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Aim to the limitations of grey relational analysis of interval grey number, based on the generalized greyness of interval grey number, this paper tries to construct a grey angle cosine relational degree model from the perspective of proximity and similarity.

Firstly, the algorithms of the generalized greyness of interval grey number and interval grey number vector are given, and its properties are analyzed. Then, based on the grey relational theory, the grey angle cosine relational model is proposed based on the generalized greyness of interval grey number, and the relationship between the classical cosine similarity model and the grey angle cosine relational model is analyzed. Finally, the validity of the model in this paper is illustrated by the calculation examples and an application example of related factor analysis of maize yield.

The results show that the grey angle cosine relational degree model has strict theoretical basis, convenient calculation and is easy to program, which can not only fully utilize the information of interval grey numbers but also overcome the shortcomings of greyness relational degree model. The grey angle cosine relational degree is an extended form of cosine similarity degree of real numbers. The calculation examples and the related factor analysis of maize yield show that the model proposed in this paper is feasible and valid.

The research results not only further enrich the grey system theory and method but also provide a basis for the grey relational analysis of the sequences in which the interval grey numbers coexist with the real numbers.

The paper succeeds in realizing the algorithms of the generalized greyness of interval grey number and interval grey number vector, and the grey angle cosine relational degree, which provide a new method for grey relational analysis.

In order to make full use of the generalized greyness of interval grey number, this paper analyzes the properties and its applications of generalized greyness.

Firstly, the static properties of generalized greyness in bounded background domain, infinite background domain and infinitesimal background domain are analyzed. Then, this paper gives the dynamic properties of generalized greyness in bounded background domain, infinite background domain and infinitesimal background domain and explains the dialectical principle contained in it. Finally, the generalized greyness is used to judge the effectiveness of interval grey number transformation.

The results show that the generalized greyness of interval grey number has relativity, normativity, unity, eternity and conservation. The static and dynamic properties of generalized greyness are the same in the infinite and infinitesimal background domain, while they are different in the bounded background domain. The generalized greyness can be used as an index to judge whether the grey number transformation is greyness or information preservation.

The research shows that the generalized greyness can be used as an index to judge the validity of the grey number transformation and also can be applied in grey evaluation, grey decision-making and grey prediction and so on.

The paper succeeds in realizing the mathematical principle of “white is black”, the “greyness clock-slow effect” of the value domain of interval grey number and the generalized greyness conservation principle, which provides a theoretical basis for the rational use of generalized greyness of interval grey number.

Given the regional heterogeneity of economic development, electricity consumption in various regions exhibits a discrepant growth pattern. The purpose of this study is to construct a grey system forecasting model with intelligent parameters for predicting provincial electricity consumption in China.

First, parameter optimization and structural expansion are simultaneously integrated into a unified grey system prediction framework, enhancing its adaptive capabilities. Second, by setting the minimum simulation percentage error as the optimization goal, the authors apply the particle swarm optimization (PSO) algorithm to search for the optimal grey generation order and background value coefficient. Third, to assess the performance across diverse power consumption systems, the authors use two electricity consumption cases and select eight other benchmark models to analyze the simulation and prediction errors. Further, the authors conduct simulations and trend predictions using data from all 31 provinces in China, analyzing and predicting the development trends in electricity consumption for each province from 2021 to 2026.

The study identifies significant heterogeneity in the development trends of electricity consumption systems among diverse provinces in China. The grey prediction model, optimized with multiple intelligent parameters, demonstrates superior adaptability and dynamic adjustment capabilities compared to traditional fixed-parameter models. Outperforming benchmark models across various evaluation indicators such as root mean square error (RMSE), average percentage error and Theil’s index, the new model establishes its robustness in predicting electricity system behavior.

Acknowledging the limitations of traditional grey prediction models in capturing diverse growth patterns under fixed-generation orders, single structures and unadjustable background values, this study proposes a fractional grey intelligent prediction model with multiple parameter optimization. By incorporating multiple parameter optimizations and structure expansion, it substantiates the model’s superiority in forecasting provincial electricity consumption.

This paper considers the brand awareness and anchor influence on consumers' live-streaming purchases, and explores the existence of “free-riding” behavior, the comparison of brand promotion effect and active live-streaming effect and the optimal strategic combination between the brand and the anchor. The authors investigate the evolutionary stabilization strategies of the bounded rational brand and anchor, and explore the conditions for the realization of the optimal strategy. Management suggestions for the development of live streaming commerce can be provided in this paper.

Two significant models are used in this paper. The Stackelberg model is used to study the “free-riding” behavior, the comparison of brand promotion effect and active live-streaming effect and the optimal strategic combination between the brand and the anchor. Using evolutionary game theory to get the evolutionary stable equilibrium strategies and analyze the binary equilibrium strategy of the bounded rational brand and anchor. In addition, relevant simulation analysis is conducted using realistic data to verify the conclusions and for further analysis, making the conclusions of the paper have realistic significance.

The study shows that “free-riding” behavior exists and the positive effect of brand promotion is greater than that of active live-streaming. The brand and the anchor take active actions as the optimal strategy. As the sensitivity coefficient of consumers to live-streaming effort and the sensitivity coefficient of consumers to brand promotion change, various evolutionary stabilization strategies will appear. When the two sensitivity coefficients are below a certain threshold, the game sides will reach the optimal strategic combination to obtain the maximum benefits. When they rise above this threshold, it is counterproductive instead. The system achieves the optimal strategic combination when the difference factor between effort cost and promotion cost must be higher than a certain value, but when it takes the smallest possible value, the game sides tend to take active actions. This study can provide management suggestions for the sustainable development of the live-streaming model.

This paper shows that under certain conditions, the brand and the anchor can evolve into the optimal strategy to maximize the profits of both parties, which has certain practical significance for the prosperous development of live streaming commerce. In future research, the authors will consider the regulatory role of the government and construct a more realistic game model to provide constructive suggestions for the sustainable prosperity of live streaming commerce. Meanwhile, there are also games between multiple brands and multiple anchors, as well as games among brands-anchors-the live streaming platforms, and the authors will conduct more in-depth research in the future.

So far, the co-impact of anchor influence and brand awareness has not been considered simultaneously in published articles. This paper provides theoretical guidance for the behavioral choices of the brand and the anchor under the live streaming commerce, which is conducive to the prosperous development of live streaming commerce.

This paper aims to develop an efficient algorithm combining straightforward response surface functions with Monte Carlo simulation to conduct seismic reliability analysis in a systematical way.

The representative slip surfaces are identified and based on to calibrate multiple response surface functions with acceptable accuracy. The calibrated response surfaces are used to determine the yield acceleration in Newmark sliding displacement analysis. Then, the displacement-based limit state function is adopted to conduct seismic reliability analysis.

The calibrated response surface functions have fairly good accuracy in predicting the yield acceleration in Newmark sliding displacement analysis. The seismic reliability is influenced by such factors as PGA, spatial variability and threshold value. The proposed methodology serves as an effective tool for geotechnical practitioners.

The multiple sources of a seismic slope response can be effectively determined using the multiple response surface functions, which are easily implemented within geotechnical engineering.

This paper aims to develop a novel grey Bernoulli model with memory characteristics, which is designed to dynamically choose the optimal memory kernel function and the length of memory dependence period, ultimately enhancing the model's predictive accuracy.

This paper enhances the traditional grey Bernoulli model by introducing memory-dependent derivatives, resulting in a novel memory-dependent derivative grey model. Additionally, fractional-order accumulation is employed for preprocessing the original data. The length of the memory dependence period for memory-dependent derivatives is determined through grey correlation analysis. Furthermore, the whale optimization algorithm is utilized to optimize the cumulative order, power index and memory kernel function index of the model, enabling adaptability to diverse scenarios.

The selection of appropriate memory kernel functions and memory dependency lengths will improve model prediction performance. The model can adaptively select the memory kernel function and memory dependence length, and the performance of the model is better than other comparison models.

The model presented in this article has some limitations. The grey model is itself suitable for small sample data, and memory-dependent derivatives mainly consider the memory effect on a fixed length. Therefore, this model is mainly applicable to data prediction with short-term memory effect and has certain limitations on time series of long-term memory.

In practical systems, memory effects typically exhibit a decaying pattern, which is effectively characterized by the memory kernel function. The model in this study skillfully determines the appropriate kernel functions and memory dependency lengths to capture these memory effects, enhancing its alignment with real-world scenarios.

Based on the memory-dependent derivative method, a memory-dependent derivative grey Bernoulli model that more accurately reflects the actual memory effect is constructed and applied to power generation forecasting in China, South Korea and India.

Component positioning is an important part of aircraft assembly, aiming at the problem that it is difficult to accurately fall into the corresponding ball socket for the ball head connected with aircraft component. This study aims to propose a ball head adaptive positioning method based on impedance control.

First, a target impedance model for ball head positioning is constructed, and a reference positioning trajectory is generated online based on the contact force between the ball head and the ball socket. Second, the target impedance parameters were optimized based on the artificial fish swarm algorithm. Third, to improve the robustness of the impedance controller in unknown environments, a controller is designed based on model reference adaptive control (MRAC) theory and an adaptive impedance control model is built in the Simulink environment. Finally, a series of ball head positioning experiments are carried out.

During the positioning of the ball head, the contact force between the ball head and the ball socket is maintained at a low level. After the positioning, the horizontal contact force between the ball head and the socket is less than 2 N. When the position of the contact environment has the same change during ball head positioning, the contact force between the ball head and the ball socket under standard impedance control will increase to 44 N, while the contact force of the ball head and the ball socket under adaptive impedance control will only increase to 19 N.

In this paper, impedance control is used to decouple the force-position relationship of the ball head during positioning, which makes the entire process of ball head positioning complete under low stress conditions. At the same time, by constructing an adaptive impedance controller based on MRAC, the robustness of the positioning system under changes in the contact environment position is greatly improved.

This paper aims to focus on the accurate analysis of the fractional heat transfer in a two-dimensional (2D) rectangular monolayer tissue with three different kinds of lateral boundary conditions and the quantitative evaluation of the degree of thermal damage and burn depth.

A symplectic method is used to analytically solve the fractional heat transfer dual equation in the frequency domain (s-domain). Explicit expressions of the dual vector can be constructed by superposing the symplectic eigensolutions. The solution procedure is rigorously rational without any trial functions. And the accurate predictions of temperature and heat flux in the time domain (t-domain) are derived through numerical inverse Laplace transform.

Comparison study shows that the maximum relative error is less than 0.16%, which verifies the accuracy and effectiveness of the proposed method. The results indicate that the model and heat source parameters have a significant effect on temperature and thermal damage. The pulse duration (Δt) of the laser heat source can effectively control the time to reach the peak temperature and the peak slope of the thermal damage curve. The burn depth is closely correlated with exposure temperature and duration. And there exists the delayed effect of fractional order on burn depth.

A symplectic approach is presented for the thermal analysis of 2D fractional heat transfer. A unified time-fractional heat transfer model is proposed to describe the anomalous thermal behavior of biological tissue. New findings might provide guidance for temperature prediction and thermal damage assessment of biological tissues during hyperthermia.

This study investigates the credit supervision issue in rural e-commerce. By studying the trading strategies of buyers and sellers under different credit supervision measures and the impact of different pricing strategies on the trading strategies of both parties, this paper proposes regulatory suggestions for the increasingly severe credit problems in rural e-commerce.

In the online agricultural product transaction between farmers and consumers, both parties' decision-making is a dynamic process. Using the copying dynamic model of the evolutionary game, this study establishes two evolutionary game models to explore the factors affecting credit supervision in the rural e-commerce transaction process. Then, the study provides corresponding countermeasures and suggestions.

First, credit supervision measures implemented by rural e-commerce platforms and the Government's legal system construction and infrastructure construction guarantees influence both parties' trust choices in rural e-commerce transactions. Second, price is a key factor affecting both parties' trading strategies. In the case of relatively fair prices, the higher the proportion of farmers who choose “low price” and “honest transaction” strategies, the easier that is for consumers to choose to trust farmers. In contrast, the higher the price, the higher the proportion of consumers who choose the “trust farmers” strategy, and the more willing farmers are to choose honest transactions.

This work develops a new approach for analyzing rural e-commerce credit supervision. Moreover, this study helps establish and improve the credit supervision mechanism of rural e-commerce and further realize the long-term sustainable development of the rural economy.