Narrow Results

Quantum gravitational effects, on the one hand, lead to a limitation in the accuracy of measuring spatial and time intervals, and, on the other hand, they generate a discrete of spacetime structure (quantum foam). The common source of both measurement limitations and discreteness of spacetime is quantum fluctuations, so their characteristics must be related to each other. We study such a relationship using Barrow entropy as a source of fractal spacetime structure. The minimum inaccuracy in measuring spacetime intervals is expressed through the Barrow entropy parameter. The connection between the level of fractality and the speed of information processing is considered.

Fractal structures appear very often in open Hamiltonian systems, and can be identified in the deposition of chaotic magnetic field line on the plates of a tokamak divertor. Indeed, tokamaks with divertors are used to control the magnetic confinement of plasmas, such that the field lines created by electric currents have escape channels, through which plasma particles can be diverted out of the tokamak wall and redirected to divertor plates. In this work, we use a symplectic map to investigate the deposition patterns on the plates of a divertor. We show that the pattern of magnetic footprints on divertor plates (deposition patterns) underlying the chaotic orbits involves a number of fractal structures related to the existence of a nonattractive invariant chaotic set. In order to investigate the fractal characteristic of magnetic footprints, we analyze quantitatively the degree of stability of the fractal structure by calculating the entropy of the basin. These numerical analyses indicate that the fractal pattern on the divertor plates depends sensibly on the magnetic field structure. We show qualitative evidences of the Wada property.

The multifractal analysis of data on nuclear fragments obtained from ²⁸Si-AgBr interactions at 14.5 A GeV is performed using three different methods (the factorial moments, G-moments and Takagi moments). The generalized fractal dimensions D_q is determined from all these methods. Data reflects multifractal geometry for the nuclear fragments. From the knowledge of D_q, the multifractal specific heat is calculated for this data and also for ¹⁶O-AgBr interactions at 60 A GeV and ³²S-AgBr interactions at 200 A GeV.

This paper presents our study of dynamics of fractal solids. Concepts of fractal continuum and time had been used in definitions of a fractal body deformation and motion, formulation of conservation of mass, balance of momentum, and constitutive relationships. A linearized model, which was written in terms of fractal time and spatial derivatives, has been employed to study the elastic vibrations of fractal circular cylinders. Fractal differential equations of torsional, longitudinal and transverse fractal wave equations have been obtained and solution properties such as size and time dependence have been revealed.

We conducted a longitudinal study involving $240$ patients grouped according to the classification of periodontal diseases agreed in the World Workshop by the different groups of specialists gathered there. We proceed to select images of Cone Beam Computed Tomography (CBCT) that were used to perform a study of bone density through a precise algorithm allowing an accurate calculation of the fractal dimensions of such images. A detailed anthropometric analysis was also carried out. Our objective was to demonstrate that there exists a direct relationship between either the loss of bone or the changes related to its height and diameter and the variations in bone density. Our results highlight significant differences among the initial and moderate periodontal groups with respect to both the control and the periodontal groups, where patients experience a severe and controlled periodontal disease. We conclude that there is a variation in the architecture of patients with periodontal disease that have an acute component and have not been treated or their treatment is not effective and their bone loss does not slow down.

Hierarchical networks as fundamental models to describe the complex networks, have many applications in networks science, engineering technology and so on. In this paper, we first propose a new class of hierarchical networks with fractal structure, which are the networks with triangles compared to traditional hierarchical networks. Second, we study the precise results of some structural properties to derive small-world effect and scale-free feature. Third, it is found that the constructed network is sparse through the average degree and density. Fourth, it is also demonstrated that the degree distributions of hub nodes and the bottom nodes are the power law and exponential, respectively. Finally, we prove that clustering coefficient with a definite value $z$ tends to stabilize at a lower bound as $t$ iterates to a certain number, and the average distance of $G_t^z$ has an increasing relationship along with the value of $ln{N}_t$.

The model of ($2+1$)-dimensional breaking soliton equation is proposed, and the novel soliton solutions are derived by using the variable separation and extended mapping approach of Riccati equation and then, the propagation dynamics of the solutions which include multi-dromions, ring breather, fusion solitons and other multi-solitons are also constructed. Furthermore, the rich local excitations and fractal structures are investigated. The simulation results verify the proposed scheme. It can get more abundant local excitation structures than the ($1+1$)-dimensional nonlinear systems and it has potential values for certain applications in optical field.

The fractal alkylketene dimer (AKD) surface is an artificial super water-repellent one with a high contact angle of 174°, therefore, may provide special surface circumstances for studies of biological cells such as cell cultures. The experimental results indicated that the distribution of F-actin in the astrocytes cultured on the fractal AKD-coated dishes showed the stellate shape, while that in the astrocytes cultured on the poly-L-lysine-coated coverslips showed the formation of long alignment. The morphological change of astrocytes is induced by the fractal AKD surface, and the result suggests that astrocyte differentiation is stimulated by the fractal AKD surface.

This study examines the structure of the universe during its high-energy era. In this context, we examine Barrow’s proposal of entropy, which forecasts a fractal configuration for the geometric properties of the black hole’s horizon. By applying the principles of thermodynamics, we get the equation of motion using the Barrow entropic consideration. A quasi-de Sitter phase, in which fractal and non-fractal states may coexist, is determined. This is an uncertainty state (coexistence of fractal and non-fractal states), and so a deformation for de Sitter spacetime, which contains a high-energy scale in the framework of the COBE normalization. With the kinetic energy term of the inflation field, we remove this ambiguity owing to the field oscillations. However, the oscillatory behavior of the inflation field continues and causes the universe to enter a new phase state. At this point, in which the inflation field oscillations are over, we obtain a solution that provides the universe transition to the radiation and dust phases.

We suggest a new scenario in which the Universe starts its evolution with a fractal topological structure. This structure is described by a gas of cosmological wormholes. It is shown that the polarization of such a gas in external fields possesses the spatial dispersion, which results in a modification of the Newton’s law. The dependence on scales is determined by the distribution of distances between throat ends. The observed in galaxies logarithmic correction confirms that the distribution has fractal properties. We also discus the possibility of restoring such a distribution from observations.