Fractal Geometry, Complex Dimensions and Zeta Functions: by Bruce E. Larock, Roland W. Jeppson, Gary Z. Watters

By Bruce E. Larock, Roland W. Jeppson, Gary Z. Watters

Number idea, spectral geometry, and fractal geometry are interlinked during this in-depth research of the vibrations of fractal strings, that's, one-dimensional drums with fractal boundary.

Key good points:

- The Riemann speculation is given a usual geometric reformulation within the context of vibrating fractal strings

- complicated dimensions of a fractal string, outlined because the poles of an linked zeta functionality, are studied intimately, then used to appreciate the oscillations intrinsic to the corresponding fractal geometries and frequency spectra

- particular formulation are prolonged to use to the geometric, spectral, and dynamic zeta features linked to a fractal

- Examples of such formulation comprise best Orbit Theorem with blunders time period for self-similar flows, and a tube formula

- the strategy of diophantine approximation is used to check self-similar strings and flows

- Analytical and geometric equipment are used to procure new effects concerning the vertical distribution of zeros of number-theoretic and different zeta functions

Throughout new effects are tested. the ultimate bankruptcy supplies a brand new definition of fractality because the presence of nonreal complicated dimensions with confident genuine parts.

The major reports and difficulties illuminated during this paintings can be used in a school room environment on the graduate point. Fractal Geometry, advanced Dimensions and Zeta services will entice scholars and researchers in quantity thought, fractal geometry, dynamical platforms, spectral geometry, and mathematical physics.

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Extra info for Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings

Example text

4), p. 169] using a key result of Besicovich and Taylor [BesTa]; see also [LapPo2, LapMa2]. 3: we refer to [Da, Edw, In, Ivi, Pat, Ti] for the classical theory of the Riemann zeta function. 19 was established in [Lap2, Eqs. 3), p. 169] and used, in particular, in [Lap2–3, LapPo1–3, LapMa1–2, HeLap1–2]. 1], or [Lap1, esp. Chapter 2 and pp. 510–511], as well as the relevant references therein for more information on spectral geometry. 4: the notion of fractal spray was introduced in [LapPo3]. 1b], [FlVa, Ger, GerSc1–2, LapPo3, LeVa], are special cases of ordinary fractal sprays.

In Chapter 10, we will study generalized Cantor strings, which can have any sequence {D + inp}n∈Z (for arbitrary D ∈ (0, 1) and p > 0) as their complex dimensions. We note that such strings can no longer be realized geometrically as subsets of Euclidean space. 4 Higher-Dimensional Analogue: Fractal Sprays Fractal sprays were introduced in [LapPo3] (see also [Lap2, §4] announcing some of the results in [LapPo3]) as a natural higher-dimensional analogue of fractal strings and as a tool to explore various conjectures about the spectrum (and the geometry) of drums with fractal boundary10 in Rd .

2: The Cantor string. 037-tubular neighborhood of the Cantor string. 2). Thus CS = ( 13 , 23 )∪( 19 , 29 )∪( 79 , 89 )∪ 1 2 27 , 27 ∪ 7 8 27 , 27 ∪ 19 20 27 , 27 ∪ 25 26 27 , 27 ∪ s, ˙ so that l1 = 1/3, l2 = l3 = 1/9, l4 = l5 = l6 = l7 = 1/27, . . , or alternatively, the lengths are the numbers 3−n−1 with multiplicity w3−n−1 = 2n , for n = 0, 1, 2, . . We note that by construction, the boundary ∂Ω of the Cantor string is equal to the ternary Cantor set. In general, the volume of the tubular neighborhood of the boundary of L is given by (see [LapPo2, Eq.

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