Symmetry, Volume 16, Issue 5 (May 2024) – 129 articles

This paper gives its contributions in four stages. First, we propose the analytical expressions of power spectrum density (PSD) responses and cross-PSD responses to seven classes of fractional vibrators driven by fractional Gaussian noise (fGn). Second, we put forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by fractional Brownian motion (fBm). Third, we present the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators driven by the fractional Ornstein–Uhlenbeck (OU) process. Fourth, we bring forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by the von Kármán process. We show that the statistical dependences of the responses to seven classes of fractional vibrators follow those of the excitation of fGn, fBm, the OU process, or the von Kármán process. We also demonstrate the obvious effects of fractional orders on the responses to seven classes of fractional vibrations. In addition, we newly introduce class VII fractional vibrators, their frequency transfer function, and their impulse response in this research. Full article

In this paper, we introduce a new three-parameter inverse Rayleigh distribution that extends the inverse Rayleigh distribution, constructed based on the generalized transmuted family of distributions proposed by Alizadeh, Merovci, and Hamedani. We explore statistical properties such as the quantile function, moments, harmonic mean, mean deviation, stress–strength reliability, and entropy. Parameter estimation is performed using various methods, including maximum likelihood, least squares, the method of the maximum product of spacings, and the method of Cramér–von Mises. The usefulness of the new three-parameter inverse Rayleigh distribution is illustrated by modeling a real dataset, demonstrating its superior fit compared to several other distributions. Full article

The traditional stratification weight is widely used in survey sampling for estimation under stratified random sampling (StRS). A neutrosophic calibration approach is proposed under neutrosophic statistics for the first time with the aim of improving conventional stratification weight. This addresses the challenge of estimating the empirical cumulative distribution function (CDF) of a finite population using the neutrosophic technique. The neutrosophic technique extends traditional statistics, dealing with indeterminate, vague, and uncertain values. Thus, using additional information, we are able to obtain an effective estimate of the neutrosophic CDF. The suggested estimator yields an interval range in which the population empirical CDF is likely to exist rather than a single numerical value. The proposed family of neutrosophic estimators will be defined under suitable calibration constraints. A simulation study is also computed in order to assess the effectiveness of the suggested and adapted neutrosophic estimators using real-life symmetric and asymmetric datasets. Full article

Systems with a multistable energy landscape are widespread in physics, biophysics, technology, and materials science. They are strongly influenced by thermal fluctuations and external mechanical actions that can be applied at different rates, moving the system from equilibrium to non-equilibrium regimes. In this paper, we focus on a simple system involving a single breaking phenomenon to describe the various theoretical approaches used to study these problems. To begin with, we propose the exact solution at thermodynamic equilibrium based on the calculation of the partition function without approximations. We then introduce the technique of spin variables, which is able to simplify the treatment even for systems with a large number of coordinates. We then analyze the energy balance of the system to better understand its underlying physics. Finally, we introduce a technique based on transition state theory useful for studying the non-equilibrium dynamical regimes of these systems. This method is appropriate for the evaluation of rate effects and hysteresis loops. These approaches are developed for both the Helmholtz ensemble (prescribed extension) and the Gibbs ensemble (applied force) of statistical mechanics. The symmetry and duality of these two ensembles is discussed in depth. While these techniques are used here for a simple system with theoretical purposes, they can be applied to complex systems of interest for several physical, biophysical, and technological applications. Full article

The pseudo-spin symmetry (PSS) provides an important angle to understand nuclear microscopic structure and the novel phenomena found in unstable nuclei. The relativistic Hartree–Fock (RHF) theory, that takes the important degrees of freedom associated with the $\pi$-meson and $\rho$-tensor ($\rho$-T) couplings into account, provides an appropriate description of the PSS restoration in realistic nuclei, particularly for the pseudo-spin (PS) doublets with high angular momenta ($\tilde{l}$). The investigations of the PSS within the RHF theory are recalled in this paper by focusing on the effects of the Fock terms. Aiming at common artificial shell closures appearing in previous relativistic mean-field calculations, the mechanism responsible for the PSS restoration of high-$\tilde{l}$ orbits is stressed, revealing the manifestation of nuclear in-medium effects on the PSS, and thus, providing qualitative guidance on modeling the in-medium balance between nuclear attractions and repulsions. Moreover, the essential role played by the $\rho$-T coupling, that contributes mainly via the Fock terms, is introduced as combined with the relations between the PSS and various nuclear phenomena, including the shell structure and the evolution, novel halo and bubble-like phenomena, and the superheavy magicity. As the consequences of the nuclear force in complicated nuclear many-body systems, the PSS itself and the mechanism therein can not only deepen our understanding of nuclear microscopic structure and relevant phenomena, but also provide special insight into the nature of the nuclear force, which can further enrich our knowledge of nuclear physics. Full article

In this study, we proposed a model for modular robots in which autonomous decentralized modules adaptively organize their behavior. The phototaxis of Gonium pectorale, a species of volvocine algae, was modeled as a modular system, and a fault-tolerant modular control method of phototaxis was proposed for it. The proposed method was based on the rotation phase of the colony and adaptively adjusted an internal response-related parameter to enhance the fault tolerance of the system. Compared to a constant parameter approach, the simulation results demonstrated a significant improvement in the phototaxis time for positive and negative phototaxis during module failures. This method contributes to achieving autonomous, decentralized, and purposeful mediation of the modules necessary for controlling modular robots. Full article

Quantum mechanics of unitary systems is considered in quasi-Hermitian representation and in the dynamical regime in which one has to take into account the ubiquitous presence of perturbations, random or specific. In this paper, it is shown that multiple technical obstacles encountered in such a context can be circumvented via just a mild amendment of the so-called Rayleigh–Schrödinger perturbation–expansion approach. In particular, the quasi-Hermitian formalism characterized by an enhancement of flexibility is shown to remain mathematically tractable while, on the phenomenological side, opening several new model-building horizons. It is emphasized that they include, i.a., the study of generic random perturbations and/or of multiple specific non-Hermitian toy models. In parallel, several paradoxes and open questions are shown to survive. Full article

Accurate short-term electrical load forecasting is crucial for the stable operation of power systems. Given the nonlinear, periodic, and rapidly changing characteristics of short-term power load forecasts, this paper introduces a novel forecasting method employing an Extreme Learning Machine (ELM) enhanced by an improved Dwarf Mongoose Optimization Algorithm (Local escape Dwarf Mongoose Optimization Algorithm, LDMOA). This method addresses the significant prediction errors of conventional ELM models and enhances prediction accuracy. The enhancements to the Dwarf Mongoose Optimization Algorithm include three key modifications: initially, a dynamic backward learning strategy is integrated at the early stages of the algorithm to augment its global search capabilities. Subsequently, a cosine algorithm is employed to locate new food sources, thereby expanding the search scope and avoiding local optima. Lastly, a “madness factor” is added when identifying new sleeping burrows to further widen the search area and effectively circumvent local optima. Comparative analyses using benchmark functions demonstrate the improved algorithm’s superior convergence and stability. In this study, the LDMOA algorithm optimizes the weights and thresholds of the ELM to establish the LDMOA-ELM prediction model. Experimental forecasts utilizing data from China’s 2016 “The Electrician Mathematical Contest in Modeling” demonstrate that the LDMOA-ELM model significantly outperforms the original ELM model in terms of prediction error and accuracy. Full article

In this study, we prove the existence and uniqueness of a best proximity point in the setting of non-Archimedean modular metric spaces via the concept of simulation functions. A non-Archimedean metric modular is shaped as a parameterized family of classical metrics; therefore, for each value of the parameter, the positivity, the symmetry, the triangle inequality, or the continuity is ensured. Also, we demonstrate how analogous theorems in modular metric spaces may be used to generate the best proximity point results in triangular fuzzy metric spaces. The utility of our findings is further demonstrated by certain examples, illustrated consequences, and an application to fuzzy fractional differential equations. Full article

In the Standard Model, ad hoc hypotheses assume the existence of three generations of point-like leptons and quarks, which possess a point-like structure and follow the Dirac equation involving four anti-commutative matrices. In this work, we consider the sedenion hypercomplex algebra as an extension of the Standard Model and show its close link to SU(5), which is the underlying symmetry group for the grand unification theory (GUT). We first consider the direct-product quaternion model and the eight-element octonion algebra model. We show that neither the associative quaternion model nor the non-associative octonion model could generate three fermion generations. Instead, we show that the sedenion model, which contains three octonion sub-algebras, leads naturally to precisely three fermion generations. Moreover, we demonstrate the use of basis sedenion operators to construct twenty-four 5 × 5 generalized lambda matrices representing SU(5) generators, in analogy to the use of octonion basis operators to generate Gell-Mann’s eight 3 × 3 lambda-matrix generators for SU(3). Thus, we provide a link between the sedenion algebra and Georgi and Glashow’s SU(5) GUT model that unifies the electroweak and strong interactions for the Standard Model’s elementary particles, which obey SU(3)SU(2)U(1) symmetry. Full article

The creating of novel models essentially stems from the requirement to appropriate describe survival cases. In this study, a novel lifetime model with two parameters is proposed and studied for modeling more types of data used in different study cases, including symmetric, asymmetric, skewed, and complex datasets. The proposed model is obtained by compounding the exponential and XLindley distributions, and it is regarded as a strong competitor for the widely applied symmetrical and non-symmetrical models. Several characteristics and statistical properties are investigated. The unknown parameters of the recommended model for the complete sample are estimated using two estimation methods; notably, maximum likelihood estimation and Bayes techniques based on several loss functions as well as an approximate tool are used to construct the confidence intervals for the unknown parameters of the suggested model. The estimation procedures are compared using a Monte Carlo simulation experiment to demonstrate their effectiveness. In the end, the applicability and flexibility of the recommended model are conducted using two real lifetime datasets. In our illustration, we compare the practicality of the recommended model with several well-known competing distributions. Full article

In this article, we report a very efficient method for the determination of the enantiopurity of 1,2-diacyl-sn-glycero-3-phosphocholine by ¹H NMR analysis using a readily available chiral derivatizing boronic acid (CDA), (R)-(2-(((1-phenylethyl)amino)methyl)phenyl)boronic acid. After the removal of the acyl groups of 1,2-diacyl-sn-glycero-3-phosphocholine via methanolysis and washing fatty acid byproducts with CHCl₃, the obtained sn-glycero-3-phosphocholine (GPC) with the free diol moiety is derivatized by the chiral boronic acid and analyzed by ¹H NMR analysis. The choline methyl resonance of each diastereomer is observed at distinctive chemical shifts in the ¹H NMR spectrum. Integration of the respective resonances allows direct determination of the enantiomeric purity. The procedure was tested successfully using 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) with different enantiomeric purities and with commercially available 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) and 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC). Full article

This paper introduces an enhanced variant of the Adam optimizer—the BGE-Adam optimization algorithm—that integrates three innovative technologies to augment the adaptability, convergence, and robustness of the original algorithm under various training conditions. Firstly, the BGE-Adam algorithm incorporates a dynamic $\beta$ parameter adjustment mechanism that utilizes the rate of gradient variations to dynamically adjust the exponential decay rates of the first and second moment estimates (${\beta }_1$ and ${\beta }_2$), the adjustment of ${\beta }_1$ and ${\beta }_2$ is symmetrical, which means that the rules that the algorithm considers when adjusting ${\beta }_1$ and ${\beta }_2$ are the same. This design helps to maintain the consistency and balance of the algorithm, allowing the optimization algorithm to adaptively capture the trending movements of gradients. Secondly, it estimates the direction of future gradients by a simple gradient prediction model, combining historic gradient information with the current gradient. Lastly, entropy weighting is integrated into the gradient update step. This strategy enhances the model’s exploratory nature by introducing a certain amount of noise, thereby improving its adaptability to complex loss surfaces. Experimental results on classical datasets, MNIST and CIFAR10, and gastrointestinal disease medical datasets demonstrate that the BGE-Adam algorithm has improved convergence and generalization capabilities. In particular, on the specific medical image gastrointestinal disease test dataset, the BGE-Adam optimization algorithm achieved an accuracy of 69.36%, a significant improvement over the 67.66% accuracy attained using the standard Adam algorithm; on the CIFAR10 test dataset, the accuracy of the BGE-Adam algorithm reached 71.4%, which is higher than the 70.65% accuracy of the Adam optimization algorithm; and on the MNIST dataset, the BGE-Adam algorithm’s accuracy was 99.34%, surpassing the Adam optimization algorithm’s accuracy of 99.23%. The BGE-Adam optimization algorithm exhibits better convergence and robustness. This research not only demonstrates the effectiveness of the combination of these three technologies but also provides new perspectives for the future development of deep learning optimization algorithms. Full article

With the continuous advancement of information technology, a growing number of works, including articles, paintings, and music, are being digitized. Digital content can be swiftly shared and disseminated via the Internet. However, it is also vulnerable to malicious plagiarism, which can seriously infringe upon the rights of creators and dampen their enthusiasm. To protect creators’ rights and interests, a sophisticated method is necessary to authenticate digital intellectual property rights. Traditional authentication methods rely on centralized, trustworthy organizations that are susceptible to single points of failure. Additionally, these methods are prone to network attacks that can lead to data loss, tampering, or leakage. Moreover, the circulation of copyright information often lacks transparency and traceability in traditional systems, which leads to information asymmetry and prevents creators from controlling the use and protection of their personal information during the authentication process. Blockchain technology, with its decentralized, tamper-proof, and traceable attributes, addresses these issues perfectly. In blockchain technology, each node is a peer, ensuring the symmetry of information. However, the transparent feature of blockchains can lead to the leakage of user privacy data. Therefore, this study designs and implements an Ethereum blockchain-based intellectual property authentication scheme with privacy protection. Firstly, we propose a method that combines elliptic curve cryptography (ECC) encryption with digital signatures to achieve selective encryption of user personal information. Subsequently, an authentication algorithm based on Zero-Knowledge Succinct Non-Interactive Argument of Knowledge (zk-SNARK) is adopted to complete the authentication of intellectual property ownership while encrypting personal privacy data. Finally, we adopt the InterPlanetary File System (IPFS) to store large files, solving the problem of blockchain storage space limitations. Full article

The aim of this research is to investigate the frequency of conical-shaped shells, consisting of different materials, based on higher-order shear deformation theory (HSDT). The shells are of non-uniform thickness, consisting of two to six symmetric cross-ply layers. Simply supported boundary conditions were used to analyse the frequency of conical-shaped shells. The differential equations, consisting of displacement and rotational functions, were approximated using spline approximation. A generalised eigenvalue problem was obtained and solved numerically for an eigenfrequency parameter and associated eigenvector of spline coefficients. The frequency of shells was analysed by varying the geometric parameters such as length of shell, cone angle, node number in circumference direction and number of layers, as well as three thickness variations such as linear, sinusoidal and exponential. It was also evident that by varying geometrical parameters, the mechanical parameters such as stress, moment and shear resultants were affected. Research results concluded that for three different thickness variations, as the number of layers of conical shells increases, the frequency values decrease. Moreover, by varying length ratios and cone angles, shells with variable thickness had lower frequency values compared to shells of constant thickness. The numerical results obtained were verified through the already existing literature. It is evident that the present results are very close to the already existing literature. Full article

In previous research simulating steelmaking ladles using cold water models, the dosage/volume of the salt tracer solution is one of the factors that has been overlooked by researchers to a certain extent. Previous studies have demonstrated that salt tracers may influence the flow and measured mixing time of fluids in water models. Based on a water model scaled down from an industrial 130-ton ladle by a ratio of 1:3, this study investigates the impact of salt tracer dosage on the transport and mixing of tracers in the water model of gas-stirred ladle with a moderate gas flow rate. A preliminary uncertainty analysis of the experimental mixing time is performed, and the standard deviations were found to be less than 15%. It was observed in the experiments that the transport paths of tracers in the ladle can be classified into two trends. A common trend is that the injected salt solution tracer is asymmetrically transported towards the left sidewall of the ladle by the main circulation. In another trend, the injected salt solution tracer is transported both by the main circulation to the left side wall and by downward flow towards the gas column. The downward flow may be accelerated and become a major flow pattern when the tracer volume increases. For the dimensionless concentration curve, the sinusoidal type, which represents a rapid mixing, is observed at the top surface monitoring points, while the parabolic type is observed at the bottom monitoring points. An exception is the monitoring point at the right-side bottom (close to the asymmetric gas nozzle area), where both sinusoidal-type and parabolic-type curves are observed. Regarding the effect of tracer volume on the curve and mixing time, the curves at the top surface monitoring points are less influenced but curves at the bottom monitoring points are noticeably influenced by the tracer volume. A trend of decreasing and then increasing as the tracer volume increases was found at the top surface monitoring points, while the mixing times at the bottom monitoring points decrease with the increase in the tracer volume. Full article

Based on the Hamilton canonical equations for ocean surface waves with four-five-six-wave resonance conditions, the determinate dynamical equation of four-five-six-wave resonances for ocean surface gravity waves in water with a finite depth is established, thus leading to the elimination of the nonresonant second-, third-, fourth-, and fifth-order nonlinear terms though a suitable canonical transformation. The four kernels of the equation and the 18 coefficients of the transformation are expressed in explicit form in terms of the expansion coefficients of the gravity wave Hamiltonian in integral-power series in normal variables. The possibilities of the existence of integrals of motion for the wave momentum and the wave action are discussed, particularly the special integrals for the latter. For ocean surface capillary–gravity waves on a fluid with a finite depth, the sixth-order expansion coefficients of the Hamiltonian in integral-power series in normal variables are concretely provided, thus naturally including the classical fifth-order kinetic energy expansion coefficients given by Krasitskii. Full article

In general, the extended Kalman filter (EKF) has a wide range of applications, aiming to minimize symmetric loss function (mean square error) and improve the accuracy and efficiency of state estimation. As the nonlinear model complexity increases, rounding errors gradually amplify, leading to performance degradation. After multiple iterations, divergence may occur. The traditional extended Kalman filter cannot accurately estimate the nonlinear model, and these errors still have an impact on the accuracy. To improve the filtering performance of the extended Kalman filter (EKF), this paper proposes a new extended Kalman filter (REKF) method that utilizes the statistical properties of the rounding error to enhance the estimation accuracy. After establishing the state model and measurement model, the residual term is used to replace the higher-order term in the Taylor expansion, and the least squares method is applied to identify the residual term step by step. Then, the iterative process of updating the extended Kalman filter is carried out. Within the Kalman filter framework, a higher-order rounding error-based extended Kalman filter (REKF) is designed for the joint estimation of rounding error and random variables, and the solution method for the rounding error is considered for the multilevel approximation of the original function. Through numerical simulations on a general nonlinear model, the higher-order rounding error-based extended Kalman filter (REKF) achieves better estimation results than the extended Kalman filter (EKF) and improves the filtering accuracy by utilizing the higher-order rounding error information, which also proves the effectiveness of the proposed method. Full article

Petrographic prediction is crucial in identifying target areas and understanding reservoir lithology in oil and gas exploration. Traditional logging methods often rely on manual interpretation and experiential judgment, which can introduce subjectivity and constraints due to data quality and geological variability. To enhance the precision and efficacy of lithology prediction, this study employed a Savitzky–Golay filter with a symmetric window for anomaly data processing, coupled with a residual temporal convolutional network (ResTCN) model tasked with completing missing logging data segments. A comparative analysis against the support vector regression and random forest regression model revealed that the ResTCN achieves the smallest MAE, at 0.030, and the highest coefficient of determination, at 0.716, which are indicative of its proximity to the ground truth. These methodologies significantly enhance the quality of the training data. Subsequently, a Transformer–long short-term memory (T-LS) model was applied to identify and classify the lithology of unexplored wells. The input layer of the Transformer model follows an embedding-like principle for data preprocessing, while the encoding block encompasses multi-head attention, Add & Norm, and feedforward components, integrating the multi-head attention mechanism. The output layer interfaces with the LSTM layer through dropout. A performance evaluation of the T-LS model against established rocky prediction techniques such as logistic regression, k-nearest neighbor, and random forest demonstrated its superior identification and classification capabilities. Specifically, the T-LS model achieved a precision of 0.88 and a recall of 0.89 across nine distinct lithology features. A Shapley analysis of the T-LS model underscored the utility of amalgamating multiple logging data sources for lithology classification predictions. This advancement partially addresses the challenges associated with imprecise predictions and limited generalization abilities inherent in traditional machine learning and deep learning models applied to lithology identification, and it also helps to optimize oil and gas exploration and development strategies and improve the efficiency of resource extraction. Full article

This study formulates a multi-objective, multi-item solid transportation issue with parameters that are neutrosophic Z-number fuzzy variables such as transportation costs, supplies, and demands. This work covers two scenarios where uncertainty in the problem can arise: the fuzzy solid transportation problem and the interval solid transportation problem. The first scenario arises when we represent data problems as intervals instead of exact values, while the second scenario arises when the information is not entirely clear. We address both models when the uncertainty alone impacts the constraint set. In order to find a solution for the interval case, we generate an additional problem. Since this auxiliary problem is typical of solid transportation, we can resolve it using the effective techniques currently in use. In the fuzzy scenario, a parametric method is used to discover a fuzzy solution to the earlier issue. Parametric analysis identifies that the best parameterized approaches to complementary problems are characterized by the application of parametric analysis. We present a suggested algorithm for determining the stability set. Finally, we provide a numerical example and sensitivity analysis for the transportation problem, which is both symmetrical and asymmetrical. Full article

Compared to conventional energy sources, wind power is a clean energy source with high intermittence and uncertainty. As a system that converts wind energy into electricity, wind farms inevitably face severe reliability issues. In this paper, based on reliability theory, a new reliability modeling method for wind farms is proposed. Firstly, a belief reliability model for wind farms is constructed. Then, a power generation model based on two-directional rotor equivalent wind speed is established to represent the wind farm performance in the belief reliability model. Finally, several numerical studies are conducted to verify the power generation model under different wind speeds and directions, to demonstrate the belief reliability model with different levels of uncertainty, and to compare the belief reliability considering two-directional rotor equivalent wind speed with other methods. Full article

The present research introduces a novel concept termed “neutrosophic fuzzy metric space”, which extends the traditional metric space framework by incorporating the notion of neutrosophic fuzzy sets. A thorough investigation of various structural and topological properties within this newly proposed generalization of metric space has been conducted. Additionally, counterparts of well-known theorems such as the Uniform Convergence Theorem and the Baire Category Theorem have been established for this generalized metric space. Through rigorous analysis, a detailed understanding of its fundamental characteristics has been attained, illuminating its potential applications and theoretical significance. Full article

In order to deal with complex risk scenarios involving claims, uncertainty, and investments, we consider the ruin problems in a compound Poisson risk model with liquid reserves and proportional investments and study the expected discounted penalty function under threshold dividend strategies. Firstly, the integral differential equation of the expected discounted penalty function is derived. Secondly, since the closed-form solution of the equation cannot be obtained, a sinc method is used to obtain the numerical approximation solution of the equation. Finally, the feasibility and superiority of the sinc method are illustrated by error analysis. In addition, based on a symmetric jump risk market, we discuss the influence of some parameters on the ruin probability with some examples. This study can help actuaries develop more robust risk management strategies and ensure the long-term stability and profitability of insurance companies. It provides a theoretical basis for actuaries to carry out risk management. Full article

Symmetrical as well as asymmetrical statistical models play a prominent role in describing and predicting the real-world phenomena of nature. Among other fields, these models are very useful for modeling data in the sector of civil engineering. Due to the applicability of the statistical models in civil engineering and other related sectors, this paper offers a statistical methodology to improve the distributional flexibility of traditional models. The suggested method/approach is called the extended-X family of distributions. The proposed method has the ability to generate symmetrical and asymmetrical probability distributions. Based on the extended-X family approach, an updated version of the Weibull model, namely, the extended Weibull model, is studied. The proposed model is very flexible and has the ability to capture the symmetrical and asymmetrical shapes of its density function. For the extended-X method, the estimation of parameters, a simulation study, and some mathematical properties are derived. Finally, the practical illustration/usefulness of the suggested model is shown by analyzing two data sets taken from the field of engineering. Both data sets represent the fracture toughness of alumina (Al₂O₃). Full article

Unit distributions, exhibiting inherent symmetrical properties, have been extensively studied across various fields. A significant challenge in these studies, particularly evident in parameter estimations, is the existence and uniqueness of estimators. Often, it is challenging to demonstrate the existence of a unique estimator. The major issue with maximum likelihood and other estimator-finding methods that use iterative methods is that they need an initial value to reach the solution. This dependency on initial values can lead to local extremes that fail to represent the global extremities, highlighting a lack of symmetry in solution robustness. This study applies a very simple, and unique, estimation method for unit Weibull and unit Burr XII distributions that both attain the global maximum value. Therefore, we can conclude that the findings from the obtained propositions demonstrate that both the maximum likelihood and graphical methods are symmetrically similar. In addition, three real-world data applications are made to show that the method works efficiently. Full article

The Klausmeier–Gray–Scott model of vegetation dynamics consists of a system of two partial differential equations relating plant growth and soil water. It is capable of reproducing the characteristic spatial patterns of vegetation found in plant ecosystems under water limitations. Recently, a discrete delay was incorporated into this model to account for the lag between water infiltration into the soil and the following water uptake by plants. In this work, we consider a more ecologically realistic distributed delay to relate plant growth and soil water availability and analyse the effects of different delay types on the dynamics of both mean-field and spatial Klausmeier–Gray–Scott models. We consider distributed delays based on Gamma kernels and use the so-called linear chain trick to analyse the stability of the uniformly vegetated equilibrium. It is shown that the presence of delays can lead to the loss of stability in the constant equilibrium and to a reduction of the parameter region where steady-state vegetation patterns can arise through symmetry-breaking by diffusion-driven instability. However, these effects depend on the type of delay, and they are absent for distributed delays with weak kernels when vegetation mortality is low. Full article

The examination of new (3 + 1)-dimensional wave equations is undertaken in this study. Initially, the identification of the Lie symmetries of the model is carried out through the utilization of the Lie symmetry approach. The commutator and adjoint table of the symmetries are presented. Subsequently, the model under discussion is transformed into an ordinary differential equation using these symmetries. The construction of several bright, kink, and dark solitons for the suggested equation is then achieved through the utilization of the new auxiliary equation method. Subsequently, an analysis of the dynamical nature of the model is conducted, encompassing various angles such as bifurcation, chaos, and sensitivity. Bifurcation occurs at critical points within a dynamical system, accompanied by the application of an outward force, which unveils the emergence of chaotic phenomena. Two-dimensional plots, time plots, multistability, and Lyapunov exponents are presented to illustrate these chaotic behaviors. Furthermore, the sensitivity of the investigated model is executed utilizing the Runge–Kutta method. This analysis confirms that the stability of the solution is minimally affected by small changes in initial conditions. The attained outcomes show the effectiveness of the presented methods in evaluating solitons of multiple nonlinear models. Full article

A modified Nambu–Jona-Lasinio Model with lattice structure is very instructive. It shows several similar problems and their solutions as the Lattice QCD. We study the limits of the large box size, small cell size and realistic pion mass. In particular, we study the relation of the discrete (bound state) solutions to the physical scattering states, for example the pion–pion scattering. Full article

This article concentrates on the problem of fixed-time tracking control for a certain class of nonlinear systems with unmodeled dynamics. Unmodeled dynamics are prevalent in practical engineering systems, such as axially symmetric systems like robotic arms, spacecraft, and missiles. In this paper, the fuzzy-logic systems (FLSs) are implemented to address the challenge of accurately approximating the unknown nonlinear terms that arise during the derived control algorithm process. By employing fixed-time command filters (FTCF), the “explosion of complexity” issues encountered in traditional backstepping methods will be effectively resolved. Moreover, error compensation mechanisms are derived to effectively mitigate the filtering errors that may arise from the FTCFs. The computational burden associated with FLSs is reduced through the utilization of the weight vector estimation method based on the maximal norm and an adaptive approach. A fixed-time adaptive fuzzy tracking controller is developed within the backstepping control framework to ensure the boundedness of all signals and achieve fixed-time convergence of the tracking error for the controlled system. Illustrative examples are conducted to illustrate the viability of the derived controller. Full article