Narrow Results

This is an analytical analysis of a previously published research for a percolation simulation. In that research the effect of mutations on adaptability was investigated in a bit-string model of invading species in a random environment. However, analytical analysis was missing which will be the topic here. The Hausdorff dimensions are calculated for the fractals and the conditions on invasion are analyzed analytically by manipulation of partial differential equations. Thus, various conclusions may be reached without having to run long simulations.

The main goal of our research is to find the connection between micro particles and microorganisms motion in the Nature, considered as Brownian’s Motion within the fractal’s nature. For ceramics and generally material science it is important to clarify the particles motion and other phenomena, especially for grains and pores. Our idea is to establish control over the relation order–disorder on particle motion and their collision effects by Brownian motion phenomena in the frame of fractal nature matter. We performed some experiments and got interesting results based on microorganism motion initiated by different outer energetic impulses. This is practically the idea of biomimetic correlation between particles and microorganisms Worlds, what is very original and leads towards biunivocal different phenomena’s understanding. Another idea is to establish some controlling effects for electro ceramic particle motion in chemical-materials sciences consolidation by some phenomena in the nature. These important research directions open new frontiers with very specific reflections for future of microelectronics materials.

Today in the age of advanced ceramic civilization, there are a variety of applications for modern ceramics materials with specific properties. Our up-to date research recognizes that ceramics have a fractal configuration nature on the basis of different phenomena. The key property of fractals is their scale-independence. The practical value is that the fractal objects’ interaction and energy is possible at any reasonable scale of magnitude, including the nanoscale and may be even below. This is a consequence of fractal scale independence. This brings us to the conclusion that properties of fractals are valid on any scale (macro, micro, or nano). We also analyzed these questions with experimental results obtained from a comet, here 67P, and also from ceramic grain and pore morphologies on the microstructure level. Fractality, as a scale-independent morphology, provides significant variety of opportunities, for example for energy storage. From the viewpoint of scaling, the relation between large and small in fractal analysis is very important. An ideal fractal can be magnified endlessly but natural morphologies cannot, what is the new light in materials sciences and space.

The goal of our research is to establish the direction of coronavirus chaotic motion to control corona dynamic by fractal nature analysis. These microorganisms attaching the different cells and organs in the human body getting very dangerous because we don’t have corona antivirus prevention and protection but also the unpredictable these viruses motion directions what resulting in very important distractions. Our idea is to develop the method and procedure to control the virus motion direction with the intention to prognose on which cells and organs could attach. We combined very rear coronavirus motion sub-microstructures images from worldwide experimental microstructure analysis. The problem of the recording this motion is from one point of view magnification, but the other side in resolution, because the virus size is minimum 10 times less than bacterizes. But all these images have been good data to resolve by time interval method and fractals, the points on the motion trajectory. We successfully defined the diagrams on the way to establish control over Brownian chaotic motion as a bridge between chaotic disorder to control disorder. This opens a very new perspective to future research to get complete control of coronavirus cases.

Hydroxyapatite scaffold is a type of bio-ceramic. Its cellular design has similarities with the morphologies in nature. Therefore, it is very important to control the structure, especially the porosity, as one of the main features for bio-ceramics applications. According to some literature, freeze casting can form the shape of dendrites and remain a foam structure after ice sublimation. Ice nucleation became more heterogeneous with the aid of printing materials during freeze casting. This procedure can even improve the issue of crack formation. In this paper, we studied the mechanical properties of hydroxyapatite scaffold. We also analyzed the porosity by fractal nature characterization, and successfully reconstructed pore shape, which is important for predicting ceramic morphology. We applied SEM analysis on bio-ceramic samples, at four different magnifications for the same pore structure. This is important for fractal analysis and pores reconstruction. We calculated the fractal dimensions based on measurements. In this way, we completed the fractal characterization of porosity and confirmed possibilities for successful porous shapes reconstruction. In this paper, we confirmed, for the first time, that fractal nature can be successfully applied in the area of porous bio-ceramics.

Forensic photography, also referred to as crime scene photography, is an activity that records the initial appearance of the crime scene and physical evidence in order to provide a permanent record for the court. Nowadays, we cannot imagine a crime scene investigation without photographic evidence. Crime or accident scene photographs can often be reanalyzed in cold cases or when the images need to be enlarged to show critical details. Fractals are rough or fragmented geometric shapes that can be subdivided into parts, each of which is a reduced copy of the whole. Fractal dimension (FD) is an important fractal geometry feature. There are many applications of fractals in various forensic fields, including image processing, image analysis, texture segmentation, shape classification, and identifying the image features such as roughness and smoothness of an image. Fractal analysis is applicable in forensic archeology and paleontology, as well. The damaged image can be reviewed, analyzed, and reconstructed by fractal nature analysis.

Cell-works which are three-dimensional cyclic edge-label controlled OL-systems, were introduced by Lindenmayer. Fracchia and Prusinkiewicz provided cell-work systems using markers to model three-dimensional cellular structures. In this paper, cell-work systems with fins are proposed, generalizing the notion of map with handles. We allow each cell of a cell-work to divide into finitely many cells at any instant of time. This generalization enables us to describe three-dimensional images and fractals. A comparison of various systems with the system introduced in this paper is made.

We generalize the concept of one-dimensional decimation invariant sequences, i.e. sequences which are invariant under a specific rescaling, to dimension N. After discussing the elementary properties of decimation-invariant sequences, we focus our interest on their periodicity. Necessary and sufficient conditions for the existence of periodic decimation invariant sequences are presented.

We have proposed a process of generating fractals not from the results of chaotic dynamics, but from the switching of ordinary differential equations. This paper experimentally and numerically analyzes the dynamics of an electronic circuit driven by stochastically switching inputs. The following two results are obtained. First, the dynamics is characterized by a set Γ(C) of trajectories in the cylindrical phase space, where C is a set of initial states on the Poincaré section. Γ(C) and C are attractive and unique invariant fractal sets that satisfy specific equations. The second result is that the correlation dimension of C is in inverse proportion to the interval of the switching inputs. These two findings move beyond the conventional theory based on contraction maps. It should be noted that the set C is constructed by noncontraction maps.

We consider an unbiased random walk on a finite, nth generation Sierpinski gasket (or "tower") in d = 3 Euclidean dimensions, in the presence of a trap at one vertex. The mean walk length (or mean number of time steps to absorption) is given by the exact formula

The generalization of this formula to the case of a tower embedded in an arbitrary number d of Euclidean dimensions is also found, and is given by

This also establishes the leading large-n behavior that may be expected on general grounds, where N_n is the number of sites on the nth generation tower and is the spectral dimension of the fractal.

Rescaled evolution sets of linear cellular automata on a lattice ℤ ⊕ G, where G is a finite Abelian group, with states in the finite field are defined and investigated.

Noise-free stochastic resonance is investigated in two chaotic maps with periodically modulated control parameter close to a boundary crisis: the Hénon map and the kicked spin model. Response of the maps to the periodic signal at the fundamental frequency and its higher harmonics is examined. The systems show noise-free stochastic multiresonance, i.e. multiple maxima of the signal-to-noise ratio at the fundamental frequency as a function of the control parameter. The maxima are directly related to the fractal structure of the attractors and basins of attraction colliding at the crisis point. The signal-to-noise ratios at higher harmonics show more maxima, as well as dips where the signal-to-noise ratio is zero. This opens a way to use noise-free stochastic resonance to probe the fractal structure of colliding sets by a method which can be called "fractal spectroscopy". Using stochastic resonance at higher harmonics can reveal smaller details of the fractal structures, but the interpretation of results becomes more difficult. Quantitative theory based on a model of a colliding fractal attractor and a fractal basin of attraction is derived which agrees with numerical results for the signal-to-noise ratio at the fundamental frequency and at the first two harmonics, quantitatively for the Hénon map, and qualitatively for the kicked spin model. It is also argued that the maps under study belong to a more general class of threshold-crossing stochastic resonators with a modulated control parameter, and qualitative discussion of conditions under which stochastic multiresonance appears in such systems is given.

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-similar two-dimensional substructures manifest themselves, ad infinitum, as the number of cells (I + 1) → ∞.

In addition to a complete gallery of time-1 characteristic functions for all 256 local rules, an accompanying table of explicit formulas is given for generating these characteristic functions directly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potential applications of these fundamental formulas, we prove rigorously that the "right-copycat" local rule is equivalent globally to the classic "left-shift" Bernoulli map. Similarly, we prove the "left-copycat" local rule is equivalent globally to the "right-shift" inverse Bernoulli map.

Various geometrical and analytical properties have been identified from each characteristic function and explained rigorously. In particular, two-level stratified subpatterns found in most characteristic functions are shown to emerge if, and only if, b₁ ≠ 0, where b₁ is the "synaptic coefficient" associated with the cell differential equation developed in Part I.

Gardens of Eden are derived from the decimal range of the characteristic function of each local rule and tabulated. Each of these binary strings has no predecessors (pre-image) and has therefore no past, but only the present and the future. Even more fascinating, many local rules are endowed with binary configurations which not only have no predecessors, but are also fixed points of the characteristic functions. To dramatize that such points have no past, and no future, they are henceforth christened "Isles of Eden". They too have been identified and tabulated.

A new method for constructing recurrent bivariate fractal interpolation surfaces through points sampled on rectangular lattices is proposed. This offers the advantage of a more flexible fractal modeling compared to previous fractal techniques that used affine transformations. The compression ratio for the above mentioned fractal scheme as applied to real images is higher than other fractal methods or JPEG, though not as high as JPEG2000. Theory, implementation and analytical study are also presented.

The features of processes of chaotic discrete map in reverse time are considered. The fractal distribution of reverse iteration points is indicated. The approach to investigating reverse points manifold based on special ordering methods is proposed.

Our scientific odyssey through the theory of 1-D cellular automata is enriched by the definition of quasi-ergodicity, a new empirical property discovered by analyzing the time-1 return maps of local rules. Quasi-ergodicity plays a key role in the classification of rules into six groups: in fact, it is an exclusive characteristic of complex and hyper Bernoulli-shift rules. Besides introducing quasi-ergodicity, this paper answers several questions posed in the previous chapters of our quest. To start with, we offer a rigorous explanation of the fractal behavior of the time-1 characteristic functions, finding the equations that describe this phenomenon. Then, we propose a classification of rules according to the presence of Isles of Eden, and prove that only 28 local rules out of 256 do not have any of them; this result sheds light on the importance of Isles of Eden. A section of this paper is devoted to the characterization of Bernoulli basin-tree diagrams through modular arithmetic; the formulas obtained allow us to shorten drastically the number of cases to take into consideration during numerical simulations. Last but not least, we present some theorems about additive rules, including an analytical explanation of their scale-free property.

We studied a hybrid dynamical system composed of a higher module with discrete dynamics and a lower module with continuous dynamics. Two typical examples of this system were investigated from the viewpoint of dynamical systems. One example is a nonfeedback system whose higher module stochastically switches inputs to the lower module. The dynamics was characterized by attractive and invariant fractal sets with hierarchical clusters addressed by input sequences. The other example is a feedback system whose higher module switches in response to the states of the lower module at regular intervals. This system converged into various switching attractors that correspond to infinite switching manifolds, which define each feedback control rule at the switching point. We showed that the switching attractors in the feedback system are subsets of the fractal sets in the nonfeedback system. The feedback system can be considered an automaton that generates various sequences from the fractal set by choosing the typical switching manifold. We can control this system by adjusting the switching interval to determine the fractal set as a constraint and by adjusting the switching manifold to select the automaton from the fractal set. This mechanism might be the key to developing information processing that is neither too soft nor too rigid.

Chaotic scattering in open Hamiltonian systems is a problem of fundamental interest with applications in several branches of physics. In this paper we analyze the effects of adding external perturbations such as dissipation and noise in chaotic scattering phenomena. Our main result is the exponential decay rate of the particles in the scattering region when the system is affected by dissipation and noise. In the case of dissipation the particles escape more slowly from the scattering region than in the conservative case. However, in the noisy case, the particles escape faster from the scattering region as compared to the noiseless case. Moreover, we analyze the fractal dimension of the set of singularities of the scattering function for the dissipative and the conservative cases. As a result of our analysis we have found that a scaling law exists between the exponential decay rate of the particles and the dissipative parameter, and that the fractal dimension for the noisy case is the unity.

Starting from the well-known Newton's fractal which is formed by the basin of convergence of Newton's method applied to a cubic equation in one variable in the field ℂ, we were able to find methods for which the corresponding basins of convergence do not exhibit a fractal-like structure. Using this approach we are able to distinguish reliable and robust methods for tackling a specific problem. Also, our approach is illustrated here for methods for computing periodic orbits of nonlinear mappings as well as for fixed points of the Poincaré map on a surface of section.

The paper is inspired by a spectral decomposition and fractal eigenvectors for a class of piecewise linear maps due to Tasaki et al. [1994] and by an ad hoc explicit derivation of the Heisenberg uncertainty relation based on a Peano–Hilbert planar curve, due to El Nashie [1994]. It is also inspired by an elegant generalization by Zhang [2008] of the exact solution by Onsager [1944] to the problem of description of the Ising lattices [Ising, 1925]. This generalization involves, in particular, opening the knots by a rotation in a higher dimensional space and studying important commutators in the corresponding algebra. The investigations of Onsager and Zhang, involving quaternion matrices of order being a power of two, can be reformulated with the use of the "quaternionic" sequence of Jordan algebras implied by the fundamental paper of Jordan et al. [1934]. It is closely related to Heisenberg's approach to quantum theories, as summarized by him in his essay dedicated to Bohr on the occasion of Bohr's seventieth birthday (1955). We show that the Jordan structures are closely related to some types of fractals, in particular, fractals of the algebraic structure. Our study includes fractal renormalization and the renormalized Dirac operator, meromorphic Schauder basis and hyperfunctions on fractal boundaries, and a final discussion.