Fractal Objects Research Articles

Large-Scale Structure of Chromatin: A Fractal Globule or a Logarithmic Fractal?

Two physical models are considered to describe the large-scale structure of chromatin in the nucleus of a biological cell in the interphase state: a fractal globule model and a logarithmic fractal model. Based on the classification of fractal objects developed by the small-angle neutron scattering (SANS) method, it is shown that the fractal globule model does not satisfy the experimental data on small-angle neutron scattering by the nuclei of biological cells. Conversely, the logarithmic fractal model well describes the experimental data on SANS and, hence, provides a good approximation to describe the large-scale structure of chromatin. The logarithmic fractal model predicts that the nuclear space is exactly half-filled with chromatin, and the second half consists of interchromatin voids filled with nucleoplasma in which various nuclear processes occur. Thus, two opposing trends are balanced in the structural organization of chromatin: an increase in the surface area of chromatin in the cell nucleus (accessibility to external agents) and a decrease in the volume occupied by chromatin (compactness of the nucleus).

On critical exponents for impact fragmentation of spherical solids

In the present paper, we consider the high-velocity impact fragmentation of two spherical solid bodies in a three-dimensional (3D) setting. Computer simulations are carried out by smoothed particle hydrodynamics method applied to the equations of mechanics of deformable solids for five different system sizes varying the impact velocity in a broad range. The material behavior is described by the Mie–Gruneisen equation of state and the Johnson–Cook model for the yield strength. We consider the transition from the damage to the fragmentation at the impact velocity variation that is treated as a phase transition. As a result, the critical exponents τ, β, γ and ν of the damage-fragmentation transition are calculated and their compatibility is checked. By calculating the gyration radius of fragments, we show that they are fractal objects with dimension depending on the fragments mass. It is shown that the fractal dimension for the small masses is consistent with the found critical exponents.

The structure of graphene oxide in polymer matrix and its effect on reinforcement degree of nanocomposite

The methodics of characterization of 2D-nanofiller tactoids structure in polymer matrix of nanocomposite was proposed, which shown that the indicated tactoids are fractal objects with dimension of 1.75-2.16. This variation of tactoids structural state is due to interfacial interactions. It has been shown within the framework of reinforcement percolation theory, that structure of tactoids is single factor, defining reinforcement efficiency is characterized by its ability to create interfacial regions.

Structure of Graphene Oxide in a Polymer Matrix and Its Effect on the Degree of Reinforcement of a Nanocomposite

A method for the structural characterization of 2D-nanofiller tactoids in the polymer matrix of a nanocomposite has been proposed to demonstrate that the mentioned tactoids are fractal objects with a dimension of 1.75–2.16. This variation of the structural state of tactoids is due to interfacial interactions. It has been demonstrated within the framework of percolation reinforcement theory that the structure of tactoids is the only factor governing the degree of reinforcement of nanocomposites. The efficiency of a nanocomposite is characterized by its ability to form interfacial regions.

Les sciences de la forme : une épistémologie heuristique pour la psychanalyse

ContextPsychoanalytic epistemology must develop, according to Anzieu, through “thinkers by images.” Recently, the sciences of form (morphogenesis, catastrophe theory, fractal objects) have attracted researchers from multiple scientific fields. The present paper attempts to synthesize psychoanalysis's grappling with this perspective. ObjectivesThe aim of this paper is to examine the use of the theories of form in the main French-language psychoanalytic publications, and to determine which new epistemological approach is emerging from them. MethodIt is essentially a theoretical investigation synthesizing French-language psychoanalytic publications. ResultsMost of the authors cited have used the theories of René Thom (catastrophe theory, semiophysics) or those describing Mandelbrot's fractal objects, foregoing earlier work on morphogenesis. The synthesis presented here allows us to perceive the emergence of an original and complex theoretical corpus, with strong heuristic value. ConclusionsThe significant number of publications seems to show that a new approach in the science of forms could be the sign of a new form of science, within psychoanalysis as well. The psychic system could be described, decoded, and analyzed by these theories as a complex, non-continuous system with highly irregular characteristics.

Demarcation of Coastal Municipalities in Spain

Pina-Garcia, F.; de Luis-Ruiz, J.M.; Castillo-Lopez, E.; Pereda-Garcia, R., and Perez-Alvarez, R., 2019. Demarcation of coastal municipalities in Spain. Journal of Coastal Research, 35(2), 305–313. Coconut Creek (Florida), ISSN 0749-0208.The demarcation of municipalities is essential for an adequate functioning of local administrations and the exercise of other public functions, such as jurisdiction, spatial planning, and urbanism. This research emphasizes the definition of municipalities in coastal areas, understanding that the beaches and the maritime–terrestrial zones are part of them. The limits of the beaches and the maritime–terrestrial zones are hard to determine, and the hydrographic zero is defined as an abstract line that separates terrestrial from marine. This line is difficult to define and even worse to materialize as it is a fractal object (too irregular for a geometric definition) and, in some cases, undergoes great changes through time. By analogy with what is done with normal baselines, which allow straight lines, or with the demarcation of the inner limit of the maritime–terrestrial public domain, it is proposed to identify the border of the municipality as a polygonal line whose vertices are defined by coordinates. A strictly administrative demarcation is thus proposed, which should neither modify the maritime–terrestrial public domain nor affect the inner waters or the territorial sea. As a result of the application of this methodology, the boundaries of a municipality would be defined and, therefore, its area. The proposal is applied to a practical case: the municipality of Ribamontan al Mar (Cantabria, Spain). Both the ambiguity of data and the imprecisions among the several sources of information regarding its boundary with respect to the sea are analyzed. A polygonal line defined by coordinates is proposed. This procedure could be generalized for the boundaries of any coastal municipality.

SELF-SIMILARITY OF SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS WITH RESPECT TO A FRACTAL MEASURE

In this paper, we study solutions of a variation of a classical integral equation (based on the Picard operator) in which Lebesgue measure is replaced by a self-similar measure [Formula: see text]. Our main interest is in the fractal nature of the solutions and we use Iterated Function System (IFS) tools to investigate the behavior and self-similarity of these solutions. Both the integral and differential forms of the equation are discussed since each brings useful insights. Several convergence results are provided along with illustrative examples that show the applications of the theory when the underlying fractal object is the celebrated Cantor set. Additionally we show that the solution to our integral equation inherits self-similarity from the defining measure [Formula: see text].

Fraktale – powrót do źródeł

Fractals have become fashionable. Therefore, in recent years there have been many articles in which the authors support something that they call a fractal account, including, for example, the sum of fractal dimensions. This paper is a recapitulation of what a fractal account is in the Earth sciences, what are its uses and boundaries. The definition of fractals is: it has a non trite structure in all scales, it is very hard to describe fractal structure in the Euclidean geometry, it is self-similar (directly or statistical), its Hausdorf dimension is greater than its topological dimension, it is described by recurrent formula, its dimension is not an integral number. In the face of such a wide and imprecise formula, various fields of science have introduced their definitions of fractal. It only has to meet most of the conditions included in the definition. In the analysis of geological objects in Earth sciences and in oil and gas industry, fractals are defined by the recurrence formula with its range of applicability, fractal dimension share a part of the space occupied by the fractal object, so the highest value of fractal dimension is equal to 3. Fundamental work in which the name of fractals for self- similar objects were introduced was The Fractal Geometry of Nature by Mandelbrot (1977). In the Earth sciences, statistical fractals (pseudofractals) are used. The straight line in the log-log plot is the indicator of fractal structure. In other words, the fractal structures are associated with the power patterns obtained during the analysis of geological objects. Generally, in the analysis of geological objects the Menger sponge and box methods of fractal dimension calculations are used. Fractals provide a unique opportunity to characterize complicated objects with the use of a single number, nevertheless, in order for the obtained results to be applicable and comparable with the results of other analyzes, both the model of the analyzed object and the method of calculation of the fractal dimension should be given, as well as the scope of applicability of this dimension.

Molecular Complexity Calculated by Fractal Dimension

Molecular complexity is an important characteristic of organic molecules for drug discovery. How to calculate molecular complexity has been discussed in the scientific literature for decades. It was known from early on that the numbers of substructures that can be cut out of a molecular graph are of importance for this task. However, it was never realized that the cut-out substructures show self-similarity to the parent structures. A successive removal of one bond and one atom returns a series of fragments with decreasing size. Such a series shows self-similarity similar to fractal objects. Here we used the number of distinct fragments to calculate the fractal dimension of the molecule. The fractal dimension of a molecule is a new matter constant that incorporates all features that are currently known to be important for describing molecular complexity. Furthermore, this is the first work that reveals the fractal nature of organic molecules.

Simple Model of Fractal Networks Formed by Self-Organized Critical Dynamics

In this paper, a simple dynamical model in which fractal networks are formed by self-organized critical (SOC) dynamics is proposed; the proposed model consists of growth and collapse processes. It has been shown that SOC dynamics are realized by the combined processes in the model. Thus, the distributions of the cluster size and collapse size follow a power-law function in the stationary state. Moreover, through SOC dynamics, the networks become fractal in nature. The criticality of SOC dynamics is the same as the universality class of mean-field theory. The model explains the possibility that the fractal nature in complex networks emerges by SOC dynamics in a manner similar to the case with fractal objects embedded in a Euclidean space.

Механизмы роста и структура кластеров 2D-нанонаполнителя в полимерных средах

AbstractThe estimations made within the fractal analysis have shown that 2D-nanofiller sheets in polymer medium form clusters (tactoids) that are fractal objects. This conclusion is confirmed by means of the irreversible aggregation model. The major factor determining the dimension of these clusters is the dimension of the random walk of initial nanoparticles. The structure of the 2D-nanofiller tactoids definitely determines the elastic modulus of the nanocomposite at the fixed nanofiller content.

The Mechanisms of Growth and the Structure of 2D-Nanofiller Clusters in Polymer Media

The estimations made within the fractal analysis have shown that 2D-nanofiller sheets in polymer medium form clusters (tactoids) that are fractal objects. This conclusion is confirmed by means of the irreversible aggregation model. The major factor determining the dimension of these clusters is the dimension of the random walk of initial nanoparticles. The structure of the 2D-nanofiller tactoids definitely determines the elastic modulus of the nanocomposite at the fixed nanofiller content.

BASIC STRATEGIC TECHNOLOGY OF INTELLECTUAL DUALITY OF HUMANITY IN INFORMATION TECHNOLOGY

The article includes new conceptual explanation of civilization content as an information process. Civilization is revealed as a global sociocultural quantum based on a specific picture for civilization and alien to others. The analysis of civilizational “quanta” historically reveals spiritual synergy of the past humanity with reconstructing goal of its entire positive present. This analysis is a product of our reflection thinking on those fractal objects of civilization as a quantum - invariants that are extracted from the space of its worldviews and events. According to the exponential growth of information production in all civilizational “quanta”, basic strategic technologies in information space are revealed as a result of ever-increasing intellectual activity of mankind in its processing, storage and protection. The basic strategic technologies include: coding, symbolization and digitalization. Coding is disclosed as a dialectical process of the relationship of thought and sign, the translation of thought into a sign; the epistemological nature of the sign and the sign situation is explained. The value-cognitive status of symbolization is analyzed. The symbol is characterized as a form of representing the ideal expression of things, phenomena and processes in image or sign form. Digital technologies, as an integral feature of the third industrial revolution, in unfolding context of its new stage - the fourth, are developing in basic strategiccontext of the entire information space, covering all areas of civilizations from the standpoint of its innovative development.

FRACTAL CHARACTERISTICS ANALYSIS OF MINE WORKINGS GEOMETRY FOR THE ESTIMATION OF SMOOTH WALL BLASTING QUALITY

The work examines the ways of estimating the smooth wall blasting quality when excavating mine workings. One of the main criteria is to achieve “smooth” mine working contour. It is shown that the mine working contour line represents a fractal object; its geometry is properly estimated by fractional fractal dimension. The definition of fractal dimension is based on the Richardson’s law, which connects the length of infinitely broken line and measurement interval. The calculation algorithm of fractal dimension in computer program is introduced. The analysis of real mine working contours which are formed in the process of drilling and blasting operations, allowed to determine statistically reliable connection of roughness coefficient of the mine working surface and the value of its fractal dimension. Quality criterion for smooth wall blasting is suggested – fractal coefficient of mine working shape. Its value represents the relation of equivalent cross-section area of a mine working to its perimeter, which is determined by fractal dimension of a contour. Obtained ratios allow to reliably estimate the quality of smooth wall blasting when excavating mine workings.

Fractal analysis of river newtworks based on remote sensing data: an example from Kelani River Basin, Sri Lanka

River network patterns are important components of basin geomorphology. The geomorphological complexity of river networks has an impact on various natural processes and hazardous events. In order to find those correlations and establish quantitative relationships, a comprehensive textural examination on river network patterns is crucial. In this study, the fractal geometry approach is applied to analyse river network by the means of their geometric distinctiveness. Since river networks are fractal objects formed by repetition of certain natural processes over a long period of time, fractal geometry provides the most sensitive method to analyse their complex branching structure. Five river networks from Kelani river basin (1. Ambalanpiti oya, 2. Gurugoda oya, 3. Pugoda oya, 4. Pusweli oya, 5. Wakoya) are subjected to fractal analysis. Fixed-size box counting algorithm is applied to obtain fractal measures. The recursive algorithm is applied to the same river network twice; allocating weighted-lines for different orders and then allocating non-weighted equal width line for all tributaries, to test the best suitable option to model their geomorphological complexity. Non-plane-filling behaviour of river networks is confirmed by present values which are greater than 1 and lesser than 2. The largest fractal dimension value is for Gurugodaoya tributary, confirming that it has the most complex geomorphologic pattern. Smallest value is from Pugodaoya concluding the least geomorphological complexity. Multifractal spectra f(α) are constructed for each river network and detailed investigation is required (considering lithological features of the basin) to link f(α) to the physical characteristics of river network.

The Caudal Part of Putamen Represents the Historical Object Value Information.

The basal ganglia, especially the circuits originating from the putamen, are essential for controlling normal body movements. Notably, the putamen receives inputs not only from motor cortical areas but also from multiple sensory cortices. However, how these sensory signals are processed in the putamen remains unclear. We recorded the activity of tentative medium spiny neurons in the caudal part of the putamen when the monkey viewed many fractal objects. We found many neurons that responded to these objects, mostly in the ventral region. We called this region "putamen tail" (PUTt), as it is dorsally adjacent to "caudate tail" (CDt). Although PUTt and CDt are mostly separated by a thin layer of white matter, their neurons shared several features. Almost all of them had receptive fields in the contralateral hemifield. Moreover, their responses were object selective (i.e., variable across objects). The object selectivity was higher in the ventral region (i.e., CDt > PUTt). Some neurons above PUTt, which we called the caudal-dorsal putamen (cdPUT), also responded to objects, but less selectively than PUTt. Next, we examined whether these visual neurons changed their responses based on the reward outcome. We found that many neurons encoded the values of many objects based on long-term memory, but not based on short-term memory. Such stable value responses were stronger in PUTt and CDt than in cdPUT. These results suggest that PUTt, together with CDt, controls saccade/attention among objects with different historical values, and may control other motor actions as well.SIGNIFICANCE STATEMENT Although the putamen receives inputs not only from motor cortical areas but also from sensory cortical areas, how these sensory signals are processed remains unclear. Here we found that neurons in the caudal-ventral part of the putamen (putamen tail) process visual information including spatial and object features. These neurons discriminate many objects, first by their visual features and later by their reward values as well. Importantly, the value discrimination was based on long-term memory, but not on short-term memory. These results suggest that the putamen tail controls saccade/attention among objects with different historical values and might control other motor actions as well.

The use of the perimeter-area method to calculate the fractal dimension of aggregates

Complicated geometrical objects like aggregate clusters can be characterised by a simple parameter: the fractal dimension. There are many ways to measure this fractal dimension. For most methods, it is found by the exponent of a relationship between an intrinsic value, such as mass or area, to a characteristic value, such as bulk aggregate length. The perimeter-area method was derived by Mandelbrot to measure the fractal dimension of chips of ore. In this method, there is no distinction of whether the perimeter and area values are characteristic or intrinsic.Various methods of measuring the fractal dimension are used on different known fractal objects to demonstrate these issues in an idealised setting.We show that while the distinction is not important for Mandelbrot's ore chips, which are island-type objects, it is very important in the analysis of cluster-type objects. The perimeter-area method is a valid tool in the fractal characterisation of aggregates and clusters, however, researchers must be careful to take the appropriate intrinsic and characteristic measurements.

TEOREMA DE PITÁGORAS E O FRACTAL ÁRVORE PITAGÓRICA: UM EXPERIMENTO NO ENSINO FUNDAMENTAL

This article approaches a qualitative research that had as objective to use some notions of euclidean geometry of students of a nineth year of Elementary School to realize the need to know some aspects of fractal geometry to understand the world in which they live. As a teaching methodology was used Van Hiele Theory for the development of reasoning in geometry with the software Geogebra in the construction of the fractal Pythagorean Tree. The students realized activities of classification of geometric figures and elements of nature, that allowed them to group them in properties or characteristics in two geometries and, with exploration of the photography resource, it was possible, for example, to identify the self-similarity characteristic of the fractal objects. The results of the research showed the efficiency of the Van Hiele Theory and Geogebra in the understanding of properties of the two geometries, in particular, on the Pythagorean theorem.

Experimental behaviour of an acoustic wave in a medium comprising self-similar (fractal) objects

After the pioneering work of one of the authors (Sapoval et al. J. Acoust. Soc. Am. 104, 1997), the study of acoustic propagation over fractal boundaries came up against the difficulty of experimenting with self-similar objects immersed in two- and three-dimensional space. The idea of one of us to use the capabilities offered by three-dimensional printing has made it possible to overcome this difficulty and produce numerous objects, as small or large as desired, in large numbers. We will present the experimental results obtained for acoustic propagation, reflection coefficient, attenuation and trapping in the case of an half fractal octahedron, isolated as well as in the case of a flat paving consisting of an important number of these objects.

Мультиплицирование спектров пространственных частот черно-белых изображений плоских фрактальных объектов.

Мультиплицирование спектров пространственных частот черно-белых изображений плоских фрактальных объектов.