Fractals

Density distributions of populations with self-diffusion and interaction in a spatial domain are dynamically visualized with coupled nonlinear reaction–diffusion equations. Incorporating self-diffusion terms creates a more pragmatic modeling paradigm and provides meaningful descriptions of influences on spatiotemporal pattern formation phenomena. This paper examines the effect of self-diffusion in a food chain system with a Holling type-IV functional response and the type of spatial structures forms on a geographical scale due to the random movement of species. We discussed the existence and uniqueness of a positive equilibrium solution and obtained the Turing instability conditions for the self-diffusive food chain model. Moreover, weakly nonlinear analysis close to the Turing bifurcation boundary is used to derive the amplitude equations. The stability of the amplitude equations and sufficient conditions for the emanation of spatiotemporal patterns (such as spots, stripes, and blended patterns) are investigated. The analytical results are verified with numerical simulations. The results are applicable to all environments and can be used to understand the effects of self-diffusion in other food chain models both qualitatively and quantitatively.

In this paper, we study the generalized Cantor-integers ${\{C_n\}}_{n\ge 1}$ with the base conversion function $f:\{0,\dots ,m\}\to [0,p]$ being strictly increasing and satisfying $f(0)=0$ and $f(m)=p$. We show that the sequence ${\{\frac{C_n}{n^{\alpha }}\}}_{n\ge 1}$ with $\alpha ={log}_{m+1}(p+1)$ is dense in the closed interval with the endpoints being its inferior and superior, respectively. Moreover, every homogeneous Cantor set $ℭ$ satisfying open set condition can be induced by some generalized Cantor-integers, we get the exact point which attains the maximal interval density of the form $[0,x]$ with respect to the self-similar probability measure supported on $ℭ$. This result partially confirms a conjecture of E. Ayer and R. S. Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets, Trans. Am. Math. Soc. 351(9) (1999) 3725–3741].

In this paper, we study Vicsek polygons and extended Vicsek networks, which are an extension of Vicsek fractal. Our research indices are some Fermat-type indices, including the Fermat eccentricity and the Fermat eccentric distance sum. Fermat-type indices are novel graph invariants with vast potential in research on structure–activity and quantitative structure–property. By the approach of finite pattern, we solve some integrals to gain their asymptotic formulas on Fermat eccentricity and Fermat eccentric distance sum.

The aim of this paper is to illustrate a strategy for how to get the exact formula of the distance sums of fractal networks (for example, the Sierpiń ski skeleton networks) with the technique of finite pattern [S. Wang, Z. Yu and L. Xi, Average geodesic distance of Sierpinski gasket and Sierpinski networks, Fractals 25(5) (2017) 1750044].

All known planets orbit the Sun so harmoniously that the entire Universe is believed to behave in a rhythmical and fractal way. Titius–Bode law is the main theoretical tool for planetary orbit description. Here we show that planetary sequence follows the Fibonacci-like prime sequence, and we introduce two gravitational spirals to predict orbits for solar members orbiting the Sun. Furthermore, we predict two hypothetical planets as far as 1500 AU and 2300 AU from the Sun, respectively, in our solar system, and some unknown planets in extrasolar planetary systems. Our planetary sequence demonstrates the possible frontier of our solar system, and the theory is also valid for other star systems and planetary systems, giving a simple way to finding the most possible adjacent unknown planets or moons near a known one. We anticipate our sequence can be more sophisticatedly used to search for unknown Universes and to depict the cosmos fully and exclusively, and we conclude that our whole Universe is of fractal property and all celestial bodies are distributed according to Fibonacci sequence.

Predicting the electrical conductivity of porous media is important for oil reservoirs, rock physics, and fuel cells. In this work, a cuboid embedded with a damaged tree-like network is employed to denote a portion of porous media. Analytical expressions for the electrical conductivity are then derived with the fractal theory. Various structural parameters have been examined in detail for the influence of electrical conductivity. It is found that the increased number of damaged channels means more difficult ion migration and lower electrical conductivity of porous media. In addition, a decreasing length ratio or an increasing diameter ratio will increase electrical conductivity. Moreover, both the channel distribution fractal dimension and tortuosity fractal dimension result in a decrease in electrical conductivity. A comparison of the results predicted by other models yields a good agreement, validating our proposed model. These results may further interpret the transport properties of porous media.

Based on the $q$-shifting operator, we introduce the concept of Riemann–Liouville fractional quantum integration for interval-valued functions (IVFs), and establish new Riemann–Liouville fractional $q$-Hermite–Hadamard ($q$-HH) and $q$-HH-Fejér inequalities for left and right $h$-convex IVFs (LR-$\hslash$-convex-IVFs). The findings obtained generalize known results in the literature and serve as a foundation for future studies in inequalities for interval-valued functions and fractional quantum calculus. The results are illustrated with examples.

This paper disrupts mistaken applications of multifractal approaches in financial time series. Specifically, we have examined the nonlinear cross-correlation between the São Paulo time series of the weekly price of ethanol and the other 14 Brazilian capitals’ time series of the weekly price of the same biofuel using the Multifractal Detrended Cross-Correlation Analysis (MF-DCCA). Given the statistical peculiars of stationary and non-stationary financial time series, we suggest two possibilities for employing multifractal approaches to these time series. Our findings shed light and promote alignment between basic time series analysis techniques and multifractal dynamics. Also, we discover that the use of MF-DCCA is highly impacted by choice of time series (stationary or non-stationary).

Our study focuses on the analysis of asymptotic stability and the development of asymptotic stabilization control methods for fractional vehicle suspension systems (FVSS). We begin by constructing a mathematical model for FVSS using the state-space equations of Caputo fractional calculus. Initially, we utilize the fractional Routh–Hurwitz criterion to derive the necessary conditions for asymptotic stability and instability in the open-loop system of the fractional vehicle suspension. Subsequently, we propose a novel control strategy for FVSS and establish an associated asymptotic stabilization criterion by combining a new vector Lyapunov function with the $M$-matrix method. Moreover, we extend the fractional-order vehicle suspension model to include time delay resulting from the interactions between different variables in the real system, thus creating a FVSS with time delay. Based on the vector Lyapunov function, $M$-matrix measure, and Razumikhin interpretation, we develop a control strategy specifically tailored for FVSSs with time delay. Lastly, we compare two numerical simulations of the FVSS, one with time delay and one without, to demonstrate the accuracy, effectiveness, and applicability of the proposed method presented in our paper.

We show that the centered Hausdorff measure, $C^s(S),$ with $s=\frac{log3}{log2},$ of the Sierpiński gasket $S$, is $C$-computable (continuous-computable), in the sense that its value is the solution of the minimization problem of a continuous function on a compact domain. We also show that $C^s(S)$ is $A$-computable (algorithmic-computable) in the sense that there is an algorithm that converges to $C^s(S),$ with explicit error bounds. Using this algorithm we show that $C^s(S)\sim 1.0049$.

I construct perforated, “take-away” fractals that support short-distance power-law scaling with complex exponents for Dirac (spin-1/2) propagators. The construction relies on a fortuitous ansatz for Dirac boundary conditions at the surfaces of spherical voids in a three-dimensional embedding space, and requires that boundary conditions vary from void to void and are distributed statistically. The appropriate distribution can ensure that nonzero charges (but not dipoles, which drive nontrivial power-law scaling) induced in voids cancel out in spatial averaging.

In this paper, we use the properties of Atangana–Baleanu (AB) fractional calculus and Prabhakar fractional calculus to construct some novel Hermite–Hadamard-type fractional integral inequalities for harmonically convex functions. And these inequalities are represented by the Mittag-Leffler functions. Finally, several special inequalities are established to illustrate the applications of our conclusions in special means.

The Calogero–Bogoyavlenskii–Schiff equation is an important nonlinear evolution model to describe the propagation of Riemann waves. A fractional Calogero–Bogoyavlenskii–Schiff is described based on the conformable derivative for the first time. Some new soliton solutions are acquired with the aid of the extended fractional $csc{h}_{\alpha }$ function method and fractional variable method. The two novel mathematical methods are very efficient and concise, which can also be utilized to solve other fractional evolution equations. Furthermore, these derived soliton solutions are illustrated by some 3D and 2D graphs with different fractal parameters and fractal dimensions, which might be helpful to study in plasma physics.

This study proposes a new fractal modified equal width-Burgers equation (MEWBE) with the local fractional derivative (LFD) for the first time. By defining the Mittag-Leffler function (MLF) on the Cantor set (CS), two special functions, namely, the $T{H}_{\upsilon }({\mu }^{\upsilon })$ and $C{H}_{\upsilon }({\mu }^{\upsilon })$ functions, are derived for constructing the auxiliary function to seek the non-differentiable (ND) exact solutions. And 16 groups of the ND exact solutions are successfully established. The solutions on the CS are depicted graphically to interpret the nonlinear dynamic behaviors. Furthermore, the comparative results of the fractal MEWBE and the classical MEWBE are also discussed. The obtained results confirm that the proposed method is effective and powerful, and can provide a promising way to find the ND exact solutions of the local fractional PDEs.

Concrete is an important part of submarine building structure, which is eroded by hydrostatic pressure and chloride ion. In this study, concrete is regarded as a dually-porous structure composed of pores and fractures. Considering hydrostatic pressure and fractal characteristics of concrete structure, a fractal model is established for chloride ion diffusion under the dual drive of concentration and pressure. Considering the effective stress caused by water pressure in pores of concrete, a fluid–solid model for chloride ion migration is proposed. The fluid–solid coupling fractal model for chloride ion migration was verified by comparing with the experimental data. The numerical simulation results show that: (1) the influence of fractal dimension on the chloride ion diffusion coefficient and chloride ion concentration is nonlinear; (2) compared with low hydrostatic pressure, high hydrostatic pressure has more significant effects on chloride ion concentration and diffusion coefficient; (3) when the effective stress and water pressure in pores are not considered, the diffusion coefficient of chloride ion is always a constant.

In this paper, we explore whether the activity of SARS-CoV-2 was associated with seasonality. MF-DFA model is utilized to calculate multifractal strength and multifractal complexity to evaluate the change state of SARS-CoV-2 activity. We select 10 countries with serious epidemic in the world, which are distributed in different latitudes of the northern and southern hemispheres. The study utilized the time series data of daily new cases and daily new deaths recorded in these countries. We regard May to October as the “high temperature season” for countries in the northern hemisphere, November to April as the “low temperature season”, and the southern hemisphere is just the opposite. By comparing the multifractal intensity $\mathrm{\Delta }H$ and multifractal complexity $\mathrm{\Delta }\alpha$ of the two time series in the two seasons, we draw a conclusion that, for both the sequence of the daily newly diagnosed persons and the daily newly increased number of deaths, in the countries of both the northern and southern hemispheres, $\mathrm{\Delta }H$ and $\mathrm{\Delta }\alpha$ are weaker in the “low temperature season”. That is, in the low temperature environment, SARS-CoV-2 can survive for a long time and be more infectious. In addition, we also observe that in the northern hemisphere, Iran is at a lower latitude, and although the SARS-CoV-2 activity in the low temperature season is higher than that in the high temperature season, the difference is not significant. Therefore, the lower latitude may resist this phenomenon. However, most of the countries in the southern hemisphere are within 30° of south latitude, with low latitude, and other meteorological characteristics such as humidity in the countries in the southern hemisphere are also relatively unique. Although SARS-CoV-2 is characterized by high activity in low temperature seasons, no direct evidence related to the characteristics of latitude distribution has been found.

The Gaussian function was initially adopted to model the tunneling-induced ground surface settlement, but it could not describe the porosity’s effect on the settlement. To solve the problem, Yu’s fractal theory is implemented to model the porous grand’s geometric property, and the surface settlement profile is modeled by a fractal solitary wave, furthermore, the effects of the maximal surface settlement, the porosity and Yu’s fractal dimension on the settlement’s profile are discussed. The new model offers a new view to predict the morphology of the surface settlement.

In this paper, we investigate the Zagreb eccentricity index of the level-$n$ Sierpinski gasket $K_n$. Based on the self-similarity and finite pattern, we compute the Zagreb eccentricity index of $K_n,$ which is the integral of square of eccentricity in terms of the self-similar singular measure.

Memory is an imperfect record of past experiences that enables us to operate in the present and think about the future. Although various factors may give a chance to a false recollection of information that may not occur. These false memories are formed based on various neuro-cognitive processes the underlying mechanism still needs to be well understood. Considering the extended searching when no memory trace is found, we hypothesized that the self-similarities in the brain activations must be higher during false memory recalls. Therefore, a language-free task based on autobiographical brand images was designed using the Deese–Roediger–McDermott (DRM) paradigm. The task was then tested on 24 healthy participants while the brain activities during the test were recorded using a 32-channel EEG system. Subsequently, the self-similarities in the brain activity pattern were estimated by taking the fractal dimension (FD) of the cleaned EEG data. Statistical analysis showed a significant increase in complexity during false memory recalls as compared to true memory recalls prominent in the frontal regions. Interestingly, the EEG findings were consistent in both genders and significantly correlated with subjects’ accuracy rates and reaction times (RTs) to recall.

The Fermat problem is a crucial topological issue corresponding to fractal networks. In this paper, we discuss the average Fermat distance (AFD) of the Vicsek polygon network and analyze structural properties. We construct the Vicsek polygon network based on Vicsek fractal in an iterative way. Given the structure of network, we present an elaborate analysis of the Fermat point under various situations. The special network structure allows a way to calculate the AFD based on average geodesic distance (AGD). Moreover, we introduce the Vicsek polygon fractal and calculate its AGD and AFD. Its relationship with the network enables us to deduce the above two indices of the network directly. The results show that both in network and fractal, the ratio of AFD and AGD tends to 3/2, which demonstrates that both of them can serve as indicators of small-world property of complex networks. In fact, in Vicsek polygon network, the AFD grows linearly with network order, implying that our evolving network does not possess the small-world property.

In this paper, we have explored the local structure and fractal characteristics of fractal functions with certain fractal dimensions. The conclusion that points with inconsistent oscillation amplitudes with the upper Box dimension of the corresponding fractal functions have been proved to be nowhere dense. This will play an important supporting role in exploring the fractal dimension estimation of the combination of fractal functions.

In this work, the coupled fractional nonlinear Hirota equation is defined by using a powerful fractional derivative sense, which is M-truncate derivative. We explore the fractional functional method and fractional simple equation method to investigate the structure of the solutions of the coupled fractional nonlinear Hirota equations, and some new periodic solutions and solitary wave solutions are successfully acquired. The two proposed approaches are simple, effective and direct. Moreover, some 3D and 2D graphs are sketched to elaborate the behavior of these solutions. These obtained solitary wave and periodic solutions are helpful to improve the understanding of the physical behavior of the corresponding mathematical model.

The fractal calculus has gained more widespread attention in the last years. The fractal variational principle plays a major role in the fractal travelling wave theory of the fractal PDEs. This paper develops the fractal generalized variational principle (GVP) of the fractal Gardner equation by virtue of the Semi-inverse method (SIM) for the first time. On the other hand, we also discuss and verify the fractal GVP via the fractal two-scale transform (FTST) from another dimension field. The extracted fractal GVP shows the conservation laws through the energy form in the fractal space, and can be manipulated to explore the fractal solitary wave properties.

We explore the synergic interplay between entropy (disorder), predictability, and informational efficiency of the daily closing price time series of 13 sectoral economics components of the Shanghai index letter considering three non-overlapping periods (before and during COVID-19 and Russia–Ukraine war). Our findings reveal that the telecom services, financials, and consumer discretionary sectors are marked by higher informational efficiency. Otherwise, the industrials, utilities, and transportation sectors exhibit lower informational efficiency. These insights are relevant for financial agents to make informed decisions, manage risk, and seek opportunities in an ever-changing market environment.

The dynamic pull-in instability of a microstructure is a vast research field and its analysis is of great significance for ensuring the effective operation and reliability of micro-electromechanical systems (MEMS). A fractal modification for the traditional MEMS system is suggested to be closer to the real state as a practical application in the air with impurities or humidity. In this paper, we establish a fractal model for a class of electrostatically driven microstructure resonant sensors and find the phenomenon of pull-in instability caused by DC bias voltage and AC excitation voltage. The variational iteration method has been extended to obtain approximate analytical solutions and the pull-in threshold value for the fractal MEMS system. The result obtained from this method shows good agreement with the numerical solution. The simple and efficient operability is demonstrated through theoretical analysis and results comparisons.

The C/SiC composite is a promising material for ablation-resistant thermal protection in near-space hypersonic environments. The formation of an SiO₂ oxide layer through passive oxidation on the surface of the composite is a significant factor influencing its performance. It is essential to accurately predict the thickness of the SiO₂ oxide layer and the recession and mass loss of the C/SiC composite during passive oxidation. The SiO₂ oxide layer is a typical porous media exhibiting self-similarity and thus fractal theory can be applied to establish the relation between the oxygen flow rate and microstructural parameters of the oxide layer. The Weierstrass–Mandelbrot (WM) function is employed to simulate the rough interfaces between the SiO₂ oxide layer and the C/SiC composite to evaluate the influence of the fractal dimensions of the oxide layer on the performance of thermal protection of the C/SiC composite. The results show that the C/SiC composite exhibits improved thermal protection performance when accompanied by a lower tortuosity fractal dimension and a higher pore area fractal dimension of the oxide layer. Conversely, the composite demonstrates enhanced ablation resistance with a higher tortuosity fractal dimension and a lower pore area fractal dimension of the oxide layer. The predictions of the calculation model show good agreement with the experimental data and demonstrate the critical influence of microstructural parameters of the oxide layer on passive oxidation of the composite, providing practical implications for designing materials with desired thermal protection or ablation resistance properties.

Analysis of brain activity for patients with brain disorders is an important research area. Mild cognitive impairment (MCI) is a condition in which patients have more memory or thinking problems compared to healthy people of the same age. In this work, we studied the alterations in brain activity among control subjects and patients with MCI. Three complexity techniques, namely sample entropy, approximate entropy, and fractal dimension, were employed to study electroencephalogram (EEG) signals recorded from 102 control (healthy) subjects, and seven subjects with MCI in a comfortable position, on a bed, with their eyes closed. The results showed that the EEG signals of patients with MCI show greater complexity than the EEG signals of healthy subjects. This analysis method can be applied to compare brain activity among healthy subjects and patients with other brain diseases.

In this work, we have adopted an advanced Julia-based function that helps not only in sorting out hidden wings in the fractals of chaotic systems, but can also generate an extra wing in chaotic systems based on a single wing. For verification, two examples Lorenz and modified stretch-twist-fold (STF) systems based on more than one wing, whereas chemical reaction-based chaotic system with a unique wing is considered. The existence of another wing in chaotic systems based on a single wing was a big question mark on the creation of multi-wings in the theory of fractals, but with the aid of advanced Julia functions, we have elaborated in detail that the existence of a second wing in such systems is also possible. The stretching and squeezing in the trajectories of fractals are also integral parts of our findings. Moreover, our study has solved another problem related to fractals. In the past, authors have shown multi-wings in chaotic systems with empty space inside all the time. In this study, we have shown for the first time that these inner spaces have special meaning due to the existence of hidden wings. Furthermore, we have shown that fractals can be divided into outer and inner wings, where the inner wings reflect the answer to the question about the empty space in between the outer wings of chaotic systems. For convenience, an extra file named as MiddleSpace.pdf is attached as a supplementary file to better understand the concept of covering empty space inside fractals.

In this paper, we first propose two Chebyshev-type inequalities associated with the general fractional-order (Yang–Abdel–Aty–Cattani) integrals with the Rabotnov fractional-exponential kernel under the condition that $\mu$ and $\nu$ are synchronous functions. What is more, by the mathematical induction, we prove a new Chebyshev-type inequality in the case that ${({\mu }_i)}_{i=1,\dots ,n}$ be $n$ positive increasing functions. Finally, we introduce a novel Chebyshev-type inequality via the general fractional-order integrals with the Rabotnov fractional-exponential kernel under the condition that $\mu$ and $\nu$ are monotonic functions.

We obtain the Hausdorff dimension of the skeleton networks of Sierpiński carpet by the self-similarity and induction, which will not touch networks.

In this paper, we introduce $(s,m)$-convex function, and obtain a new identity by the method called integrating by parts. Based on the identity, many Ostrowski type inequalities are presented through the Hölder’s inequality and the well-known power-mean inequality. Under certain conditions, the results we obtained can be transformed into the classical results. Of course, at the end of the paper, some examples are given to support the main results.

Aiming to solve the problem of blind image inpainting, this study proposed a blind image inpainting model integrated with rational fractal interpolation information. First, wavelet decomposition and closed operations were adopted to obtain masks and transform blind inpainting into non-blind inpainting. Then, on the basis of similar structural groups, rational fractal interpolation functions were introduced to complete the restoration. On the one hand, this model can sufficiently express the texture features of the image with high fidelity. On the other hand, it can better represent the structural features of the image, avoid serrated edges, enhance the restoration effect, and approximate the original image. The experimental results show that the restoration effect of this model can reserve texture details and ensure edges without distortion, possessing great practical application value.

Clarifying the control mechanism of structures in water injection coal on dynamic water transport is fundamental for improving the effect of pre-wetting and dust reduction. Using nuclear magnetic resonance experimental system, we simulated realistic water injection coal seam conditions. We obtained the real mechanical environment of water injection coal seam, the $T_2$ spectrum curves of the pore structure at different scales, realized a fine measurement of liquid permeability, proposed a method for calculating the fractal dimension of the specific surface area which combined a tortuosity fractal dimension algorithm, delineated micro-holes and medium–large holes, clarified the quantitative relationships between the structural porosity, the specific surface area, tortuosity, and permeability, and investigated the control mechanism of structural characteristics on the dynamic water transport process. The results showed that, under different mechanical environments, the fractal dimension of the specific surface area of micro–small pores was close to 2, while that of medium–large pores was close to 3. The former fractal dimension increased more obviously, indicating that medium–large pores are more easily compressed than micro–small ones. Under such conditions, the complexity of pore size distribution increased. Additionally, the fractal dimension increase of tortuosity was higher for medium–large pores than for micro–small pores, indicating that the hydraulic reconstruction process was more obvious at a large scale. According to the quantitative relationship between structural porosity, the specific surface area, tortuosity, and permeability, medium–large pores control most of the dynamic water migration in the coal bodies of Daliuta Coal Mine, while the whole pore size range has an important influence on dynamic water migration in the coal bodies of Luling Coal Mine. Our findings provide a solid scientific basis for the development of hydraulic permeability technologies for difficult-to-penetrate coal seams.

The microspatial structure of porous media affects the electrical properties of reservoir rocks significantly. In this work, a dual-porosity model is established to investigate the electrical properties of porous media, in which tree-like networks and capillary channels represent fractures and pores. By using fractal theory, we established an analytical equation for the conductivity of water-saturated dual-porosity media. The analytical equation, devoid of any empirical constants, expresses the electrical properties of the porous media as a function of some structural parameters ($D_g$, $D_f$, $D_{\tau }$, $\alpha$, $\beta$, $m$, $N$, $𝜃$, $\partial$, $\mathrm{\Gamma }$, $l_0$, $d_0)$. We also examine the impact of various matrix structural parameters on conductivity. It is found that increasing the length of mother channel ($l_0)$, length ratio ($\alpha )$, the number of branching layers ($m)$, and tortuosity fractal dimension ($D_{\tau })$ leads to a decrease in conductivity, whereas increasing the diameter of mother channel ($d_0)$, diameter ratio ($\beta )$, the cross-sectional porosity (${\phi }_g$, ${\phi }_f)$, and the channel bifurcation number ($N)$ enhances conductivity. Furthermore, we validated this analytical model by comparing it with the experimental data available, and the results demonstrate good agreement. This research has proposed an advanced conductivity model that enables us to better understand the underlying physical mechanisms of the electrical properties in porous media.

Complex time series appear commonly in a large diversity of the science, engineering, economy, financial and social fields. In many instances, complex time series exhibit scaling behavior over a wide range of scales. The traditional rescaled-range (R/S) analysis and the detrended fluctuation analysis (DFA) are commonly used to characterize the scaling behavior via the Hurst exponent. These methods perform well for long-time series. However, the performance may be poor for short times resulting from scarce measurements (e.g. less than a hundred). This work proposes an approach based on singular value decomposition (SVD) entropy for estimating the Hurst exponent for short-time series. In the first step, synthetic time series were used to find the relationship between Hurst exponent and SVD entropy. In the second step, an empirical relationship was proposed to estimate the Hurst exponent from SVD entropy computations of the time series. The performance of the approach was illustrated with two examples of real-time series (consumer price index (CPI) and El Niño Oceanic Index), showing that the estimated Hurst exponent provides valuable insights into the physical mechanisms involved in the generation of the time series.

As a multi-scale system featuring fractal hierarchical branching structure, the quantitative characterization of natural river networks is of fundamental significance for the assessment of the hydrological and ecological issues. However, as already evidenced, the fractal behavior cannot be uniquely inverted by fractal dimension, which induces a challenge in accurately describing the arbitrary scale-invariance properties in natural river networks. In this work, as per fractal topography theory, an open mathematical framework for the description of arbitrary fractal river networks is proposed by clarifying the assembly mechanisms of complexity types (i.e. the original and behavioral complexities) in a river network. On this basis, a general algorithm for the characterization of complexities is developed, and the effects of the original and behavioral complexities on the structure of a river network are systematically explored. The results indicate that the original complexity determines the tortuosity and spatial coverage of a river network, and the behavioral complexity dominates the river patterns, heterogeneity, and scale-invariance properties. Our investigation lays a foundation for assessing and predicting accurately the effect on environments, ecology and humans from river networks.

The fractal $(m,h)$-preinvex mappings are put forward and their properties are investigated firstly. Meanwhile, some fractal Hermite–Hadamard-type ($\mathrm{\text{HH}}$) and Fejér–Hermite–Hadamard-type ($\mathrm{\text{FHH}}$) inequalities concerning $(m,h)$-preinvexity are popularized. Then, two weighted parameterized $2\tilde{\alpha }$-fractal identities are proposed, which involve twice the local fractional differentiable mappings. Based upon these identities and taking advantage of the fractal $(m,h)$-preinvex mappings as well as $(\phi ,{\rho }^{\tilde{\alpha }})$-Lipschitzian mappings, a range of error estimations are deduced in the fractal domains. Finally, certain fractal inequalities with relation to the weighted formula and random variable are correspondingly presented as applications.

The exploration of nonlinear phenomena entails the representation of intricate systems with space-time variables, and across this line, Itô calculus, as the stochastic calculus version of the change pertaining to the variables formula and chain rules, involves the second derivative of f, coming from the property that Brownian motion has non-zero quadratic variation. To this end, the extended ($2+1$)-dimensional two-component Itô equation, as an applicable mathematical tool employed in this study for enhancing our understanding of the complex dynamics inherent in multidimensional physical systems, serves the purpose of modeling and understanding dynamic phenomena pervading various disciplines. For modeling complex phenomena, fractional differential equations (FDEs), ordinary differential equations (ODEs), partial differential equations (PDEs) as well as the other ones provide benefits in terms of accuracy and tractability. Accordingly, our study provides the analysis of the two-component nonlinear extended ($2+1$)-dimensional Itô equation using the Hirota bilinear method to derive multiple soliton solutions, including novel variations along with their dispersion coefficients, which shed light into the intriguing attributes of the Itô equation. The investigation further encompasses diverse soliton types, such as the general first-order soliton, second-order soliton with fission and bifurcation, third-order soliton, and fourth-order soliton with fission and bifurcation. Besides these, the study also explores the rogue wave and lump solutions by varying parameters across distinct planes. Consequently, these results validate the characteristics and utility of the two-component nonlinear extended ($2+1$)-dimensional Itô equation and its relevance to related systems. The novel findings based on the Itô calculus systems revealed, through the analyses, theoretical and experimental aspects in combination with the graphical presentation of the parameter effects on solitons in line with the analyses obtained. These have enhanced the understanding of the dynamics of intricate attributes governed by the two-component nonlinear extended ($2+1$)-dimensional Itô equation, particularly concerning chaotic patterns, fission and bifurcation soliton nonlinear complexities.

Analysis of the changes in brain activity between rest and various conditions is an important area of research. We investigated the changes in brain activity among rest and multitask workload (SIMKAP multitasking test) by quantifying the complexity of EEG signals. The results showed that EEG signals have smaller fractal dimensions, sample entropy, and approximate entropy during the SIMKAP multitasking test than the rest. Therefore, the complexity of EEG signals was lower during the SIMKAP multitasking test than the rest. In further research, we can study the changes in brain activity among other conditions, which has great benefits in decoding brain activity in various conditions.