Nonequilibrium Time Reversibility with Maps and Walks

Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt’s and Zermélo’s paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the irreversible Second Law of Thermodynamics (Loschmidt’s) as well as periodic in the time (Zermélo’s, illustrating Poincaré recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals’ information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here.

A Fractal Description of Fluvial Networks in Chile: a Geography not as Crazy as Thought

Chilean geography is highly variable, not only from a climatic and hydrological point of view, but also a morphological one, showing unpredictable natural patterns with marked contrasts throughout the country, for which sometimes it is considered as a "crazy" geography. In this paper we have investigated this apparent disorganized character by exploring the fractal properties of fluvial networks extracted from basins distributed across the continental territory. Analytical and semi-empirical methods were applied, finding striking patterns of organization in the distributions of Horton parameters and the fractal dimension of the drainage networks. Fractal dimension reveals to be quite dependent on the drainage area of each unit, showing clear groupings by tectonic and climatological factors. Such dimension reveals to be an important geomorphic parameter, if not the only one able to capture the real morphology of a fluvial network. From our results and despite the diversity of landforms, hydrological, climatic and tectonic conditions, Chilean’s geography is perhaps not as crazy and disorganized as believed.

Fractal functions of exponential type that is generated by the $\mathbf{Q_2^*}$-representation of argument

We consider function $f$ which is depended on the parameters $0<a\in R$, $q_{0n}\in (0;1)$, $n\in N$ and convergent positive series $v_1+v_2+...+v_n+...$, defined by equality $f(x=\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=a^{\varphi(x)}$, where $\alpha_n\in \{0,1\}$, $\varphi(x=\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=\alpha_1v_1+...+\alpha_nv_n+...$, $q_{1n}=1-q_{0n}$, $\Delta^{Q_2^*}_{\alpha_1...\alpha_n...}=\alpha_1q_{1-\alpha_1,1}+\sum\limits_{n=2}^{\infty}\big(\alpha_nq_{1-\alpha_n,n}\prod\limits_{i=1}^{n-1}q_{\alpha_i,i}\big)$.In the paper we study structural, variational, integral, differential and fractal properties of the function $f$.

Modified Catalysts and Their Fractal Properties

Obtaining high-area catalysts is in demand in heterogeneous catalysis as it influences the ratio between the number of active surface sites and the number of total surface sites of the catalysts. From this point of view, fractal theory seems to be a suitable instrument to characterize catalysts’ surfaces. Moreover, catalysts with higher fractal dimensions will perform better in catalytic reactions. Modifying catalysts to increase their fractal dimension is a constant concern in heterogeneous catalysis. In this paper, scientific results related to oxide catalysts, such as lanthanum cobaltites and ferrites with perovskite structure, and nanoparticle catalysts (such as Pt, Rh, Pt-Cu, etc.) will be reviewed, emphasizing their fractal properties and the influence of their modification on both fractal and catalytic properties. Some of the methods used to compute the fractal dimension of the catalysts (micrograph fractal analysis and the adsorption isotherm method) and the computed fractal dimensions will be presented and discussed.

The search for statistical patterns of pathological activity in human EEG signals in focal epilepsy

Modern data science faces a lot of challenges, one of which is the search for diagnostic criteria for neurological diseases. New methods of statistical analysis are actively applied in the field of biophysics to solve this issue. In this paper we apply the Memory Functions Formalism to analyze electroencephalogram signal recordings in the sleeping state of 8 healthy subjects and 19 patients with nocturnal lobe epilepsy. We observe the considerable difference of statistical memory effects and fractal properties at the pathology in comparison with the control group. Furthermore, we reveal significant alterations in brain rhythms at power spectra of statistical memory functions for two groups of subjects. As a result, we show that the application of the statistical analysis methodology of bioelectrical brain cortex activity recordings, after appropriate verification, can be useful in the search for diagnostic criteria of nocturnal frontal lobe epilepsy.

Studying the fractal properties of Ceres

Currently, the asteroid Ceres belongs to small celestial bodies with the most well-known physical parameters. The study of the structural and real properties of Ceres is an urgent and modern task, the solution of which will make it possible to develop the evolutionary theory of a minor planet. In this work, the fractal properties of the dwarf planet Ceres were analyzed using data from the Dawn space mission. Using the expansion in a harmonic series in spherical functions the height parameters of the structural model of Ceres, a 3D model of Ceres was constructed. The analysis showed that the resulting system has a complex multiparameter fractal configuration. The study of such objects requires the use of harmonic multiparameter methods. Multivariate fractal analysis allows to represent systems similar to the Ceres model in the form of a spectrum of fractal dimensions. The advantage of fractal analysis is the ability to explore local areas of the physical surface. In this work, the Minkowski algorithm was used for this purpose. At the final stage, an overdetermined system was solved for various local areas of topocentric information in order to postulate a model that takes into account external measures. Fractal dimensions D are determined for local regions and the entire model of the planet. Fractal dimensions vary from 1.37 to 1.92 depending on the longitude and latitude of Ceres. The main results are as follows: 1) the structure of the Ceres surface varies more strongly in longitude; 2) the structure of Ceres is smoother in latitude; 3) the coefficient of self-similarity changes rather quickly in longitude, which indicates that different local regions of the minor planet were formed under the influence of various physical processes. It is necessary to emphasize that the resulting fractal dimensions are significantly scattered both in longitude and latitude of Ceres. This fact confirms the presence of a complex structure in the spatial model of a minor planet. This also applies to the actual physical surface of Ceres. The results of the work allow us to conclude that fractal modeling can give independent values of the fractal dimension both for the entire model of Ceres and for its local macrostructural regions.

Fundamentals of metric theory of real numbers in their $\overline{Q_3}$-representation

In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by $Q_3$-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties. We have been studied the functions that store the "tails" of $\overline{Q_3}$-representation of numbers and the set of such functions,some metric problems and some problems of probability theory are connected with $\overline{Q_3}$-representation.

Peculiarities and Applications of Stochastic Processes with Fractal Properties

In this paper, the fractal properties of stochastic processes and objects in different areas were specified and investigated. These included: measuring systems and sensors, navigation and motion controls, telecommunication systems and networks, and flaw detection technologies. Additional options that occur through the use of fractality were also indicated and exemplified for each application. Regarding the problems associated with navigation information processing, the following fractal nature processes were identified: errors of inertial sensors based on the microelectromechanical systems called MEMS, in particular gyroscopic drift and accelerometer bias, and; the trajectory movement of mobile objects. With regard to navigation problems specifically, the estimation problem statement and its solution are given by way of the Bayesian approach for processing fractal processes. The modified index of self-similarity for telecommunication series was proposed, and the self-similarity of network traffic based on the R/S method and wavelet analysis was identified. In failure detection, fractality manifested as porosity, wrinkles, surface fractures, and ultrasonic echo signals measured using non-destructive sensors used for rivet compound testing.