Box Dimension Research Articles

Study on the freeze-thaw damage characteristics of skarn based on CT three-dimensional reconstruction

To study the mesoscopic damage evolution characteristics of skarn under freeze-thaw cycles, based on CT technology, the skarn samples under freeze-thaw action were scanned by CT, and the image data of skarn were segmented by Avizo software. The digital model of the three-dimensional structure of skarn was established, and the evolution law of the internal structure of skarn during the freeze-thaw cycle was quantitatively analyzed. The box dimension algorithm calculates the fractal dimension of the pore structure under freeze-thaw conditions. The relationship between fractal dimension, pore volume fraction, and freeze-thaw cycles was studied. According to the statistical results of the pore size distribu-tion of skarn, the change characteristics of pore structure in the rock under the influence of freeze-thaw were studied. Based on the theory of rock damage mechanics, the damage variable of skarn was defined using the concept of the effective bearing zone, and the freeze-thaw damage evolution of skarn was studied. The results show that the three-dimensional reconstructed model can directly show each medium’s meso-pore structure and spatial distribution. There is a positive correlation between pore volume fraction and fractal dimension under freeze-thaw conditions. The fractal dimension satisfies the exponential growth law. The skarn damage variable increases with the increase of freeze-thaw cycles, which is consistent with the development trend of pore structure. The exponential function can better reflect the damage evolution of skarn under freezing and thawing.

BOX DIMENSION OF FRACTAL INTERPOLATION SURFACES WITH VERTICAL SCALING FUNCTION

In this paper, we first present a simple lemma which allows us to estimate the box dimension of graphs of given functions by the associated oscillation sums and oscillation vectors. Then we define vertical scaling matrices of generalized affine fractal interpolation surfaces (FISs). By using these matrices, we establish relationships between oscillation vectors of different levels, which enables us to obtain the box dimension of generalized affine FISs under certain constraints.

RESEARCH ON FRACTAL DIMENSIONS AND THE HÖLDER CONTINUITY OF FRACTAL FUNCTIONS UNDER OPERATIONS

Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given.

CONSTRUCTION OF A WEIGHTED FRACTAL INTERPOLATION SURFACE BASED ON MATKOWSKI CONTRACTIONS

In this paper, we construct a new kind of weighted recursive iteration function system (IFS) and prove the existence of the unique attractor for the kind of IFS based on the Matkowski fixed point theorem. We confirm that the attractor is a bivariate fractal interpolation surface (FIS), which interpolates a given set of data. In addition, we also provide an upper error estimate of such FISs caused by changes of weights. Finally, we give their box dimension estimates for a specific type of the FISs.

Box Dimension Estimationfor a Class of MonotonicallyDecreasing Positive Sequence

Box Dimension Estimationfor a Class of MonotonicallyDecreasing Positive Sequence

A STUDY OF FRACTAL DUAL MOMENTUM INVESTMENT STRATEGY UNDER THE CONSTRAINT OF MULTI-FRACTAL CHARACTERISTICS OF STOCK MARKET

Since the discovery of momentum effect, people have started the journey of using the momentum effect to construct momentum strategies. As a result of coupling cross-sectional and time-series momentum strategy, dual momentum strategy (DM strategy) has been widely used in practice and closely followed by academics. To address the shortcoming of the classical DM strategy that has not considered the multi-fractal characteristics of the stock market, we construct the fractal dual momentum strategy (FDM strategy) from the two aspects of optimizing the ranking index of the cross-sectional momentum strategy by using fractal statistical measures and improving the timing selection of the time-series momentum strategy by using the trend entropy dimension. The empirical results show that the FDM strategy outperforms the DM strategy. Both in terms of the size and stability of the strategy returns, the FDM strategy shows an optimization effect compared with the DM strategy, which is beneficial to provide investors with better decision-making references.

Pullback Exponential Attractors with Explicit Fractal Dimensions for Non-Autonomous Partial Functional Differential Equations

The aim of this paper is to propose a new method to construct pullback exponential attractors with explicit fractal dimensions for non-autonomous infinite-dimensional dynamical systems in Banach spaces. The approach is established by combining the squeezing properties and the covering of finite subspace of Banach spaces, which generalize the method established for autonomous systems in Hilbert spaces (Eden A, Foias C, Nicolaenko B, and Temam R Exponential attractors for dissipative evolution equations, Wiley, New York, 1994). The method is especially effective for non-autonomous partial functional differential equations for which phase space decomposition based on the exponential dichotomy of the linear part or variation techniques are available for proving squeezing property. The theoretical results are illustrated by applications to several specific non-autonomous partial functional differential equations, including a retarded reaction–diffusion equation, a retarded 2D Navier–Stokes equation and a retarded semilinear wave equation. The constructed exponential attractors possess explicit fractal dimensions which do not depend on the entropy number but only on some inner characteristics of the studied equations including the spectra of the linear part and the Lipschitz constants of the nonlinear terms and hence do not require the smooth embedding between two spaces in the previous work.

The Impact of Turkish Economic News on the Fractality of Borsa Istanbul: A Multidisciplinary Approach

This study explores the connection between the fractal dimensions of time series representing sentiments regarding economic news and the fractal dimensions of correlation networks among companies listed in the Borsa Istanbul star section. While there have been many studies on the correlation between different time series, the investigation into the impact of fractal dimensions on correlation networks’ dynamics has been somewhat restricted. This study investigates the correlation networks among companies listed in the Borsa Istanbul Stars segment, employing distance and topological filters. The network fractional dimensions are evaluated using the box counting and information dimension techniques. A convolutional neural network is employed to perform analysis of sentiments regarding on 2020 Turkish economic news. The network is trained on user comments and specifically built to identify fluctuations in news editorials. The Zemberek natural language processing framework is beneficial for data preprocessing. Identical analytical methods are employed to quantify the fractal dimensions of each sentiment time series. Experiments are performed on these measurements using various sliding window widths to ascertain both independence and causality. The findings indicate a substantial correlation between market behavior and the feelings expressed in economic news.

Pencirian Isyarat Data Muzik yang Terhasil dari Alat Muzik Tradisional Menggunakan Ciri Fraktal

Musical instruments are usually distinguished by their sound produced through human perception which may lead to misinterpretation due to auditory perception bias and other disturbances. Therefore, recognition using music signals is carried out to help characterize signals from different musical instruments. Fractal analysis is a mathematical tool used to study complex and irregular patterns in various systems. In this study, fractal analysis was used to study and analyze musical notes signal data from different instruments. The fractal analysis method used is the box counting method. The traditional musical instruments involved in this study are the seruling, cak lempong, kompang, and gambus. Matlab software was used to analyze the musical data signal. First, the fractal dimension of the music data signal in the form of time domain is calculated. Then, the mean and standard deviation of the fractal dimension values were determined to recognize different musical instruments. Additionally, different image resolutions and box sizes are also used to calculate the fractal dimension of musical instrument from the time domain data. The error of best fit line, E will also be calculated to ensure the reliability of box counting dimensions using leastsquares regression method. The results show that the value of the fractal dimension for all 18 data is between 1.2636 to 1.7543. In terms of musical instrument recognition, the recognition of seruling and kompang using mean fractal dimension is successful. However, this method is less suitable for identification for cak lempong and gambus due to the high standard deviation. By using different image resolutions and box sizes, the accuracy of fractal dimension values will be affected. Increasing these parameters increases the accuracy of the results. The results of the study show that an image with a resolution of 1024 x 1024 pixels and a scaling factor of 9 is suitable to analyze the fractal characteristics of musical instrument data signals by using the box counting method.

Numerical study for trihybrid nanomaterial flow by convectively heated curved sheet

Trihybrid nanofluid has vast applications including cooling systems, heat exchangers, electronic chips, automobile radiators, slurries, and biomedicine. Flow of MHD hybrid nanomaterial by convectively heated surface with entropy formulation is explored. The trihybrid nanofluid model is composed of three distinct nano-sized particles along with base material. Here 2 % of ferric oxide (Fe3O4), 2 % of silver (Ag) and 2 % of copper (Cu) nanoparticles are taken with ethylene glycol (C2H6O2) as base fluid. Nonlinear differential system is constructed. Adequate transformation is employed in order to compute the problem numerically by applying the finite difference method. When compared to regular fluid, nanofluid and hybrid fluid, the trihybrid nanofluid exhibits superior thermal properties. It is perceived that adding nanoparticles in traditional fluids improve the ability to transmit heat. The temperature field and velocity are shown visually in relation to a number of emerging parameters. The coefficient of skin friction, Bejan number and rates of entropy and heat transfer have been explored physically. The results revealed that with magnetic effects the velocity field reduces while temperature improves when volume fraction for trihybrid nanoparticles is taken as 2 %. Moreover, the Bejan number is also improved by altering the radiation parameter. Excellent agreement is found when current numerical results for curved surfaces are compared to previous published literature for regular flat surface for large K.

On two special classes of fractal surfaces with certain Hausdorff and Box dimensions

In this paper, using two special types of rise-dimensional operators based on existing fractal functions, we construct new fractal surfaces with any value of the Hausdorff and Box dimension between two and three. Further, we demonstrate that the lower and upper Box dimension of such fractal surfaces may be unequal to each other. This result could be useful to the research on creating various fractal surfaces with the required fractal dimensions in the future.

Computational innovation of in situ metallic elements with zirconia as a novel possible carrier for chemotherapeutic medication.

Electronic sustainable behavior on the material surface and in situ metal configuration were accompted with some metal atoms like Li, Na, and K elements. Metal-doped ZrO2 crystal exported modified characteristics related to electronic conduction and exhibited some dynamic modification around the surface of the metal oxide. Computational perturbations were considered to discuss the modification behavior in addition to the studied Li, Na, and K metals. Optimization of the three doping systems was achieved followed by generating DOS and electronic band structure maps. A dynamic simulation was performed with temperature over 2000 k: the presence of the metal on the surface and prediction of its ZrO2 inclusion leading to access adsorption behavior, besides generating predictive designed models described the adsorption affinity on the solid-state surface. It cannot be neglected the importance of various metals as a main role in chemotherapy. Molecular docking investigation was considered to predict the binding behavior of the studied metal ZrO2 carrier system as an anticancer agent. Also, docking affinity was helpful in comparing the active sites binding for the studied metals, resulting in a notable binding affinity for both Li- and Na-zirconia incorporation. The program PWSCF, which is a component of the quantum ESPRESSO suite for quantum simulation of materials, was used to construct geometric systems. The generalized gradient approximation in the Perdew-Burke-Ernzerhof (GGA/PBE) function with D3 correction (Becke-Jonson damping) was applied to the exchange-correlation energy. As the last step in the DFT postulation and design, adsorption locator annealing was carried out on the convergent models using the Materials Studio simulation package. The main roles played by metal atoms are in protein binding and the suppression of bio-active regions. For the docking process, the protein was produced using AutoDock 4.2 and Discovery Studio software in accordance with the usual methodology. Chimera and Discovery Studio were used to examine the docking data that was processed after generating specific grid box dimensions for 7BTN.

RF fingerprint extraction and device recognition algorithm based on multi-scale fractal features and APWOA-LSSVM

RF fingerprints can be used for device identification and network access authentication. An RF fingerprint extraction and device identification algorithm based on multi-scale fractal features and APWOA-LSSVM is proposed. First, a Hilbert transform is performed on the original RF signal, and differential operations are performed on the I/Q signal. Then, the variational modal decomposition is performed on the differentially processed I/Q signals separately to obtain a number of band-limited Intrinsic Modal Function (IMF) components. The fractal box dimension of each IMF component is calculated separately as the first-dimensional RF fingerprint. The multi-fractal spectrum of the original RF signal is further computed by Multi-fractal Detrended Fluctuation Analysis (MFDFA) as the second-dimensional RF fingerprint. The RF fingerprint feature vector consists of a combination of the first- and second-dimensional RF fingerprints and is used to train the LSSVM model. Since the penalty factor and kernel parameter width directly affect the recognition accuracy and generalization ability of the LSSVM model, APWOA is used to optimize the hyperparameters of the LSSVM model. Finally, the LSSVM model is constructed using the optimal hyperparameters and the performance of the model is verified on a test dataset. The experimental results show that the proposed model achieves an average recognition accuracy of 99.13%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} for 16 Bluetooth devices. Multi-scale fractal features consisting of fractal box dimensions and multiple fractal dimensions are more beneficial in the recognition of wireless devices as compared to single features. In addition, the proposed model possesses superior recognition capabilities when compared to classical machine learning algorithms.

Coupled effects of variable permeability and adiabatic undulating walls on natural convective flow in a trapezoidal cavity: Finite element analysis

Investigation of hydrothermal characteristics and entropy produced by change in diameter of circular cylinder positioned at the center of trapezoidal enclosure with undulated upper wall is conducted. Natural convection phenomenon is considered along with physical aspects of inclining magnetic field and permeable medium. Envisioning all these factors combined no such study is available which makes current problem novel. So, the premier intends to commence this artifact is to fill the existing gap by comprehensively disclosing the influence of governing transport parameters attained after mathematical modeling in dimensionless format. Entropy phenomenon is analyzed by estimating total entropy and average Bejan number. The numerical solution of modeled mathematical framework is attained by utilizing finite element based open-source software renowned as COMSOL. Wide range of flow concerning parameters are considered for instance, diameter of cylinder (0.10≤D1≤0.20), Prandtl number (0.015≤Pr≤7.00), Rayleigh number (103≤Ra≤105), Darcy number (10-5≤Da≤10-3) and Hartmann number (0≤Ha≤10) while, remaining parameters like angle of inclination (γ) and irreversibility ratio (ω) are fixed at 45° and 10-2 respectively. The influence of these parameters on distributions is interpreted by displaying streamlines and isotherms. After scrutinization of outcomes, it is deduced that average Nusselt number, total entropy and average Bejan number enhances up to 3.49 %, 1.86 % and 2.70 % respectively for situation when diameter of cylinder is increased from 0.10 to 0.20. Furthermore, 28.35 %, 43.06 % and 59.55 % decrement in previously stated quantities is revealed in comparative manner for hydrodynamic (Ha =0) and hydromagnetic (Ha ≠0) cases.

Study on the Difference of Fracture Extension in Hydraulic Fracturing of Marine and Continental Shales Based on Optical Scanning Technology

After hydraulic fracturing, the geometric characteristics of rock morphology is a crucial means for evaluating the effectiveness of fracture stimulation in enhancing production. In order to quantitatively analyze the surface undulations of marine and continental shale morphology further following hydraulic fracturing, a method for calculating the three-dimensional fractal dimension of fracture surfaces based on optical scanning technology is proposed. This method involves the acquisition of point cloud data using a 3D surface scanner. The obtained data are subsequently subjected to smoothing processes, followed by the reconstruction of the three-dimensional representation of the fractures. The box-counting dimension algorithm is employed to calculate the fractal characteristics of post-fracture morphology. The research results indicate that marine shale, due to its higher proportion of brittle minerals such as quartz and calcite, predominantly exhibits vertically oriented longitudinal fractures, perpendicular to the minimum horizontal stress. The average initiation pressure is 8% higher compared to continental shale, with an average fractal dimension of 2.24397. In contrast, continental shale, characterized by its high clay content and the development of natural fractures and bedding planes, is more prone to capturing hydraulic fracture features, resulting in predominantly single transverse shear fractures. The average fractal dimension is 2.087065, which is 7% lower than that of marine shale. These research findings offer a certain degree of guidance for the optimization of fracturing process parameters for different types of reservoirs.

Impacts of entropy generation in second-grade fuzzy hybrid nanofluids on exponentially permeable stretching/shrinking surface

The present investigation aims to use entropy analysis to analyze the unsteady Magnetohydrodynamic (MHD) flow in a second-grade fuzzy hybrid Al2O3-Cu/SA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {{\ ext{Al}}_{{2}} {\ ext{O}}_{{3}} - {\ ext{Cu/SA}}} \\right)$$\\end{document} nanofluid over an exponentially shrinking/stretching surface. The model for hybridization of the mixture of alumina Al2O3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( {{\ ext{Al}}_{{2}} {\ ext{O}}_{{3}} } \\right)$$\\end{document} and copper (Cu) nanoparticles in the sodium alginate (SA) base fluid under heat source/sink, nonlinear thermal radiation, and viscous dissipation. The fundamental partial differential equations (PDEs) are simplified using an appropriate similarity conversion to generate the ordinary differential equations (ODEs). The analytical computation occurs in the MATHEMATICA program implementing the homotopy analysis method (HAM). In terms of code validity, our results are preferable to previous findings. The features of several parameters against the velocity, surface friction coefficient, entropy, temperature, and Nusselt number are described through graphs. According to our findings, the rise in the Brinkman and Reynolds numbers enhanced the total entropy of the system. Furthermore, the nanoparticle volume fraction and viscus dissipation magnifies the fluid temperature while retards the flow profile throughout the domain. Fluid velocity declined due to the Lorentz force using magnetic impact applications. The imprecision of nanofluid and hybrid nanofluid volume fractions was modelled as a triangular fuzzy number (TFN) [0%, 1%, 2%] for comparison. The double parametric approach was applied to deal with the fuzziness of the associated fuzzy parameters. The nonlinear ODEs convert into fuzzy differential equations (FDEs) and use HAM for the fuzzy solution. From our observation, the hybrid nanofluid displays the maximum heat transfer compared to nanofluids. This important contribution will support industrial growth, particularly in the processing and manufacturing sectors. The percentage increase in skin friction factor is 18.3 and 15.0 when α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} take input in the ranges of 0 ≤ α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} ≤ 0.8 and 0 ≤ β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document} ≤ 1, respectively.

Generalized approximation and estimation of entropy numbers

Generalized approximation and estimation of entropy numbers

FRACTAL DIMENSIONS OF THE LOGARITHM OF CONTINUOUS FUNCTIONS

This paper investigates the variation of fractal dimensions of continuous functions under operations. It is found that the Box dimension of the logarithm of positive continuous functions remains constant, and any nonzero real power of a positive continuous function can keep fractal dimensions variation closed. The study also discusses the fractal dimension of logarithmic function with bases of fractal functions, and the corresponding fractal dimension is determined by the function with the larger fractal dimension.

The Relationship between the Box Dimension of Continuous Functions and Their (k,s)-Riemann–Liouville Fractional Integral

This article is a study on the (k,s)-Riemann–Liouville fractional integral, a generalization of the Riemann–Liouville fractional integral. Firstly, we introduce several properties of the extended integral of continuous functions. Furthermore, we make the estimation of the Box dimension of the graph of continuous functions after the extended integral. It is shown that the upper Box dimension of the (k,s)-Riemann–Liouville fractional integral for any continuous functions is no more than the upper Box dimension of the functions on the unit interval I=[0,1], which indicates that the upper Box dimension of the integrand f(x) will not be increased by the σ-order (k,s)-Riemann–Liouville fractional integral ksD−σf(x) where σ>0 on I. Additionally, we prove that the fractal dimension of ksD−σf(x) of one-dimensional continuous functions f(x) is still one.

Influencer identification of dynamical networks based on an information entropy dimension reduction method

Identifying critical nodes or sets in large-scale networks is a fundamental scientific problem and one of the key research directions in the fields of data mining and network science when implementing network attacks, defense, repair and control. Traditional methods usually begin from the centrality, node location or the impact on the largest connected component after node destruction, mainly based on the network structure. However, these algorithms do not consider network state changes. We applied a model that combines a random connectivity matrix and minimal low-dimensional structures to represent network connectivity. By using mean field theory and information entropy to calculate node activity, we calculated the overlap between the random parts and fixed low-dimensional parts to quantify the influence of node impact on network state changes and ranked them by importance. We applied this algorithm and the proposed importance algorithm to the overall analysis and stratified analysis of the C. elegans neural network. We observed a change in the critical entropy of the network state and by utilizing the proposed method we can calculate the nodes that indirectly affect muscle cells through neural layers.