Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics

Hambly, Ben; Lapidus, Michel L.

doi:10.1090/s0002-9947-05-03646-9

Cited by 31 publications

(31 citation statements)

References 19 publications

(37 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many applications and extensions of fractal string theory and/or of the corresponding theory of complex fractal dimensions can be found throughout the books La5] and in [La1,La2,La3,La4,LaPo1,LaPo2,LaPo3,LaMa1,LaMa2,HeLa,HamLa,Tep,LaPe,LaPeWi,LaLeRo,ElLaMaRo,LaLu1,LaLu2,LalLa1,LalLa2,LaRaZu,HerLa1,HerLa2,HerLa3,HerLa4,La6]. These include, in particular, applications to various aspects of number theory and arithmetic geometry, dynamical systems, spectral geometry, geometric measure theory, noncommutative geometry, mathematical physics and nonarchimedean analysis.…”

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Herichi¹,

Lapidus²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

Self Cite

View full text Add to dashboard Cite

A Christophe Soulé, avec une profonde amitié et admiration,à l'occasion de ses 60 ans Abstract. Nous rappelons quelques unes des principales propriétés d'universalité de la fonction zêta de Riemann ζ(s). De plus, nous expliquons comment obtenir une quantification naturelle du théorème d'universalité de Voronin (et de ses généralizations). Notre travail est basé sur la théorie des cordes fractales développée par le deuxiéme auteur et M. van Frankenhuijsen dans . Nous utilisonségalement la théorie développée dans [HerLa1-3] par les auteurs de cet article pourétudier de façon rigoreuse l'opérateur spectral (qui relie la géométrie et le spectre des cordes fractales généralizées). Cet opérateur spectral est representée comme le composé de la fonction zêta de Riemann du 'shift infinitesimal' ∂ : a = ζ(∂). Dans le processus du quantification du théorème d'universalité de la fonction zêta de Riemann, le rôle joué par la variable s (au sens classique du théoréme d'universalité) (dans le théorème classique d'universalité) est joué par la famille des shifts infinitésimaux tronqués afin d'étudier l'opérateur spectral en lien avec la reformulation spectrale de l'hypothése de Riemann vue comme un problème spectral inverse pour les cordes fractales. Ce dernier résultat fournit une version opératorielle de la reformulation spectrale obtenue par le second auteur et H. Maier dans [LaMa2]. Notre présent travail au long terme, ainsi que [La5, La6], a en partie pour but d'obtenir une quantification naturelle de divers aspects de la théorie analytiques des nombres et de la géométrie arithmétiques.2010 Mathematics Subject Classification. Primary 11M06, 11M26, 11M41, 28A80, 32B40, 47A10, 47B25, 65N21, 81Q12, 82B27. Secondary 11M55, 28A75, 34L05, 34L20, 35P20, 47B44, 47D03, 81R40.Key words and phrases. Riemann zeta function, Riemann zeros, Riemann hypothesis, spectral reformulations, fractal strings, complex dimensions, explicit formulas, geometric and spectral zeta functions, geometric and spectral counting functions, inverse spectral problems, infinitesimal shift, truncated infinitesimal shifts, spectral operator, truncated spectral operators, universality of the Riemann zeta function, universality of the spectral operator.

show abstract

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Herichi¹,

Lapidus²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

Self Cite

View full text Add to dashboard Cite

show abstract

“…We recall the notion of "fractal string" and "fractal spray from [30], see also [25,26,20,18,19,27,10] …”

Section: 1mentioning

confidence: 99%

Tube Formulas and Complex Dimensions of Self-Similar Tilings

Lapidus

Pearse

2010

Acta Appl Math

Self Cite

View full text Add to dashboard Cite

Abstract. We use the self-similar tilings constructed in [32] to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubular zeta function and hence develop a tube formula for self-similar tilings in R d . The resulting power series in ε is a fractal extension of Steiner's classical tube formula for convex bodies K ⊆ R d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i = 0, 1, . . . , d − 1, just as Steiner's does. However, our formula also contains a term for each complex dimension. This provides further justification for the term "complex dimension". It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in [30].

show abstract

“…However, in recent years, advances in the theory of chaos and fractals revealed relationships with fractional derivatives and integrals, leading to renewed interest in this field [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

Fractional order calculus: historical apologia, basic concepts and some applications

David

Linares

Pallone

2011

Rev. Bras. Ensino Fís.

View full text Add to dashboard Cite

Fractional order calculus (FOC) deals with integrals and derivatives of arbitrary (i.e., non-integer) order, and shares its origins with classical integral and differential calculus. However, until recently, it has been investigated mainly from a mathematical point of view. Advances in the field of fractals have revealed its subtle relationships with fractional calculus. Nonetheless, fractional calculus is generally excluded from standard courses in mathematics, partly because many mathematicians are unfamiliar with its nature and its applications. This area has emerged as a useful tool among researchers. One of the objectives of this paper is to discuss the usefulness of fractional calculus in applied sciences and engineering. In view of the increasing interest in the development of the new paradigm, another objective is to encourage the use of this mathematical idea in various scientific areas by means of a historical apologia for the development of fractional calculus. Keywords: fractional order calculus, non-integer order systems, dynamical systems.O cálculo de ordem fracionária (COF) lida com derivadas e integrais de ordem arbitrária (ou seja, não-inteiros) e tem as mesmas origens do clássico cálculo diferencial e integral. No entanto, até recentemente, tem sido investigado principalmente do ponto de vista matemático. Progressos nasáreas de fractais revelaram suas sutis relações com o cálculo fracionário. No entanto, o cálculo fracionárioé geralmente excluído dos cursos tradicionais de Matemática, em parte porque muitos matemáticos não estão familiarizados com a sua natureza e suas aplicações. O cálculo fracionário tem se mostrado uma ferramentaútil entre os pesquisadores. Um dos objetivos deste trabalhoé discutir a utilidade do cálculo fracionário nas ciências aplicadas e na engenharia. Além disso, outro propósitoé incentivar o uso dessa idéia matemática como ferramenta para pesquisadores das mais diversaś areas por meio de uma apologia histórica envolvendo o desenvolvimento do cálculo fracionário. Palavras-chave: cálculo fracionário, sistemas de ordem não-inteira, sistemas dinâmicos.

show abstract

Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics

Cited by 31 publications

References 19 publications

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Tube Formulas and Complex Dimensions of Self-Similar Tilings

Fractional order calculus: historical apologia, basic concepts and some applications

Contact Info

Product

Resources

About