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

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

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

Key good points:

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

- advanced 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

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

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

- the tactic of diophantine approximation is used to review self-similar strings and flows

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

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

The major stories and difficulties illuminated during this paintings can be utilized in a school room surroundings on the graduate point. Fractal Geometry, complicated Dimensions and Zeta features will attract scholars and researchers in quantity conception, fractal geometry, dynamical structures, spectral geometry, and mathematical physics.

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

Sample text

However, for simplicity, we do not indicate this explicitly in our notation. 18. 35) f which converges for Re s sufficiently large. 35), the sum is extended over all distinct frequencies of L. 9 According to footnote 1, the frequency f = x should be counted with the weight 1/2. 24 1. Complex Dimensions of Ordinary Fractal Strings ∞ Let ζ(s) be the Riemann zeta function defined by ζ(s) = n=1 n−s for Re s > 1. It is well known that ζ(s) has an extension to the whole complex plane as a meromorphic function, with one simple pole at s = 1, with residue 1.

13. 10, ζL (s) is holomorphic in the half-plane Re s > D and hence DL = DL (W ) ⊂ {s ∈ W : Re s ≤ D} . 26) Moreover, since it is the set of poles of a meromorphic function, DL (W ) is a discrete subset of C. Hence its intersection with any compact subset of C is finite. When L consists of finitely many lengths, we have DL = ∅, since then, ζL (s) is an entire function. 14. , from [Ser, Proposition 7, p. 18) has positive coefficients). Thus s = DL is always a singularity of ζL (s), but not necessarily a pole.

9). 1 The Multiplicity of the Lengths Another way of representing a fractal string L is by listing its different lengths l, together with their multiplicity wl : wl = #{j ≥ 1 : lj = l}. 7) Thus, for example, NL (x) = wl . 8) l−1 ≤x In Chapter 4, we will introduce a third way to represent a fractal string, similar to this one, namely, by a measure. 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, .

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