A Characterization of Self-similar Symbolic Spaces

In this paper we use fractal structures to study self-similar sets and self-similar symbolic spaces. We show that these spaces have a natural fractal structure, justifying the name of fractal structure, and we characterize self-similar symbolic spaces in terms of fractal structures. We prove that self-similar symbolic spaces can be characterized in a similar way, in the form, to the definition of classical self-similar sets by means of iterated function systems. We also study when a self-similar symbolic space is a self-similar set. Finally, we study relations between fractal structures with “pieces” homeomorphic to the space and different concepts of self-homeomorphic spaces. Along the paper, we propose several methods in order to construct self-similar sets and self-similar symbolic spaces from a geometrical approach. This allows to construct these kind of spaces in a very easy way.     

Self-similar p-adic fractal strings and their complex dimensions

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings. The text was submitted by the authors in English.     

Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics

In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.

     

Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets

We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corresponding functionals for self-similar sets. In particular, we characterize the situation, when these functionals coincide. In this case, the Minkowski content and the fractal curvatures of a self-similar set can be expressed completely in terms of the volume function or curvature data, respectively, of the generator of the tiling. In special cases such formulas have been obtained recently using tube formulas and complex dimensions or as a corollary to results on self-conformal sets. Our approach based on the classical Renewal Theorem is simpler and works for a much larger class of self-similar sets and tilings. In fact, generator type formulas are obtained for essentially all self-similar sets, when suitable volume functions (and curvature functions, respectively) related to the generator are used. We also strengthen known results on the Minkowski measurability of self-similar sets, in particular on the question of non-measurability in the lattice case.     

A trace theorem for Dirichlet forms on fractals

We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on Sierpinski gaskets and Sierpinski carpets to their boundaries, where the boundaries are represented by triangles and squares that confine the gaskets and the carpets. As an application, we construct diffusion processes on a collection of fractals called fractal fields. These processes behave as an appropriate fractal diffusion within each fractal component of the field.     

Student-like models for risky asset with dependence

We present a new construction of the Student and Student-like fractal activity time model for risky asset. The construction uses the diffusion processes and their superpositions and allows for specified exact Student or Student-like marginal distributions of the returns and for flexible and tractable dependence structure. The fractal activity time is asymptotically self-similar, which is a desired feature seen in practice.     

Estimate the shortest paths on fractal m-gons

The self-similar sets seem to be a class of fractals which is most suitable for mathematical treatment. The study of their structural properties is important. In this paper, we estimate the formula for the mean geodesic distance of self-similar set (denote fractal m-gons). The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractal considered. Out of result, obtained exact solution exhibits that the mean geodesic distance approximately increases as a exponential function of the number of nodes (small copies with the same size) with exponent equal to the reciprocal of the fractal dimension.     

Open set condition for graph directed self-similar structure

Graph directed self-similar structure generalizes the concept of self-similar set and contains some important instants of fractal sets. We characterize the open set condition (OSC), which is fundamental in the study of self-similar set, for graph directed self-similar structure in terms of the post critical set. Using this characterization, we establish the relations between OSC and other separation conditions including post-critically finite, finitely ramified condition and finite preimage property. It turns out that whether the intrinsic metric is doubling makes difference. In particular, finitely ramified condition implies OSC in case of doubling metric but does not in case of non-doubling metric.     

Existence of self-similar energies on finitely ramified fractals

A self-similar energy on finitely ramified fractals can be constructed starting from an eigenform, i.e., an eigenvector of a special operator defined on the fractal. In this paper, we prove two existence results for regular eigenforms that consequently are existence results for self-similar energies on finitely ramified fractals. The first result proves the existence of a regular eigenform for suitable weights on fractals, assuming only that the boundary cells are separated and the union of the interior cells is connected. This result improves previous results and applies to many finitely ramified fractals usually considered. The second result proves the existence of a regular eigenform in the general case of finitely ramified fractals in a setting similar to that of P.C.F. self-similar sets considered, for example, by R. Strichartz in [11]. In this general case, however, the eigenform is not necessarily on the given structure, but is rather on only a suitable power of it. Nevertheless, as the fractal generated is the same as the original fractal, the result provides a regular self-similar energy on the given fractal.     

A LOW-FREQUENCY ELECTROMAGNETIC NEAR-FIELD INVERSE PROBLEM FOR A SPHERICAL SCATTERER

The interior low-frequency electromagnetic dipole excitation of a dielectric sphere is uti- lized as a simplified but realistic model in various biomedical applications. Motivated by these considerations, in this paper, we investigate analytically a near-field inverse scatter- ing problem for the electromagnetic interior dipole excitation of a dielectric sphere. First, we obtain, under the low-frequency assumption, a closed-form approximation of the series of the secondary electric field at the dipole＇s location. Then, we utilize this derived approx- imation in the development of a simple inverse medium scattering algorithm determining the sphere＇s dielectric permittivity. Finally, we present numerical results for a human head model, which demonstrate the accurate determination of the complex permittivity by the developed algorithm.     

Zeta functions of discrete self-similar sets

In this paper we study a class of countable and discrete subsets of a Euclidean space that are “self-similar” with respect to a finite set of (affine) similarities. Any such set can be interpreted as having a fractal structure. We introduce a zeta function for these sets, and derive basic analytic properties of this “fractal” zeta function. Motivating examples that come from combinatorial geometry and arithmetic are given particular attention.     

Homogenization of time harmonic Maxwell equations: the effect of interfacial currents

We consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time‐harmonic current source. We perform the two‐scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. Copyright © 2016 John Wiley & Sons, Ltd.     

The necessary and sufficient conditions for various self-similar sets and their dimension

We have given several necessary and sufficient conditions for statistically self-similar sets and a.s. self-similar sets and have got the Hausdorff dimension and exact Hausdorff measure function of any a.s. self-similar set in this paper. It is useful in the study of probability properties and fractal properties and structure of statistically recursive sets.     

A normal inverse Gaussian model for a risky asset with dependence

We present a new construction of the normal inverse Gaussian (NIG) fractal activity time model for a risky asset. The construction uses superpositions of diffusion processes and allows for specified exact NIG marginal distributions of the returns and flexible and tractable dependence structure including short or long range dependence. In the case of finite superposition, the fractal activity time is asymptotically self-similar, which is a desired feature seen in practice. The support for the distributional and dependence features of the risky asset model is provided by the data of currency exchange rates.     

An algorithm for calculating the frequency dependence of the complex dielectric permittivity and the complex compliance

Conclusions 1. The algorithm developed for a numerical transform of the kernel in the Boltzmann — Volterra integral equation from the time domain to the frequency domain is suitable for calculating the complex dielectric permittivity from the polarization current or for calculating the complex compliance from the creep kernel.2. The algorithm is applicable to any distribution of relaxation times and has been based on changing the readings of the given function with time according to a geometrical progression. Its use requires that a certain number of coefficients be known and that readings of the given function be taken with the corresponding time coordinates.3. Calculations using expressions where the distribution of readings of the given function is shifted with respect to the time corresponding to the selected frequency makes it possible to extend the frequency range within which the sought function (complex dielectric permittivity or complex compliance) is to be determined.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 524–530, May–June, 1977.     

Eigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlaps

Under the assumption that a self-similar measure defined by a one-dimensional iterated function system with overlaps satisfies a family of second-order self-similar identities introduced by Strichartz et al., we obtain a method to discretize the equation defining the eigenvalues and eigenfunctions of the corresponding fractal Laplacian. This allows us to obtain numerical solutions by using the finite element method. We also prove that the numerical eigenvalues and eigenfunctions converge to the true ones, and obtain estimates for the rates of convergence. We apply this scheme to the fractal Laplacians defined by the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the open set condition or the post-critically finite condition; we use second-order self-similar identities to analyze the measures.     

The concept of fractal experiments: New possibilities in quantitative description of quasi-reproducible measurements

In this paper the authors suggest a new conception of the so-called fractal (self-similar) experiment. Under the fractal experiment (FE) one can imply a cycle of measurements that are subjected by the scaling transformations F(z) → F(zξ^m) in contrast with conventional scheme F(z) → F(z + mT) (m = 0,1,…, M–1), where z defines the controllable (input) variable and can be associated with time, complex frequency, wavelength and etc., T – mean period of time between successive measurements and m defines a number of successive measurements. One can connect a fractal experiment with specific memory effect that arises between successive measurements. The general theory of experiment for quasi-periodic measurements proposed in [1] after some transformations can be applied for the set of the FE, as well. But attentive analysis shown in this paper allows generalizing the previous results for the case when the influence of uncontrollable factors becomes significant. The theory developed for this case allows to consider more real cases when the influence of dynamic (unstable) processes taking place during the cycle of measurements corresponding to some FE is becoming essential. These experiments we define as quasi-reproducible (QR) fractal experiments.The proposed concept opens new possibilities in theory of measurements and numerous applications, especially in different nanotechnologies, when the influence of the scaling factor plays the essential role. This concept allows also to introduce the so-called intermediate model (IM) which can serve as an unified platform for reconciliation of the proposed microscopic theory with reliable experiments “refined” from the influence of the random noise and apparatus function. We forced to consider a modified model experiment in order to demonstrate some common peculiarities that can be appeared in real cases. We know only couple of similar examples of experiments that are close to the proposed concept. Mechanical relaxation and dielectric spectroscopy (based on measurements of the complex susceptibility ε(jω)) represent the branches of physics related to consideration of mechanical and electric relaxation phenomena in different heterogeneous materials. The dielectric spectroscopy can be considered as an instructive example for better understanding of the proposed concept.In cases, when the microscopic model is absent the results of measurements can be expressed in terms of the fitting parameters associated with the generalized Prony spectrum (GPS) belonging to the IM. The authors do hope that this new approach will find an interesting continuation in various applications of different nanotechnologies.     

Self-similar and oscillating solutions of Einstein's equation and other relevant consequences of a stochastic self-similar and fractal Universe via El Naschie's ε(∞) Cantorian space–time

In this paper we make some suggestion regarding the unification of the fundamental forces and the age of the Universe in the context of the a stochastic self-similar and fractal Universe using El Naschie's ε^(∞) Cantorian space–time. We also show how Einstein's equation can admit for the scale factor a(t) a self-similar solution in agreement with our stochastic self-similar, fractal Universe and El Naschie's ε^(∞) Cantorian space–time. In addition, this solution is found to be oscillating one. Thanks to the first quantization it is possible to recast the equations in a Schrödinger-like form. Consequently, the presently observed large scale structure reflects the phenomenology of the Early Universe or of the microscopic world. Again it appears clear that the Universe and the structures inside must have a memory of its quantum origin as conjectured sometime ago.     

Generating hierarchial scale-free graphs from fractals

Motivated by the hierarchial network model of E. Ravasz, A.-L. Barabási, and T. Vicsek, we introduce deterministic scale-free networks derived from a graph directed self-similar fractal Λ. With rigorous mathematical results we verify that our model captures some of the most important features of many real networks: the scale-free and the high clustering properties. We also prove that the diameter is the logarithm of the size of the system. We point out a connection between the power law exponent of the degree distribution and some intrinsic geometric measure theoretical properties of the underlying fractal. Using our (deterministic) fractal Λ we generate random graph sequence sharing similar properties.     

Self-similarity and spectral asymptotics for the continuum random tree

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application, we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.