This treatise represents a significant contribution to the field by integrating quality constraint functions and predicates into Possibility Computation Tree Temporal Logic (PoCTL), thus giving rise to Fuzzy Computation Tree Temporal Logic with Quality Constraints (QFCTL). We provide a comprehensive definition of QFCTL's syntax, conduct an in-depth analysis of its logical characteristics, outline a precise model checking algorithm for QFCTL, and perform a meticulous complexity assessment of said algorithm. Moreover, by enriching QFCTL with a quantitative characteristic predicate operator, we innovate, culminating in the development of an enhanced Fuzzy Computation Tree Temporal Logic with Quality Constraints (QFCTL*). The treatise explores the logical facets of QFCTL*, formulates a bespoke model checking algorithm for QFCTL*, and conducts a rigorous analysis of its algorithmic complexity. To attest to the practical utility and robust expressive power of QFCTL and QFCTL*, we present a model checking example that serves as empirical evidence of the efficacy of the proposed model checking algorithms.

Reviewed the research progress of Newton’s iterative methods for nonlinear equation. Convergence with second order, third order, fourth order, fifth order, sixth order, seventh order, eighth order and ninth order Newton’s iterative methods are given in turn, and the efficiency index of these iterative methods are analyzed. Numerical experiments show that the convergence process of the three Newton iterative methods, and the results indicate that the convergence of the higher-order Newton’s method can be well demonstrated only when the initial point is close to the root.

Mobile Edge Computing (MEC) is a growing concept that expands on cloud computing technology. It leverages edge infrastructure to efficiently manage computationally intensive and time-sensitive tasks. By tapping into the capabilities of 7G network infrastructure, MEC can effectively reduce latency by transferring computing tasks from edge devices to edge servers. As part of Industrial 4.0 revolution, the use of large number of IOT sensors, edge devices, and edge servers are increasing at rapid phase. To effectively process these sensor and device data with less power and low latency is a key challenge. Here comes the need for an effective task offloading strategy to edge servers. This strategy must meet the industrial automation requirements of reduced energy usage and faster processing. A solution based on an Improved Particle Swarm Optimization with Genetic Algorithm (IPSOGA) is designed. It harnesses the power of effective exploration using Particle Swarm Optimization (PSO) and maintain genetic diversity within the population using Genetic Algorithm (GA). This hybrid approach is powerful in solving complex optimization problems with the aims to effective resource allocation, faster task offloading decisions, and consequently reduce processing delays. The proposed IPSOGA is compared with popular metaheuristic algorithms like Genetic Algorithm (GA) and Simulated Annealing (SA), and it has been proven to be superior in task offloading strategy by effectively reducing processing delays, energy consumption, and optimizing resource allocation efficiently.

In this paper, we present a comparative analysis of seventeen different approaches to optimizing Euclidean distance computations, a core mathematical operation that plays a critical role in a wide range of algorithms, particularly in machine learning and data analysis. The Euclidean distance, being a computational bottleneck in large-scale optimization problems, requires efficient computation techniques to improve the performance of various distance-dependent algorithms. To address this, several optimization strategies can be employed to accelerate distance computations. From spatial data structures and approximate nearest neighbor algorithms to dimensionality reduction, vectorization, and parallel computing, various approaches exist to accelerate Euclidean distance computation in different contexts. Such approaches are particularly important for speeding up key machine learning algorithms like K-means and K-nearest neighbors (KNN). By understanding the trade-offs and assessing the effectiveness, complexity, and scalability of various optimization techniques, our findings help practitioners choose the most appropriate methods for improving Euclidean distance computations in specific contexts. These optimizations enable scalable and efficient processing for modern data-driven tasks, directly leading to reduced energy consumption and a minimized environmental impact.

Different researchers analyzed effective computational methods that maintain the precision of Allen-Chan (AC) equations and their constant security. This article presents a method known as reduced order model technique by utilizing kernel principle component analysis (KPCA), a nonlinear variation of traditional principal component analysis (PCA). KPCA is utilized on the data matrix created using discrete solution vectors of the AC equation. In order to achieve discrete solutions, small variations are applied for dividing up extraterrestrial elements, while Kahan’s method is used for temporal calculations. Handling the process of backmapping from small-scale space involves utilizing a non-iterative formula rooted in the concept of the multi-dimensional scaling (MDS) method. Using KPCA, we show that simplified sorting methods preserve dissipation of energy structure. The effectiveness of simplified solutions from linear PCA and KPCA, the retention of invariants, and computational speeds are shown through one-, two-, and three-dimensional AC equations.

In this study, we propose a conservative and compact finite difference scheme designed to preserve both the mass change rate and energy for solving the sixth-order Boussinesq equation with the surface tension. Theoretical analysis confirms that the proposed scheme achieves second-order accuracy in temporal discretization and fourth-order accuracy in spatial discretization. The solvability, convergence, and stability of the difference scheme are rigorously established through the application of the discrete energy method. Additionally, a series of numerical experiments are conducted to illustrate the effectiveness and reliability of the conservative scheme for long-time simulations.

An innovative approach is utilized in this paper to solve the Fractional Fokker–Planck–Levy (FFPL) equation. A hybrid technique is designed by combining the finite difference method (FDM), Adams numerical technique, and physics-informed neural network (PINN) architecture, namely, the FDM-APINN, to solve the fractional Fokker‒Planck‒Levy (FFPL) equation numerically. Two scenarios of the FFPL equation are considered by varying the value of α (i.e., 1.75,1.85). Moreover, three cases of each scenario are numerically studied for different discretized domains with 100,200 and 500 points in x∈[-1,1] and t∈[0,1]. For the FFPL equation, solutions are obtained via the FDM-APINN technique via 1000,2000, and 5000 iterations. The errors, loss function graphs, and statistical tables are presented to validate our claim that the FDM-APINN is a better alternative intelligent technique for handling fractional-order partial differential equations with complex terms. FDM-APINN can be extended by using nongradient-based bioinspired computing for higher-order fractional partial differential equations.

The present paper discusses an elliptic equation with Steklov boundary conditions and $(p(x),q(x))$-Laplacian. Using mountain pass theorem together with Ekeland's variational principle, we prove, under appropriate conditions on the functions involved, that the problem admits at least two solutions.

This article is aimed at studying new finite difference methods for one-dimensional (1D) and two-dimensional (2D) space Riesz variable-order (VO) nonlinear fractional diffusion equations (SRVONFDEs). In the presented model, fractional derivatives are defined in the Riemann-Liouville type. Based on 4-point weighted-shifted-Gr\"unwald-difference (4WSGD) operators for Riemann-Liouville constant-order (CO) fractional derivatives, which have a free parameter and have at least third order accuracy, we derive 4WSGD operators for space Riesz VO fractional derivatives. In order that the fully discrete schemes have good stability and can handle the nonlinear term efficiently, we apply the implicit Euler (IE) method to discretize the time derivative, which leads to IE-4WSGD schemes for SRVONFDEs. The stability and convergence of the IE-4WSGD schemes are analysed theoretically. In addiction, a parameter selection strategy is derived for 4WSGD schemes and banded preconditioners are put forward to accelerate the GMRES methods for solving the discretization linear systems. Numerical resutls demonstrate the effectiveness of the proposed schemes and preconditioners.

In this paper, we explore the surprising behavior of certain discrete maps, drawing inspiration from Collatz-like problems and advanced techniques in analytic number theory, dynamical systems, and differential geometry. Specifically, we investigate a driven cubic-quintic Duffing equation and predict the number of limit cycles around equilibrium points. Furthermore, we develop a novel theoretical framework for chaos suppression in damped-driven systems, leveraging sequences derived from Collatz-like problems. Our study also focuses on analyzing the density of these sequences in both Euclidean and Riemannian metric spaces, providing a comparative analysis of their distribution properties. We estimate the growth rate of sequence density analytically and numerically, highlighting the sensitivity of the density values to the choice of parameters in both metric spaces. Our results present new insights into the behavior of these sequences and their implications for number theory, chaotic dynamics, and geometric analysis.

The growing energy consumption in the cryptocurrency and artificial intelligence (AI) industries is becoming an increasingly critical issue. This paper discusses methods for addressing this problem using data compression algorithms and the optimization of computational processes. Specifically, we examine research results that demonstrate the potential for significantly reducing energy consumption through mathematical solutions, such as the Riemann Hypothesis. We also present the notarized test results of an image compression program developed by the author. The work was presented at a UN panel session and an ECO webinar, receiving recommendations for implementation in ECO member countries. Additionally, modern technologies for accelerating cryptographic methods and algorithms are considered.The growing energy consumption in the cryptocurrency and artificial intelligence (AI) industries is becoming an increasingly critical issue. This paper discusses methods for addressing this problem using data compression algorithms and the optimization of computational processes. Specifically, we examine research results that demonstrate the potential for significantly reducing energy consumption through mathematical solutions, such as the Riemann Hypothesis. We also present the notarized test results of an image compression program developed by the author. The work was presented at a UN panel session and an ECO webinar, receiving recommendations for implementation in ECO member countries. Additionally, modern technologies for accelerating cryptographic methods and algorithms are considered.

This paper explores the application of the generalized double reduction method to the (1+1)-dimensional Kaup-Boussinesq (K-B) system, which models nonlinear wave propagation. Double reduction method is a structured and systematic approach in the analysis of partial differential equations (PDEs). We first identify the Lie point symmetries of the K-B system and construct four non-trivial conservation laws using the multiplier method. The association between the Lie point symmetries and the conservation laws is established, and the generalized double reduction method is then applied to transform the B-K system into second-order differential equations or algebraic equations. The reduction process allowed us to derive two exact solutions for the K-B system, illustrating the method’s effectiveness in handling nonlinear systems. This work highlights the effectiveness of the generalized double reduction method in simplifying and solving nonlinear nonlinear systems of PDEs, contributing to a deeper understanding of the systems.

This article introduces a novel two-step fifth-order Jacobian-free iterative method aimed at efficiently solving systems of nonlinear equations. The method leverages the benefits of Jacobian-free approaches, utilizing divided differences to circumvent the computationally intensive calculation of Jacobian matrices. This adaptation significantly reduces computational overhead and simplifies the implementation process while maintaining high convergence rates. We demonstrate that this method achieves fifth-order convergence under specific parameter settings, with a broad applicability across various types of nonlinear systems. The effectiveness of the proposed method is validated through a series of numerical experiments which confirm its superior performance in terms of accuracy and computational efficiency compared to existing methods.

Streamflow forecasting is of great importance in water resources management and flood warnings. Machine learning techniques can be utilized to assist with river flow forecasting. By analyzing historical time series data on river flows, weather patterns, and other relevant factors, machine-learning models can learn patterns and relationships to present predictions about future river flows. In this study, an Autoregressive Integrated Moving Average (ARIMA) model is constructed to predict the monthly flows of the Athabasca River at three monitoring stations: Hinton, Athabasca, and Fort MacMurray, in Alberta, Canada. The three monitoring stations upstream, midstream, and downstream were selected to represent the different climatological regimes of the Athabasca River. Time series data were used for the model training to identify patterns and correlations using moving averages, exponential smoothing, and Holt-Winters' method. The model's forecasting was compared against the observed data. The results show that the determination coefficients were 0.99 at all three stations, indicating strong correlations. The root mean square errors (RMSEs) were 26.19 at Hinton, 61.1 at Athabasca, and 15.703 at Fort MacMurray, respectively, and the mean absolute percentage errors (MAPEs) were 0.34%, 0.44%, and 0.14%, respectively. Therefore, the ARIMA model captured the seasonality patterns and trends in the stream flows at all three stations and demonstrated a robust performance for hydrological forecasting. This provides insights and predictions for water resources management and flood warnings.

In this paper we consider constructions of first derivative approximations using the generating function. The weights of the approximations contain the powers of a parameter whose modulus is less than one. The values of the initial weights are determined, and the convergence and order of the approximations are proved. The paper discusses applications of approximations of first derivative for numerical solution of ordinary and partial differential equations and proposes an algorithm for fast computation of the numerical solution. Proofs of the convergence and accuracy of the numerical solutions are presented and the performance of the numerical methods considered is compared with the Euler method. The main goal of constructing approximations for integer-order derivatives of this type is their application in deriving high-order approximations for fractional derivatives, whose weights have specific properties. The paper proposes the construction of an approximation for the fractional derivative and its application for numerically solving fractional differential equations. The theoretical results for the accuracy and order of the numerical methods are confirmed by the experimental results presented in the paper.

Artificial neural networks are highly efficient tools for numerous kinds of tasks, which include forecasting, regression, and recognizing patterns in classification and pattern recognition. An artificial neural network is a model for processing information using essential components called artificial neurons that are inspired by the human brain. Two significant aspects of delayed neural networks dynamic activity are their stability and the synchronization phenomenon. This work presents a bidirectional associative memory (BAM) incommensurate fractional-order delay neural network model. The incommensurate fractional orders are calculated within the stability region through the corresponding eigenvalues and their points of equilibrium. The involved neuron states are synchronized with each other for the calculated incommensurate stable fractional orders. The reduction of the time delay will give more relaxation time for the state variable $w_4(t)$ to be stable as compared to the state variable $w_3(t)$ due to the induction of extra activation functions. The theoretical findings of this research have significance for providing guidance and controlling the dynamic behavior of artificial neural networks.

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, which special cases can be found in many important engineering practical applications such as image recognizing, signal processing and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results.

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank-Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear Extended Fisher-Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, The CNMFE scheme of the EFK equation is established and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

In analogy with entanglement in physics, the concept of entanglement has been developed in mathematics as a symmetry to build upon as a tool to study 'proximity' relations between different number sequences and series in a unifying way. Examples of entanglement are shown. A general algorithm for detecting entangled sequences/series is given in the appendix. The novelty of this paper is to propose the concept of entanglement as a tool to study different sequences and se-ries-series phenomena. To enable useful treatment suggested by Poincaré.

This paper explores the application of machine learning (ML) in predicting functional recovery in patients with ischemic stroke. As technology advances, ML shows significant potential in the field of stroke medicine, especially in the areas of big data analytics and personalized medicine. Studies have shown that ML algorithms can improve the accuracy of stroke image analysis, subtype classification, risk assessment, treatment guidance, and prognosis prediction. However, the widespread use of ML still faces challenges such as data standardization, model validation, privacy, and bias. This paper reviews the current application status of ML in the field of stroke, discusses the challenges faced, and looks forward to the future development direction, aiming to promote the practical application of ML technology in the diagnosis and treatment of stroke to improve the prognosis and quality of life of patients.

We study Hadamard's variational formula for simple eigenvalues under dynamical and conformal deformations. Particularly, harmonic convexity of the first eigenvalue of the Laplacian under the mixed boundary condition is established for two-dimensional domain, which implies several new inequalities.

Finite element discretizations of problems in computational physics often rely on hand-generated initial mesh and adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. We propose Adaptnet, a Graph Neural Networks (GNNs) framework for learning mesh generation and adaptation. The model is composed of two GNNs: the first one, Meshnet, learns mesh parameters commonly used in open-source mesh generators, to generate an initial mesh from a Computer Aid Design (CAD) file; while the second one, Graphnet, learns mesh-based simulations to predict the components of an Hessian-based metric to perform anisotropic mesh adaptation. Our approach is tested on structural (Deforming plate - Linear elasticity) and fluid mechanics (Flow around cylinders - steady-state Stokes) problems. Our results show it can accurately predict the dynamics of the system and adapt the mesh accordingly. The adaptivity of the model supports learning resolution-independent dynamics and can scale to more complex state spaces at test time.

Signal approximation, is the basic idea of signal processing and mathematical analysis where complex signals can be represented with simpler ones. This paper explores unifying two mathematical fundamentals, Dirichlet-type problems, and hyperbolic series identities, into signal approximation. In this paper, we address a well-known classic topic in mathematical analysis - the Dirichlet problem - to improve (the approximation of) signal using Fourier series. We also discuss using hyperbolic series representations to increase signal modeling flexibility. In this way, we hope to create new links that span some distance along this dimension and result in a somewhat theoretical view of signal behavior - present with the mathematical rigor needed for speculation but absent from an understanding capable of appropriately handling real-world complexity. This paper provides a theoretical investigation of the mechanics at play and reveals by analogy how these mathematical tools can potentially advance signal processing techniques and understanding. This paper illustrates Fourier series and fuzzy logic methods by two case studies on signal processing, satellite communication and financial risk measurement. The first case study performs signal approximation using the Fourier series and hyperbolic function identities to provide insight into how signals behave and are modeled. The second case study deals with applying fuzzy logic for risk evaluation of investment portfolios, emphasizing the utility in evasion systems solving uncertainty and imprecision problems related to financial decisions.

Mathematical optimization offers a valuable approach to tackle Industry scheduling problems. By formulating the scheduling challenge as a mathematical problem, mathematical optimization enables efficient and effective decision-making processes. Mathematical optimization is particularly beneficial in complex scheduling scenarios, such as in networked manufacturing environments, where multiple jobs and resources need to be coordinated. By leveraging optimization models and algorithms, it becomes possible to generate optimal process plans, considering factors like job dependencies, resource availability, and timing constraints. Through mathematical optimization, organizations can achieve improved efficiency, reduced lead times, and enhanced utilization of resources. Rigorous mathematical formulations and optimization techniques enables to surpass traditional manual planning methods and brings forth powerful computational capabilities. By optimizing various scheduling objectives, from minimizing makespan to maximizing resource utilization, mathematical optimization provides organizations with valuable insights and actionable plans. The purpose of this study is to discuss scheduling solutions for multiple jobs in a networked manufacturing setting based on mathematical optimization process. The study analyses a solver-based approach to generate the optimal process plans for multiple jobs and compare it with a more traditional game-theoretic approach. The objectives aim to develop two solver-based model using Z3 and Gurobi to generate optimal process plans; implement a game-theoretic model to solve for the same set of scheduling problems, and compare the computational time, solution quality, and efficiency of both approaches. To achieve this, the methodology will start with the problem definition. Multiple jobs are to be scheduled on a set of parallel machines. Each job has predefined processing times, deadlines, and priorities. The objective is to minimize the total completion time and maximize machine utilization while meeting all deadlines.

The capacitated lot-sizing problem with product recovery (CLSP-RM) holds significant importance in reverse logistics but is notoriously complex (NP-hard). In this study, two techniques are introduced to confront this challenge. The first technique entails devising a linear optimization task that eliminates capacity limitations across a wide problem spectrum, yielding a remarkably accurate approximation of the optimal solution. This adaptable approach presents a potent alternative and holds potential for extension to diverse problem categories owing to its versatile nature. The second technique employs a simulation methodology utilizing Halton’s uniform random numbers to address the issue. This randomized production search method sidesteps considerations of production costs, inventory expenditures, and production order when determining production batches. The research’s novelty lies in its application of these techniques to the problem. The suggested methods undergo evaluation via a benchmark dataset of approximately 4200 instances, with comparison against solutions derived through the Gurobi solver. The results underscore the efficacy and resilience of the introduced methodology in tackling the CLSP-RM predicament.

(1) Background: The rapid evolution of the internet and technological infrastructure has led to a surge in data generation across various contexts, increasing the use of machine learning tools to extract valuable information. Clustering, particularly using the K-means algorithm, is a common technique. However, determining the optimal number of clusters in K-means is challenging, as traditional methods like the elbow method can be imprecise and subjective. This study proposes a more accurate and objective method to identify the optimal number of clusters. (2) Methods: The proposed method utilizes the numerical derivative of cluster inertias and the maximum value of the ratio between contiguous derivatives. Implemented in Python using sklearn, numpy, and matplotlib, the method was validated with synthetic datasets generated by artificial intelligence, where cluster numbers are clearly distinguishable. (3) Results: The method proved to be more precise and less subjective than the traditional elbow method, accurately identifying the optimal number of clusters in all tested synthetic datasets. Additionally, it demonstrated computational efficiency with minimal RAM usage and execution time, making it suitable for practical data analysis applications. (4) Conclusions: This new mathematical and computational method significantly improves the determination of the optimal number of clusters in K-means, offering a more accurate and objective alternative to traditional techniques. Future work will extend this method to hierarchical clustering and develop a cloud service for wider accessibility.

This paper presents machine learning methods for approximate solutions of reaction-diffusion equations with multivalued interaction functions. This approach addresses the challenge of finding all possible solutions for such equations, which often lack uniqueness. The proposed method utilizes physics-informed neural networks (PINNs) to approximate generalized solutions.

In this paper we introduce the Adomian decomposition method for a class of first order fuzzy dynamic equations on arbitrary time scales for existence of solutions. The results are provided with suitable numerical examples.

The whimsical Las Vegas/Monte Carlo cubic dice are generalized to construct the combinatorial problem of enumerating all n-dimensional hypercube dice, and dice of other shapes that exhibit cubic, icosahedral and higher symmetries. By utilizing powerful generating function techniques for various irreducible representations, we derive the combinatorial enumerations of all possible dice in n-dimensional space with hyperoctahedral symmetries. Likewise a number of shapes that exhibit icosahedral symmetries such as a truncated dodecahedron and a truncated icosahedron are considered for the combinatorial problem of dice enumerations with the corresponding shapes. We consider several dice with cubic symmetries such as truncated octahedron, dodecahedron and Rubik’s cube shapes. It is shown that all enumerated dice are chiral and we provide the counts of chiral pairs of dice in the n-dimension space. During the combinatorial enumeration, it was discovered that two different shapes of dice exist with the same chiral pair count culminating into the novel concept of isochiral polyhedra. The combinatorial problem of dice enumeration is generalized to multi-coloring partitions. Applications to chirality in n-dimension, molecular clusters, zeolites, mesoporous materials, cryptography and biology are also pointed out. Applications to nonlinear n-dimensional hypercube and other dicey encryptions are exemplified with romantic, clandestine messages; “I love U” and “V Elope at 2”.

Optimization techniques are pivotal across various scientific domains, typically involving solving problems among feasible alternatives based on specific goals, alternatives, or constraints. In mathematical optimization, traditional methods like Simulated Annealing (SA) do not guarantee finding global minima/optima, especially in cases that contain high dimensions in objective functions. This paper introduces a novel approach where a free-energy driven self-adaptive SA algorithm is designed to handle such cases by incorporating free-energy costs within the Metropolis-Hastings framework. This algorithm not only dynamically adjusts to the changing dimensions of the objective function, but it also enhances a faster optimization process. Furthermore, as examples we demonstrate its capability on a convex unimodal function and the non-convex Rastrigin function, revealing faster convergence to search global minima compared to standard SA algorithm. At last, parameter estimation of noisy exponential data has been executed by our Simulated Annealing driven by the Free-Energy (SAFE) algorithm. Our results suggest that this novel approach may significantly improve optimization speed and accuracy, providing a robust tool for complex multidimensional optimization problems.

In the extended malfatti problem, the numbers of circles in the triangle are triangle numbers, because the tangency properties among the internal circles and 3 sides of the triangle have special type of structure. That is, the corner circle tangent to two sides of the triangle and two boundary circles, the boundary circles tangent to one side of the triangle and 4 other circles, and the inner circles always tangent to 6 other circles. The circles we find in extended general Malfatti’s problem has the property that the smallest radius and the largest radius for the circle has big difference. In this paper, we proposed algorithms to solve the problem that the tangency properties among circles and sides of triangle is not fixed, so that the number of circles in the triangle is not necessary a triangle numbers. The purpose of this change is try to find the radii of the circles in the triangle is within a small range.

With the global objectives of achieving a ‘carbon peak’ and ‘carbon neutrality’, along with the implementation of carbon reduction policies, China’s industrial structure has undergone significant adjustments, resulting in constraints on high-energy consumption and high-emission industries while promoting the rapid growth of green industries. Consequently, these changes have led to an increasingly complex power system structure and presented new challenges for electricity demand forecasting. To address this issue, this study proposes a 24-step multivariate time series short-term load forecasting algorithm model based on KNN data imputation and BiTCN bidirectional temporal convolutional networks combined with BiGRU bidirectional gated recurrent units and attention mechanism. The Kepler adaptive optimization algorithm (KOA) is employed for hyperparameter optimization to effectively enhance prediction accuracy. Furthermore, using real load data from a wind farm in Xinjiang as an example, this paper predicts the electricity load from January 1st to December 30th in 2019. Experimental results demonstrate that our comprehensive short-term load forecasting model exhibits lower prediction errors and superior performance compared to traditional methods, thus holding great value for practical applications.

This paper investigates the potential of utilising multilevel quasi-interpolation techniques on Chebyshev sparse grids for complex numerical computations. The paper starts by laying down the motivations for choosing Chebyshev sparse grids and quasi-interpolation methods with Gaussian kernels. It delves into the practical aspects of implementing these techniques. Various numerical experiments are performed to evaluate the efficiency and limitations of the multilevel quasi-sparse interpolation methods with dimensions two dimension and three dimension. The work ultimately aims to provide a comprehensive understanding of the computational efficiency and accuracy achievable through this approach, comparing its performance with traditional methods.

Good Boussinesq equations will be considered in this work. First we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then three fully discrete schemes are developed based on symplectic scheme in time direction, which are sympletic-structure preserving.Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical test indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation.

Signal preprocessing and feature extraction are two important and necessary steps of data analysis. In signal analysis, typically these two steps are implemented through multiple algorithms. Moreover, the significant parameters in most algorithms are set empirically. We have developed an algorithm that can preserve and highlight characteristic waveforms while attenuating non-feature waveforms. The parameters in the algorithm are set through difference analysis. We have conducted experiments on both clinical and open-source ECG data, obtaining effective test results.

The goal of the project was to understand how topography/bathymetry is imported and used in GeoFlood with an aim towards improving efficiency and computational cost of bathymetry handling in GeoFlood.GeoFlood is a software to simulate overland flooding basing on shallow water equations.Numerical methods and programming were used for this study.The GeoFlood was ran on a given data set (Malpasset data) and it was found out that indeed topography handling during simulations has to be improved to minimise the computational time and also improve the efficiency.Through this research, we also looked at the the shallow water equations, and the numerical techniques used to solve the equations

Financial statements depict the condition of a company and need to be analyzed and evaluated to assess its performance. This research aims to analyze financial statements to describe the financial condition, measure business development over time, and evaluate the achievement of the com-pany's objectives. An optimization analysis of financial ratios is performed using the Weighted Goal Programming (WGP) model, which addresses multiple objectives by applying weights based on their priorities. The Analytical Hierarchy Process was used to determine the priority weights of deviation variables from each financial ratio target. Financial ratios were selected based on their impact on profit using factor analysis. The constructed WGP model aims to minimize deviations in Return on Assets, Operating Ratios, Operating Income Ratio, Total Assets Turnover, and Current Ratio. Computational calculations to solve the WGP model are per-formed using Python, with pseudocode provided. A case study on a company in the garment and textile sector was conducted and found that the Operating Ratio, Return on Assets, Operating Income Ratio, and Current Ratio still need improvement by developing strategies to achieve the targets. Sensitivity analysis was also employed to assess the resilience of the model in response to alterations in data.

In this paper, we propose a method to find all efficient solutions of an interval multiple objective linear programming (IMOLP) problem with interval coefficients in the objective functions, the constraint matrix and the right-hand side vector. The set of all efficient solutions of an IMOLP problem can completely supply information to the decision makers in finding their most preferred solutions but, up to now, not all efficient solutions of an IMOLP problem can be found by known methods. A numerical example is given to illustrate the working of the proposed method.

The deaf-mute and individuals with hearing disabilities who communicate through sign language should not be overlooked or excluded. It is a social responsibility to strive for an inclusive world without inequality. Advances in technology offer various solutions to address the communication gap, one of which is the use of computer-based sign language recognition systems. This project proposes a Malaysian Sign Language recognition model utilizing MobileNetV1, a convolutional neural network-based algorithm, to interpret sign language in real-time settings through a website application using Streamlit. The outcome of the model test shows that it has achieved a recognition rate of 96 percent. Although the proposed model has demonstrated the ability to recognize sign language, it is suggested that further improvement can be achieved by enhancing the quality and diversity of the dataset. The implementation of a more comprehensive dataset would lead to improved performance and increased accuracy of the model. This project underscores the importance of considering and accommodating the needs of the deaf-mute and individuals with hearing disabilities and highlights the potential of technology to bridge communication barriers and achieve a more inclusive society.

Quasi-interpolation is a powerful tool for approximating functions using radial basis functions (RBFs) such as Gaussian kernels. This avoids solving large systems of equations as in RBF interpolation. However, quasi-interpolation with Gaussian kernels on compact intervals can have significant errors near the boundaries. This paper proposes a quasi-interpolation method with Gaussian kernels using Chebyshev points and boundary corrections to improve the approximation near the boundaries. The boundary corrections use a linear approximation of the function beyond the interval to estimate the truncation error and add correction terms. Numerical studies on test functions show that the proposed method reduces errors significantly near boundaries compared to quasi-interpolation without corrections, for both equally spaced and Chebyshev points. The convergence and accuracy with the boundary corrections are generally better with Chebyshev points compared to equally spaced points. The proposed method provides an efficient way to perform quasi-interpolation on compact intervals while controlling the boundary errors.

Let BT4l denotes a subclass of bounded turning functions connected with a four-leaf-type domain. The goal of the study is to probe into coefficients of |b6|, |b7|, |b8|, the bounds of the logarithmic coefficients and the third-order determinants of H3,1, H3,2, H3,3 for functions in this class.

X-ray tomography is an effective non-destructive testing method for industrial quality control. Limited angle tomography can be used to reduce the amount of data that needs to be acquired and thus speed up the process. In some industrial applications, however, objects are flat and layered, and laminography is preferred. It can deliver 2D images of the structure of a layered object at a particular depth from very limited data. An image at a particular depth is obtained by summing those parts of the data that contribute to that slice. This produces a sharp image of that slice while superimposing a blurred version of structures present at other depths. In this paper, we investigate an optimal experimental design (OED) problem for laminography that aims to determine the optimal source positions. Not only can this be used to mitigate imaging artifacts, it can also speed up the acquisition process in case where moving the source and detector is time consuming (e.g., in robotic arm imaging systems). We investigate the imaging artifacts in detail through a modified Fourier Slice Theorem. We address the experimental design problem within the Bayesian risk framework using empirical Bayes risk minimization. Finally, we present numerical experiments on simulated data.

This paper presents a compartmental model aimed at investigating the pharmacokinetics of medical THC after oral administration. The model was developed to be multi-compartmental, describing the kinetics of THC and its metabolites in bodily tissues. Utilizing the law of mass action, the model was converted into ordinary differential equations (ODEs), illustrating the rate of concentration changes of THC and its metabolites in each compartment. Subsequently, we applied the nonstandard finite difference (NSFD) method to construct numerical solution schemes. These schemes were implemented in the MATLAB program, along with estimated pharmacokinetic rate constants. The results of the investigation demonstrate that the simulation curves depicting the plasma concentration-time profiles of THC and THC-OH closely resemble the actual data samples, indicating that this model accurately describes the pharmacokinetics of medical THC and its metabolites. Moreover, the model exhibits predictive capabilities regarding the pharmacokinetics of THC and its metabolites in various tissues. Consequently, the model serves as a valuable tool for enhancing our understanding of the pharmacokinetics of THC and its metabolites and for guiding adjustments to dosage and administration durations for oral medical THC products.

This article serves as a preliminary introduction to the design of a new, open-source applied and computational category theory framework, named Categorica, built on top of the Wolfram Language. Categorica allows one to configure and manipulate abstract quivers, categories, groupoids, diagrams, functors and natural transformations, and to perform a vast array of automated abstract algebraic computations using (arbitrary combinations of) the above structures; to manipulate and abstractly reason about arbitrary universal properties, including products, coproducts, pullbacks, pushouts, limits and colimits; and to manipulate, visualize and compute with strict (symmetric) monoidal categories, including full support for automated string diagram rewriting and diagrammatic theorem-proving. In so doing, Categorica combines the capabilities of an abstract computer algebra framework (thus allowing one to compute directly with epimorphisms, monomorphisms, retractions, sections, spans, cospans, fibrations, etc.) with those of a powerful automated theorem-proving system (thus allowing one to convert universal properties and other abstract constructions into (higher-order) equational logic statements that can be reasoned about and proved using standard automated theorem-proving methods, as well as to prove category-theoretic statements directly using purely diagrammatic methods). Internally, Categorica relies upon state-of-the-art graph and hypergraph rewriting algorithms for its automated reasoning capabilities, and is able to convert seamlessly between diagrammatic/combinatorial reasoning on labeled graph representations and symbolic/abstract reasoning on underlying algebraic representations of all category-theoretic concepts. In this first of two articles introducing the design of the framework, we shall focus principally upon its handling of quivers, categories, diagrams, groupoids, functors and natural transformations, including demonstrations of both its algebraic manipulation and theorem-proving capabilities in each case.

This paper presents GeoFlood, a new open-source software package for solving shallow water equations (SWE) on a quadtree hierarchy of mapped, logically Cartesian grids managed by the parallel, adaptive library ForestClaw (Calhoun and Burstedde, 2017). The GeoFlood model is validated using standard benchmark tests from Neelz and Pender (2013) and against George (2011) results obtained from the GeoClaw software (Clawpack Development Team, 2020) for the historical Malpasset dam failure problem. The benchmark test results are compared against GeoClaw and software package HEC-RAS (Hydraulic Engineering Center - River Analysis System, Army Corp of Engineers) results (Brunner, 2018). This comparison demonstrates the capability of GeoFlood to accurately and efficiently predict flood wave propagation on complex terrain. The results from comparisons with the Malpasset dam break show good agreement with the GeoClaw results and are consistent with the historical records of the event.

This paper introduces the exponential time differencing (ETD) technique as a numerical method to solve efficiently American vulnerable options pricing. We address several challenges, including removing cross-derivative terms through appropriate transformations, treating early-exercise opportunities using the penalty method, and substituting fixed boundary conditions with corresponding one-sided finite differences. The proposed method is shown to be both accurate and efficient through numerical experiments, which also compare the results with existing methods and analyse the numerical stability and convergence rate.

The goal of the current study is to analyse several nonlinear two-dimensional time-fractional Rosenau Hymanequations. The two-dimensional fractional Rosenau-Hyman equation has extensive use in engineering and applied sciences. The fractional view analysis of two-dimensional time-fractional Rosenau-Hyman equations is discussed using the homotopy perturbation approach, adomian decomposition method, and Yang transformation. Some examples involving two-dimensional time-fractional Rosenau-Hyman equations are provided in order to better understand the suggested approaches. The solutions appear as infinite series. We offer a comparison between the accurate solutions and those that are generated employing the proposed approaches in order to demonstrate the effectiveness and applicability of the proposed techniques. The results are graphically illustrated using 2D and 3D graphs. It has been noted that the obtained results and the targeted problems real solutions are quite similar. Calculated solutions at various fractional levels describe some of the problems useful dynamics. Other fractional problems that arise in other fields of science and engineering can be solved using a modified version of the current techniques.

In this paper, a distributed algorithm is proposed to solve a consensus convex optimization problem. It is a Jacobi-proximal alternating direction method of multipliers with a damping parameter $\gamma$ in the iteration of multiplier. Compared with existing algorithms, it has the following nice properties: (1) The restriction on proximal matrix is relaxed substantively, thus alleviating the weight of the proximal term. Therefore, the algorithm has a faster convergence speed. (2) The convergence analysis of the algorithm is established for any damping parameter $\gamma\in(0,2]$, which is larger ones in the literature. In addition, some numerical experiments and an application to a logistic regression problem are provided to validate the effectiveness and the characteristics of the proposed algorithm.

This is the second in a series of two articles introducing the Gravitas computational general relativity framework, in which we now focus upon the design and capabilities of Gravitas's numerical subsystem, including its ability to perform general 3+1 decompositions of spacetime via the ADM formalism, its support for the definition and construction of arbitrary Cauchy surfaces as initial data, its support for the definition and enforcement of arbitrary gauge and coordinate conditions, its various algorithms for ensuring the satisfaction of the ADM Hamiltonian and momentum constraints, and its unique (totally-unstructured) adaptive refinement algorithms based on hypergraph rewriting via Wolfram model evolution. Particular attention will be paid to the seamless integration between Gravitas's symbolic and numerical subsystems, its ability to configure, run, analyze and visualize complex numerical relativity simulations and their outputs within a single notebook environment, and its capabilities for handling generic curvilinear coordinate systems and spacetimes with general (and often highly non-trivial) topologies using its specialized and highly efficient hypergraph-based numerical algorithms. We also provide illustrations of Gravitas's functionality for the visualization of hypergraph geometries and spacetime embedding diagrams in two and three dimensions, the ability for Gravitas's symbolic and numerical subsystems to be used in concert for the extraction of gravitational wave signals and other crucial simulation data, and Gravitas's in-built library of standard initial data, matter distributions and gauge conditions. We conclude by demonstrating how the numerical subsystem can be used to set up, run, visualize and analyze a standard yet nevertheless reasonably challenging numerical relativity test case: a binary black hole collision and merger within a vacuum spacetime (including the extraction of its outgoing gravitational wave profile).

There are many problems based on solving non-autonomous differential equations of the form x¨(t)+ω2(t)x(t)=0, where x(t) represents the coordinate of a material point and ω is the angular frequency. The inverse problem involves finding the bounded coefficient ω. Continuity of the function ω(t) is not required. The trajectory x(t) is unknown, but the initial and final values of the phase variables are given. The variation principle of minimum time for the entire dynamic process allows for the determination of the optimal solution {x(t),ω(t)}. Thus, the inverse problem is an optimal control problem. No simplifying assumptions were made.

In this paper, we propose an efficient numerical method to solve the diffusive logistic model with a free boundary which is often used to simulate the spreading of the new or invasive species. The boundary movement is tracked by the level-set method, where the Hamilton-Jacobi weighted essentially nonoscillatory (HJ-WENO) scheme is utilized to capture the boundary curve embedded by the Cartesian grids via the embedded boundary method. Then the radial basis function-finite difference (RBF-FD) method is adopted to the spatial discretization and the IMEX scheme is considered for time integration. A variety of numerical examples are carried out to demonstrate the evolution of diffusive logistic model in different initial boundaries.

A primary challenge in isogeny-based cryptography lies in the substantial computational cost associated to computing and evaluating prime-degree isogenies. This computation traditionally relied on Vélu’s formulas, an approach with time complexity linear in the degree but which was further enhanced by Bernstein, De Feo, Leroux, and Smith to a square-root complexity. The improved square-root Vélu’s formulas exhibit a degree of parallelizability which has not been exploited in major implementations. In this study, we introduce a theoretical framework for parallelizing isogeny computations and provide a proof-of-concept implementation in C with OpenMP. While the parallelization effectiveness exhibits diminishing returns with the number of cores, we still obtain strong results when using a small number of cores. Concretely, our implementation shows that for large degrees it is easy to achieve speedup factors of up to 1.74, 2.54 and 3.44 for 2, 4 and 8 cores, respectively.

In this work we study the intersection properties of a finite disk system in the euclidean space. We accomplish this by utilizing subsets of spheres with varying dimensions and analyze specific points within them, referred to as poles. Additionally, we introduce two applications: estimating the common scale factor for the radii that makes the re-scaled disks intersects in a single point, this is the \v{C}ech scale, and constructing the minimal Axis-Aligned Bounding Box (AABB) that encloses the intersection of all disks in the system.

Homotopy perturbation and analysis methods have been widely used to obtain both approximate and exact so- lutions to nonlinear problems. In general, these two methods are based on the Taylor series with respect to an embedding parameter. Many researchers have compared the two methods and raised more concerns on the homo- topy perturbation method (HPM) because the homotopy analysis method (HAM) contains a convergence-control parameter ~: For this reason, in this article, a more general form of HPM is introduced as the -homotopy per- turbation method (-HPM), which contains a control parameter : The introduction of parameter  in this new modication gives a better way to adjust and control the convergence region and the rate of the series solution. We conrm through the given examples in this study that the HPM is a special case of the -HPM. The error and convergence analysis of this proposed method are also presented

This paper provides the numerical investigation of two-dimensional stochastic Navier-Stokes equation driven by additive white noises. The vorticity-stream function method is presented to calculate the incompressible fluid flow. In order to solve the formulation of steam function equation, we propose a Crank-Nicolson Fourier pseudo-spectral method. Numerical results obviously show that the presence of stochastic forcing can have an impact upon the shapes of vortex in fluid flow. To a certain extent, the phenomenon of eddy shedding and the formation of some new vortex in a random medium can be observed, which make it possible for us to conduct optimal control experiments in the development of vortex structures.

Finding the roots of an equation is a fundamental problem in diverse fields. Optimization problem leads to non-linear equations for calculating the roots of the equation efficiently. Numerical iterative techniques are frequently applied when analytic solution is not available. Hybrid techniques to find roots of an equation is new to this fundamental problem. Recently, some attempts emerged to design hybrid algorithms for efficient solutions. A new heuristic algorithm is designed which is an intuitive approach to hybridization. In this paper, we implement two new algorithms: (1) a new BTsection algorithm blending the Bisection and Trisection algorithm, (2) a hybrid algorithm, that promises to be more efficient than the existing hybrid algorithms. The new hybrid algorithm is a blend of BTsection algorithm and Regula Falsi algorithm. The implementation results validate that the new algorithm, Hybrid4, surpasses the efficiency of existing hybrid algorithms Hybrid1, Hybrid2, and Hybrid3 algorithms. This paper contributes an essential hybrid algorithm to the repertoire of hybrid root finding algorithms.

Stratified Taylor Couette Flow (STCF) undergoes transient growth. Recent studies have shown that there exists transient amplification in the linear regime of counter-rotating STCF. It is reported that as Gr increases, the maximum amplification factor ($G_0$) initially decays before it eventually starts growing after reaching a certain critical value Gr$_{c}$. Assuming other parameters of the model are kept constant, the value Gr$_{c}$ changes with respect to different speed ratios of the cylinders. The observed decay/growth pattern of $G_0$ is attributed to the interplay between the induced shear and buoyancy. In this study, the mechanism that triggers this observed pattern is investigated. The kinetic budget of the optimal transient perturbation is analysed numerically to simulate the interaction of the shear production (SP), buoyancy production (BP) and other energy components that contributes to the total optimal transient kinetic energy. These contributions affect the total energy by influencing the perturbation to extract kinetic energy (KE) from the mean flow. The decay of $G_0$ resulted from the positive amplification of both BP and SP, while the growth is attributed to the negative and positive amplification of BP and SP, respectively. The optimal SP is positively amplified, implying that there is the possibility of constant linear growth. These findings agree with the linear growth rate for increasing values of Gr.

In the literature, the question of the amount of data necessary and sufficient to validate different models of the occurrence of risk of adverse events for patients or the classification of the presence or absence of pathological features has been repeatedly raised. In the presented study, we propose a new approach to determine the necessary and enough studies for validation of medical software based on artificial intelligence technology, whose main task is to classify medical X-rays according to the presence of normality and pathology. It is shown that for several studies in a dataset, when AUC ROC has maximum heterogeneity, it varies depending on the balance of "norm"/"pathology" classes. Thus, for a balance of "normal"/"abnormality", where 90% is "normal" and 10% is "abnormality", maximum heterogeneity is achieved for 190 studies, for a balance of 80% ("normal")/20% ("abnormality") for 80 studies, for a balance of 70% ("norm")/30% ("abnormality") - 120 trials, for a balance of 60% ("norm")/40% ("abnormality") the maximum heterogeneity is reached at 110 trials, and for a balance of 50% ("norm")/50% ("abnormality") - at 70 trials. The obtained data are in good agreement with the previous results. They allow us to determine a sufficient (necessary) number of studies in the dataset to perform an unbiased assessment of AUC ROC.

This paper introduces new improvements to the modified version of the BIRECT (BI secting RECTangles) algorithm referred to as BIRECTv. We explore various approaches, by first including a grouping strategy for hyper-rectangles having almost the same sizes by categorizing them into different classes. Therefore constraining them not to exceed a certain pre-defined threshold (a small positive value to define the tolerance level). This approach allows for more efficient computation and can be particularly useful when dealing with a large number of hyper-rectangles with varying sizes. To avoid over-sampling, and preventing redundant or excessive sampling, at some shared vertices in descendant subregions, a particular vertex database is used to limit the number of samples taken within each subregion to two. The experimental investigation shows that these improvements have a positive impact on the performance of the BIRECTv(imp.) algorithm and the proposal is a promising global optimization algorithm compared to the original BIRECTv algorithm and its variants. Additionally, the BIRECTv(imp.) algorithm showed particular efficacy in solving high-dimensional problems.

In the research we consider a structure-fluid interaction of oscillating piezodevice and acoustic liquid. Such problem requires finding resonance or anti-resonance frequencies of the solid structure to use them in the latter frequency-response analyses step. After finite element discretization of the issue, we need to solve a generalized eigenvalue problem (GEP). The most efficient numerical method for solving the partial GEP is the shift-and-invert Lanczos method. Each step of this algorithm requires solving a large scale system of linear algebraic equations (SLAEs) with special properties. Various ways of ordering grid nodes lead to coefficient matrices with different structure. Several methods are compared in terms of efficiency. The results of the numerical experiments are presented.

Software-Defined Networking (SDN) is a network implementation paradigm of great importance, as it has a profound impact on the pace of technological progress. While SDN doesn’t directly address the technical complexities of routing, congestion control, traffic engineering, security, mobility, reliability, or real-time communication, it paves the way for innovative solutions to emerge for these and similar challenges. Security is of utmost importance in SDN with Distributed Denial of Service (DDoS) being an attack which creates large scale problems. DDoS creates malicious traffic that resembles normal traffic in order to create service problems. As such, mechanisms that distinguish between benign and malicious traffic is essential, since this is the first step to mitigate the problem of DDoS. In this paper, we take a dataset onboard which exhibits benign and malicious traffic in SDN. There are 23 features that are used for classification purposes. Here we utilise classification procedure based on three methods based Grammatical Evolution applied on the aforementioned data. We provide results that show the efficiency of our approach and show that all three methods exhibits satisfactory results.

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection-reaction-diffusion equations on surfaces. The temporal direction adopted backward-Euler method, while the spatial direction employed surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, the good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is made with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

This paper introduces the complex Rayleigh-van der Pol-Duffing oscillators (RVDOs), which are hyperchaotic and can be either autonomous or non-autonomous. The fundamental dynamics of the autonomous and nonautonomous complex RVDOs including dissipation, symmetry, fixed points and their stability are studied. These oscillators are found in various important fields of physics and engineering. The paper proposes a scheme to achieve phase synchronization (PS) and antiphase synchronization (APS) for different dimensional models. These kinds of synchronization are considered as generalization of several other types of synchronization. For this scheme we use the active control method based on Lyapunov stability theory. By analytically determining the control functions, the scheme achieved PS and APS. Our scheme is applied to study PS of hyperchaotic behaviours for two distinct hyperchaotic non-autonomous and autonomous complex RVDOs. Additionally, the scheme is employed to achieve APS of chaotic real non-autonomous RVDO and hyperchaotic complex autonomous RVDO, including those with different dimensions. Our work presents numerical results that plot the amplitudes and phases of these hyperchaotic behaviours, demonstrating the successful achievement of PS and APS.

The idea of strong and weak adjacencies between vertices has been generalized into fuzzy graphs and intuitionistic fuzzy graphs (IFGs) and it is an important part of making decisions. However, one or two membership degrees are not always sufficient for making decisions on real-world problems that need an answer of types “yes, neutral, and no". Consequently, in previous work, we generalized the concept into picture fuzzy graphs (PFGs) where each element in the PFG has membership, neutral, and non-membership degrees. Moreover, we constructed the notion of the coloring of PFGs based on strong and weak adjacencies between vertices. In this paper, we investigate some properties of the chromatic number of PFGs based on the concept of strong and weak adjacencies between vertices. According to the properties, we construct an algorithm to find the chromatic number of PFGs. The algorithm is usefull when we work with large PFGs. Further, we improve the method to implement the PFG’s coloring for determining traffic signal phasing at an intersection. A case study has also been done to evaluate the method.

Partial differential equations (PDEs) are used to describe a wide range of phenomena, such as heat transfer, fluid dynamics, and quantum mechanics. By solving PDEs, we can ob- tain insights into the behavior of the system and make predictions about its future evolution. Conventional numerical methods for obtaining the approximate solutions of PDEs may re- quire extensive computational resources and time, especially for complex PDEs and large- scale problems. The recently developed physics-informed neural network (PINN) has shown promise in a variety of scientific and engineering fields by incorporating physical laws into the loss functions of the neural network (NN). In addition, the training of PINN does not require ground truth data, but it has poor generalization ability to unseen domains. On the other hand, a convolutional neural network (CNN) has fast inference and better generalization ability, but it requires a large amount of training data. Taking the advantages of PINN and CNN by using Legendre multiwavelets (LMWs) as basis functions, we introduce a new method to approach the PDEs in this paper, namely Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), in order to continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any pre- computed training data, simultaneously providing fast and continuous solutions that gener- alize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, incompressible Navier-Stokes (N- S) equation, and two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs.

A novel approach was developed that combined LP-based row generation with optimization-based sorting to tackle computational challenges posed by budget allocation problems with combinatorial constraints. The proposed approach dynamically generated constraints using row generation and prioritized them using optimization-based sorting to ensure a high-quality solution. Computational experiments and case studies revealed that as the problem size increased, the proposed approach outperformed simplex solutions in terms of solution search time. Specifically, for a problem with 50 projects (N = 50) and 2,251,799,813,685,250 constraints, the proposed approach found a solution in just 1.4 seconds, while LP failed due to the problem size. The proposed approach demonstrated enhanced computational efficiency and solution quality compared to traditional LP methods.

The migration from MATLAB to Python (M-to-PY) has gained significant traction in recent computational research. While MATLAB has long served as a linchpin in myriad scientific endeavors, there's an emerging trend to rejuvenate these projects using Python's extensive AI tools and libraries. This study presents a semi-automated process for M-to-PY conversion, using a detailed case study of an image skeletonization project comprising fifteen MATLAB files and a 1404-image dataset. Skeletonization is foundational for ongoing 3D motion detection research using AI transformers, predominantly developed in Python. The utilization of ChatGPT-4, acting as an AI co-programmer, is pivotal in this conversion. By leveraging the public OpenAI API, we developed an M-to-PY converter prototype, evaluated its efficacy using test cases from the Bard bot, and subsequently employed the converted code in an AI application. The dual contributions encompass a well-tested M-to-PY converter and a Skeleton App capable of sketching and skeletonizing any given image, enriching the AI toolset. This study accentuates how AI resources, like ChatGPT-4, can simplify code transitions, opening doors for innovative AI implementations using primarily MATLAB-coded scientific research.

This paper presents three types of tensor Conjugate-Gradient methods for solving large-scale linear discrete ill-posed problems based on the t-product between third-order tensors. An automatical determination strategy of a suitable regularization parameter is proposed for the tensor conjugate gradient (tCG) method. A truncated version and a preprocessed verion of the tCG method are further presented. The discrepancy principle is employed to determine a suitable regularization parameter. Several numerical examples are given to show the effectiveness of the proposed tCG methods in image and video restoration.

Quantum computing is an emerging field that have given a significant impact on optimization. Among the diverse quantum algorithm, the quantum gradient descent has become a prominent technique for solving unconstrained optimization problems. In this paper, we propose a quantum-spectral Polak-Ribi{\'e}re-Polyak (PRP) conjugate gradient approach. The technique is considered as a generalization of the spectral PRP method which employs a $q$-gradient that approximate the classical gradient with quadratically better dependence on the quantum variable $q$. Additionally, the proposed method reduces to the classical variant as quantum variable $q$ gets closer to $1$. The quantum search direction always satisfies the sufficient descent condition and does not depend on any line search. This approach is globally convergent with the standard Wolfe line search without any convexity assumption. Numerical experiments are conducted and compared with the existing approach to demonstrate the improvement of the proposed strategy.

This study focuses on the formulation and analysis of a numerical method to compute singularly perturbed functional differential equations (SP-FDEs) involving large mixed shifts. SP-FDEs are a class of differential equations with both shifts and small perturbation parameter. Such equations arise in various scientific fields to model phenomena with memory effects and perturbed dynamics. In this particular study, the considered SP-FDEs involve large negative shift (delay) and large positive shift (advance). Dealing with large mixed shifts in SP-FDEs gives rise to challenges in terms of stability analysis, numerical methods, and convergence of a solution. In this paper, the influence of the large mixed shifts is treated by choosing a uniform mesh discretized in such a way that the term with the shifts lie on the nodal points. Then, using the Numerov’s finite difference method, the continuous problem is transformed into a finite difference scheme and the influence of the perturbation parameter is handled by introducing a fitting factor in the difference scheme. Stability estimate and convergence analysis of the developed scheme are investigated and proved. Numerical experiments are carried out to demonstrate the validity and applicability of the developed scheme. It is shown that the numerical scheme obtained in this work is parameter-robust of second order convergence rate.

New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order r+1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in Hp for any p≥1. We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used.

Randomized block Kaczmarz Method and randomized extended block Kaczmarz Method are proposed for solving matrix equation AXB=C, where the matrix A and B may be full rank or rank deficient. These methods are iterative methods without matrix multiplication, and are especially suitable for solving large-scale matrix equations. It is theoretically proved these methods converge to the solution or least square solution of the matrix equation. The numerical results show that these methods are more efficient than the existing algorithms for high-dimensional matrix equations.

This paper presents a novel approach for the recognition of multiple sclerosis (MS) using wavelet entropy and a Particle Swarm Optimization (PSO)-based neural network. MS is a complex neurological disorder with diverse clinical manifestations, often challenging to diagnose accurately. In this study, we leverage wavelet entropy (WE) as a feature extraction method to capture intricate patterns within brain imaging data. These extracted features are then employed to train a neural network model optimized through PSO to enhance classification accuracy. Experimental results on a dataset comprising MS and healthy control subjects demonstrate the effectiveness of our proposed approach, viz., WE-PSONN, which achieves a sensitivity of 91.95±1.15, a specificity of 92.36±0.88, a precision of 92.28±0.88, an accuracy of 92.16±0.90. The combination of advanced signal processing techniques and machine learning optimization holds promise for improving the early diagnosis and management of MS, offering potential benefits to both patients and healthcare providers.

This research work deals with the formulation of numerical scheme with algorithm to compute singularly perturbed partial differential equations(SP-PDEs) involving a significant negative shift. The problem involves small parameter which causes a rapidly changing boundary layers in the vicinity of a body, and the negative shift term causes interior layer. Such appearance of an abruptly varying layers causes difficulties to find the exact solution and it is not adequate to employ classical numerical methods. The technique presented in this research work simplifies these challenges and yields accurate numerical solution.The stability and convergence analyses of the methods are examined and proven. To test the developed technique, numerical experiments are carried out and confirmed with theoretical analysis.

Yijing possesses inherent principles and logic, and it has achieved consensus as an auxiliary decision-making function. Research on Yijing has been validated in the field of superpsychology, primarily focusing on individuals. However, the universal value of Yijing extends beyond individual divination or telepsychological induction, and there are untapped scientific potential values associated with it. Consequently, many scholars studying Yijing, including those in Chinese medicine, attempt to decipher its internal scientific mechanism through quantitative methods. In summary, scientific quantitative analysis of Yijing can be categorized into several approaches. These include quantification based on Yijing itself (binary and hexagram order), as well as quantification based on mathematical statistics, probability, and calculus. Another approach involves quantification based on expert knowledge, as well as the quantification of timewave curves and Yi-globle. Significant progress has been made in all of these quantitative research areas within the scientific paradigm. Nevertheless, future quantification of Yijing may explore rule-based evolutionary analysis, incorporating artificial intelligence technology, and conduct quantitative analysis based on the evolutionary process and results. There is even potential to deepen this research with tools such as ChatGPT. Additionally, the focus of Yijing divination may shift from individual events to group events, accompanied by space-time modeling and analysis through the construction of a new coordinate system.

In this paper, we analyze the convergence of a finite difference method with the implicit forward time central space (FTCS) scheme for solving the three-dimensional advection-diffusion-reaction equation (ADRE). It is proved that the method is unconditionally convergent. We apply the scheme to obtain numerical solutions for the transport of pollutants in street tunnel problems with various reaction coefficients and various rates of change of concentrations of sources or sinks of pollution.

We consider a stochastic production planning problem with regime switching. There are k≥1 regimes corresponding to different economic cycles. The problem is to minimize the production costs and analyze the problem by the value function approach. Our main contribution is to show that the optimal production is characterized by an exact solution of an elliptic system of partial differential equations. A verification result is given for the determined solution.

In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. Using this procedure, we achieve a remarkable increase in the order of convergence, raising it from p to p + 3 units, which results in a remarkable improvement in the overall performance. We have conducted comprehensive numerical tests in various scenarios to validate the theoretical results, showing the efficiency and effectiveness of the new Jacobian-free schemes.

The fractional derivatives are a generalization the derivatives of integer order and find applications in studying memory processes in various scientific fields. Numerical methods are used to solve and analyze fractional models of real world problems. In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic formula, whose generating function is the polylogarithmic function. In the paper, we prove the convergence of the approximation and derive an estimate for the error and order. We consider an application of the approximation for the construction of finite difference schemes for numerical solution of the two-term ordinary fractional differential equation and the time-fractional Black-Scholes equation for option pricing. The properties of the approximation are used to prove the convergence of the methods used for numerical solution of the fractional differential equations. The theoretical results for the error and order of the methods are illustrated by the results of numerical experiments.

This article is devoted to methods of processing random processes. Of particular relevance is the task of processing broadband non-stationary random processes. The processing of random processes is usually related to the assessment of their probabilistic characteristics. Very often, a non-stationary broadband random process is represented by a single implementation in a priori uncertainty about the type of distribution function. Such random processes occur in information and measuring communication systems in which information is transmitted at a real time pace (for example, radio telemetry systems of spacecraft). The use of methods of traditional mathematical statistics, for example, maximum likelihood methods to determine probability characteristics, in this case is not possible. The article discusses a method of processing non-stationary broadband random processes based on the use of non-parametric methods of decision theory. An algorithm for dividing the observation interval into stationary intervals using non-parametric Kendall statistics is considered, as well as methods for estimating probabilistic characteristics on the stationary interval using ordinal statistics. The article presents the results of statistical modeling using the Mathcad program.

Among the various generalizations of persistent topology, the one based on rank functions and leading to indexing-aware functions appears to be particularly suited to catch graph-theoretical properties without the need for a simplicial construction and a homology computation. This paper defines and studies "simple" and "single-vertex" features in directed and undirected graphs, by which several indexing-aware persistence functions are produced, within the scheme of steady and ranging sets. The implementation of the "sink" feature and its application to trust networks provide an example of the ease of use and meaningfulness of the method.

This paper is concerned with developing an efficient numerical algorithm for fast implementation of the sparse grid method for computing the $d$-dimensional integral of a given function. The new algorithm, called the MDI-SG ({\em multilevel dimension iteration sparse grid}) method, implements the sparse grid method based on a dimension iteration/reduction procedure, it does not need to store the integration points, nor does it compute the function values independently at each integration point, instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order $O(Nd^3 )$ or better, compared to the exponential order $O(N(\log N)^{d-1})$ for the standard sparse grid method, where $N$ denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration.

To assess the impact of tourism on the incidence of HIV and AIDS using Malaysian epidemiological data over the period of 1986-2011 with additional consideration for newborns infected with HIV. A population-level mathematical model was used to investigate: i) the role of tourism in the spread of HIV and measures used to reduce HIV spread in Malaysia; ii) whether the stability of infectious disease transmission is dependent on the flow of visiting tourists. We first derived an equation for the reproduction number (R0) threshold to quantify the contagiousness of HIV in Malaysia. Sensitivity analyses were used to determine the effect of various parameters on HIV transmission with respect to the increase in tourism. Our findings suggest that a stable disease-free state is sustainable based on the low value of R0 was 0.0017. This result is encouraging from a public health perspective. Approximately 14% of outbound tourists who leave the country return infected with HIV and the difference between the rate at which tourists move to the susceptible category and the rate at which tourists leave the susceptible is category is 12%. Estimated parameters for the influx of tourist rates, δ=1.1540x10-3[1.1477x10-3 - 1.15954x10-3], δ1=7.7901x10-4[7.7867x10-4 - 7.79418x10-4], and δ2=1.4030x10-8[-7.2287x10-7-7.5096x10-7], significantly impacted the spread of HIV in Malaysia. Some significant adjustments were made to the expected parameters. The methods used are helpful to public health analyses and provide a framework for epidemiological modeling of HIV spread among tourists. The trend and magnitude of tourist inflows may be determinants in the incidence of HIV and AIDS in Malaysia.

In this paper, we consider the one-dimensional Ising model (shortly, 1D-MSIM) having mixed spin-(s,(2t−1)/2) with nearest neighbors and the external magnetic field. We establish the partition function of the model by means of the transfer matrix. Under a null boundary condition, we compute certain thermodynamic quantities for the 1D-MSIM. We find some precise formulas to determine the model’s free energy, entropy, magnetization, and susceptibility. By examining the model’s associated iterative equations, we use the cavity approach to investigate the phase transition problem. We numerically determine the model’s periodicity.

This paper investigates the ion-acoustic wave structures in fluid ions for the Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation. The various types of wave structures, are extracted including the three-wave hypothesis, breather wave, lump periodic, mixed-type wave, periodic cross-kink, cross-kink rational wave, M-shaped rational wave, M-shaped rational wave solution with one kink wave and M-shaped rational wave with two kink wave solutions. The Hirota bilinear transformation is a powerful tool that allows us to accurately solutions and predict the behavior of these wave structures. Through our analysis, we gain a better understanding of the complex dynamics of ion-acoustic waves and their potential applications in various fields. Moreover, our findings contribute to the ongoing research in plasma physics that utilize ion-acoustic wave phenomena. To show the physical behavior of the solutions, some 3D plots have been performed and their respective contour level, choosing different values of the parameters.

TThis paper delves into the extension and characterization of radial positive definite functions into non-integer dimensions. We provide a thorough investigation by employing the Riemann-Liouville fractional integral and fractional Caputo derivatives, enabling a comprehensive understanding of these functions. Additionally, we introduce a secondary characterization based on the Bernstein characterization of completely monotone functions. The practical significance of our study is showcased through an examination of the positivity of the fundamental solution of the space-fractional Bessel diffusion equation, highlighting the real-world applicability of the developed concepts. Through this work, we contribute to the broader understanding of radial positive definite functions and their utility in diverse mathematical and applied contexts.

With technological transformations such as big data, blockchain, artificial intelligence, and cloud computing, digital techniques are penetrating the field of finance. Digital finance is a resource-saving and environmentally friendly innovative financial service. It shows great green attributes and can drive the flow of financial resources towards environmentally friendly enterprises, thereby promoting green and low-carbon development. This paper investigates the effects of digital finance on green and low-carbon cyclic development and the mechanism. First, the level of green and low carbon cyclic development in China is estimated from multiple aspects based on the panel data of 31 provinces by using the spatio-temporal range entropy weight method. Then, the working mechanism of digital finance on green and low carbon cyclic development is revealed by performing empirical tests based on panel regression models, mediating effect models, and threshold models. Finally, the effects of digital finance and conventional variables on green and low-carbon cyclic development are investigated by using a random forest model and a CatBoost model in based on the machine learning. The results show that digital finance has significant positive effects on green and low-carbon cyclic development, and technological innovation plays a key role in the effects of digital finance on green and low-carbon cyclic development; meanwhile, the driving effect of digital finance on green and low carbon cyclic development shows nonlinear features with increasing “marginal effect”; besides, both digital finance and conventional factors have significant impacts on green and low carbon cyclic development. This study provides an empirical basis and path reference for digital finance to achieve “carbon peak, carbon neutralization” in China.

The rapidly evolving field of artificial intelligence (AI) continues to witness the introduction of innovative open-source pre-trained models, fostering advancements in various applications. One such model is Llama 2, an open-source pre-trained model released by Meta, which has garnered significant attention among early adopters. In addition to exploring the foundational elements of the Llama v2 model, this paper investigates how these early adopters leverage the capabilities of Llama 2 in their AI projects. Through a qualitative study, we delve into the perspectives, experiences, and strategies employed by early adopters to leverage Llama 2's capabilities. The findings shed light on the model's strengths, weaknesses, and areas of improvement, offering valuable insights for the AI community and Meta to enhance future model iterations. Additionally, we discuss the implications of Llama 2's adoption on the broader open-source AI landscape, addressing challenges and opportunities for developers and researchers in the pursuit of cutting-edge AI solutions. The present study constitutes an early exploration of the Llama 2 pre-trained model, holding promise as a foundational basis for forthcoming research investigations.

This paper presents a numerical method for solving a two-dimensional subdiffusion equation with a Caputo fractional derivative. The problem considered assumes symmetry in both the equation’s solution domain and the boundary conditions, allowing for a reduction of the two-dimensional equation to a one-dimensional one. The proposed method is an extension of the fractional Crank-Nicolson method, based on the discretization of the equivalent integral-differential equation. To validate the method, the obtained results were compared with a solution obtained through Laplace transform. The analytical solution in the image of the Laplace transform was inverted using the Gaver-Wynn-Rho algorithm implemented in the specialized mathematical computing environment, Wolfram Mathematica. The results clearly show the mutual convergence of the solutions obtained by the two methods.

In this paper we have studied some properties for determinant-calculating for cubic-matrix of order 2 and order 3. These properties are analogous to some properties for determinants of square matrix we have proved and noted that these properties also are applicable (or not in some details) on this concept for cubic-matrix of orders 2 and 3. All results in this paper, are presented in detail during the theorem proofs.

In this paper, we introduce as a pedagogical strategy an internal division-free, straightforward, and symmetrically progressing algorithm in manually computing matrix inverse and solving systems of linear equations by revisiting the application of elementary row operations in the Gauss-Jordan reduction method and connecting it to the determinant method. The proposed cross-multiplication-flip (CMF) algorithm employs cross-multiplication similar to the butterfly movement in computing determinants as a strategic application of elementary row operations to efficiently reduce the rows and then applies flipping of rows and entries to put an upper triangular matrix into lower triangular form to continue the reduction process.

In this paper, a new multi parameter iterative algorithm is proposed to address the PageRank problem based on the multi-splitting iteration method described by Gu et al . The proposed method in each iteration needs to solve two linear subsystems by splitting the coefficient matrix, therefore, we consider inner and outer iteration to find the approximate solutions of these linear subsystems. It can be shown that the iterative sequence generated by the multi parameter iterative algorithm finally converges to the PageRank vector when the parameters satisfy the certain conditions. Numerical experiments show that the proposed algorithm has better convergence and numerical stability than the existing algorithms.

It is shown that for any integers k≥2, q≥2k and N≥k+q+2 there exists a real solvable Lie algebra of rank one with a maximal torus of derivations t possessing the eigenvalue spectrum spec(t)=1,2,⋯,k,q,q+1⋯,N and a nilradical of nilpotence index N−k and characteristic sequence (N−k,1k).

The (numerical) solution of Ordinary Differential Equations (ODEs) problems is of paramount relevance, ODEs system being an ubiquitous mathematical formulation of many physical phenomena (such as those involved in fluid dynamics, chemistry, biology, evolutionary-anthropology, ...): almost every dynamici phenomenon can be modeled by means of an ODEs system. The present paper is the first manifesto of FOODIE, a library aimed to numerically solve ODEs problems by means of a clear, concise and efficient abstract interface. FOODIE, meaning Fortran Object oriented Ordinary Differential Equations integration library, has manifolds aims: to provide a set to built-in numerical schemes that are accurate, robust, validated and efficient and to allow easy application of these schemes to (almost) all ODEs problems by means of an effective Abstract Calculus Pattern. The key idea is to allow the same solver-implementation to be applied to all ODEs problems thus avoiding the re-implementation of the ODEs solver for each different ODEs problem: code re-usability is consequently maximized, FOODIE being a general robust framework. Besides, the same framework also allows rapid development of new ODEs solvers due to the high abstraction level of the library itself. The present paper is the first announcement of FOODIE project: the current implementation is extensively discussed and its capabilities are proved by means of tests and examples.

This paper applied the space-time generalized finite difference scheme for solving the nonlinear dispersive shallow water waves as the modified Camassa−Holm equation, modified Degasperis-Procesi equation, Fornberg-Whitham equation, and its modified form. The proposed meshless numerical scheme was composed of the space-time generalized finite difference method, the two-step Newton-Raphson method, and the time-marching method. The space-time approach can treat the temporal derivative as one of the spatial derivatives. This numerical technique enables all the partial derivatives in the governing equation can be discretized by a spatial discretization method, and the mixed derivative can efficiently deal with using the proposed meshless numerical scheme. The space-time generalized finite difference method is advanced from the Taylor series expansion and the moving-least square method. The numerical discretization process is only related to functional data and weighting coefficients on the central node and its nearby nodes. Thus, the matrix system composed of nonlinear algebraic equations will be a sparse matrix and can be efficiently solved by the two-step Newton-Raphson method. Furthermore, the time-marching method was utilized to proceed with the space-time domain along the time axis. In this paper, several numerical examples were tested to verify the capability of the proposed space-time generalized finite difference scheme.

New analytical solutions of the heat conduction equation are presented in cylindrical and spherical coordinates. Then these solutions are reproduced with high accuracy by recent explicit and unconditionally stable finite difference methods. After this, real experimental data from the literature regarding a heated cylinder are reproduced by the explicit numerical methods as well as by Finite Element Methods (FEM) ANSYS workbench. Convection and nonlinear radiation are also considered on the boundary of the cylinder, and it is shown that the explicit methods are much more accurate than the commercial software.

Quantum computing arrives. Preceded by Mādhava of Sangamagrāma and other Hindu mathematicians, around 1,500 CE, and our previous work, we show at least 4 quantum properties that are indeed universal in some number systems; here we use them to enable quantum computing. Using the sets B={0,1}, B^n, N, Z, Q, and Q*, one has +4 quantum properties that one can trust, as archetypes of easy communication to friends and foes, to ignoranti or cognoscenti, and for quantum consciousness. Therefore, this work applies to computer calculations, apps in cell phones, algebra, atoms, mathematics, physics, and the human mind.

This paper proposed a time two-mesh (TT-M) finite difference numerical scheme to improve the efficiency of solving the symmetric regularized long wave (SRLW) equation. The TT-M Crank-Nicolson discretization and finite difference method are employed in time and space approximation respectively. The scheme involves three main steps: firstly, the time interval is divided into coarse and fine time meshes, then the nonlinear system is solved on the coarse time mesh; secondly, coarse numerical solutions on the fine time mesh are computed using an interpolation formula based on the solutions derived in the step one; lastly, the TT-M finite difference numerical solutions can be obtained through constructing the linearized fine time mesh system using Taylor’s formula. Compared to the currently existing TT-M numerical methods, the novelty of this study is that the nonlinear term including derivatives is linearized by Taylor’s formula for a function with three variables, whose error analysis is more complex. Finally, some numerical examples, including computational time and accuracy, preservation of conservation laws, are given to verify the efficiency of the scheme. By comparing it with the standard nonlinear finite difference scheme, this method can reduce CPU time without sacrificing accuracy.

Due to nonlinear components in the charging piles of electric vehicles, harmonics and non-stationary signals in the electric vehicle charging load brings voltage and current distortion, seriously affecting the accuracy of the power-related calculation in non-sinusoidal environments. This paper proposed a new approach to calculate the active power and root mean square values from decomposed components using the adaptive chirp mode decomposition (ACMD) method on voltage and current. The advantage of the ACMD-based method is that it correctly provides the power-related quantities of harmonics or non-stationary components for the electric vehicle charging load. The performance of the proposed method was verified using synthetic signals and simulation tests. The experimental results presented better estimations for each quantity defined in IEEE Standard 1459-2010, compared with the discrete wavelet transform approach.

In this paper, we consider the fourth-order parabolic equations with integer and fractional order time derivatives with Neumann boundary conditions. The integer order time derivatives are approximated by backward Euler difference quotients, and the fractional order time derivatives are approximated by L1 interpolation. We propose the block-centered finite difference scheme for fourth-order parabolic equations of integer and fractional order time derivatives. We prove the stability of the block-centered finite difference scheme and the second-order convergence of the discrete L2 norms of the approximate solution and its derivatives of every order. Numerical examples are given to verify the effectiveness of the block-centered finite difference scheme.