Symmetry, Volume 16, Issue 5 (May 2024) – 104 articles

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

Researchers have created cryptography algorithms that encrypt data using a public or private key to secure it from intruders. It is insufficient to protect the data by using such a key. No research article has identified an algorithm capable of protecting both the data and the associated key, nor has any mechanism been developed to determine whether access to the data is permissible or impermissible based on the authentication of the key. This paper presents a WEDEx-Kerberotic Framework for data protection, in which a user-defined key is firstly converted to a cipher key using the “Secure Words on Joining Key (SWJK)” algorithm. Subsequently, a WEDEx-Kerberotic encryption mechanism is created to protect the data by encrypting it with the cipher key. The first reason for making the WEDEx-Kerberotic Framework is to convert the user-defined key into a key that has nothing to do with the original key, and the length of the cipher key is much shorter than the original key. The second reason is that each ciphertext and key value are interlinked. When an intruder utilizes the snatching mechanism to obtain data, the attacker obtains data or a key unrelated to the original data. No matter how efficient the algorithm is, an attacker cannot access the data when these methods and algorithms are used to protect it. Finally, the proposed algorithm is compared to the previous approaches to determine the uniqueness of the algorithm and assess its superiority to the previous algorithms. Full article

This pioneering investigation introduces two innovative estimators crafted to evaluate the finite population distribution function of a study variable, employing auxiliary variables within the framework of stratified random sampling and post-stratification while emphasizing symmetry in the sampling process. The derivation of mathematical expressions for bias and the mean square error up to the first degree of approximation fortifies the credibility of the proposed estimators. Drawing from three distinct datasets, including real-world data capturing student behaviors and exam performances from 500 students, this research highlights the superior efficiency of the proposed estimators compared to existing methods across both sampling schemes. Employing the proposed estimator, we effectively forecast students’ exam scores based on their study hours, backed by empirical evidence showcasing its precision in terms of mean square error and percentage relative efficiency. This study not only introduces inventive solutions to enduring challenges in survey sampling but also provides practical insights into enhancing predictive accuracy in educational assessments. Full article

The number of failures serves as a critical indicator that dynamically impacts the reliability of self-announcing failure products, making it highly practical to incorporate the failure count into reliability management throughout the entire product life cycle. This paper investigates comprehensive methodologies for effectively managing the reliability of self-announcing failure products throughout both the warranty and post-warranty stages, taking into account factors such as the failure count, mission cycles, and limited time duration. Three renewable warranty strategies are introduced alongside proposed models for post-warranty replacements. By analyzing variables like the failure number, mission cycles, and time constraints, these proposed warranties provide practical frameworks for efficient reliability management during the warranty stage. Additionally, the introduced warranties utilize cost and time metrics to extract valuable insights that inform decision making and enable effective reliability management during the warranty stage. Moreover, this study establishes cost and time metrics for key post-warranty replacements, facilitating the development of individual cost rates and model applications in other post-warranty scenarios. Analyses of the renewable free-repair–replacement warranties demonstrate that establishing an appropriate number of failures as the replacement threshold can effectively reduce warranty-servicing costs and extend the coverage duration. Full article

Nowadays, rumors spread more rapidly than before, leading to more panic and instability in society. Therefore, it is essential to seek out propagation law in order to prevent rumors from spreading further and avoid unnecessary harm. There is a connection between rumor models and symmetry. The consistency of a system or model is referred to as the level of symmetry under certain transformations. For this purpose, we propose a fractional-order Ignorant–Latent–Spreader–Remover (ILSR) rumor propagation model that incorporates media reports and a nonlinear inhibition mechanism. Firstly, the boundedness and non-negativeness of the solutions are derived under fractional differential equations. Secondly, the threshold is used to evaluate and illustrate the stability both locally and globally. Finally, by utilizing Pontryagin’s maximum principle, we obtain the necessary conditions for the optimal control in the fractional-order rumor propagation model, and we also obtain the associated optimal solutions. Furthermore, the numerical results indicate that media reports can decrease the spread of rumors in different dynamic regions, but they cannot completely prevent rumor dissemination. The results are also exhibited and corroborated by replicating the model with specific hypothetical parameter values. It can be inferred that fractional order yields more favorable outcomes when rumor permanence in the population is higher. The presented method facilitates the acquisition of profound insights into the dissemination dynamics and subsequent consequences of rumors within a societal network. Full article

In this paper, we investigate Ricci solitons on spacelike hypersurfaces in a special Lorentzian warped product manifold, the so-called generalized Robertson–Walker (GRW) spacetimes. Such spacetimes admit a natural form of symmetry which is represented by the conformal vector field $f\partial t$, where f is the warping function and $\partial t$ is the unit timelike vector field tangent to the base (which is here a one-dimensional manifold). We use this symmetry to introduce some fundamental formulas related to the Ricci soliton structures and the Ricci curvature of the fiber, the warping function, and the shape operator of the immersion. We investigate different rigidity results for Ricci solitons on the slices, in addition to the totally umbilical spacelike supersurfaces of GRW. Furthermore, our study is focused on significant GRW spacetimes such as Einstein GRW spacetimes and those which obey the well-known null convergence condition (NCC). Full article

The estimated ultimate recovery (EUR) of a single well must be predicted to achieve scale-effective shale gas extraction. Accurately forecasting EUR is difficult due to the impact of various geological, engineering, and production factors. Based on data from 200 wells in the Weiyuan block, this paper used Pearson correlation and mutual information to eliminate the factors with a high correlation among the 31 EUR influencing factors. The RF-RFE algorithm was then used to identify the six most important factors controlling the EUR of shale gas wells. XGBoost, RF, SVM, and MLR models were built and trained with the six dominating factors screened as features and EUR as labels. In this process, the model parameters were optimized, and finally the prediction accuracies of the models were compared. The results showed that the thickness of a high-quality reservoir was the dominating factor in geology; the high-quality reservoir length drilled, the fracturing fluid volume, the proppant volume, and the fluid volume per length were the dominating factors in engineering; and the 360−day flowback rate was the dominating factor in production. Compared to the SVM and MLR models, the XG Boost and the RF models based on integration better predicted EUR. The XGBoost model had a correlation coefficient of 0.9 between predicted and observed values, and its standard deviation was closest to the observed values’ standard deviation, making it the best model for EUR prediction among the four types of models. Identifying the dominating factors of shale gas single-well EUR can provide significant guidance for development practice, and using the optimized XGBoost model to forecast the shale gas single-well EUR provides a novel idea for predicting shale gas well production. Full article

This paper describes a solution for the image density classification problem (DCP) using an entirely distributed system with only local processing of information named cellular automata (CA). The proposed solution uses two cellular automata’s features, density conserving and translation of the information stored in the cellular automata’s cells through the lattice, in order to obtain the solution for the density classification problem. The motivation for choosing a bio-inspired technique based on CA for solving the DCP is to investigate the principles of self-organizing decentralized computation and to assess the capabilities of CA to achieve such computation, which is applicable to many real-world decentralized problems that require a decision to be taken by majority voting, such as multi-agent holonic systems, collaborative robots, drones’ fleet, image analysis, traffic optimization, forming and then separating clusters with different values. The entire application is coded using the C# programming language, and the obtained results and comparisons between different cellular automata configurations are also discussed in this research. Full article

In this study, ten questions measuring distributive justice on classical Likert and neutrosophic Likert scales consisting of two subdimensions—distributive and procedural justice—were used. Participants responded to the same questions for both the classical Likert and neutrosophic Likert scales within a single survey, with the neutrosophic method applied, for the first time, to the questions included in the scale. The neutrosophic scale responses were answered in percentages to resemble natural language, and the answers received for each question were reduced to the range [−1, 1] to grade the agreement approach through a score function used in neutrosophic decision-making theory. In this study, the neutrosophic scale, a scaling method with strong theoretical foundations, was compared with the traditional Likert scale. The results of the statistical analyses (exploratory factor analysis, reliability analysis, neural network analysis, correlation analysis, paired samples t-test, and one-way and two-way ANOVAs) and evaluations of the scales were compared to measure organizational justice within a single study. In this article, the symmetric and non-symmetric properties of statistical analysis that are specific to this paper in addition to general symmetric and non-symmetry properties are discussed. These symmetric and non-symmetric features are conceptualized according to the features on which each statistical analysis focuses. Finally, although this study presents a new area of research in the social sciences, we believe that the neutrosophic Likert scale and survey approach will contribute to collecting detailed and sensitive information on many topics, such as economics, health, audience perceptions, advertising responses, and product, market, and service purchase research, through the use of score functions. Full article

In this paper, we carry out some research on a predator–prey system with maturation delay, a stage structure for generalist predators and a Holling type-II functional response, which has already been proposed. First, for the delayed model, we obtain the conditions for the occurrence of stability switches of the positive equilibrium and possible Hopf bifurcation values owing to the growth of the value of the delay by applying the geometric criterion. It should be pointed out that when we suppose that the characteristic equation has a pair of imaginary roots $\lambda =\pm i\omega (\omega >0)$, we just need to consider $i\omega (\omega >0)$ due to the symmetry, which alleviates the computation requirements. Next, we investigate the nature of Hopf bifurcation. Finally, we conduct numerical simulations to verify the correctness of our findings. Full article

The Schlieren technique is widely adopted for visualizing supersonic jets owing to its non-invasiveness to the flow field. However, extending the classical Schlieren method for quantitative refractive index measurements is cumbersome, especially for three-dimensional supersonic flows. Background-oriented Schlieren has gained increasing popularity owing to its ease of implementation and calibration. This study utilizes multi-view-based tomographic background-oriented Schlieren (TBOS) to reconstruct axisymmetric and weakly non-axisymmetric supersonic jets, highlighting the impact of flow axisymmetry breaking on TBOS reconstructions. Several classical TBOS reconstruction algorithms, including FDK, SART, SIRT, and CGLS, are compared quantitatively regarding reconstruction quality. View spareness is identified to be the main cause of degraded reconstruction quality when the flow experiences axisymmetry breaking. The classic visual hull approach is explored to improve reconstruction quality. Together with the CGLS tomographic algorithm, we successfully reconstruct the weakly non-axisymmetric supersonic jet structures and confirm that increasing the nozzle bevel angle leads to wider jet spreads. Full article

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm of analytic functions, whether they are univalent or nonunivalent. In this paper, we rigorously derive bounds for specific Toeplitz determinants involving logarithmic coefficients pertaining to classes of convex and starlike functions concerning symmetric points. Furthermore, we present illustrative examples showcasing the sharpness of these established bounds. Our findings represent a substantial contribution to the advancement of our understanding of logarithmic coefficients and their profound implications across diverse mathematical contexts. Full article

Hox gene clusters are crucial in embryogenesis. It was observed that some Hox genes are located in order along the telomeric to centromeric direction of the DNA sequence: Hox1, Hox2, Hox3…. These genes are expressed in the same order in the ontogenetic units of the Drosophila embryo along the anterior–posterior axis. The two entities (genome and embryo) differ significantly in linear size and in-between distance. This strange phenomenon was named spatial collinearity (SP). Later, it was observed that, particularly in the vertebrates, a temporal collinearity (TC) coexists: first Hox1 is expressed, later Hox2, and later on Hox3…. According to a biophysical model (BM), pulling forces act at the anterior end of the cluster while a cluster fastening applies at the posterior end. Hox clusters are irreversibly elongated along the force direction. During evolution, the elongated Hox clusters are broken at variable lengths, thus split clusters may be created. An empirical rule was formulated, distinguishing development due to a complete Hox cluster from development due to split Hox clusters. BM can explain this empirical rule. In a spontaneous mutation, where the cluster fastening is dismantled, a weak pulling force automatically shifts the cluster inside the Hox activation domain. This cluster translocation can probably explain the absence of temporal collinearity in Drosophila. Full article

Smartphones, digital cameras, and other imaging devices generate vast amounts of high-resolution colored images daily, stored on devices equipped with multi-core central processing units or on the cloud. Safeguarding these images from potential attackers has become a pressing concern. This paper introduces a set of six innovative image ciphers designed to be stronger, faster, and more efficient. Three of these algorithms incorporate the State-Based Tent Map (SBTM) Chaotic Pseudo Random Number Generator (CPRNG), while the remaining three employ a proposed modified variant, SBTMPi. The Grayscale Image Cipher (GIC), Colored Image Cipher Single-Thread RGB (CIC1), and Colored Image Cipher Three-Thread RGB (CIC3) showcase the application of the proposed algorithms. By incorporating novel techniques in the confusion and diffusion phases, these ciphers demonstrate remarkable performance, particularly with large colored images. The study underscores the potential of SBTM-based image ciphers, contributing to the advancement of secure image encryption techniques with robust random number generation capabilities. Full article

This paper presents a combined analysis of muon neutrino and antineutrino charged-current cross sections at kinematics of relevance for the T2K, MINERvA and MicroBooNE experiments. We analyze the sum, difference and asymmetry of neutrino versus antineutrino cross sections in order to get a better understanding of the nuclear effects involved in these processes. Nuclear models based on the superscaling behavior and the relativistic mean field theory are applied, covering a wide range of kinematics, from hundreds of MeV to several GeV, and the relevant nuclear regimes, i.e., from quasileastic reactions to deep inelastic scattering processes. The NEUT neutrino-interaction event generator, used in neutrino oscillation experiments, is also applied to the analysis of the quasielastic channel via local Fermi gas and spectral function approaches. Full article

The presence of sex-based asymmetry in the behaviours of youths with Autism Spectrum Disorder (ASD) is currently under research scrutiny. ASD is characterised by challenging behaviour (CB) and is often accompanied by anxiety, both of which often exacerbate social interaction difficulties. The present study examined the presence of sex-based asymmetry in the prevalence of CB and anxiety and in the association between CB and anxiety in a sample including 32 male autistic youths (M age = 10.09, SD = 3.83, range = 6–18 yr) and 32 female autistic youths (M age = 10.31, SD = 2.57, range = 6–15 yr) matched for age, IQ, and ASD severity (p > .101). While the prevalence and severity of behavioural characteristics across males and females with ASD were similar (p = .767), representing symmetry, there was asymmetry in the ways that CBs and anxiety were associated with each other across the two sexes. Specifically, there were 3 instances of symmetry (r > .3, p < .05)), but there were also 10 occurrences of sex-based asymmetry (r < .3, p > .05) in the association between five aspects of CB and five anxiety disorders. These findings emphasise the underlying sex-based symmetry in the prevalence of ASD-related behaviours, also highlighting unique sex-based asymmetry in the association between CBs and anxiety in autistic youths. Full article

Existing direct and inverse kinematic models of planar parallel robots assume that the robot’s active joints are all at the bases. However, this approach becomes excessively complex when modeling a planar parallel robot in which the active joints are within one single kinematic chain. To address this problem, our article unveils an alternative for a 3RRR symmetric planar robot modeling technique for the derivation of the robot workspace and the analysis of its direct and inverse kinematics. The workspace was defined using a system of inequalities, and the direct and inverse kinematics models were generated using vectorial analysis and an optimized geometrical approach, respectively. The resulting models are systematically presented and validated. Two final model renditions are delivered supplying a thorough equation analysis and an applicability discussion based on the importance of the robot’s mobile platform orientation. The advantages of this model are discussed in comparison to the traditional modeling approach: whereas conventional techniques require the solution of complex eighth-degree polynomials for the analysis of the active joint configuration of these robots, these models provide an efficient back-of-the-envelope analysis approach that requires the solution of a simple second-degree polynomial. Full article

Network traffic classification is crucial for identifying network applications and defending against network threats. Traditional traffic classification approaches struggle to extract structural features and suffer from poor interpretability of feature representations. The high symmetry between network traffic classification and its interpretable feature representation is vital for network traffic analysis. To address these issues, this paper proposes a traffic classification and feature representation model named the attention mechanism autoencoder (AMAE). The AMAE model extracts the global spatial structural features of network traffic through attention mechanisms and employs an autoencoder to extract local structural features and perform dimensionality reduction. This process maps different network traffic features into one-dimensional coordinate systems in the form of spectra, termed FlowSpectrum. The spectra of different network traffic represent different intervals in the coordinate system. This paper tests the interpretability and classification performance of network traffic features of the AMAE model using the ISCX-VPN2016 dataset. Experimental results demonstrate that by analyzing the overall distribution of attention weights and local weight values of network traffic, the model effectively explains the differences in the spectral representation intervals of different types of network traffic. Furthermore, our approach achieves the highest classification accuracy of up to 100% for non-VPN-encrypted traffic and 99.69% for VPN-encrypted traffic, surpassing existing traffic classification schemes. Full article

Underwater images often exhibit detail blurring and color distortion due to light scattering, impurities, and other influences, obscuring essential textures and details. This presents a challenge for existing super-resolution techniques in identifying and extracting effective features, making high-quality reconstruction difficult. This research aims to innovate underwater image super-resolution technology to tackle this challenge. Initially, an underwater image degradation model was created by integrating random subsampling, Gaussian blur, mixed noise, and suspended particle simulation to generate a highly realistic synthetic dataset, thereby training the network to adapt to various degradation factors. Subsequently, to enhance the network’s capability to extract key features, improvements were made based on the symmetrically structured blind super-resolution generative adversarial network (BSRGAN) model architecture. An attention mechanism based on energy functions was introduced within the generator to assess the importance of each pixel, and a weighted fusion strategy of adversarial loss, reconstruction loss, and perceptual loss was utilized to improve the quality of image reconstruction. Experimental results demonstrated that the proposed method achieved significant improvements in peak signal-to-noise ratio (PSNR) and underwater image quality measure (UIQM) by 0.85 dB and 0.19, respectively, significantly enhancing the visual perception quality and indicating its feasibility in super-resolution applications. Full article

In biomedicine, the critical task is to decode Drug–Drug Interactions (DDIs) from complex biomedical texts. The scientific community employs Knowledge Graph Embedding (KGE) methods, enhanced with advanced neural network technologies, including capsule networks. However, existing methodologies primarily focus on the structural details of individual entities or relations within Biomedical Knowledge Graphs (BioKGs), overlooking the overall structural context of BioKGs, molecular structures, positional features of drug pairs, and their critical Relational Mapping Properties. To tackle the challenges identified, this study presents HSTrHouse an innovative hierarchical self-attention BioKGs embedding framework. This architecture integrates self-attention mechanisms with advanced neural network technologies, including Convolutional Neural Network (CNN) and Graph Neural Network (GNN), for enhanced computational modeling in biomedical contexts. The model bifurcates the BioKGs into entity and relation layers for structural analysis. It employs self-attention across these layers, utilizing PubMedBERT and CNN for position feature extraction, and a GNN for drug pair molecular structure analysis. Then, we connect the position and molecular structure features to integrate them into the self-attention calculation of entity and relation. After that, the output of the self-attention layer is combined with the connected vectors of the position feature and molecular structure feature to obtain the final representation vector, and finally, to model the Relational Mapping Properties (RMPs), the representation vector is embedded into the complex vector space using Householder projections to obtain the BioKGs model. The paper validates HSTrHouse’s efficacy by comparing it with advanced models on three standard BioKGs for DDIs research. Full article

One of the limitations of the dung beetle optimization (DBO) is its susceptibility to local optima and its relatively low search accuracy. Several strategies have been utilized to improve the diversity, search precision, and outcomes of the DBO. However, the equilibrium between exploration and exploitation has not been achieved optimally. This paper presents a novel algorithm called the ODBO, which incorporates cat map and an opposition-based learning strategy, which is based on symmetry theory. In addition, in order to enhance the performance of the dung ball rolling phase, this paper combines the global search strategy of the osprey optimization algorithm with the position update strategy of the DBO. Additionally, we enhance the population’s diversity during the foraging phase of the DBO by incorporating vertical and horizontal crossover of individuals. This introduction of asymmetry in the crossover operation increases the exploration capability of the algorithm, allowing it to effectively escape local optima and facilitate global search. Full article

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes. Full article

We reveal the continuous change of information dynamics patterns in anyonic-PT symmetric systems that originates from the continuity of anyonic-PT symmetry. We find there are three information dynamics patterns for anyonic-PT symmetric systems: damped oscillations with an overall decrease (increase) and asymptotically stable damped oscillations, which are three-fold degenerate and are distorted using the Hermitian quantum Rényi entropy or distinguishability. It is the normalization of the non-unitary evolved density matrix that causes the degeneracy and distortion. We give a justification for non-Hermitian quantum Rényi entropy being negative. By exploring the mathematics and physical meaning of the negative entropy in open quantum systems, we connect negative non-Hermitian quantum Rényi entropy and negative quantum conditional entropy, paving the way to rigorously investigate negative entropy in open quantum systems. Full article

The study of dynamic singularity formation in spacetime, focusing on scalar field collapse models, is analyzed. We revisit key findings regarding open spatial topologies, concentrating on minimal conditions necessary for singularity and apparent horizon formation. Moreover, we examine the stability of initial data in the dynamical system governed by Einstein’s equations, considering variations in parameters that influence naked singularity formation. We illustrate how these results apply to a family of scalar field models, concluding with a discussion on the concept of genericity in singularity studies. Full article

With the continuous development and progress of wireless self-organizing network communication technology, how to carry out long-distance cooperative control of multiple intelligences under the framework of an air–ground integrated wireless high-mobility self-organizing network has become a hot and difficult topic that needs to be solved urgently. This paper takes the air–ground integrated wireless high-mobility self-organizing network system as the basic framework and focuses on solving the long-distance cooperative fault-tolerant control of multi-intelligent bodies and the topological stability of a wireless mobile self-organizing network. To solve the above problems, a direct neural network with a robust adaptive fault-tolerant controller is designed in this paper. By constructing a symmetric population neural network model and combining it with the Lyapunov stabilization criterion, the system feedback matrix K has the ability of autonomous adaptive learning, and symmetrically distorts, rotates, or scales the training data to instantly adjust the system’s fault-tolerant corrections and adaptive adjusting factors to resist the unknown disturbances and faults, to achieve the goals of multi-intelligent body stable control and the stable operation of a wireless high-mobility self-organizing network topology. Simulation results show that with the feedback adjustment of the multi-system under the designed controller, the multi-system as a whole has good fault-tolerant performance and autonomous learning approximation performance, and the tracking error asymptotically converges to zero. The experimental results show that the multi-flight subsystems fly stably, the air–ground integrated wireless high-mobility self-organizing network topology has good stability performance, and the maximum relative improvement of the topology stability performance is 50%. Full article

This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements. Full article