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Traditional methods do some very important work on complete sets in normed spaces. But to this day, complete sets are still not well understood. With the development of big data, multi-scale theory has been introduced into the field of data mining. For the above problems, we have combined the general multi-scale data mining theory with normed space. Firstly, based on the concept hierarchy theory, the definition of teaching data scale division and data scale, and the relationship between the upper and lower scale datasets between multi-scale datasets are given. Secondly, the methods for classification of multi-scale data mining are clarified. Finally, we propose the Multi-scale Data Mining Algorithm (MSDMA), its theoretical basis is given, and this framework is applied to association rule mining, and a multi-scale association rule mining algorithm is proposed to realize the cross-knowledge between multi-scale datasets. We experiment and analyze the MSDMA algorithm using the IBMTl014DlooK dataset and the real dataset of the entire population of H province. Experimental results show that our algorithm has high coverage, accuracy and low support estimation error.

The materials with a perovskite phase have been in the limelight due to their power conversion efficiency (PSC) in solar cells. New perovskite materials are essential to predict the abundant availability of efficient materials for technological applications. Mechanical properties can predict the mechanical stability of crystals. Therefore, it is very important to know their mechanical parameters. So, in this work, the elastic constants $C_{11}$, $C_{12}$ and $C_{44}$ of the cubic chloride perovskites (ABCl${}_3)$ have been determined through first principles study using density functional theory by using the Chirpan method integrated with WIEN2k. After calculating the elastic constants, we have also calculated different moduli like Shear, Bulk and Young moduli, different parameters like Kleinman’s constant, Lame constants, Chung–Buessem anisotropy index, universal anisotropic index, acoustic behavior and its anisotropy, hardness, melting temperature, Poisson ratios by using different formulas in connection with the elastic constants. It has been found that the studied compounds possess low resistance to the plastic deformations. It has also been found that the majority of the materials possess a central type of force because Poisson’s ratio is greater than 0.25. It has been studied that six out of eighteen new perovskites were brittle and the rest were ductile. The anisotropy of the materials was checked and found that all the materials are anisotropic elastically. This work is useful for the synthesis of these new perovskites.

The stability in stock markets is the theme of this work. It has been demonstrated that the random walk theory alone is insufficient to explain the dynamic behavior of asset or stock prices. Deviations from the random walk theory reveal collective behaviors that produce waves and patterns. Research has shown that the Fibonacci sequence and the golden ratio emerge in such dynamic systems, representing states of minimal stability. Their sustained stability is ensured by the presence of Landau damping within the system. The complex dynamics of the stock market can be analyzed by considering the exchanged shares as fluctuating and the unexchanged shares as nonfluctuating entities. The traders exhibit a kind of group oscillations that resemble the waves in physical plasma. At steady state, the waves can be expressed by a cosine term, and at the least stable state, the dynamics involves the golden ratio, such that the cosine of $36^{\circ }$ is equal to half of the golden ratio. Using the trigonometric cosine formula, it is possible to obtain other angles which are the multiples of $9^{\circ }$. They can be expressed in terms of the golden ratio, and they stand as Fibonacci angles. The stabilization is achieved by a mechanism so-called Landau damping, and the waves thus created are called Elliott waves, and they keep the system near the instability border. It was found that these angles appear quite often in the motive and corrective Elliott waves in the weekly price change of crude oil between January 2001 and June 2023. The percent occurrence of these angles increases through oscillations in motive waves with peaks at 18°, $45^{\circ }$ and 72°. The corrective wave has a maximum peak at 36°, and the percent occurrence decreases through oscillations having smaller peak values at 54° and 72°. The highest values are such that the motive waves appear at 50% in the bull market, and the corrective waves at 27.8% in the bear market.

This paper focuses on developing and analyzing a new AR-iterative scheme designed for estimating fixed points of generalized non-expansive mappings of the Suzuki kind. The scheme’s stability and convergence properties are rigorously proven, highlighting its superior convergence rate for non-expansive mappings when compared to existing schemes in the literature. Additionally, an application of the AR-iterative scheme to differential equations is presented. These findings underscore the effectiveness and versatility of the proposed scheme in numerical analysis and computational mathematics.

This paper discusses a fractional-order prey–predator system with Gompertz growth of prey population in terms of the Caputo fractional derivative. The non-negativity and boundedness of the solutions of the considered model are successfully analyzed. We utilize the Mittag-Leffler function and the Laplace transform to prove the boundedness of the solutions of this model. We describe the topological categories of the fixed points of the model. It is theoretically demonstrated that under certain parametric conditions, the fractional-order prey–predator model can undergo both Neimark–Sacker and period-doubling bifurcations. The piecewise constant argument approach is invoked to discretize the considered model. We also formulate some necessary conditions under which the stability of the fixed points occurs. We find that there are two fixed points for the considered model which are semi-trivial and coexistence fixed points. These points are stable under some specific constraints. Using the bifurcation theory, we establish the Neimark–Sacker and period-doubling bifurcations under certain constraints. We also control the emergence of chaos using the OGY method. In order to guarantee the accuracy of the theoretical study, some numerical investigations are performed. In particular, we present some phase portraits for the stability and the emergence of the Neimark–Sacker and period-doubling bifurcations. The biological meaning of the given bifurcations is successfully discussed. The used techniques can be successfully employed for other models.

This paper investigates the bifurcation problem in a fractional-order delayed food chain model that incorporates a fear effect. We observe that the fractional order significantly impacts the delayed system, influencing its stability in the presence of fear. Both the fractional order and the fear effect play crucial roles in determining the system’s stability. Furthermore, we observe stability switching induced by the fear effect while keeping the delay fixed. We identify the stability condition of the proposed model and precisely establish bifurcation points by utilizing delay as a bifurcation parameter. The system exhibits robust stability performance with smaller control parameters, and Hopf bifurcation arises as the control parameter surpasses a critical value. Additionally, through theoretical analysis and numerical simulations, we investigate the effects of fractional order, the fear effect, and time delay on the system’s stability.

In this paper, a mathematical model is proposed to study the combined effects of media awareness and fear-induced behavioral changes on the dynamics of infectious diseases. It is considered that in comparison to the unaware individuals, the aware individuals have a lower contact with infected ones. The number of media advertisements is assumed to increase at a rate proportional to the number of infected persons and declines as the number of aware individuals increases. The stability analysis of the model shows that an increase in the growth rate of media advertisements leads to generation of periodic oscillations in the system due to occurrence of Hopf-bifurcation at interior equilibrium. The fear factor and the decline in advertisements due to an increase in the number of aware individuals are found to have stabilizing effect on dynamics of system and their high values can eliminate the limit cycle oscillations present in the system. The rate at which awareness spreads among susceptible individuals and the behavioral response of the aware population are found to be the critical parameters which shape the overall impact of awareness on disease dynamics. It has been observed that the increase in contact rate of aware individuals with infected ones and the dissemination rate of awareness can result into emergence of multiple stability switches via double Hopf-bifurcation.

The principal parameter resonance of a ferromagnetic functionally graded (FG) cylindrical shell under the action of axial time-varying tension in magnetic and temperature fields is investigated. The temperature dependence of physical parameters for functionally graded materials (FGMs) is considered. Meanwhile, the tension bending coupling effect is eliminated by introducing the physical neutral surface. The kinetic and strain energies are gained with the Kirchhoff–Love shell theory. Based on the nonlinear magnetization characteristics of ferromagnetic materials, the electromagnetic force acting on the shell is calculated. The nonlinear vibration equations are obtained through Hamilton’s principle. Afterward, the vibration equations are discretized and solved by Galerkin and multiscale methods, respectively. The stability criterion for steady-state motion is established utilizing Lyapunov stability theory. After example analysis, the effects of magnetic field intensity, temperature and power law index on the static deflection are elucidated. Subsequently, the impacts of these parameters, as well as axial tension, on the amplitude-frequency characteristics, resonance amplitude, and multiple solution regions are discussed explicitly. Results indicate that the stiffness can be enhanced due to the generation of static deflection. The amplitude decreases with increasing magnetic field intensity, temperature, and power law index. When the magnetic field intensity surpasses a threshold, the resonance phenomenon disappears.

We study the greedy algorithms for $m$-term approximation. We propose a modification of the Weak Rescaled Pure Greedy Algorithm (WRPGA) — Approximate Weak Rescaled Pure Greedy Algorithm (AWRPGA) — with respect to a dictionary of a Banach space $X$. By using a geometric property of the unit sphere of $X$, we obtain a general error estimate in terms of some $K$-functional. This estimate implies the convergence condition and convergence rate of the AWRPGA. Furthermore, we obtain the corresponding error estimate for the Vector Approximate Weak Rescaled Pure Greedy Algorithm (VAWRPGA). We show that the AWRPGA (VAWRPGA) performs as well as the WRPGA (VWRPGA) when the noise amplitude changes relatively little. Finally, by using the wavelet bases and trigonometric system of Lebesgue spaces, we show that the convergence rate of the AWRPGA is optimal.

A magnetic quantum walk is a quantum walk coupling to (or perturbed by) a magnetic field, which plays an active role in describing the motion of a particle in a magnetic field. In this paper, we introduce and investigate a discrete-time magnetic quantum walk in the framework of Bernoulli functionals. We first introduce magnetic shift operators in terms of the annihilation and creation operators on Bernoulli functionals, and then use them to construct the desired walk. After proving some technical results, we establish a formula for determining probability distributions of the walk. Finally, we show that the walk has probability distributions that are completely independent of the magnetic potential when its initial state meets some mild conditions.

In this paper, a new Milstein-type derivative-free numerical method is proposed for Itô stochastic differential equations (SDEs). The mean-square stability of the method is thoroughly analyzed, and its error estimation is rigorously examined. The method is subsequently extended twice using the concepts of balanced methods and drift-split-step schemes to achieve the highest possible mean-square stability while preserving the derivative-free structure. Detailed numerical simulations are provided to support the theoretical discussions.

The objective of this study is to examine the possible existence of traversable wormhole geometries within the context of $f(R,L_m,T)$ gravity. To meet this objective, we employ the Karmarkar condition to construct the shape function that aids in identifying the wormhole configurations. This developed function is found to satisfy the essential conditions and provides a link between two asymptotically flat spacetime regions. We then assume the Morris–Thorne line element that expresses the wormhole configuration and formulate the anisotropic gravitational equations for a particular minimal matter-spacetime coupled model of the modified theory. Afterward, we develop three solutions and determine their viability by analyzing whether they violate the null energy conditions. Different stability methods are applied to the resulting geometries to explore the acceptance of the considered modified model. We conclude that the developed wormhole structures potentially fulfill the required criteria and thus exist in this modified gravity under all choices of the matter Lagrangian density.

In this paper, the charged acoustic black hole with a cosmological constant has been assumed. We have taken into account the negative cosmological constant as thermodynamic pressure in the extended phase space. Then we derived the thermodynamic quantities and investigated their behavior. We have studied the critical values of temperature and pressure. By calculating the specific heat capacity, we have analyzed the thermal stability of the charged acoustic black hole. Then we studied the heat engine phenomena of the black hole. We discovered the Carnot engine’s efficiency and the brand-new engine phenomenon of the black hole.

This study explores the interaction between Euler–Heisenberg black holes and perfect fluid dark matter by focusing on their stability, time evolution, and radiative properties. By using exponential entropy corrections, we derive the specific heat capacity to examine thermodynamic stability, instability, and phase transition points. Our results reveal how quantum electrodynamics effects and dark matter parameters significantly influence the stability landscape, driving critical changes in stability transitions. Additionally, we analyze how these parameters affect oscillation frequencies and damping rates, offering fresh insights into the dynamical behavior of Euler–Heisenberg black holes. Last, we demonstrate that these parameters substantially reshape greybody factor bounds by altering Hawking radiation escape channels and significantly impacting the observable energy spectra.

In this paper, we aim to investigate several anisotropic spherical distributions of celestial bodies in the framework of $f(R)$ gravity, where $R$ is known as Ricci scalar. By using the Karmarkar–Tolman space–time and two different mathematical gravity models, we investigate the characteristics of particular compact objects. The structural parameter behavior is assessed using the graphic methods. We then examine energy constraints to determine how well our results fit into the Karmarkar–Tolman space–time model. In the framework of stellar structure modeling, we evaluated the behavior of material variables, energy conditions, equations of state parameters, anisotropy measurements, and the Tolman–Oppenheimer–Volkoff equation for different values of the curvature term. We presented graphical representations of compact star configurations. We accomplished this by making use of well-known $f(R)$ models of gravity, and we presented the results graphically in order to reinforce the veracity and stability of the stars.

New Julolidine substituted phthalocyanine dyes developed in this work exhibit being cost-effective and simple to produce. FTIR, MALDI-TOF, UV-visible, and NMR analysis verified the structural properties of the dyes. The dyes did not clump and were soluble in a wide range of organic solvents. The dyes ZnPc, NiPc, CoPc and CuPc are completely transparent in solution. The dye’s thermal stability and transmittance were measured using thermogravimetric analysis (TGA) and UV-visible spectroscopy. Among the other produced dyes, ZnPc (17%) spin-coated films had the lowest light scattering process and the maximum transmittance (83%) in the blue area. Finally, the combined results revealed that ZnPc dye might be employed as an LCD color filter.

In this paper, we investigate the stability of the corresponding nonlocal Sobolev inequality

The aim of this paper is to study the stability of the solutions of some uncertain delay (integro) differential equations under different conditions. In this regard, applying the Lyapunov direct method and fixed point technique, some sufficient conditions will be presented for the stability of a linear scalar uncertain delay differential equation. In some other cases that the conditions make it difficult to define a Lyapunov function, the stability problem will be studied only by fixed point method.

Pine wilt disease is a destructive forest disease with strong infectivity, a wide spread range and high difficulty in prevention and control. Since controlling Monochamus alternatus, the vector of pine wood nematode (Bursaphelenchus xylophilus) can reduce the occurrence of pine wilt disease efficiently, the parasitic natural enemy of M. alternatus, Dastarcus helophoroides, is introduced in this paper. Considering the influence of parasitic time of D. helophoroides on the control effect, based on the mutualistic symbiosis and parasitic relationship among pine wood nematode, M. alternatus and D. heloporoides, this paper establishes a pine wood nematode prevention and control model with delay. Then, the stability of positive equilibrium and the existence of Hopf bifurcation are discussed. Besides, we obtain the normal form of Hopf bifurcation by applying the multiple time scales method. Finally, numerical simulations with two sets of meaningful parameters selected by means of statistical analysis are carried out to support the theoretical findings. Through the comparative analysis of numerical simulations, the factors affecting the control effect of pine wilt disease are obtained, and some suggestions are put forward for practical control in the forest.

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.