WorldCat Identities

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
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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 in-depth study of the vibrations of fractal strings, that is, one-dimensional 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 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 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 self-contained 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 self-contained, 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 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
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

Dynamical, spectral, and arithmetic zeta functions : AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions, January 15-16, 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

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 graduate-level description of how ideas from non-commutative 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

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

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

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

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

Complex dimensions and oscillatory phenomena, with applications to the gemetry of fractal strings and to the critical zeros of zeta-functions by Michel L Lapidus( Book )

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

Dynamical, spectral, and arithmetic zeta functions : AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions, January 15-16, 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

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

Brief van van Frankenhuysen aan Johannes Christoffel Schultz Jacobi (1806-1865) by Machiel Van Frankenhuysen( )

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 zeta-functions by Michel L Lapidus( Book )

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

 
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Audience level: 0.63 (from 0.51 for Fractal ge ... to 0.96 for Complex di ...)

Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions, with 26 illustrations
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-

Languages
English (129)

Covers
Fractal geometry and number theory : complex dimensions of fractal strings and zeros of zeta functions, with 26 illustrationsFractal geometry and applications : a jubilee of Benoît MandelbrotDynamical, spectral, and arithmetic zeta functions : AMS Special Session on Dynamical, Spectral, and Arithmetic Zeta Functions, January 15-16, 1999, San Antonio, TexasFractal Geometry, Complex Dimensions and Zeta Functions