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Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions
 
Christina Q. He University of California, Riverside, Riverside, CA
Michel L. Lapidus University of California, Riverside, Riverside, CA
Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions
eBook ISBN:  978-1-4704-0193-1
Product Code:  MEMO/127/608.E
List Price: $46.00
MAA Member Price: $41.40
AMS Member Price: $27.60
Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions
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Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions
Christina Q. He University of California, Riverside, Riverside, CA
Michel L. Lapidus University of California, Riverside, Riverside, CA
eBook ISBN:  978-1-4704-0193-1
Product Code:  MEMO/127/608.E
List Price: $46.00
MAA Member Price: $41.40
AMS Member Price: $27.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1271997; 97 pp
    MSC: Primary 35; 11; 28; 34; 58; Secondary 26; 47

    This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of “fractal drums” (and especially of “fractal strings”).

    In this work, the authors extend previous results in this area by using the notion of generalized Minkowski content which is defined through some suitable “gauge functions” other than power functions. (This content is used to measure the irregularity (or “fractality”) of the boundary of an open set in \(R^n\) by evaluating the volume of its small tubular neighborhoods.) In the situation when the power function is not the natural “gauge function”, this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators.

    Readership

    Graduate students and research mathematicians interested in dynamical systems, fractal geometry, partial differential equations, analysis, measure theory, number theory or spectral geometry. Physicists interested in fractal geometry, condensed matter physics or wave propagation in random or fractal media.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. Statement of the main results
    • 3. Sharp error estimates and their converse when $n$ = 1
    • 4. Spectra of fractal strings and the Riemann zeta-function
    • 5. The complex zeros of the Riemann zeta-function
    • 6. Error estimates for $n \geq 2$
    • 7. Examples
    • Appendix. Examples of gauge functions
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1271997; 97 pp
MSC: Primary 35; 11; 28; 34; 58; Secondary 26; 47

This memoir provides a detailed study of the effect of non power-like irregularities of (the geometry of) the fractal boundary on the spectrum of “fractal drums” (and especially of “fractal strings”).

In this work, the authors extend previous results in this area by using the notion of generalized Minkowski content which is defined through some suitable “gauge functions” other than power functions. (This content is used to measure the irregularity (or “fractality”) of the boundary of an open set in \(R^n\) by evaluating the volume of its small tubular neighborhoods.) In the situation when the power function is not the natural “gauge function”, this enables the authors to obtain more precise estimates, with a broader potential range of applications than in previous papers of the second author and his collaborators.

Readership

Graduate students and research mathematicians interested in dynamical systems, fractal geometry, partial differential equations, analysis, measure theory, number theory or spectral geometry. Physicists interested in fractal geometry, condensed matter physics or wave propagation in random or fractal media.

  • Chapters
  • 1. Introduction
  • 2. Statement of the main results
  • 3. Sharp error estimates and their converse when $n$ = 1
  • 4. Spectra of fractal strings and the Riemann zeta-function
  • 5. The complex zeros of the Riemann zeta-function
  • 6. Error estimates for $n \geq 2$
  • 7. Examples
  • Appendix. Examples of gauge functions
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.