We put the theory of Dirichlet series and integrals in the geometric setting of`fractal strings' (one-dimensional drums with fractal boundary). The poles of a Dirichlet series thus acquire the geometric meaning of`complex dimensions' of the associated fractal string, and they describe the geometric and spectral oscillations of this string by means of anèxplicit formula'. We deene thèspectral op...

A direct determination of the fractal dimensions of a fracture surface is essential for a better understanding of its complete topographic characteristics. In this paper, a laser profilometer is employed to measure the topography of a rock’s fracture surface. With the use of the triangular prism surface area and projective covering methods, the resultant data set enables us to directly determin...

OBJECTIVE The purpose of this study was to determine whether the results of fractal analysis can be used as criteria for midpalatal suture maturation evaluation. METHODS The study included 131 subjects aged over 18 years of age (range 18.1-53.4 years) who underwent cone-beam computed tomography. Skeletonized images of the midpalatal suture were obtained via image processing software and used ...

Multiple fractal dimensions can depict the geometry characteristic of the signals from different levels, thus it can extract the features of different signals. The paper proposed an improved algorithm of multiple fractal dimension, it changed the calculation method of traditional multiple fractal dimension which accumulated the q dimensions characteristics. It increased the regularity and gathe...

It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a newly proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that th...

Fractals are measurable metric sets with non-integer Hausdorff dimensions. If electric and magnetic fields are defined on fractal and do not exist outside of fractal in Euclidean space, then we can use the fractional generalization of the integral Maxwell equations. The fractional integrals are considered as approximations of integrals on fractals. We prove that fractal can be described as a sp...

A general technique is developed to retrieve the fractal dimension of self-similar soils through microwave (radar) scattering. The technique is based on a mathematical model relating the fractal dimensions of the georadargram to that of the scattering structure. Clear and different fractal signatures have been observed over four geosystems (soils and sediments) compared in this work.