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Generalized Minkowski Content, Spectrum of Fractal Drums, Fractal Strings and the Riemann Zeta-Functions
Christina Q. He and Michel L. Lapidus, University of California, Riverside, CA
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Memoirs of the American Mathematical Society
1997; 97 pp; softcover
Volume: 127
ISBN-10: 0-8218-0597-5
ISBN-13: 978-0-8218-0597-8
List Price: US$43
Individual Members: US$25.80
Institutional Members: US$34.40
Order Code: MEMO/127/608
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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

  • Introduction
  • Statement of the main results
  • Sharp error estimates and their converse when \(n=1\)
  • Spectra of fractal strings and the Riemann zeta-function
  • The complex zeros of the Riemann zeta-function
  • Error estimates for \(n\geq 2\)
  • Examples
  • Appendix: Examples of Gauge functions
  • References
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