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

ISBN-10: 0849318068

ISBN-13: 9780849318061

Number idea, 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 positive factors:

- The Riemann speculation is given a common 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 capabilities linked to a fractal

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

- the strategy of diophantine approximation is used to review 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 provides a brand new definition of fractality because the presence of nonreal complicated dimensions with optimistic genuine parts.

The major reviews and difficulties illuminated during this paintings can be utilized in a school room environment on the graduate point. Fractal Geometry, advanced Dimensions and Zeta services will entice scholars and researchers in quantity idea, fractal geometry, dynamical platforms, 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**

**Example text**

1. 6 (Reality Principle). 14) expresses V (ε) as an inﬁnite sum of complex numbers. 13) expresses the real-valued function b−{u} as an inﬁnite sum of complex values. These sums are in fact real-valued, as can be seen by combining the terms for n and −n into one, for n ≥ 1. Indeed, these two terms are the complex conjugate of one another, and hence their sum is real-valued. Thus we ﬁnd the following alternative expression for V (ε): ∞ VCS (ε) = 1 2−D ε1−D + Re D(1 − D) log 3 log 3 n=1 (2ε)1−D−inp (D + inp)(1 − D − inp) − 2ε.

It will become very helpful when we study direct or inverse spectral problems. 19. The spectral counting function of L is given by Nν (x) = NL (x) + NL x x + NL + ... 38) where ζ(s) is the Riemann zeta function. Thus ζν (s) is holomorphic for Re s > 1. 19)). Moreover, it has a meromorphic extension to a neighborhood of the window W of L. Proof. For the spectral counting function, this follows from the following computation: ∞ ∞ Nν (x) = #{j : lj−1 ≤ x/k} = 1= k=1 j: k·l−1 ≤x j k=1 ∞ NL k=1 x . k Observe that this is a ﬁnite sum, since NL (y) = 0 for y < l1−1 .

4 (Independence of the Geometric Realization). Observe that the Minkowski dimension of a self-similar set coincides with its Hausdorﬀ dimension. The basic reason for using the Minkowski dimension is that it is invariant under displacements of the intervals of which a fractal string is composed. 1, pp. 512–514], [LapPo2]). 22 below for further comparison between the various notions of fractal dimensions and for additional justiﬁcation of the choice of the notion of Minkowski dimension in the context of fractal strings.

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

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