Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings (Springer Monographs in Mathematics)
De (autor) Michel L. Lapidus, Machiel van Frankenhuijsenen Limba Engleză Carte Hardback – 20 Sep 2012
Key Features of this Second Edition:
The Riemann hypothesis is given a natural geometric reformulation in the context of vibrating fractal strings
Complex dimensions of a fractal string, defined as the poles of an associated zeta function, are studied in detail, then 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 selfsimilar flows, and a geometric tube formula
The method of Diophantine approximation is used to study selfsimilar strings and flows
Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic and other zeta functions
Throughout, 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.
The significant studies and problems illuminated in this work may be used in a classroom setting at the graduate level. Fractal Geometry, Complex Dimensions and Zeta Functions, Second Edition will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics.
Review of the First Edition:
"In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding 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 actual and has many applications."  NicolaeAdrian Secelean for Zentralblatt MATH
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Specificații
ISBN13: 9781461421757
ISBN10: 1461421756
Pagini: 596
Ilustrații: 73 schwarzweiße Abbildungen, 3 schwarzweiße Tabellen
Dimensiuni: 155 x 235 x 40 mm
Greutate: 1.05 kg
Ediția: 2nd ed. 2013
Editura: Springer
Colecția Springer
Seria Springer Monographs in Mathematics
Locul publicării: New York, NY, United States
ISBN10: 1461421756
Pagini: 596
Ilustrații: 73 schwarzweiße Abbildungen, 3 schwarzweiße Tabellen
Dimensiuni: 155 x 235 x 40 mm
Greutate: 1.05 kg
Ediția: 2nd ed. 2013
Editura: Springer
Colecția Springer
Seria Springer Monographs in Mathematics
Locul publicării: New York, NY, United States
Public țintă
ResearchCuprins
Preface.
Overview.
Introduction.
1.
Complex
Dimensions
of
Ordinary
Fractal
Strings.
2.
Complex
Dimensions
of
SelfSimilar
Fractal
Strings.
3.
Complex
Dimensions
of
Nonlattice
SelfSimilar
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
SelfSimilar
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.
Recenzii
“This
interesting
volume
gives
a
thorough
introduction
to
an
active
field
of
research
and
will
be
very
valuable
to
graduate
students
and
researchers
alike.”
(C.
Baxa,
Monatshefte
für
Mathematik,
Vol.
180,
2016)
“In this research monograph the authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a selfcontained manner the results … are completely proved. I appreciate that the book is useful for mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying the fractals and dimension theory.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, April, 2013)
“The authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a selfcontained manner, the results (including some fundamental ones) are completely proved. … the book will be useful to mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying fractals and dimension theory.” (NicolaeAdrian Secelean, Zentralblatt MATH, Vol. 1261, 2013)
"In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding 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 actual and has many applications."  NicolaeAdrian Secelean for Zentralblatt MATH
"This highly original selfcontained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style."  Mathematical Reviews (Review of previous book by authors)
"It is the reviewera (TM)s opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is selfcontained, intelligent and well paced."  Bulletin of the London Mathematical Society (Review of previous book by authors)
"The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics."  Simulation News Europe (Review of previous book by authors)
“In this research monograph the authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a selfcontained manner the results … are completely proved. I appreciate that the book is useful for mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying the fractals and dimension theory.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, April, 2013)
“The authors provide a mathematical theory of complex dimensions of fractal strings and its many applications. … The book is written in a selfcontained manner, the results (including some fundamental ones) are completely proved. … the book will be useful to mathematicians, students, researchers, postgraduates, physicians and other specialists which are interested in studying fractals and dimension theory.” (NicolaeAdrian Secelean, Zentralblatt MATH, Vol. 1261, 2013)
"In this book the author encompasses a broad range of topics that connect many areas of mathematics, including fractal geometry, number theory, spectral geometry, dynamical systems, complex analysis, distribution theory and mathematical physics. The book is self containing, the material organized in chapters preceding 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 actual and has many applications."  NicolaeAdrian Secelean for Zentralblatt MATH
"This highly original selfcontained book will appeal to geometers, fractalists, mathematical physicists and number theorists, as well as to graduate students in these fields and others interested in gaining insight into these rich areas either for its own sake or with a view to applications. They will find it a stimulating guide, well written in a clear and pleasant style."  Mathematical Reviews (Review of previous book by authors)
"It is the reviewera (TM)s opinion that the authors have succeeded in showing that the complex dimensions provide a very natural and unifying mathematical framework for investigating the oscillations in the geometry and the spectrum of a fractal string. The book is well written. The exposition is selfcontained, intelligent and well paced."  Bulletin of the London Mathematical Society (Review of previous book by authors)
"The new approach and results on the important problems illuminated in this work will appeal to researchers and graduate students in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics."  Simulation News Europe (Review of previous book by authors)
Textul de pe ultima copertă
Number
theory,
spectral
geometry,
and
fractal
geometry
are
interlinked
in
this
indepth
study
of
the
vibrations
of
fractal
strings;
that
is,
onedimensional
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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian 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 selfsimilar flows, and a geometric tube formula
· The method of Diophantine approximation is used to study selfsimilar strings and flows
· Analytical and geometric methods are used to obtain new results about the vertical distribution of zeros of numbertheoretic 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."
—NicolaeAdrian Secelean, Zentralblatt
Caracteristici
The
Riemann
hypothesis
is
given
a
natural
geometric
reformulation
in
the
context
of
vibrating
fractal
strings
Number theory, spectral geometry, and fractal geometry are interlinked in this indepth study of the vibrations of fractal strings, that is, onedimensional drums with fractal boundary
Numerous theorems, examples, remarks and illustrations enrich the text
Number theory, spectral geometry, and fractal geometry are interlinked in this indepth study of the vibrations of fractal strings, that is, onedimensional drums with fractal boundary
Numerous theorems, examples, remarks and illustrations enrich the text