# Fractal Geometry and Number Theory

## Michel L. Lapidus; Machiel van Frankenhuysen

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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.

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van Frankenhuysen, Institut des Hautes Etudes Scientifiques, France.

Lapidus, University of California, Riverside.

Review:

"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 (Review of First Edition)

"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 (Review of First Edition)

"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 First Edition)

Le informazioni nella sezione "Su questo libro" possono far riferimento a edizioni diverse di questo titolo.

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## 1.Fractal Geometry and Number Theory

Editore: Birkhauser Boston (1999)
ISBN 10: 0817640983 ISBN 13: 9780817640989
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Descrizione libro Birkhauser Boston, 1999. HRD. Condizione libro: New. New Book.Shipped from US within 10 to 14 business days.THIS BOOK IS PRINTED ON DEMAND. Established seller since 2000. Codice libro della libreria IP-9780817640989

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## 2.Fractal Geometry and Number Theory. Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

Editore: BirkhÇÏuser (1999)
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Descrizione libro BirkhÇÏuser, 1999. Hardback. Condizione libro: NEW. 9780817640989 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Codice libro della libreria HTANDREE0285653

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## 3.Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

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Descrizione libro BirkhÃ¤user, 2016. Paperback. Condizione libro: New. PRINT ON DEMAND Book; New; Publication Year 2016; Not Signed; Fast Shipping from the UK. No. book. Codice libro della libreria ria9780817640989_lsuk

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## 4.Fractal Geometry and Number Theory

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Descrizione libro Birkhauser Boston, 1999. HRD. Condizione libro: New. New Book. Delivered from our US warehouse in 10 to 14 business days. THIS BOOK IS PRINTED ON DEMAND.Established seller since 2000. Codice libro della libreria IP-9780817640989

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## 5.Fractal Geometry and Number Theory

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Descrizione libro SPRINGER VERLAG GMBH. soft. Condizione libro: New. Can one hear the shape of a drum? The relationship between the shape (geometry) of a drum and its. . . *** Nota: EL COSTE DE ENVÍO A CANARIAS ES 8 euros. Si ha realizado un pedido con destino a CANARIAS no podemos hacer el envío con un coste de 3,5 euros . Nos pondremos en contacto con usted para comunicar el coste total del envío a Canarias y si está de acuerdo, Abebooks le efectuará el cargo adicional. Codice libro della libreria 888761

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## 6.Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Hardback)

Editore: Birkhauser Boston, Germany (1999)
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Descrizione libro Birkhauser Boston, Germany, 1999. Hardback. Condizione libro: New. 1999 ed.. Language: English . Brand New Book ***** Print on Demand *****.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. Codice libro della libreria APC9780817640989

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## 7.Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions (Hardback)

Editore: Birkhauser Boston, Germany (1999)
ISBN 10: 0817640983 ISBN 13: 9780817640989
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Descrizione libro Birkhauser Boston, Germany, 1999. Hardback. Condizione libro: New. 1999 ed.. Language: English . Brand New Book ***** Print on Demand *****. 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. Codice libro della libreria APC9780817640989

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## 8.Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions

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Descrizione libro Birkhauser. Hardcover. Condizione libro: New. Hardcover. 280 pages. Dimensions: 9.3in. x 6.2in. x 0.9in.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. jl 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. This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Hardcover. Codice libro della libreria 9780817640989

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## 9.Fractal Geometry And Number Theory: Complex Dimensions Of Fractal Strings And Zeros Of Zeta Functions

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Descrizione libro Condizione libro: New. This item is Print on Demand - Depending on your location, this item may ship from the US or UK. Codice libro della libreria POD_9780817640989

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## 10.Fractal Geometry and Number Theory

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Descrizione libro Birkhäuser, 1999. Hardcover. Condizione libro: New. 1999. Codice libro della libreria DADAX0817640983

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