Symmetry

In this paper, we first study the uniqueness and symmetry of solution of nonlinear Schrödinger–Kirchhoff equations with constant coefficients. Then, we show the uniqueness of the solution of nonlinear Schrödinger–Kirchhoff equations with the polynomial potential. In the end, we investigate the asymptotic behaviour of the positive least energy solutions to nonlinear Schrödinger–Kirchhoff equations with vanishing potentials. The vanishing potential means that the zero set of the potential is non-empty. The uniqueness results of Schrödinger equations and the scaling technique are used in our proof. The elliptic estimates and energy analysis are applied in the proof of the asymptotic behaviour of the above Schrödinger–Kirchhoff-type equations. Full article

We consider the backward shift operator on a sequence Banach space in the context of two infinite-dimensional phenomena: the existence of topologically transitive operators, and the existence of entire analytic functions of the unbounded type. It is well known that the weighted backward shift (for an appropriated weight) is topologically transitive on $1\le p<\infty$ and on $c_0.$ We construct some generalizations of the weighted backward shift for non-separable Banach spaces, which remains topologically transitive. Also, we show that the backward shift, in some sense, generates analytic functions of the unbounded type. We introduce the notion of a generator of analytic functions of the unbounded type on a Banach space and investigate its properties. In addition, we show that, using this operator, one can obtain a quasi-extension operator of analytic functions in a germ of zero for entire analytic functions. The results are supported by examples. Full article

In this research, random censoring is employed as a methodology for parameter estimation within the context of an exponential distribution. These parameter estimations are conducted using both the Bayesian and maximum likelihood approaches. In the Bayesian framework, Lindley’s approximation method is applied to derive estimates, which are subsequently assessed under three distinct balanced loss functions. To gauge the efficacy of different estimation techniques, simulation-based investigations are conducted. Additionally, a real-world data analysis is executed to illustrate the practical applicability of these methodologies. The findings consistently underscore the superiority of Bayesian parameter estimates in comparison with their maximum likelihood counterparts across all analyzed methodologies. Full article

In recent years, with the variety of digital objects around us becoming a source of information, the fields of artificial intelligence (AI) and machine learning (ML) have experienced very rapid development. Processing and converting the information around us into data within the framework of the information processing theory is important, as AI and ML techniques need large amounts of reliable data in the training and validation stages. Even though information naturally contains uncertainty, information must still be modeled and converted into data without neglecting this uncertainty. Mathematical techniques, such as the fuzzy theory and the intuitionistic fuzzy theory, are used for this purpose. In the intuitionistic fuzzy theory, membership and non-membership functions are employed to describe intuitionistic fuzzy sets and intuitionistic fuzzy numbers (IFNs). IFNs are characterized by the mathematical statements of these two functions. A more general and inclusive definition of IFN is always a requirement in AI technologies, as the uncertainty introduced by various information sources needs to be transformed into similar IFNs without neglecting the variety of uncertainty. In this paper, we proposed a general and inclusive mathematical definition for IFN and called this IFN a non-linear pentagonal intuitionistic fuzzy number (NLPIFN), which allows its users to maintain variety in uncertainty. We know that AI technology implementations are performed in computerized environments, so we need to transform the IFN into a crisp number to make such IFNs available in such environments. Techniques used in transformation are called defuzzification methods. In this paper, we proposed a short-cut formula for the defuzzification of a NLPIFN using the intuitionistic fuzzy weighted averaging based on levels (IF-WABL) method. We also implemented our findings in the minimum spanning tree problem by taking weights as NLPIFNs to determine the uncertainty in the process more precisely. Full article

To accurately model user preference information and ensure the symmetry or similarity between real user preference and the estimated value in product optimization design, an interactive estimation of a distribution algorithm integrated with surrogate-assisted fitness evaluation (SAF-IEDA) is proposed in this paper. Firstly, taking the evaluation information of a few individuals as training data, a similarity evaluation method between decision variables is proposed. Following that, a preference probability model is built to estimate the distribution probability of decision variables. Then, the preference utility function of individuals is defined based on the similarity of decision variables. Finally, the surrogate-assisted fitness evaluation is realized by optimizing the weight of the decision variables’ similarities. The above strategies are incorporated into the interactive estimation of the distribution algorithm framework and applied to address the optimal product design problem and the indoor lighting optimization problem. The experimental results demonstrate that the proposed method outperforms the comparative method in terms of search efficiency and fitness prediction accuracy. Full article

This paper introduces a novel collocation scheme based on an extended cubic B-spline for approximating the solution of a second-order partial integro-differential equation. The proposed scheme employs new extended cubic B-splines to discretize the second-order derivatives in the spatial domain, while discretization of spatial derivatives of lower orders is achieved using extended cubic B-spline functions. Temporal derivatives are discretized using the forward difference formula. The stability of the algorithm is assessed using the von Neumann stability method to ensure that error magnification is avoided. Furthermore, convergence analysis of the scheme is provided. Numerical experiments are conducted to validate the effectiveness and efficiency of the proposed scheme. The free parameter is optimized using $L_2$ and $L_{\infty }$ norms. The computed results are compared with those obtained from various standard numerical schemes found in the literature. Mathematical 12 is used to obtain numerical results. Full article

Inconsistent lighting phenomena in digital images, such as underexposure and overexposure, pose challenges in computer vision. Many studies have developed to address these issues. However, most of these techniques cannot remedy both exposure problems simultaneously. Meanwhile, existing methods that claim to be capable of handling these cases have not yielded optimal results, especially for images with blur and noise distortions. Therefore, this study proposes a system to improve underexposed and overexposed photos, consisting of two different residual attention convolution networks with the CIELab color space as the input. The first model working on the L-channel (luminance) is responsible for recovering degraded image illumination by using residual memory block networks with self-attention layers. The next model based on dense residual attention networks aims to restore degraded image colors using ab-channels (chromatic). A properly exposed image is produced by fusing the output of these models and converting them to RGB color space. Experiments on degraded synthetic images from two public datasets and one real-life exposure dataset demonstrate that the proposed system outperforms the state-of-the-art algorithms in optimal illumination and color correction outcomes for underexposed and overexposed images. Full article

To alleviate the data imbalance problem caused by subjective and objective factors, scholars have developed different data-preprocessing algorithms, among which undersampling algorithms are widely used because of their fast and efficient performance. However, when the number of samples of some categories in a multi-classification dataset is too small to be processed via sampling or the number of minority class samples is only one or two, the traditional undersampling algorithms will be less effective. In this study, we select nine multi-classification time series datasets with extremely few samples as research objects, fully consider the characteristics of time series data, and use a three-stage algorithm to alleviate the data imbalance problem. In stage one, random oversampling with disturbance items is used to increase the number of sample points; in stage two, on the basis of the latter operation, SMOTE (synthetic minority oversampling technique) oversampling is employed; in stage three, the dynamic time-warping distance is used to calculate the distance between sample points, identify the sample points of Tomek links at the boundary, and clean up the boundary noise. This study proposes a new sampling algorithm. In the nine multi-classification time series datasets with extremely few samples, the new sampling algorithm is compared with four classic undersampling algorithms, namely, ENN (edited nearest neighbours), NCR (neighborhood cleaning rule), OSS (one-side selection), and RENN (repeated edited nearest neighbors), based on the macro accuracy, recall rate, and F1-score evaluation indicators. The results are as follows: of the nine datasets selected, for the dataset with the most categories and the fewest minority class samples, FiftyWords, the accuracy of the new sampling algorithm was 0.7156, far beyond that of ENN, RENN, OSS, and NCR; its recall rate was also better than that of the four undersampling algorithms used for comparison, corresponding to 0.7261; and its F1-score was 200.71%, 188.74%, 155.29%, and 85.61% better than that of ENN, RENN, OSS, and NCR, respectively. For the other eight datasets, this new sampling algorithm also showed good indicator scores. The new algorithm proposed in this study can effectively alleviate the data imbalance problem of multi-classification time series datasets with many categories and few minority class samples and, at the same time, clean up the boundary noise data between classes. Full article

In this study, we construct symmetric explicit symplectic schemes for the non-rotating Konoplya and Zhidenko black hole spacetime that effectively maintain the stability of energy errors and solve the tangent vectors from the equations of motion and the variational equations of the system. The fast Lyapunov indicators and Poincaré section are calculated to verify the effectiveness of the smaller alignment index. Meanwhile, different algorithms are used to separately calculate the equations of motion and variation equations, resulting in correspondingly smaller alignment indexes. The numerical results indicate that the smaller alignment index obtained by using a global symplectic algorithm is the fastest method for distinguishing between regular and chaotic cases. The smaller alignment index is used to study the effects of parameters on the dynamic transition from order to chaos. If initial conditions and other parameters are appropriately chosen, we observe that an increase in energy E or the deformation parameter $\eta$ can easily lead to chaos. Similarly, chaos easily occurs when the angular momentum L is small enough or the magnetic parameter Q stays within a suitable range. By varying the initial conditions of the particles, a distribution plot of the smaller alignment in the X–Z plane of the black hole is obtained. It is found that the particle orbits exhibit a remarkably rich structure. Researching the motion of charged particles around a black hole contributes to our understanding of the mechanisms behind black hole accretion and provides valuable insights into the initial formation process of an accretion disk. Full article

This investigation is related to this study of entropy generation during Carreau nanofluid flow under variable thermal conductivity conditions. The heat and mass transfer phenomena are observed in the presence of thermal radiation and activation energy. The flow is induced by a porous stretching surface. Appropriate variables are used in order to simplify the problem into dimensionless form. The numerical simulations are performed by using the shooting technique. The physical aspects of the problem in view of different flow parameters are reported. It is observed that consideration of variable fluid thermal conductivity enhances heat transfer. An enhancement in heat and mass transfer phenomena is observed with increasing the Weissenberg number. Moreover, entropy generation increases with Weissenberg and Brinkman numbers. Current results present applications in thermal processes, heat exchangers, energy systems, combustion and engine design, chemical processes, refrigeration systems, etc. Full article

This work considers the influence of concrete creep on track irregularities and establishes the dynamic motion equation of the train-track-bridge coupling system. The track irregularity is obtained by superposition of the initial geometric irregularity and additional geometric irregularity of the steel rail caused by creep. When high-speed railway trains pass through bridges; the vertical acceleration and vertical displacement of continuous beam bridges are related to the train’s operating speed, and the influence of creep camber is relatively small. At the same time, considering the randomness of track irregularities, the dynamic responses of the train track bridge coupling system under the action of random track irregularities are analyzed, and the dynamic responses of trains at different operating speeds are obtained. The deterministic and uncertain dynamic responses of the train track bridge system were compared and analyzed to verify the accuracy of the Karhunen Loéve expansion (KLE)-Point estimate method (PEM) calculation results. The results indicate that the random characteristics of track irregularities have a significant impact on train dynamic response. Based on the random system vibration analysis and considering the safety and comfort indicators of high-speed railway trains, the creep deformation limit of a continuous beam bridge with a length of 48 m + 80 m + 48 m is obtained to be 19 mm. This is the first time that the dynamic responses of train-symmetry-bridge system are calculated by considering concrete creep and the creep-induced track irregularity, which has certain significance for understanding the dynamics of train -bridge system. In addition, the proposed creep threshold is also of great significance to ensure the safety of traveling. Full article

This paper proposes a novel robust model predictive control (RMPC) scheme for constrained linear discrete-time systems with bounded disturbance. Firstly, the adjustable error tube set, which is affected by local error and error variety rate, is introduced to overcome uncertainties and disturbances. Secondly, the auxiliary control rate associated with the cost function is designed to minimize the discrepancy between the actual system and the nominal system. Finally, a constrained deep neural network (DNN) architecture with symmetry properties is developed to address the optimal control problem (OCP) within the constrained system while conducting a thorough convergence analysis. These innovations enable more flexible adjustments of state and control tube cross-sections and significantly improve optimization speed compared to the homothetic tube MPC. Moreover, the effectiveness and practicability of the proposed optimal control strategy are illustrated by two numerical simulations. In practical terms, for 2-D systems, this approach achieves a remarkable 726.23-fold improvement in optimization speed, and for 4-D problems, it demonstrates an even more impressive 7218.07-fold enhancement. Full article

The present paper examines the plane wave’s reflection in a semiconducting magneto-thermoelastic rotating nonlocal half-space medium, which is stress-free, thermally insulated, and with a diffusion transport process at the boundaries. The novel mathematical model of extended three-phase-lag (TPL) heat transfer law with memory-dependent derivative (MDD) and two temperatures (2T) is used to model this problem. The investigated 2D model shows that a longitudinal wave, when striking the surface z = 0, produces four reflected waves. The characteristics of the plane wave such as phase velocity, amplitude ratios, penetration depth, attenuation-coefficients-specific loss, and energy ratios of various reflected waves are obtained. The symmetric and asymmetric tensor representations of all physical quantities are used. The effects of various theories of thermoelasticity, two temperatures, and rotation on wave characteristics are illustrated graphically using MATLAB software. Full article

In this paper, we investigate the behavior of dust ion acoustic solitary waves (DIASWs) with arbitrary amplitudes in a magnetized anisotropic dusty plasma that includes inertial hot ion fluid, electrons following a Kappa distribution, and negatively charged dust particles in the background. An ambient magnetic field aligns with the x-direction, while the wave propagation occurs obliquely to the ambient magnetic field. In the linear regime, two distinct modes, namely fast and slow modes, are observed. We employ the Sagdeev pseudo-potential method to analyze the fundamental properties of arbitrary amplitude DIASWs. Additionally, we examine how various physical parameters influence the existence and characteristics of symmetric planar dust ion acoustic solitary structures (DIASs). The characteristics of the solitary structures are greatly influenced by the dust concentration, the electrons superthermality (spectral) index, the obliquity parameter, the magnetic field, the parallel ion pressure and the perpendicular ion pressure. The results show that the amplitude and width of both compressive and rarefactive DIASWs are sensitive to the degree of electron superthermality and dust concentration. Additionally, it is shown that the propagation features of DIASWs are highly affected by the parallel component of ion pressure as compared to perpendicular component of ion pressure. Full article

We use power series with rational exponents to find exact solutions to initial value problems for fractional differential equations. Certain problems that have been previously studied in the literature can be solved in a closed form, and approximate solutions are derived by constructing recursions for the relevant expansion coefficients. Full article

In this manifestation, we explain the geometrisation of $\eta$-Ricci–Yamabe soliton and gradient $\eta$-Ricci–Yamabe soliton on Riemannian submersions with the canonical variation. Also, we prove any fiber of the same submersion with the canonical variation (in short $CV$) is an $\eta$-Ricci–Yamabe soliton, which is called the solitonic fiber. Also, under the same setting, we inspect the $\eta$-Ricci–Yamabe soliton in Riemannian submersions with a $\phi (\mathcal{Q})$-vector field. Moreover, we provide an example of Riemannian submersions, which illustrates our findings. Finally, we explore some applications of Riemannian submersion along with cohomology, Betti number, and Pontryagin classes in number theory. Full article

In this study, a target-network update of deep reinforcement learning (DRL) based on mutual information (MI) and rewards is proposed. In DRL, updating the target network from the Q network was used to reduce training diversity and contribute to the stability of learning. If it is not properly updated, the overall update rate is reduced to mitigate this problem. Simply slowing down is not recommended because it reduces the speed of the decaying learning rate. Some studies have been conducted to improve the issues with the t-soft update based on the Student’s-t distribution or a method that does not use the target-network. However, there are certain situations in which using the Student’s-t distribution might fail or force it to use more hyperparameters. A few studies have used MI in deep neural networks to improve the decaying learning rate and directly update the target-network by replaying experiences. Therefore, in this study, the MI and reward provided in the experience replay of DRL are combined to improve both the decaying learning rate and the target-network updating. Utilizing rewards is appropriate for use in environments with intrinsic symmetry. It has been confirmed in various OpenAI gymnasiums that stable learning is possible while maintaining an improvement in the decaying learning rate. Full article

We introduce a systematic way to obtain expressions for computing the amount of fundamental quantities such as helicity and angular momentum contained in static matter, given its charge and magnetization densities. The method is based on a scalar product that we put forward which is invariant under the ten-parameter conformal group in three-dimensional Euclidean space. This group is obtained as the static restriction (frequency $\omega =0$) of the symmetry group of Maxwell equations: The fifteen-parameter conformal group in 3+1 Minkowski spacetime. In an exemplary application, we compute the helicity and angular momentum squared stored in a magnetic Hopfion. Full article

We consider the Banach space $C[0,1]$, which is a symmetric Banach space, and prove the existence and approximation of numerical solutions for a broad class of third-order BVPs. Our approach is based on an integral operator that is constructed using Green’s function. The Banach contraction principle (BCP) is applied to guarantee a unique solution to our problem. Moreover, in order to find the value of the numerical solution, this new operator is embedded within the three-step Noor iterative scheme; we named this new iterative scheme the Noor–Green iterative scheme. We provide a convergence theorem for the proposed scheme by employing suitable restrictions on the parameters involved in the problem and in the scheme. The results of the stability of our scheme are also reported. It is worth mentioning that unlike the concept of stability in the classical sense, our result for stability is based on the concept of weak $w^2$ stability. In order to support our findings, we carried out various numerical experiments using different third-order BVPs. Finally, we report on the application of our iterative scheme to solve a class of fractional BVPs in the same symmetric Banach space. Our results are essentially new in the present literature and extend several of the results found in the current literature. Full article