Narrow Results

In procedural programming languages, the order of executing the statements may follow a regular pattern, including sequences of statements, conditional and branching statements and loops. On the other hand, regular languages can be represented by finite state acceptors (finite automata), by regular expressions and by (a special form of) railroad diagrams (syntax diagrams) allowing alternatives, option, concatenation and iteration. Context-free languages can also be described by (the general form of) railroad diagrams allowing also recursion. Based on the analogy of finite automata and railroad diagrams, the transformation between node-labelled and edge-labelled graphs, special infinite state automata, namely the fractal automata, are established to characterize the class of context-free languages. Deterministic and linear variants are also investigated. To establish these connections, we also define such variants of pushdown automata that accept the deterministic context-free and linear languages by empty stack. A transformation between the pushdown automata and fractal automata is also shown. The proposed model gives some new insights and a new view of context-free languages, deterministic context-free languages, linear context-free and deterministic linear context-free languages.

We study the greedy algorithms for $m$-term approximation. We propose a modification of the Weak Rescaled Pure Greedy Algorithm (WRPGA) — Approximate Weak Rescaled Pure Greedy Algorithm (AWRPGA) — with respect to a dictionary of a Banach space $X$. By using a geometric property of the unit sphere of $X$, we obtain a general error estimate in terms of some $K$-functional. This estimate implies the convergence condition and convergence rate of the AWRPGA. Furthermore, we obtain the corresponding error estimate for the Vector Approximate Weak Rescaled Pure Greedy Algorithm (VAWRPGA). We show that the AWRPGA (VAWRPGA) performs as well as the WRPGA (VWRPGA) when the noise amplitude changes relatively little. Finally, by using the wavelet bases and trigonometric system of Lebesgue spaces, we show that the convergence rate of the AWRPGA is optimal.

In this paper, the classical one-dimensional Dirac equation is considered under the framework of fractal calculus. First, the maximal and minimal operators corresponding to the problem are defined. Then the symmetric operator is obtained, the Green’s function corresponding to the problem is constructed, and the eigenfunction expansion is given. Finally, some examples are given.

The characteristic of the fixed points of the Carotid–Kundalini (C–K) map is investigated and the boundary equation of the first bifurcation of the C–K map in the parameter plane is given. Based on the studies of the phase graph, the power spectrum, the correlation dimension and the Lyapunov exponents, the paper reveals the general features of the C–K map transforming from regularity. Meanwhile, using the periodic scanning technology proposed by Welstead and Cromer, a series of Mandelbrot–Julia (M–J) sets of the complex C–K map are constructed. The symmetry of M–J set and the topological inflexibility of distributing of periodic region in the Mandelbrot set are investigated. By founding the whole portray of Julia sets based on Mandelbrot set qualitatively, we find out that Mandelbrot sets contain abundant information of structure of Julia sets.

The problem of determining total distance ascended during a mountain bike trip is addressed. Altitude measurements are obtained from GPS receivers utilizing both GPS-based and barometric altitude data, with data averaging used to reduce fluctuations. The estimation process is sensitive to the degree of averaging, and is related to the well-known question of determining coastline length. Barometric-based measurements prove more reliable, due to their insensitivity to GPS altitude fluctuations.

A model of fission gas migration in nuclear fuel pellet is proposed. Diffusion process of fission gas in granular structure of nuclear fuel with presence of inter-granular bubbles in the fuel matrix is simulated by fractional diffusion model. The Grunwald–Letnikov derivative parameter characterizes the influence of porous fuel matrix on the diffusion process of fission gas. A finite-difference method for solving fractional diffusion equations is considered. Numerical solution of diffusion equation shows correlation of fission gas release and Grunwald–Letnikov derivative parameter. Calculated profile of fission gas concentration distribution is similar to that obtained in the experimental studies. Diffusion of fission gas is modeled for real RBMK-1500 fuel operation conditions. A functional dependence of Grunwald–Letnikov derivative parameter with fuel burn-up is established.

We have performed a Monte Carlo (MC) study of the classical XY-model on a Sierpiński carpet, which is a planar fractal structure with infinite order of ramification and fractal dimension 1.8928. We employed the Wolff cluster algorithm in our simulations and our results, in particular those for the susceptibility and the helicity modulus, indicate the absence of finite-temperature Berezinskii–Kosterlitz–Thouless (BKT) transition in this system.

Two aspects of fractal networks are considered: the community structure and the class structure, where classes of nodes appear as a consequence of a local symmetry of nodes. The analyzed systems are the networks constructed for two selected symmetric fractals: the Sierpinski triangle and the Koch curve. Communities are searched for by means of a set of differential equations. Overlapping nodes which belong to two different communities are identified by adding some noise to the initial connectivity matrix. Then, a node can be characterized by a spectrum of probabilities of belonging to different communities. Our main goal is that the overlapping nodes with the same spectra belong to the same class.

The box-counting (BC) algorithm is applied to calculate fractal dimensions of four fractal sets. The sets are contaminated with an additive noise with amplitude $\gamma =10^{-5}-10^{-1}$. The accuracy of calculated numerical values of the fractal dimensions is analyzed as a function of $\gamma$ for different sizes of the data sample. In particular, it has been found that even in case of pure fractals ($\gamma =0$) as well as for tiny noise ($\gamma \approx 10^{-5}$) one has considerable error for the calculated exponents of order 0.01. For larger noise the error is growing up to 0.1 and more, with natural saturation limited by the embedding dimension. This prohibits the power-like scaling of the error. Moreover, the noise effect cannot be cured by taking larger data samples.

In this paper, fractal compression methods are reviewed. Three new methods are developed and their results are compared with the results obtained using four previously published fractal compression methods. Furthermore, we have compared the results of these methods with the standard JPEG method. For comparison, we have used an extensive set of image quality measures. According to these tests, fractal methods do not yield significantly better compression results when compared with conventional methods. This is especially the case when high coding accuracy (small compression ratio) is desired.

Linear fractals associated with weights are investigated. Such fractals are important from a conservation law perspective that is relevant in a variety of physical systems such as materials science, sand dune fractals, barred galaxies, as well as in temporal processes like in the electroencephalogram (EEG). The weight associated with fractals is an additional feature that may be associated with distributions consistent with the ubiquitous power law and the first digit phenomenon. These distributions form a bridge to processes and applications in natural, biological, and engineering systems and, therefore, open up the possibility of the application of linear weighted fractals to these subjects. Two linear fractal algorithms that are near optimal in the information theoretic sense are described. A mechanism for the emergence of these fractals is proposed: it is the indistinguishability amongst the particles in the evolution and transformation of physical systems. Since the fractal approach is an established method of signal processing and coding, the newly proposed weighted fractals have the potential to lead to new useful algorithms.

This paper investigates the dimensionality of genetic information from the perspective of optimal representation. Recently it has been shown that optimal coding of information is in terms of the noninteger dimension of e, which is accompanied by the property of scale invariance. Since Nature is optimal, we should see this dimension reflected in the organization of the genetic code. With this as background, this paper investigates the problem of the logic behind the nature of the assignment of codons to amino acids, for they take different values that range from 1 to 6. It is shown that the non-uniformity of this assignment, which goes against mathematical coding theory that demands a near uniform assignment, is consistent with noninteger dimensions. The reason why the codon assignment for different amino acids varies is because uniformity is a requirement for optimality only in a standard vector space, and is not so in the noninteger dimensional space. It is noteworthy that there are 20 different covering regions in an e-dimensional information space, which is equal to the number of amino acids. The problem of the visualization of data that originates in an e-dimensional space but examined in a 3-dimensional vector space is also discussed. It is shown that the assignment of the codons to the amino acids is fractal-like that is well modeled by the Zipf distribution which is a power law. It is remarkable that the Zipf distribution that holds for the letter frequencies of words in a natural language also applies to the rank order of triplets in the code for amino acids.

A methodology for the identification of homomorphisms in high level functional language specifications is presented and their role illustrated with a series of examples. These include the synthesis of a fractal-based algorithm for image decompression and a logarithmic parallel simulation of a single server, first-come-first-served queue.

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 specific medium.

We investigate the classical evolution of a ϕ⁴ scalar field theory, using in the initial state random field configurations possessing a fractal measure expressed by a noninteger mass dimension. These configurations resemble the equilibrium state of a critical scalar condensate. The measures of the initial fractal behavior vary in time following the mean field motion. We show that the remnants of the original fractal geometry survive and leave an imprint in the system time averaged observables, even for large times compared to the approximate oscillation period of the mean field, determined by the model parameters. This behavior becomes more transparent in the evolution of a deterministic Cantor-like scalar field configuration. We extend our study to the case of two interacting scalar fields, and we find qualitatively similar results. Therefore, our analysis indicates that the geometrical properties of a critical system initially at equilibrium could sustain for several periods of the field oscillations in the phase of nonequilibrium evolution.

There are a substantial number of empirical relations that began with the identification of a pattern in data; were shown to have a terse power-law description; were interpreted using existing theory; reached the level of "law" and given a name; only to be subsequently fade away when it proved impossible to connect the "law" with a larger body of theory and/or data. Various forms of allometry relations (ARs) have followed this path. The ARs in biology are nearly two hundred years old and those in ecology, geophysics, physiology and other areas of investigation are not that much younger. In general if X is a measure of the size of a complex host network and Y is a property of a complex subnetwork embedded within the host network a theoretical AR exists between the two when Y = aX^b. We emphasize that the reductionistic models of AR interpret X and Y as dynamic variables, albeit the ARs themselves are explicitly time independent even though in some cases the parameter values change over time. On the other hand, the phenomenological models of AR are based on the statistical analysis of data and interpret X and Y as averages to yield the empirical AR: 〈Y〉 = a〈X〉^b. Modern explanations of AR begin with the application of fractal geometry and fractal statistics to scaling phenomena. The detailed application of fractal geometry to the explanation of theoretical ARs in living networks is slightly more than a decade old and although well received it has not been universally accepted. An alternate perspective is given by the empirical AR that is derived using linear regression analysis of fluctuating data sets. We emphasize that the theoretical and empirical ARs are not the same and review theories "explaining" AR from both the reductionist and statistical fractal perspectives. The probability calculus is used to systematically incorporate both views into a single modeling strategy. We conclude that the empirical AR is entailed by the scaling behavior of the probability density, which is derived using the probability calculus.

By resorting to measurements of physically characterizing observables of water samples perturbed by the presence of Nafion and by iterative filtration processes, we discuss their scale free, self-similar fractal properties. By use of algebraic methods, the isomorphism is proved between such self-similarity features and the deformed coherent state formalism.

Advanced research frontiers are extended from biophysics relations on the Earth upto the discovering any type of alive matter within the whole space. Microorganisms’ motion within the molecular biology processes integrates variety of microorgnisms functions. In continuation of our Brownian motion phenomena research, we consistently build molecular-microorganisms structures hierarchy. We recognize everywhere biomimetic similarities between the particles in alive and nonalive matter. The research data are based on real experiments, without external energy impulses. So, we develop the analysis, inspired by fractal nature Brownian motion, as recognized joint parameter between particles in alive and nonalive biophysical systems. This is also in line with advance trends in hybrid submicroelectronic integrations. The important innovation in this paper is that we introduced approximation of trajectory and error calculations, using discrete mean square approximation, what cumulatively provide much more precise biophysical systems parameters. By this paper, we continue to generate new knowledge in direction to get complex relations between the particles clusters in biophysical systems condensed matter.

The natural variability in physiological structure is herein related to the geometric concept of a fractal. The average dimensions of the branches in the tracheobronchial tree, long thought to be exponential, are shown to be an inverse power law of the generation number modulated by a harmonic variation. A similar functional form is found for the power spectrum of the QRS-complex of the healthy human heart. These results follow from the assumption that the bronchial tree and the cardiac conduction system are fractal forms. The fractal concept provides a mechanism for the morphogenesis of complex structures which are more stable than those generated by classical scaling (i.e., they are more error tolerant).

We introduce a new method to generate three-dimensional structures, with mixed topologies. We focus on Multilayered Regular Hyperbranched Fractals (MRHF), three-dimensional networks constructed as a set of identical generalized Vicsek fractals, known as Regular Hyperbranched Fractals (RHF), layered on top of each other. Every node of any layer is directly connected only to copies of itself from nearest-neighbor layers. We found out that also for MRHF the eigenvalue spectrum of the connectivity matrix is determined through a semi-analytical method, which gives the opportunity to analyze very large structures. This fact allows us to study in detail the crossover effects of two basic topologies: linear, corresponding to the way we connect the layers and fractal due to the layers' topology. From the wealth of applications which depends on the eigenvalue spectrum we choose the return-to-the-origin probability. The results show the expected short-time and long-time behaviors. In the intermediate time domain we obtained two different power-law exponents: the first one is given by the combination linear-RHF, while the second one is peculiar either of a single RHF or of a single linear chain.