Van Frankenhuysen, Machiel 1967
Overview
Works:  17 works in 134 publications in 1 language and 2,551 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Author, Editor, Correspondent, Other 
Classifications:  QA614.86, 514.742 
Publication Timeline
.
Most widely held works by
Machiel Van Frankenhuysen
Fractal geometry, complex dimensions and zeta functions : geometry and spectra of fractal strings by
Michel L Lapidus(
Book
)
30 editions published between 2006 and 2013 in English and held by 352 WorldCat member libraries worldwide
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. Key Features 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. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts, and discusses several open problems and extensions. 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 will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics. From Reviews of Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, by Michel Lapidus and Machiel van Frankenhuysen, Birkhäuser Boston Inc., 2000. "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 "It is the reviewer'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
30 editions published between 2006 and 2013 in English and held by 352 WorldCat member libraries worldwide
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. Key Features 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. The final chapter gives a new definition of fractality as the presence of nonreal complex dimensions with positive real parts, and discusses several open problems and extensions. 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 will appeal to students and researchers in number theory, fractal geometry, dynamical systems, spectral geometry, and mathematical physics. From Reviews of Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, by Michel Lapidus and Machiel van Frankenhuysen, Birkhäuser Boston Inc., 2000. "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 "It is the reviewer'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
Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions, with 26 illustrations by
Michel L Lapidus(
Book
)
17 editions published between 1999 and 2012 in English and held by 324 WorldCat member libraries worldwide
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 onedimensional 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 [Berrl2, Lapl4, LapPol3, LapMal2, HeLapl2] 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 [Lapl3, LapPol3, LapMal2, HeLapl2]). 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
17 editions published between 1999 and 2012 in English and held by 324 WorldCat member libraries worldwide
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 onedimensional 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 [Berrl2, Lapl4, LapPol3, LapMal2, HeLapl2] 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 [Lapl3, LapPol3, LapMal2, HeLapl2]). 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
Fractal geometry and applications : a jubilee of Benoît Mandelbrot(
Book
)
39 editions published in 2004 in English and held by 314 WorldCat member libraries worldwide
39 editions published in 2004 in English and held by 314 WorldCat member libraries worldwide
Dynamical, spectral, and arithmetic zeta functions : AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions,
January 1516, 1999, San Antonio, Texas by spectral, and arithmetic Zeta functions AMS Special session on dynamical(
Book
)
10 editions published between 2001 and 2002 in English and held by 211 WorldCat member libraries worldwide
10 editions published between 2001 and 2002 in English and held by 211 WorldCat member libraries worldwide
The Riemann hypothesis for function fields : Frobenius flow and shift operators by
Machiel Van Frankenhuysen(
Book
)
15 editions published between 2013 and 2014 in English and Undetermined and held by 161 WorldCat member libraries worldwide
A graduatelevel description of how ideas from noncommutative geometry could provide a means to attack the Riemann hypothesis, one of the most important conjectures in mathematics. The book provides a strong foundation for further research in this area
15 editions published between 2013 and 2014 in English and Undetermined and held by 161 WorldCat member libraries worldwide
A graduatelevel description of how ideas from noncommutative geometry could provide a means to attack the Riemann hypothesis, one of the most important conjectures in mathematics. The book provides a strong foundation for further research in this area
Fractal geometry and dynamical systems in pure and applied mathematics by Fractal Geometry, Dynamical Systems and Economics PISRS International Conference on Analysis(
Book
)
4 editions published in 2013 in English and held by 97 WorldCat member libraries worldwide
4 editions published in 2013 in English and held by 97 WorldCat member libraries worldwide
Hyperbolic spaces and the abc conjecture by
Machiel Van Frankenhuysen(
Book
)
3 editions published in 1995 in English and Undetermined and held by 13 WorldCat member libraries worldwide
3 editions published in 1995 in English and Undetermined and held by 13 WorldCat member libraries worldwide
Fractal geometry and dynamical systems in pure and applied mathematics by Fractal Geometry, Dynamical Systems and Economics PISRS International Conference on Analysis(
Book
)
4 editions published in 2013 in English and held by 7 WorldCat member libraries worldwide
4 editions published in 2013 in English and held by 7 WorldCat member libraries worldwide
Fractal Geometry and Number Theory Complex Dimensions of Fractal Strings and Zeros of Zeta Functions by
Michel L Lapidus(
)
1 edition published in 2000 in English and held by 6 WorldCat member libraries worldwide
1 edition published in 2000 in English and held by 6 WorldCat member libraries worldwide
Fractal Geometry, Complex Dimensions and Zeta Functions by
Machiel Van Frankenhuysen(
Book
)
1 edition published in 2006 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2006 in English and held by 3 WorldCat member libraries worldwide
Complex dimensions and oscillatory phenomena, with applications to the gemetry of fractal strings and to the critical zeros
of zetafunctions by
Michel L Lapidus(
Book
)
2 editions published in 1997 in English and held by 2 WorldCat member libraries worldwide
2 editions published in 1997 in English and held by 2 WorldCat member libraries worldwide
Good abc examples over number fields by
Machiel Van Frankenhuysen(
Book
)
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
Dynamical, spectral, and arithmetic zeta functions : AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions,
January 1516, 1999, San Antonio, Texas by
Spectral, and Arithmetic Zeta Functions AMS Special Session on Dynamical(
)
1 edition published in 2001 in English and held by 1 WorldCat member library worldwide
1 edition published in 2001 in English and held by 1 WorldCat member library worldwide
Complex dimensions of fractal strings and oscillatory phenomena in geometry and arithmetic by
Michel L Lapidus(
Book
)
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
Brief van van Frankenhuysen aan Johannes Christoffel Schultz Jacobi (18061865) by
Machiel Van Frankenhuysen(
)
1 edition published in 1852 in Undetermined and held by 1 WorldCat member library worldwide
1 edition published in 1852 in Undetermined and held by 1 WorldCat member library worldwide
Complex dimensions of fractal strings and explicit formulas for geometric and spectral zetafunctions by
Michel L Lapidus(
Book
)
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
1 edition published in 1997 in English and held by 1 WorldCat member library worldwide
Fractal geometry and dynamical systems in pure and applied mathematics by Fractal Geometry, Dynamical Systems and Economics PISRS 2011 International Conference on Analysis(
)
in English and held by 0 WorldCat member libraries worldwide
in English and held by 0 WorldCat member libraries worldwide
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Related Identities
 Lapidus, Michel L. (Michel Laurent) 1956 Other Author Editor
 Mandelbrot, Benoit B. Honoree Dedicatee
 Carfi, David 1971 Editor
 Pearse, Erin P. J. 1975 Editor
 Special Session on Fractal Geometry in Pure and Applied Mathematics in Memory of Benoît Mandelbrot 2012 Boston, Mass
 American Mathematical Society
 American Mathematical Society
 London Mathematical Society
 Machiel
 AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions
Useful Links
Associated Subjects
Algebraic fields Differentiable dynamical systems Differential equations, Partial Ergodic theory Fractals Functional analysis Functions, Zeta Geometry Geometry, Algebraic Geometry, Riemannian Global differential geometry Hyperbolic spaces Mathematics Measure theory Noncommutative differential geometry Number theory Prediction (Logic) Riemann hypothesis Spectral geometry
Alternative Names
@Frankenhuysen, Machiel van
@Van Frankenhuijsen, Machiel
@Van Frankenhuysen, Machiel
Frankenhuijsen, Machiel van.
Frankenhuijsen, Machiel van 1967
Frankenhuysen, M. van 1967
Frankenhuysen, Machiel van.
Frankenhuysen Machiel van 1967....
Machiel van Frankenhuijsen
Machiel van Frankenhuijsen niederländischer Mathematiker
Machiel van Frankenhuysen
Van Frankenhuijsen, Machiel.
Van Frankenhuijsen, Machiel 1967
Van Frankenhuysen, Machiel
Van Frankenhuysen, Machiel 1967
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