Format:
Online-Ressource (XXVI, 567 p. 73 illus, digital)
Edition:
2nd ed. 2013
ISBN:
9781461421764
,
1283909553
,
9781283909556
Series Statement:
Springer Monographs in Mathematics
Content:
Preface -- Overview -- Introduction -- 1. Complex Dimensions of Ordinary Fractal Strings -- 2. Complex Dimensions of Self-Similar Fractal Strings -- 3. Complex Dimensions of Nonlattice Self-Similar Strings -- 4. Generalized Fractal Strings Viewed as Measures -- 5. Explicit Formulas for Generalized Fractal Strings -- 6. The Geometry and the Spectrum of Fractal Strings -- 7. Periodic Orbits of Self-Similar Flows -- 8. Fractal Tube Formulas -- 9. Riemann Hypothesis and Inverse Spectral Problems -- 10. Generalized Cantor Strings and their Oscillations -- 11. Critical Zero of Zeta Functions -- 12 Fractality and Complex Dimensions -- 13. Recent Results and Perspectives -- Appendix A. Zeta Functions in Number Theory -- Appendix B. Zeta Functions of Laplacians and Spectral Asymptotics -- Appendix C. An Application of Nevanlinna Theory -- Bibliography -- Author Index -- Subject Index -- Index of Symbols -- Conventions -- Acknowledgements.
Content:
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. This second edition of Fractal Geometry, Complex Dimensions and Zeta Functions will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, complex analysis, distribution theory, and mathematical physics. The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Key Features include: · The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt Key Features include: · The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt Key Features include: · The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt · Explicit formulas are extended to apply to the geometric, spectral, and dynamical zeta functions associated with a fractal · Examples of such explicit formulas include a Prime Orbit Theorem with error term for self-similar flows, and a geometric tube formula · The method of Diophantine approximation is used to study self-similar strings and flows · Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of number-theoretic and other zeta functions The unique viewpoint of this book culminates in the definition of fractality as the presence of nonreal complex dimensions. The final chapter (13) is new to the second edition and discusses several new topics, results obtained since the publication of the first edition, and suggestions for future developments in the field. Review of the First Edition: " The book is self contained, the material organized in chapters preceded by an introduction and finally there are some interesting applications of the theory presented. ...The book is very well written and organized and the subject is very interesting and actually has many applications." —Nicolae-Adrian Secelean, Zentralblatt Key Features include: · The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings · Complex dimensions of a fractal string are studied in detail, and used to understand the oscillations intrinsic to the corresponding fractal geometries and frequency spectra · Explicit formulas are extended to apply to the geometric, spectral, and dynami...
Note:
Description based upon print version of record
,
Fractal Geometry, Complex Dimensions and Zeta Functions; Overview; Preface; Contents; List of Figures; List of Tables; Introduction; 1 Complex Dimensions of Ordinary Fractal Strings; 1.1 The Geometry of a Fractal String; 1.1.1 The Multiplicity of the Lengths; 1.1.2 Example: The Cantor String; 1.2 The Geometric Zeta Function of a Fractal String; 1.2.1 The Screen and the Window; 1.2.2 The Cantor String (continued); 1.3 The Frequencies of a Fractal String and the Spectral Zeta Function; 1.4 Higher-Dimensional Analogue: Fractal Sprays; 1.5 Notes
,
2 Complex Dimensions of Self-Similar Fractal Strings2.1 Construction of a Self-Similar Fractal String; 2.1.1 Relation with Self-Similar Sets; 2.2 The Geometric Zeta Function of a Self-Similar String; 2.2.1 Self-Similar Strings with a Single Gap; 2.3 Examples of Complex Dimensions of Self-Similar Strings; 2.3.1 The Cantor String; 2.3.2 The Fibonacci String; 2.3.3 The Modified Cantor and Fibonacci Strings; 2.3.4 A String with Multiple Poles; 2.3.5 Two Nonlattice Examples: the Two-Three String and the Golden String; The Golden String; 2.4 The Lattice and Nonlattice Case
,
2.5 The Structure of the Complex Dimensions2.6 The Asymptotic Density of the Poles in the Nonlattice Case; 2.7 Notes; 3 Complex Dimensions of Nonlattice Self-Similar Strings: Quasiperiodic Patterns and Diophantine Approximation; 3.1 Dirichlet Polynomial Equations; 3.1.1 The Generic Nonlattice Case; 3.2 Examples of Dirichlet Polynomial Equations; 3.2.1 Generic and Nongeneric Nonlattice Equations; 3.2.2 The Complex Roots of the Golden Plus Equation; 3.3 The Structure of the Complex Roots; 3.4 Approximating a Nonlattice Equation by Lattice Equations; 3.4.1 Diophantine Approximation
,
3.4.2 The Quasiperiodic Pattern of the Complex Dimensions3.4.3 Application to Nonlattice Strings; 3.5 Complex Roots of a Nonlattice Dirichlet Polynomial; 3.5.1 Continued Fractions; 3.5.2 Two Generators; 3.5.3 More than Two Generators; 3.6 Dimension-Free Regions; 3.7 The Dimensions of Fractality of a Nonlattice String; 3.7.1 The Density of the Real Parts; 3.8 A Note on the Computations; 3.9 Notes; 4 Generalized Fractal Strings Viewed as Measures; 4.1 Generalized Fractal Strings; 4.1.1 Examples of Generalized Fractal Strings; 4.2 The Frequencies of a Generalized Fractal String
,
4.2.1 Completion of the Harmonic String: Euler Product4.3 Generalized Fractal Sprays; 4.4 The Measure of a Self-Similar String; 4.4.1 Measures with a Self-Similarity Property; 4.5 Notes; 5 Explicit Formulas for Generalized Fractal Strings; 5.1 Introduction; 5.1.1 Outline of the Proof; 5.1.2 Examples; 5.2 Preliminaries: The Heaviside Function; 5.3 Pointwise Explicit Formulas; 5.3.1 The Order of Growth of the Sum over the Complex Dimensions; 5.4 Distributional Explicit Formulas; 5.4.1 Extension to More General Test Functions; 5.4.2 The Order of the Distributional Error Term
,
5.5 Example: The Prime Number Theorem
Additional Edition:
ISBN 9781461421757
Additional Edition:
Buchausg. u.d.T. Lapidus, Michel L., 1956 - Fractal geometry, complex dimensions and zeta functions New York, NY : Springer, 2013 ISBN 9781461421757
Additional Edition:
ISBN 9781489988386
Language:
English
Subjects:
Mathematics
Keywords:
Zetafunktion
;
Geometrische Maßtheorie
;
Fraktal
;
Asymptotische Verteilung
;
Partielle Differentialgleichung
;
Zetafunktion
;
Geometrische Maßtheorie
;
Fraktal
;
Asymptotische Verteilung
;
Partielle Differentialgleichung
;
Fraktalgeometrie
;
Bibliografie
DOI:
10.1007/978-1-4614-2176-4
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Frankenhuysen, Machiel van 1967-
Author information:
Lapidus, Michel L. 1956-
