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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
Available Formats:
Electronic 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 Click above image for expanded view 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 Available Formats:  Electronic 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.

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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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.

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 reviewers who would like to review an AMS book
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.