Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta FunctionsSpringer Science & Business Media, 2000 - Broj stranica: 268 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. |
Sadržaj
Introduction | 1 |
Complex Dimensions of Ordinary Fractal Strings | 7 |
Complex Dimensions of SelfSimilar Fractal Strings | 23 |
Generalized Fractal Strings Viewed as Measures | 55 |
Explicit Formulas for Generalized Fractal Strings | 71 |
The Geometry and the Spectrum of Fractal Strings | 111 |
Tubular Neighborhoods and Minkowski Measurability | 143 |
The Riemann Hypothesis Inverse Spectral Problems | 163 |
The Critical Zeros of Zeta Functions | 181 |
Concluding Comments | 197 |
A Zeta Functions in Number Theory | 221 |
B Zeta Functions of Laplacians and Spectral Asymptotics | 227 |
References | 235 |
Conventions | 253 |
265 | |
Generalized Cantor Strings and their Oscillations | 173 |
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apply approximated associated assume asymptotic boundary bounded called Cantor string Chapter choose close coefficients complex dimensions condition Conjecture consider continuation converges corresponding counting function defined definition denotes derive Dirichlet discussion distributional drum equal Equation error term estimate example exists expansion explicit formulas expressed extend fact Figure finite fractal spray fractal string frequencies Further geometric give given Hence holds Hypothesis infinite integral interpreted Lap3 lattice string Lemma lengths meromorphic Minkowski measurable multiplicity natural Note notion obtain oscillations oscillatory particular periodic pointwise poles positive present prime problem proof Recall refer Remark residue respectively Riemann zeta function satisfies scaling screen Section self-similar string sense sequence simple spectral spectral zeta function spectrum suitable Theorem theory tion vertical line zeros zeta function
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