contemporary Mathematics, Volume 600) focuses on the more mathematical aspects of fractal geometry and dynamical systems.$

جزئیات و درخواست

By David Carfi, Michel L. Lapidus, Erin P. J. Pearse, Machiel van Frankenhuijsen
Publisher: American Mathematical Soc., 2013-10-24
Pages: 384
Language: en
ISBN_13:9780821891483
ISBN_10:0821891480
Subject: Mathematics

Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot

$This volume offers an excellent selection of cutting-edge articles about fractal geometry, covering the great breadth of mathematics and related areas touched by this subject. Included are rich survey articles and fine expository papers. The high-quality contributions to the volume by well-known researchers--including two articles by Mandelbrot--provide a solid cross-section of recent research representing the richness and variety of contemporary advances in and around fractal geometry. In demonstrating the vitality and diversity of the field, this book will motivate further investigation into the many open problems and inspire future research directions. It is suitable for graduate students and researchers interested in fractal geometry and its applications. This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.$

جزئیات و درخواست

By Michel Laurent Lapidus, Machiel Van Frankenhuysen
Publisher: American Mathematical Soc., 2004
Pages: 760
Language: en
ISBN_10:0821836374
ISBN_13:9780821836378
Subject: Mathematics

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

$This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of "fractal drums" (and especially of "fractal strings"). In this work, the authors extend previous results in this area by using the notionof generalized Minkowski content which is defined through some suitable "gauge functions" other than power functions. (This content is used to measure the irregularity (or "fractality") of the boundary of an open set in R]n by evaluating the volume of its small tubular neighborhoods). In the situation when the power function is not the natural "gauge function", this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators. This text will also be of interest to those working in mathematical physics.$

جزئیات و درخواست

By Christina Q. He, Michel Laurent Lapidus
Publisher: American Mathematical Soc., 1997
Pages: 114
Language: en
ISBN_13:9780821805978
ISBN_10:0821805975
Subject: Mathematics

Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot

جزئیات و درخواست

By Michel Laurent Lapidus, Machiel Van Frankenhuysen
Publisher: American Mathematical Soc., 2004
Pages: 534
Language: en
ISBN_13:9780821836378
ISBN_10:0821836374
Subject: Mathematics

Fractal Geometry and Applications: Analysis, number theory, and dynamical systems

جزئیات و درخواست

2004
Language: en
OTHER:LCCN:2004057084
Subject: Ergodic theory

In Search of the Riemann Zeros

Strings, Fractal Membranes and Noncommutative Spacetimes

جزئیات و درخواست

By Michel Laurent Lapidus
Publisher: American Mathematical Soc., 2008
Pages: 594
Language: en
ISBN_10:0821842226
ISBN_13:9780821842225
Subject: Mathematics

Dynamical, Spectral, and Arithmetic Zeta Functions

AMS Special Session on Dynamical, Spectral, and Arithmetic Z...

$The original zeta function was studied by Riemann as part of his investigation of the distribution of prime numbers. Other sorts of zeta functions were defined for number-theoretic purposes, such as the study of primes in arithmetic progressions. This led to the development of $L$-functions, which now have several guises. It eventually became clear that the basic construction used for number-theoretic zeta functions can also be used in other settings, such as dynamics, geometry, and spectral theory, with remarkable results. This volume grew out of the special session on dynamical, spectral, and arithmetic zeta functions held at the annual meeting of the American Mathematical Society in San Antonio, but also includes four articles that were invited to be part of the collection. The purpose of the meeting was to bring together leading researchers, to find links and analogies between their fields, and to explore new methods. The papers discuss dynamical systems, spectral geometry on hyperbolic manifolds, trace formulas in geometry and in arithmetic, as well as computational work on the Riemann zeta function. Each article employs techniques of zeta functions. The book unifies the application of these techniques in spectral geometry, fractal geometry, and number theory. It is a comprehensive volume, offering up-to-date research. It should be useful to both graduate students and confirmed researchers.$

جزئیات و درخواست

By Michel Laurent Lapidus, Machiel Van Frankenhuysen
Publisher: American Mathematical Soc., 2001
Pages: 210
Language: en
ISBN_13:9780821820797
ISBN_10:0821820796
Subject: Mathematics

Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot

جزئیات و درخواست

By Michel Laurent Lapidus, Machiel Van Frankenhuysen
Publisher: American Mathematical Soc., 2004
Pages: 592
Language: en
ISBN_13:9780821836385
ISBN_10:0821836382
Subject: Mathematics

Fractal Geometry and Number Theory

Complex Dimensions of Fractal Strings and Zeros of Zeta Func...

$A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap pendix B.) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.$

جزئیات و درخواست

By Michel L. Lapidus, Machiel van Frankenhuysen
Publisher: Springer Science & Business Media, 2013-12-01
Pages: 277
Language: en
ISBN_13:9781461253143
ISBN_10:1461253144
Subject: Mathematics

Generalized Dyson Series, Generalized Feynman Diagrams, the Feynman Integral and Feynman's Operational Calculus

جزئیات و درخواست

By Gerald W. Johnson, Michel Laurent Lapidus
Publisher: American Mathematical Soc., 1986
Pages: 88
Language: en
ISBN_13:9780821824139
ISBN_10:0821824139
Subject: Mathematics

Feynman's Operational Calculus and Beyond

Noncommutativity and Time-Ordering

جزئیات و درخواست

By Gerald W Johnson, Michel L. Lapidus, Lance Nielsen
Publisher: OUP Oxford, 2015-08-06
Pages: 385
Language: en
ISBN_13:9780191006876
ISBN_10:0191006874
Subject: Science

Progress in Inverse Spectral Geometry

جزئیات و درخواست

By Stig I. Andersson, Michel L. Lapidus
Publisher: Birkhäuser, 2012-12-06
Pages: 202
Language: en
ISBN_13:9783034889384
ISBN_10:3034889380
Subject: Mathematics

Horizons of Fractal Geometry and Complex Dimensions

$This volume contains the proceedings of the 2016 Summer School on Fractal Geometry and Complex Dimensions, in celebration of Michel L. Lapidus's 60th birthday, held from June 21–29, 2016, at California Polytechnic State University, San Luis Obispo, California. The theme of the contributions is fractals and dynamics and content is split into four parts, centered around the following themes: Dimension gaps and the mass transfer principle, fractal strings and complex dimensions, Laplacians on fractal domains and SDEs with fractal noise, and aperiodic order (Delone sets and tilings).$

جزئیات و درخواست

By Robert G. Niemeyer, Erin P. J. Pearse, John A. Rock, Tony Samuel
Publisher: American Mathematical Soc., 2019-06-26
Pages: 302
Language: en
ISBN_13:9781470435813
ISBN_10:1470435810
Subject: Fractals

The Feynman Integral and Feynman's Operational Calculus

جزئیات و درخواست

By Gerald W. Johnson, Michel L. Lapidus
Publisher: Clarendon Press, 2000-03-16
Pages: 790
Language: en
ISBN_13:9780191546266
ISBN_10:0191546267
Subject: Mathematics

Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot

جزئیات و درخواست

By Benoit B. Mandelbrot Michel L. Lapidus Machiel Van Frankenhuysen
Publisher: American Mathematical Soc., 2004
Pages: 724
Language: en
ISBN_10:0821868489
ISBN_13:9780821868485

Towards a Noncommutative Fractal Geometry?

Laplacians and Volume Measures on Fractals

جزئیات و درخواست

By Michel L. Lapidus
1997
Pages: 42
Language: en
OTHER:OCLC:50457621

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