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Multi-source information fusion is a sophisticated estimation process that generates a unified profile to assess complex situations. Dempster–Shafer evidence theory (DSET) is a practical theory in handling uncertain information in multi-source information fusion. However, highly-conflicting evidence may cause the combination rule of Dempster to provide counterintuitive results. Thus, how to effectively reconcile highly-conflicting evidence in DSET is still an open issue. To address this problem, in this paper, a novel belief divergence, higher order belief Jensen-Shannon divergence is proposed to measure the discrepancy between BPAs in DSET. The proposed higher order belief Jensen-Shannon divergence is the first method to dynamically measure the discrepancy between BPAs over the time evolution, i.e., to measure the discrepancy between BPAs with different time scale in the future. Besides, the proposed higher order belief Jensen-Shannon divergence has benefits from the perspective of measurement. It satisfies the properties of nonnegativeness and nondegeneracy, symmetry, and the triangle inequality of root form. Based on the proposed higher order belief Jensen-Shannon divergence, a novel multi-source information fusion algorithm is proposed. Eventually, the proposed algorithm is applied to a pattern classification experiment with real-world datasets. [ABSTRACT FROM AUTHOR]

Tristable piezoelectric energy harvesters (TPEHs) have been widely studied in recent years. Previous works have illustrated the persistent coexistence of high-energy and low-energy responses. Jumps among the responses can be induced by unavoidable initial-condition disturbances of TPEHs in practice, which concerns the final output of the harvested energy. For a more accurate evaluation of the TPEH performance, the disturbances of initial conditions should be taken into account. This study focuses on global dynamics of a typical TPEH with two fixed external magnets. Via static bifurcation analysis of the TPEH oscillatory system, the separation and gap distances are identified as key parameters to configure triple potential wells. The primary resonance analysis of the intra-well and inter-well responses is carried out to determine the optimum separation distance for efficient energy harvesting. Then, sequences of attractors and their basins of attraction (BAs) with the rise in the ambient excitation level and frequency is illustrated, showing that initial conditions of the TPEH system perturbed within the central well are beneficial for energy harvesting. For a low-level base excitation, BAs outside of the triple wells are severely fractal where the perturbation of the initial state of the TPEH can easily induce a jump that implies the unreliability of the TPEH performance. At a much higher ambient excitation level, the high-energy inter-well response can be globally attractive and reliable, which is preferable for practical applications of energy harvesters. • The key structural parameter of the TPEH is selected via static bifurcation analysis for configuring the potential wells. • Influence of the ambient excitation on the intra-well and inter-well resonant responses of the TPEH system is discussed. • BAs under different excitation levels and frequencies are depicted for reliably evaluating the performance of the TPEH. • Complex dynamical behaviors such as high-order responses, rare and hidden attractors, and fractal BAs are detected. [ABSTRACT FROM AUTHOR]

Chaotic image encryption is widely used in the field of information security. This paper proposes a novel chaotic image encryption method with spiral-transform-based fractal sorting matrix (STFSM). First of all, the theory of STFSM with good scrambling effect is introduced, which has good irregularity and iterative. Then, the iterative algorithm and calculation example of STFSM are introduced. STFSM can be used as the map of spatial location transformations to implement the design of image encryption. Based on the complete STFSM theory and iterative algorithm, a chaotic image cryptosystem based on STFSM is proposed to achieve a good image encryption process. To test the security of the proposed algorithm, security tests and analyses such as entropy analysis, correlation analysis, resistance to differential attacks analysis, and robustness analysis are used for the proposed algorithm. The experimental analysis illustrates that the algorithm has a better encryption effect, whether using conventional tests or the attacks simulations described in this paper and can also effectively resist the attacks. [ABSTRACT FROM AUTHOR]

A model of neuro-astrocytic coupling with electromagnetic induction has been developed to deeply investigate the genesis and propagation of Delay within networks. Delay is defined as the time neurons continue to fire after external stimulation has ceased until they reach a resting state. The findings reveal that the primary cause of Delay in the neuron-astrocyte coupling system is the role of astrocytes. We observe that within a certain range, the feedback intensity causes a fractal structure of Delay, and as the feedback from astrocytes intensifies, Delay increases indefinitely. Neurons can continue firing indefinitely post-stimulation, and we have identified the feedback threshold for this infinite Delay. Further, we observed the transmission of Delay in neuron-astrocyte networks. Subsequent neurons, which would not normally fire, begin to discharge during the transmission of Delay, a phenomenon induced by either synaptic or astrocytic coupling. Finally, with electromagnetic induction, we noted the cumulative effect of Delay in the network, leading to longer sustained firing in more distant neurons as the network extends and propagates. Our research elucidates the precise mechanisms of generation and propagation of Delay, offering insights akin to mechanisms of memory in the nervous system, potentially providing novel understandings in the field of cognitive science. • "Delay" is the period after stimulation ends when neurons keep firing before becoming inactive. • Astrocytes are key to the Delay in neurons, showing fractal traits in the Delay produced. • Electromagnetic reduces Delay but enhances its spread in network dissemination. [ABSTRACT FROM AUTHOR]

The total uncertainty measurement of basic probability assignment (BPA) in Dempster-Shafer evidence theory (DSET) has always been an open issue. Although some scholars put forward various measurements and entropies of BPA, due to the existence of discord and non-specificity, there is no method can measure BPA reasonably. In order to utilize BPA to practical decision-making, pignistic probability transformation of BPA is a significant method. In the paper, we simulate the pignistic probability transformation (PPT) process based on the fractal idea, which describes PPT process in detail and shows the process of information volume changes during transformation intuitively. Based on transformation process, we propose a new belief entropy called fractal-based belief (FB) entropy. After verification, FB entropy is superior to all existing belief entropies in terms of total uncertainty measurement and physical model consistency. [ABSTRACT FROM AUTHOR]

Rock fracture network description and characterization are of fundamental importance in rock engineering, such as coal mining, shale gas development and slope stability. Due to the complexity of rock fracture networks, it is impossible to describe all the fracture features individually. The purpose of this paper is to discuss the state of the art of the development in rock fracture network descriptions using traditional geometry and fractal methods. Advantages and disadvantages of each method are analyzed. Based on the analysis of fractal dimension shortage, a new characterization method is proposed. This method has higher measurement accuracy along with self-similar characteristics of fractal stuffs and the head/tail break method to avoid problems existing in previous fractal quantification methods. Two 2D coal fracture samples are used to examine this method. [ABSTRACT FROM AUTHOR]

• The spatial Julia set of the fractional complex system is obtained, and the dynamical characteristics of the system are studied from the fractal point of view. • The influence of dynamic noise on the fractal characteristics of fractional Lotka-Volterra system is analysed quantificationally and visualized. • Propose the Julia deviation index to characterize and analyze the effect of the noise on the systems Julia sets. The new definition has stronger immanence and wider applicability. This paper investigates the structures and properties of the spatial Julia set generated by a fractional complex Lotka-Volterra system with noise. The influence of several types of dynamic noise upon the system's Julia set is quantitatively analyzed through the Julia deviation index. Then, the symmetry of the Julia set is discussed and the symmetrical structure destruction caused by noise is studied. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a hierarchical scale-free network with special fractal structure. We study the average path length and associate the average path length with the symbol length. Then using a recursive method, we obtain the asymptotic formula for average path length of our networks. [ABSTRACT FROM AUTHOR]

Highlights • Totally disconnected and overlapping product IFSs are characterized in terms of the corresponding coordinate IFSs. • Product of condensation IFSs and their separation properties in terms of the corresponding homogenous IFSs are discussed. • A method to generate an open set which satisfies the OSC for a totally disconnected IFS is obtained. • Conditions for a product IFS to be just touching are given in terms of the corresponding coordinate IFSs. • Type 1 homogenous IFS on product space is introduced and its separation properties are obtained. Abstract Fractal theory is the study of irregularity which occurs in natural objects. It also enables us to see patterns in the highly complex and unpredictable structures resulting from many natural phenomena, using self-similarity property. The most common mathematical method to generate self-similar fractals is using an iterated function system (IFS). This paper discusses separation properties of finite products of hyperbolic IFSs. Characterizations for totally disconnected and overlapping product IFSs are obtained. A method to generate an open set which satisfies the open set condition for a totally disconnected IFS is given. Some necessary and sufficient conditions for a product IFS to be just touching are discussed. Also, Type 1 homogenous IFSs are introduced and its separation properties in terms of the separation properties of coordinate projections are explained towards the end. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamical behaviors of the classical dual-channel supply chain from the perspective of fractal theory. First, the definition of filled Julia set in supply chain price competition model is introduced. Second, according to the periodicities of the attractors, the filled Julia set is divided into several sub basins. On this basis, the importance of mixed coexisting attraction basins' connectivity is further illustrated. Then, the inner characteristics of filled Julia sets under different online preferences are analyzed, which indicates that online preference increase has negative influence on filled Julia sets' structures. Finally, for the unstable cases caused by high online preference, we adopt three control strategies and analyze their control effects by quantifying the connectivity restorations. The results suggest that the fractal viewpoint is beneficial for a global observation of the prices evolutions in different initial states, and could also evaluate the market status more comprehensively. • Adopt fractal connectivity concept to evaluate the multi-stability in price game. • Analyze the influence of channel preference on attraction basin and market stability. • Consider different control strategies to deal with excessive online preference cases. [ABSTRACT FROM AUTHOR]

Urbanization is a phenomenon of concern for planning and public health: projections are difficult because of policy changes and natural events, and indicators are multiple. There are previous studies of development that used fractals, but none for this specific problem, nor extrapolating the future trend. In the first part of this paper, we construct a theoretical framework for analyzing dynamic (changing) fractals and extrapolating their future trends based on their fractal dimension—a measure of the complexity of the fractal. We believe this approach holds enormous potential for applications in analyzing changing fractals in the real world, such as urban growth, cells, cancers, etc., all of which are invaluable to research. This theoretical framework may shed light on a factor overlooked in past research: the trend of how fractals change. In the second part of this paper, we apply this theoretical framework to study the urbanization of Boston. We compare several maps and measurements of fractal dimensions, produce code 1 1 https://github.com/yinj66/fractal-dimension. that reads the maps and divides the city into subsections, and ultimately graph the fractal dimension over time using both differential and difference equations. Finally, we postulate the logistic equation as a model to fit the evolution of the fractal dimension, as well as the total population derived from census data, which serves as a component for comparing dynamical fractals. • Formally define dynamical fractal and analyze its mathematical properties. • Develop models of extrapolating short & long-term trends of dynamical fractals. • Use the concept of curvature to compare the similarity of two dynamical fractals. • Apply all the theoretical concepts to our case study: The urbanization of Boston. • Develop new tools approximating fractal dimension and compare them with existing ways. [ABSTRACT FROM AUTHOR]

This paper proposes a simple model of anomalous diffusion, in which a particle moves with the velocity field induced by a single "dipole" (a doublet or a pair of source and sink), whose moment is modulated randomly at each time step. A motivation to introduce such a model is that it may serve as a toy model to investigate an anomalous diffusion of fluid particles in turbulence. We perform a numerical simulation of the fractal dimension of the trajectory using periodic boundary conditions in two and three dimensions. For a wide range of the dipole moment, we estimate the fractal dimension of the trajectory to be 1.5–1.9 (2D) and 1.6–2.7 (3D). • This paper proposes a simple model of anomalous diffusion, in which a particle moves with the velocity field induced by a single "dipole" (a doublet or a pair of source and sink), whose moment is modulated randomly at each time step. • A motivation to introduce such a model is that it may serve as a toy model to investigate an anomalous diffusion of fluid particles in turbulence. • We perform a numerical simulation of the fractal dimension of the trajectory using periodic boundary conditions in two and three dimensions. • For a wide range of the dipole moment, we estimate the fractal dimension of the trajectory to be 1.5–1.9 (2D) and 1.6–2.7 (3D). [ABSTRACT FROM AUTHOR]

Random permutation set (RPS) is a recently introduced set based on the Dempster–Shafer evidence theory, which considers all possible permutations of the elements within a given set. The information dimension is a significant fractal dimension which plays a vital role in the information theory. Nevertheless, how to develop the information dimension of a specific permutation mass function in RPS remains an unresolved problem. To solve this problem, we propose a new dimension named information fractal dimension of Random Permutation Set. Moreover, several properties of the proposed dimension are explored and numerical examples are provided to illustrate its effectiveness. The research discovers an interesting property related to the permutation mass function corresponding to the maximum RPS entropy: its information dimension is 2, which is equivalent to the fractal dimension of Brownian motion and Peano curve. • The information dimension of a specific permutation mass function in RPS is defined. • Some properties of the dimension are presented. • The information dimensions of several distribution modes are calculated. • The information dimension corresponding to the maximum RPS entropy is 2. [ABSTRACT FROM AUTHOR]

We investigate the RNA fractal properties of some textual viruses of the Coronaviridae family. These organisms can seriously infect people and endanger the world's health systems. The whole data follows from the NCBI databases. After investigating fractality at various scales, we created two dummy time series to compare with the walks of the coronavirus sequences. We also employ a technique to look for crossover in DNA temporal sequences. Our findings demonstrate that the sequences do not act randomly and contain more significant concentrations of adenine and guanine bases. It proves the existence of power law correlations for RNA sequences at various sizes and the persistence tendency that all species exhibit in their RNA sequences. We also observe that there are no crossover timeframes and that all sequences are monoscale. • Analyses of RNA fractal properties of viruses of coronaviridae family. • Multifractal Detrended Fluctuations Analysis and fractality of nucleotide sequences. • The sequences contain a significant concentration of adenine and guanine bases. • Power law correlations for RNA sequences at various sizes and the persistent tendency of all species. [ABSTRACT FROM AUTHOR]

Fractal theory has been widely applied in a variety of disciplines to understand the theory behind chaotic phenomena based on internal self-similarity. In this study, three ideal geological models are used to analyze the unreliability of the capacity dimension in the fractal calculation of geological bodies with different scales. Additionally, by varying the side length r of the statistical units, the geological meanings of the fractal dimension D and the correlation coefficient R 2 are discussed. The points of information (POIs) are densely filled by binarizing the geological bodies to black/white. Based on the optimized r of a geological body, an algorithm is derived that divides the grids of the statistical units to determine the probability of the POIs falling into different grids. The information dimension (D I ) and R 2 of a geological body are obtained by fitting the variable data. An example calculation of the information dimension field in the Jinhu sag is presented to demonstrate the methodology and to test its reliability. The results show that determining the appropriate side length of the statistical unit is key to evaluating the fractal calculation. Compared to the capacity dimension, D I is more reliable in the fractal calculation of multi-scale geological bodies; D I is thereby the preferred fractal dimension to use in the analyses of these types of geological bodies. [ABSTRACT FROM AUTHOR]

The fractal leaf vein model has been proposed to investigate the electrical property of porous media in this work. The characteristics and evolution of fractal leaf vein network are described in detail. Besides, the structural parameters and corresponding electrical resistance of fractal leaf vein model are addressed by analytical solutions. The influences of structural parameters, like diameter ratio, length ratio, level number and so on, on electrical resistance of fractal network have been discussed profoundly. What is more, the comprehensive effects of temperature, concentration of electrolyte solution and applied frequency on electrical conductivity of pore solution have been significantly analyzed in this work. The interactions among kinds of structure parameters of fractal leaf vein model are studied at last. [ABSTRACT FROM AUTHOR]

Global-matching models of memory argue that knowledge emerges from the interaction between presented cues and traces of past experiences. But these models generally rely on the use of independent episodic traces, unable to account for global interactions between learned situations (see Versace et al., 2009). Enactivism (Varela, 1993) could theoretically take advantage of an inter-dependent processing of traces to account for abstraction processing using only sensorimotor covariances (Hutto & Myin, 2012), but no mathematical formalization of an enactivist memory has yet been proposed. In this paper, we propose the ATHENA model as an enactivist mathematical formalization of Act-In theories (Versace et al., 2014) within MINERVA2 (Hintzman, 1986) non-specific traces: ATHENA is a fractal model which keeps track of former processes that led to the emergence of knowledge, and is therefore able to process contextual processes (abstraction manipulation). We present three simulations designed to test ATHENA’s ability to construct, learn, and manipulate emergent abstractions. [ABSTRACT FROM AUTHOR]

A new route to strange nonchaotic attractors (SNAs) is investigated in a quasiperiodically driven nonsmooth map. It is shown that the smooth quasiperiodic torus becomes nonsmooth (continuous and non-differentiable) due to the border-collision bifurcation of the torus. As the coefficients change, the nonsmooth torus gets gradually fractal, forming strange nonchaotic attractors. It is termed the border-collision bifurcation route to SNAs. A novel feature of this route is that a large number of SNAs are formed, and the parameter area of SNAs makes up about 40 % of the given parameter regions. These SNAs are identified by the largest Lyapunov exponents and the phase sensitivity exponents. They are also characterized by the distribution of finite-time Lyapunov exponents. Unlike other types of SNAs, the distribution of finite-time Lyapunov exponents has its maximum at a relatively small negative Lyapunov exponent, which contributes largely to lead to the abundance of SNAs. • Border-collision bifurcations can lead to the birth of strange nonchaotic attractors (SNAs). • This type of SNAs is abundant. • The distribution of finite-time Lyapunov exponents shows a novel feature. [ABSTRACT FROM AUTHOR]

The Hausdorff derivative partial differential equations have in recent years been found to be capable of describing complex mechanics and physics behaviors such as anomalous diffusion, creep and relaxation in fractal media. But most research is concerned with time Hausdorff derivative models, and little has been reported on the numerical solution of spatial Hausdorff derivative partial differential equations. In this study, we derive the fundamental solutions of the Hausdorff derivative Laplace, Helmholtz, modified Hemholtz, and convection-diffusion equations via a non-Euclidean metric, called the Hausdorff fractal distance. And then the singular boundary method is used to numerically simulate the steady heat transfer governed by the Hausdorff Laplace equation in comparison with the corresponding fractional Laplacian models. Numerical experiments show the validity and applicability of the derived fundamental solution of the Hausdorff Laplace equation. [ABSTRACT FROM AUTHOR]

The global structure of nonlinear response of mechanical centrifugal governor, forming in two-dimensional parameter space, is studied in this paper. By using three kinds of phases, we describe how responses of periodicity, quasi-periodicity and chaos organize some self-similarity structures with parameters varying. For several parameter combinations, the regular vibration shows fractal characteristic, that is, the comb-shaped self-similarity structure is generated by alternating periodic response with intermittent chaos, and Arnold's tongues embedded in quasi-periodic response are organized according to Stern-Brocot tree. In particular, a new type of mixed-mode oscillations (MMOs) is found in the periodic response. These unique structures reveal the natural connection of various responses between part and part, part and the whole in parameter space based on self-similarity of fractal. Meanwhile, the remarkable and unexpected results are to contribute a valid dynamic reference for practical applications with respect to mechanical centrifugal governor. [ABSTRACT FROM AUTHOR]

The fractal properties of sea clutter are proposed and applied to the maritime target detection. Calculation of the measured data shows that, the Hurst exponent of the sea clutter with targets and the sea clutter without targets are different, which enables us to detect low-observable targets within the sea clutter. This paper explains the reason why the Hurst exponent can distinguish the presence or absence of targets in theory and proposes a fractal detector based on the Hurst exponent. Comparing the fractal detector proposed in this paper to the energy detector by the detection results of 140 frames data of the real sea clutter, it is demonstrated that the fractal detection method has a better detection performance. In order to get systemic conclusions, the new sea clutter data with different signal-to-clutter ratios (SCRs) are constructed in the way that add the sea clutter data with targets to the pure sea clutter data. The results show that the fractal detection method has a better performance than the statistical method on detection of maritime targets, especially maritime weak targets with low SCR. [ABSTRACT FROM AUTHOR]

The influence of rough elements of porous materials on sound propagation characteristics in roughened porous materials is investigated. Rough surface topography on the roughened porous materials is simulated by the fractal geometry theory, in which relative roughness is defined clearly as a function of the fractal dimension, porosity, average diameter and diameter ratio of the hexagonal elements on pore wall of the porous materials. Based on the sound propagation model of the smooth porous materials, a sound propagation model of the roughened porous materials is built. The effective density and bulk modulus, acoustic impedance and propagation constant, flow resistivity and sound absorption coefficient of the roughened porous materials are derived and discussed as a function of the relative roughness. It is demonstrated that the sound absorption coefficient of the roughened porous materials is improved as the relative roughness increases. The model predictions for the sound absorption coefficient of the roughened porous materials are well agreed with that of the existing test results. [ABSTRACT FROM AUTHOR]

This paper is a pre-step in conducting a restudy for an emerging theory in applied sciences, namely Fractal interpolation. It is one of the best-fit models for capturing irregular data that arise in physical situations. On the other hand, it has fixed point theory as the staunch basis, so any inspection of it would get governed by the Hutchinson–Barnsley theory of fractals. In this regard, we classify an enormous collection of maps owned by the literature of fixed point theory into two — conventional and nonconventional. Suitably, every conventional iterated function system (IFS) has delivered fractal, but nonconventional IFSs are yet to make a mark. Therefore, the present work introduces a novel nonconventional map of the Ćirić–Reich–Rus genre to fulfill this gap. It incorporates a parameter δ ∈ (0 , ∞) , in a Ćirić–Reich–Rus condition, for the first time in the literature. Consequently, we obtain extension, improvement, and generalization of the results produced in Sahu et al. (2010), Shaoyuan et al. (2015), Dung and Petruşel (2017) and Abbas et al. (2022). In addition, a rational map and a Suzuki-type Kannan map are considered to prove the point. [ABSTRACT FROM AUTHOR]

The occupancy of the road and transition rate have a significant impact on traffic flow in real traffic dynamics. Thus, the macroscopic traffic flow has been investigated in a percolation-backbone fractal with the consideration of occupancy of the road and transition rate. The traffic flow on the fractal network is presented with the help of a cell-transmission graph. The density equations are derived using the speed-matching model. The fundamental diagrams are obtained for urban-scale traffic networks by solving the density equations numerically. The steady-state vehicular densities and traffic currents are plotted for different cases of occupancy and transition rate. It is found that the traffic flow increases significantly with the increase in occupancy of the road and transition rate. • We studied the role of occupancy and transition rate on traffic flow in a percolation-backbone fractal. • The fractal network is described with the help of cell-transmission graph. • The fundamental diagrams are obtained for urban-scale traffic networks. • It is found that traffic flow increases significantly with the increase in occupancy and transition rate. • Theoretical results are validated through simulation. [ABSTRACT FROM AUTHOR]

Asymmetric damage segregation (ADS) is ubiquitous among unicellular organisms: After a mother cell divides, its two daughter cells receive sometimes slightly, sometimes strongly different fractions of damaged proteins accumulated in the mother cell. Previous studies demonstrated that ADS provides a selective advantage over symmetrically dividing cells by rejuvenating and perpetuating the population as a whole. In this work we focus on the statistical properties of damage in individual lineages and the overall damage distributions in growing populations for a variety of ADS models with different rules governing damage accumulation, segregation, and the lifetime dependence on damage. We show that for a large class of deterministic ADS rules the trajectories of damage along the lineages are chaotic, and the distributions of damage in cells born at a given time asymptotically becomes fractal. By exploiting the analogy of linear ADS models with the Iterated Function Systems known in chaos theory, we derive the Frobenius–Perron equation for the stationary damage density distribution and analytically compute the damage distribution moments and fractal dimensions. We also investigate nonlinear and stochastic variants of ADS models and show the robustness of the salient features of the damage distributions. • Damage evolution in clonal cell populations is described by Iterative Function System • Trajectories of damage in individual lineages are chaotic • Dynamics of damage distribution obeys the solvable Frobenius–Perron equation • Stationary damage distribution in a broad class of ADS models is fractal [ABSTRACT FROM AUTHOR]

This paper is devoted to the development of local vector calculus in fractional-dimensional spaces, on fractals, and in fractal continua. We conjecture that in the space of non-integer dimension one can define two different del-operators acting on the scalar and vector fields respectively. The basic vector differential operators and Laplacian in the fractional-dimensional space are expressed in terms of two del-operators in a conventional way. Likewise, we construct Laplacian and vector differential operators associated with Fα-derivatives on fractals. The conjugacy between Fα and ordinary derivatives allow us to map the vector differential operators on the fractal domain onto the vector differential calculus in the corresponding fractal continuum. These results provide a novel tool for modeling physical phenomena in complex systems. • In non-integer dimension space del-operator acting on vector fields differs from del-operator acting on scalar fields. • Basic vector differential operators in fractional dimensional space are expressed in terms of two del-operators. • Fα-calculus is generalized for fractals with the non-integer number of effective spatial degrees of freedom. • A novel version of vector differential calculus in the fractal continuum is developed. [ABSTRACT FROM AUTHOR]

We study the macroscopic traffic flow and jamming transitions on directed Sierpinski gasket fractal which is closely connected to complex network. We propose a dynamic model for macroscopic traffic flow on directed Sierpinski gasket. The macroscopic dynamic equations of vehicular densities are derived on the directed network. It is shown where and how traffic jam occurs on directed networks. It is found that multiple transitions occur successively with increasing vehicular density. The successive transition points are derived numerically and analytically. • We studied the jamming transitions on directed Sierpinski gasket. • We presented the macroscopic traffic model in the directed fractal network. • We found that the jamming transitions occur successively with increasing mean density. [ABSTRACT FROM AUTHOR]

We propose the Fractal version of the hyper-Wiener index and we calculate the hyper-Wiener index on the Sierpiński gasket, a classical self-similar fractal, by developing the notion of pattern introduced by S. Wang et al (2017). [ABSTRACT FROM AUTHOR]

Graphics with fractal features are usually generated using the Iterated Function System (IFS). IFS can generate the entire family of Sierpinski gaskets by performing different operations on the attractors. As the most classical graphic, Sierpinski gasket can also be generated using m o d (n , 2). Dempster–Shafer Theory (DST), as a mathematical theory about evidence, models information on the all possible combination states (power set), which relates to 2 n. In this paper, we explore the relationship between the Sierpinski gasket and matrix calculus in DST, which is the first time to connect fractal theory and DST from the perspective of geometry. In addition, based on the generation process of the matrices, we propose a method to generate the Sierpinski Gasket using the Kronecker product. • The relation between belief matrices and fractal geometry are found in first time. • A connection of Sierpinski triangles, Dempster–Shafer Theory and Pascal's triangle is established. • A new iteration system based on Kronecker product to generate Sierpinski Gasket on matrix is proposed. [ABSTRACT FROM AUTHOR]

In this study, based on the φ 4 model, a new model (called the B φ 4 model) is introduced in which the potential form for the values of the field whose magnitudes are greater than 1 is multiplied by the positive number B. All features related to a single kink (antikink) solution remain unchanged and are independent of parameter B. However, when a kink interacts with an antikink in a collision, the results will significantly depend on parameter B. Hence, for kink–antikink collisions, many features such as the critical speed, output velocities for a fixed initial speed, two-bounce escape windows, extreme values, and fractal structure in terms of parameter B are considered in detail numerically. The role of parameter B in the emergence of a nearly soliton behavior in kink–antikink collisions at some initial speed intervals is clearly confirmed. The fractal structure in the diagrams of escape windows is seen for the regime B ≤ 1. However, for the regime B > 1 , this behavior gradually becomes fuzzing and chaotic as it approaches B = 3. 3. The case B = 3. 3 is obtained again as the minimum of the critical speed curve as a function of B. For the regime 3. 3 < B ≤ 10 , the chaotic behavior gradually decreases. However, a fractal structure is never observed. Nevertheless, it is shown that despite the fuzzing and shuffling of the escape windows, they follow the rules of the resonant energy exchange theory. • The B φ 4 is a modified version of the φ 4 theory in which a parameter B is multiplied over the kink sector. • The features of a single kink are independent of B but the kink interaction depends on it. • The critical velocity, fractal structure, and soliton behavior are studied as a function of B. • In the regime 0.2 ≤ B ≤ 1, the existence of a quasi-fractal structure is confirmed. • The lowest critical velocity and the peak of the chaotic behavior occur in B = 3.3. [ABSTRACT FROM AUTHOR]

Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this survey, we gather different definitions and counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the part of Fractal Theory related to finite families of (weak) contractions, called iterated function systems (IFS, herein). An attractor is a compact set which remains invariant for such a family. Thus, we consider spaces homeomorphic to attractors of either IFS or weak IFS, as well, which we will refer to as Banach and topological fractals, respectively. We present a collection of counterexamples in order to show that all the presented definitions are essential, though they are not equivalent in general. [ABSTRACT FROM AUTHOR]

Fractal geometry is mainly based on the idea of self-similar forms. To be self-similar, a shape must able to be divided into parts that are smaller copies, which are more or less similar to the whole. There are different forms of self similarity in nature and mathematics. In this paper, some of the topological properties of super self similar sets are discussed. It is proved that in a complete metric space with two or more elements, the set of all non super self similar sets are dense in the set of all non-empty compact sub sets. It is also proved that the product of self similar sets are super self similar in product metric spaces and that the super self similarity is preserved under isometry. A characterization of super self similar sets using contracting sub self similarity is also presented. Some relevant counterexamples are provided. The concepts of exact super and sub self similarity are introduced and a necessary and sufficient condition for a set to be exact super self similar in terms of condensation iterated function systems (Condensation IFS’s) is obtained. A method to generate exact sub self similar sets using condensation IFS’s and the denseness of exact super self similar sets are also discussed. [ABSTRACT FROM AUTHOR]

Understanding an electrode–electrolyte interface (EEI) behavior is a valuable tool in several areas of science. There are models based on discrete fractal structures, which explain the measurements of linear and non-linear impedance at fixed frequencies, or at determined ranges of high and low current densities. A level by level discrete calculation is needed to evaluate these models, or the use of black-box models, which affect the good understanding of the phenomenon. A continuous model based on a differential equation of an EEI is presented in this paper. It includes an electrical circuit similar to a long transmission line. It has been deduced from the discrete Liu model. [ABSTRACT FROM AUTHOR]

Hourly resolution time series of groundwater level fluctuations are analyzed after removing the seasonal cycle. It is found that fluctuations of groundwater levels have fractal scaling and a persistent behavior. We show also that groundwater level fluctuations exhibit non-Gaussian heavy tailed probability distribution that is well fitted by the Lévy stable distribution. Implications of the present results on the groundwater system modeling as a fractional Lévy motion and the connection with the anomalous diffusion inside the soil are discussed. [ABSTRACT FROM AUTHOR]

As the classical ones among the fractal sets, Julia set (abbreviated as J set) and Mandelbrot set (abbreviated as M set) have been explored widely in recent years. In this study, J set and M set under additive noise perturbation and multiplicative noise perturbation are created by equipotential point algorithm. Changes of the J set and M set under random noise perturbation as well as the close correlation between them are studied. Experimental results show that either additive noise perturbation or multiplicative noise perturbation may cause dramatic changes on J set. On the other hand, when the M set is perturbed by additive noise, it almost changes nothing but its position; when the M set is perturbed by multiplicative noise, its inner structures change with the stabilized areas shrinking, but it keeps the symmetry with respect to X axis. In addition, the J set and the M set still share the same stabilized periodic point in spite of noise perturbation. [ABSTRACT FROM AUTHOR]

The formation of three-dimensional domains in monomolecular layers (Nucleation Dynamics, ND) of four fatty acids, stearic, arachidic, behenic and lignoceric acids, containing the same carboxylic (–COOH) head and an alkyl chain with 18, 20, 22 and 24 carbon atoms, respectively, on water surface, has been studied through Specific Molecular Area ( A ) versus time ( t ) studies from Surface Pressure ( π )- A isotherms and Brewster Angle Microscopy (BAM). To investigate the fractal nature, the gray-scale Brewster angle micrographs are converted to binary images containing only two pixel values – 0 for 2D phase and 255 for 3D phase before box-counting method is employed to compute the fractal dimension. The 3D phase in the background of 2D phase is found to be fractal in nature. In fact, 3D phase is an interpenetration of two fractal structures with two different fractal dimensions – one corresponding to smaller (intra-domain) structures and other corresponding to larger (inter-domain) structures. These fractal dimensions are seen to evolve with coverage as 3D phase grows. The two fractal dimensions and their evolution in ND dynamics are identical for the longest-tailed lignoceric acid which has a porous 3D phase. [ABSTRACT FROM AUTHOR]

The present endeavor investigates an area concerning interpolative operators to approach attractors, particularly fractals; ergo, an interpolative iterated operator system (I δ -IOS) is employed. As a consequence, dull and active fractals are perceived for the first time in the literature. Moreover, fractal function thereby fractal interpolation space is ascertained using an introductory e -metric and a finite collection of modified Berinde weak operators. Further, the investigation is extended onto multivalued fractals via set-valued interpolative operators. Cut to, properties of a multi-operator associated with set-valued IOS are explored. • Interpolative maps possess isolated and limiting fixed points. • Active and Dull fractals are perceived through the inception of interpolative maps. • Fractal interpolation space is ascertained. • The cyclic multi-valued approach is determined to seek a multivalued fractal. [ABSTRACT FROM AUTHOR]

In this paper we establish a strong coupled fixed point theorem for a generalized coupling between two subsets of a metric space. These are cyclic generalizations of coupled mappings. Starting from two arbitrary points collected from the two subsets between which the coupling is defined, we construct two iterations each of which converges to the coupled fixed point. Further it is shown that such a point is unique. The main result is supported with an example which shows that our result is an actual generalization of an existing result. We also discuss an application in which we construct an iterated function system leading to the generation of a strong coupled fractal which we define here. Further we illustrate the generation of such a fractal set through an example. • A metric coupled fixed-point result is deduced with respect to φ -couplings. • Notion of strong coupled fractals is introduced. • An application of the fixed-point result is given to generate strong coupled fractals. • The main result extends a previous result. [ABSTRACT FROM AUTHOR]

We write a program to implement the Cluster–Cluster Aggregation (CCA) model with java programming language. By using the simulation program, the fractal aggregation growth process can be displayed dynamically in the form of a three-dimensional (3D) figure. Meanwhile, the related kinetics data of aggregation simulation can be also recorded dynamically. Compared to the traditional programs, the program has better real-time performance and is more helpful to observe the fractal growth process, which contributes to the scientific study in fractal aggregation. Besides, because of adopting java programming language, the program has very good cross-platform performance. Program summary Program title: CCA Catalogue identifier: AEUK_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEUK_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 390649 No. of bytes in distributed program, including test data, etc.: 83881850 Distribution format: tar.gz Programming language: Java. Computer: Any computer with Windows, java jdk and java3d jdk installed. Operating system: Windows. RAM: Determined by the number of the initial particles in the simulation test Classification: 14, 16.12. Nature of problem: The programs for the CCA model published are very rare, and are mostly in the form of a 2D figure. So, existing computer simulation programs cannot well simulate the fractal aggregation. Compared to the existing programs, our simulation program for the CCA model in the form of a 3D figure can better simulate the process dynamically. Solution method: A program is provided that well simulates CCA model in the form of a 3D figure. The program implementation is based on the java programming language. Especially, we use the java3d technology to display the simulative graphics in form of 3D. Restrictions: In order to show the process of fractal aggregation growth dynamically, this program needs to refresh the graphics of aggregation of colloidal particle frequently. So this program takes up a lot of computer memory. The storage capacity of the computer memory used to run the program will limit the number of single particles which the program can simulate. Unusual features: The program simulates the CCA model in the form of a 3D figure using the java3d technology. Additional comments: !!!!Due to the large file size, CCA is not delivered directly when download or Email is requested. Instead an html file giving details of how the program can be obtained is sent.!!!! Running time: Determined by the size of the problem. [ABSTRACT FROM AUTHOR]

In order to reveal the migration mechanism of dynamic water between coal particles in the process of coal seam water injection, in this study, nuclear magnetic resonance experiments (NMR) were carried out on wetted coal samples, the transverse relaxation time (T 2) spectra of samples presenting different degrees of metamorphism and particle size ranges were measured over different time periods of the wetting process, and the fractal dimension (D NMR) of the wetting phase distribution in the migration channel was calculated. (1) The average values of T 2 g decrease percentage Δ of the three coal samples are 84%, 71.25%, and 57.75%, indicating that the order of the wettability strength is Daliuta coal sample (DLT) > Xinglongzhuang coal sample (XLZ) > Qincheng (QC), this shows that the coal samples are more easily wetted as the degree of metamorphism decreases. (2) Sodium dodecyl sulfate (SDS) will make the wetting process between coal and water smoother, and in the range of 60–80 mesh and 100–120 mesh particle size, compared to the solution properties change, the particle size has relatively little influence on the wetting process when its change is not large. (3) The surface tension reduction caused by the addition of surfactants in the water, the narrowing of the pore space caused by the reduction of the overall size of the particles, and the decrease of the particle surface wettability caused by the increase in the metamorphic degree of coal sample will lead to the decrease of D NMR measured in the experiment. Based on the above research, it can provide a theoretical basis for improving the effect of coal seam wetting and dust reduction. • The relationship between the degree of coal metamorphism and water-wet characteristics was clarified. • The mechanism of SDS on improving wetting properties of coal particles was explained. • A calculation method of fractal dimension of the wetting phase distribution was proposed. [ABSTRACT FROM AUTHOR]

The inability of Albert Einstein and quantum physicists to resolve whether the laws of nature operate exactly or probabilistically impacted scientific methodology. Karl Popper further compounded the problem by stressing empirical falsifiability and proposing causes cannot be proven because of infinite regress. For these and other reasons, many investigators either downplayed propositions of cause or even ridiculed them. With causality vilified, metaphysics took a back seat to empirical science. In Earth sciences, however, causality remains a top priority. Thus, a significant interdisciplinary rift developed in established scientific procedures. Yet, some physicists still wanted to know why their equations worked, and some still postulated causality despite its diminished stature. Metaphysics has a rightful place in science and revive a forgotten approach, consupponibility, for testing metaphysical assumptions and theories. A set of fundamental assumptions and theories are consupponible if they exist without contradicting one another. Importantly, a single contradiction among the set falsifies the entire paradigm. To demonstrate the concepts associated with fundamental assumptions and a paradigm, this work analyses a paradigm based on infinitely fractal matter. Just like Popperian falsifiability, a large set of prohibitive fundamental assumptions allows proponents, opponents, and undecided investigators a means for falsifying the infinitely fractal universe model if the paradigm incorrectly describes the universe. That is, the goal is never to prove the paradigm, which is impossible. Instead, the goal is to attempt to disprove the paradigm by finding contradictions. Repeated failures to disprove the paradigm increase the likelihood of its correctness. [ABSTRACT FROM AUTHOR]

The exponential growth of space exploration, and of nuclear energy applications, leads to optimize the material distribution in a shield to protect against hypervelocity impacts and radiation. In order to find the optimal distribution, we study multilayer and porous structures, regular and random, by simulating straight particle trajectories crossing a rectangle filled with a constant mass according to the different structures. These trajectories are analyzed theoretically in multilayer structures and following a Monte Carlo method in porous structures. In this second case, distribution and probability estimators are used to ensure results with almost sure convergence, which indicates that the amount of matter crossed by particles is independent of the underlying structure. However, in regular porous structures, there are singularities given by fractal sets of measure zero that allow a theoretical construction. These fractal sets reveal protection gaps in regular porous structures that make them extremely dangerous for shielding purposes. Furthermore, this result allows to explain X-ray fractal patterns of certain semi-crystalline materials. • Optimal distribution of mass is studied in shields • The degree of protection is independent of the structure: multilayer or porous • Regular porous structures show a protection singularity: a fractal set of measure zero • The fractal set involves a huge protection risk in spacecraft shields • X-ray diffraction patterns can be explained by the fractal set [ABSTRACT FROM AUTHOR]

In Grossu (2022) [1] it was discussed the migration of Hyper-Fractal Analysis from Visual Basic 6 to C#.Net. The main goal of current work was developing a medical module for fractal analysis of computed tomography multi-channel images. A new tool for comparing images by RGB channels superposition was also considered. This could be of particular interest in medical image analysis (e.g. compare native with contrast CT slices). Program Title: Hyper-Fractal Analysis v06 CPC Library link to program files: https://doi.org/10.17632/z9knmny56p.2 Licensing provisions: GPLv2 Programming language: C# 7.3 /.Net Framework 4.7.1 Journal reference of previous version: Computer Physics Communications 271, (2022) 108189 Does the new version supersede the previous version?: Yes Nature of problem: Estimating the fractal dimension of images, multi-dimensional objects, and medical computed tomography images. Solution method: Optimized version of the fuzzy box-counting algorithm. Reasons for the new version: Develop a medical module for fractal analysis of computed tomography multi-channel images. Summary of revisions: • The migration of Hyper-Fractal Analysis from Visual Basic 6 to C#.Net was discussed in [1]. The new SOLID [2] application design is facilitating the development process. Thus, the main goal of current version was adding a medical module for fractal analysis [3-5] of computed tomography (CT) images [6,7]. • Hyper-Fractal Analysis v06 allows opening specific csv files (. ct.csv) containing up to three Hounsfield units matrices (CT slices) separated by empty lines. Most commonly, those three "channels" were used for storing the corresponding slices from native, arterial contrast, and venous contrast series. • Each of the previously described CT slices is represented on a different RGB channel [8], which results in highlighting the images differences (Fig. 1). • A new tab (CT Band-Pass) was added to allow specifying the Hounsfield units input parameters for the CT multi-channel fuzzy band-pass filter. [1] I.V. Grossu, Migration of hyper-fractal analysis from visual basic 6 to C#.Net, Comput. Phys. Commun., 271 (2022) 108199, https://doi.org/10.1016/j.cpc.2021.108189. [2] S. Millett, Professional ASP.NET Design Patterns, Wiley, Indianapolis, 2010. [3] R.H. Landau, M.J. Paez, C.C. Bordeianu, Computational Physics: Problem Solving with Computers, Wiley-VCH-Verlag, Weinheim, 2007. [4] J. Ruiz de Miras, J. Navas, P. Villoslada, F.J. Esteban, UJA-3DFD: A program to compute the 3D fractal dimension from MRI data, Computer Methods and Programs in Biomedicine, 104 (3) (2011) 452-460, https://doi.org/10.1016/j.cmpb.2010.08.015. [Display omitted] [5] J. Ruiz de Miras, Fractal Analysis in MATLAB: A Tutorial for Neuroscientists, in: A. Di Ieva (Eds.), The Fractal Geometry of the Brain, Springer, New York, NY, 2016, https://doi.org/10.1007/978-1-4939-3995-4%5f33. [6] E. Seeram, Computed Tomography, 4th Edition, Saunders 2015, ISBN: 9780323312882. [7] N. Verga, A.I. Miron, O. Savencu, I.V. Grossu. Structure of Computed Tomography Imaging, WAIHIP Conference, Romania, 2021. [8] I.V. Grossu, A.I. Miron, O. Savencu, C. Besliu, N. Verga, Experimental software for CT image analysis, WAIHIP Conference, Romania, 2021, https://doi.org/10.13140/RG.2.2.22757.32486. [ABSTRACT FROM AUTHOR]

We show that the almost sure θ -intermediate dimension of the image of the set F p = { 0 , 1 , 1 2 p , 1 3 p , ... } under index- h fractional Brownian motion is θ p h + θ , a value that is smaller than that given by directly applying the Hölder bound for fractional Brownian motion. In particular this establishes the box-counting dimension of these images. [ABSTRACT FROM AUTHOR]

The approximate fractality of the fractional Fourier transform (FRFr) spectrum of sea clutter and its application of the approximate fractal characteristic of the sea clutter FRFT spectrum in target detection are mainly studied. According to the scale transformational property of FRFT, the approximate fractality of the FRFI spectrum of a self-similar process is studied. Then, an analytical method of approximate fractality is introduced to analyze the FRFT spectrum of real sea clutter. It's found that approximate fractality exists among the sea clutter FRFT spectra of some specific transform orders. Moreover, target echoes can cause the descriptive parameter of approximate fractality, or the Hurst exponent in FRFT domain to change. Therefore, the Hurst exponent in FRVr domain can be used to design the CFAR detection algorithm. By verifying measured data of X-band and C-band radar, the proposed method performs much better than the classical detection method using the Hurst exponent in time domain. [ABSTRACT FROM AUTHOR]

Abstract: Mandelbrot is best appreciated for his broad attempt to describe irregular shapes in nature. He founded fractal geometry in 1975. Subsequently the whole fractal theory developed using one-step feedback systems. In 2002, an attempt was made to study and analyze fractal objects using two-step feedback systems. Researchers used superior iteration methods to implement two-step feedback systems. This was the beginning of a new iterative approach in the study of fractal models, and it seems promising to extend fractal theory. The purpose of this paper is to present a review of literature in fractal analysis using this new iterative approach and explore its potential applications. [Copyright &y& Elsevier]

Abstract: This paper builds on the recently begun extension of continuum thermomechanics to fractal media which are specified by a fractional mass scaling law of the resolution length scale R. The focus is on pre-fractal media (i.e., those with lower and upper cut-offs) through a technique based on a dimensional regularization, in which the fractal dimension D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer-order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D and R, as well as a surface fractal dimension d. While the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure capable of describing local material anisotropy. This measure allows one to grasp the anisotropy of fractal dimensions on a mesoscale and the ensuing lack of symmetry of the Cauchy stress. This naturally leads to micropolar continuum mechanics of fractal media. Thereafter, the reciprocity, uniqueness and variational theorems are established. [Copyright &y& Elsevier]

Abstract: We report the appearance of anomalous water diffusion in hydrophilic Sephadex gels observed using pulse field gradient (PFG) nuclear magnetic resonance (NMR). The NMR diffusion data was collected using a Varian 14.1 Tesla imaging system with a home-built RF saddle coil. A fractional order analysis of the data was used to characterize heterogeneity in the gels for the dynamics of water diffusion in this restricted environment. Several recent studies of anomalous diffusion have used the stretched exponential function to model the decay of the NMR signal, i.e., exp[−(bD) α ], where D is the apparent diffusion constant, b is determined the experimental conditions (gradient pulse separation, durations and strength), and α is a measure of structural complexity. In this work, we consider a different case where the spatial Laplacian in the Bloch–Torrey equation is generalized to a fractional order model of diffusivity via a complexity parameter, β, a space constant, μ, and a diffusion coefficient, D. This treatment reverts to the classical result for the integer order case. The fractional order decay model was fit to the diffusion-weighted signal attenuation for a range of b-values (0< b <4000smm−2). Throughout this range of b values, the parameters β, μ and D, were found to correlate with the porosity and tortuosity of the gel structure. [Copyright &y& Elsevier]