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The purpose of the present article is to obtain the similarity solution for the shock wave generated by a piston propagating in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows.

The Lie group theoretic method given by Sophus Lie is used to obtain the similarity solution in the present article.

Similarity solution with exponential law shock path is obtained for both ideal and nonideal gas cases. The effects on the flow variables, density ratio at the shock front and shock strength by the variation of the shock Cowling number, adiabatic index of the gas, gravitational parameter and nonidealness parameter are investigated. The shock strength decreases with an increase in the shock Cowling number, nonidealness parameter and adiabatic index, whereas the strength of the shock wave increases with an increase in gravitational parameter.

Propagation of shock wave with spherical geometry in a self-gravitating nonideal gas under the impact of azimuthal magnetic field for adiabatic and isothermal flows has not been studied by any author using the Lie group theoretic method.

This study aims to propose a consensus model that considers dynamic trust and the hesitation degree of the expert's evaluation, and the model can provide personalized adjustment advice to inconsistent experts.

The trust degree between experts will be affected by the decision-making environment or the behavior of other experts. Therefore, based on the psychological “similarity-attraction paradigm”, an adjustment method for the trust degree between experts is proposed. In addition, we proposed a method to measure the hesitation degree of the expert's evaluation under the multi-granular probabilistic linguistic environment. Based on the hesitation degree of evaluation and trust degree, a method for determining the importance degree of experts is proposed. In the feedback mechanism, we presented a personalized adjustment mechanism that can provide the personalized adjustment advice for inconsistent experts. The personalized adjustment advice is accepted readily by inconsistent experts and ensures that the collective consensus degree will increase after the adjustment.

The results show that the consensus model in this paper can solve the social network group decision-making problem, in which the trust degree among experts is dynamic changing. An illustrative example demonstrates the feasibility of the proposed model in this paper. Simulation experiments have confirmed the effectiveness of the model in promoting consensus.

The authors presented a novel dynamic trust consensus model based on the expert's hesitation degree and a personalized adjustment mechanism under the multi-granular probabilistic linguistic environment. The model can solve a variety of social network group decision-making problems.

The purpose of this paper is to exploit the multiple-Toeplitz matrices reconstruction method combined with quadratic spatial smoothing processing to improve the direction-of-arrival (DOA) estimation performance of coherent signals at low signal-to-noise ratio (SNRs).

An improved multiple-Toeplitz matrices reconstruction method is proposed via quadratic spatial smoothing processing. Our proposed method takes advantage of the available information contained in the auto-covariance matrices of individual Toeplitz matrices and the cross-covariance matrices of different Toeplitz matrices, which results in a higher noise suppression ability.

Theoretical analysis and simulation results show that, compared with the existing Toeplitz matrix processing methods, the proposed method improves the DOA estimation performance in cases with a low SNR. Especially for the cases with a low SNR and small snapshot number as well as with closely spaced sources, the proposed method can achieve much better performance on estimation accuracy and resolution probability.

The study investigates the possibility of reusing pre-existing designs for the DOA estimation of the coherent signals. The proposed technique enables achieve good estimation performance at low SNRs.

The paper includes implications for the DOA problem at low SNRs in communication systems.

The proposed method proved to be useful for the DOA estimation at low SNR.

Motivated by the wide application of procurement strategies in retailing, this paper aims to examine the effect of procurement strategies on decisions and profits and strategic inventory (SI) is considered.

The game-theoretic models are developed under a two-period fresh product supply chain (FSC), and consist of the mode of purchasing products only in the first period without SI (Scenario S), the mode of purchasing products in every period without SI (Scenario T) and the mode of purchasing products in every period with SI (Scenario TS).

Conducting the calculating and comparing, some major findings can be concluded. In general, two-period purchasing strategies (Scenarios T and TS) promote a higher freshness-keeping effort than the single buying strategy (Scenario S). Regarding the pricing strategy, SI and Scenario S can both contribute to obtaining a lower wholesale price, the retailer's pricing is relatively complicated and hinges on the consumer's sensitivity to freshness-keeping effort and the holding cost. Besides, comparing the sales quantity and the profit, the authors find that Scenario TS stimulates more demands and brings more profits for the manufacturer. However, Scenario TS is not the optimal selection for the reason that SI sometimes hurts the retailer and even the whole supply chain. Whereas, when the holding cost is in a certain range, Scenario TS will lead to a win-win situation.

The main findings of this study can give the enterprises some advice on the procurement strategies of fresh products and the decisions of pricing and the freshness-keeping effort.

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.

Motivated by consumers' concerns about water resources, this paper studies the interactive impact of advertising efforts and water-saving on corporate profits in apparel supply chains. Moreover, this study attempts to find an effective way to improve the profit of supply chain members under different game scenarios. Therefore, this study explores the game scenarios with considering the cost-sharing contract.

This paper constructs two basic game models considering different market power in the apparel supply chains, and explores the Pareto improvement combined with cost-sharing contracts based on the basic models. Furthermore, this study extends the models by considering cost-saving and non-linear demand.

In this paper, it can be found that advertising efforts and water-saving have complex interactive relationships. Counter-intuitively, the increase in advertising efforts may increase water savings. Furthermore, it presents a Pareto improvement when considering cost-sharing contracts, and both the manufacturer and the retailer's profits may improve simultaneously. Moreover, it does not affect the main conclusions when consider the effects of cost-saving and non-linear demand.

Although some important findings have been reached, this paper can be extended in many ways in the future. For example, the coordination mechanism among supply chain members can be considered and the fair distribution of profits can be studied. Moreover, the influence of the government policies on the optimal strategy, as well as changes in social welfare can be considered.

This study offers supply chain members the guidelines on coordinating water-saving investment and advertising efforts which provided new insight into the interaction of these two factors in the apparel supply chains. Moreover, it can provide a coordination mechanism for the supply chain members to improve their profits.

This paper explores the interactive relationship between water-saving and advertising efforts. It can not only save more water resources but also enable consumers to enjoy more environmentally friendly apparel products.

The current literature mainly focuses on the impact of advertising efforts on firm profit. However, this paper studies the interaction between advertising efforts and water-saving in apparel supply chains. Furthermore, this study explores the optimal pricing strategies and Pareto improvement by considering cost-sharing contracts. It can provide theoretical and practical guidance for the decision-maker in deciding on advertising and water-saving investment.

The need for precise synthesis of customized designs has resulted in the development of advanced coating processes for modern nanomaterials. Achieving accuracy in these processes requires a deep understanding of thermophysical behavior, rheology and complex chemical reactions. The manufacturing flow processes for these coatings are intricate and involve heat and mass transfer phenomena. Magnetic nanoparticles are being used to create intelligent coatings that can be externally manipulated, making them highly desirable. In this study, a Keller box calculation is used to investigate the flow of a coating nanofluid containing a viscoelastic polymer over a circular cylinder.

The rheology of the coating polymer nanofluid is described using the viscoelastic model, while the effects of nanoscale are accounted for by using Buongiorno’s two-component model. The nonlinear PDEs are transformed into dimensionless PDEs via a nonsimilar transformation. The dimensionless PDEs are then solved using the Keller box method.

The transport phenomena are analyzed through a comprehensive parametric study that investigates the effects of various emerging parameters, including thermal radiation, Biot number, Eckert number, Brownian motion, magnetic field and thermophoresis. The results of the numerical analysis, such as the physical variables and flow field, are presented graphically. The momentum boundary layer thickness of the viscoelastic polymer nanofluid decreases as fluid parameter increases. An increase in mixed convection parameter leads to a rise in the Nusselt number. The enhancement of the Brinkman number and Biot number results in an increase in the total entropy generation of the viscoelastic polymer nanofluid.

Intelligent materials rely heavily on the critical characteristic of viscoelasticity, which displays both viscous and elastic effects. Viscoelastic models provide a comprehensive framework for capturing a range of polymeric characteristics, such as stress relaxation, retardation, stretching and molecular reorientation. Consequently, they are a valuable tool in smart coating technologies, as well as in various applications like supercapacitor electrodes, solar collector receivers and power generation. This study has practical applications in the field of coating engineering components that use smart magnetic nanofluids. The results of this research can be used to analyze the dimensions of velocity profiles, heat and mass transfer, which are important factors in coating engineering. The study is a valuable contribution to the literature because it takes into account Joule heating, nonlinear convection and viscous dissipation effects, which have a significant impact on the thermofluid transport characteristics of the coating.

The momentum boundary layer thickness of the viscoelastic polymer nanofluid decreases as the fluid parameter increases. An increase in the mixed convection parameter leads to a rise in the Nusselt number. The enhancement of the Brinkman number and Biot number results in an increase in the total entropy generation of the viscoelastic polymer nanofluid. Increasing the strength of the magnetic field promotes an increase in the density of the streamlines. An increase in the mixed convection parameter results in a decrease in the isotherms and isoconcentration.

Whilst a modest number of investigations have been undertaken concerning nanofluids (NFs), the exploration of fluid flow under exponentially stretching velocities using NFs remains comparatively uncharted territory. This work presents a distinctive contribution through the comprehensive examination of heat and mass transfer phenomena in the NF ND–Cu/H₂O under the influence of an exponentially stretching velocity. Moreover, the investigation delves into the intriguing interplay of gyrotactic microorganisms and convective boundary conditions within the system.

Similarity transformations have been used on PDEs to convert them into dimensionless ODEs. The solution is derived by using the homotopy analysis method (HAM). The pictorial notations have been prepared for sundry flow parameters. Furthermore, some engineering quantities are calculated in terms of the density of motile microbes, Nusselt and Sherwood numbers and skin friction, which are presented in tabular form.

The mixed convection effect associated with the combined effect of the buoyancy ratio, bioconvection Rayleigh constant and the resistivity due to the magnetization property gives rise to attenuating the velocity distribution significantly in the case of hybrid nanoliquid. The parameters involved in the profile of motile microorganisms attenuate the profile significantly.

The current simulations have uncovered fascinating discoveries about how metallic NFs behave near a stretched surface. These insights give us valuable information about the characteristics of the boundary layer close to the surface under exponential stretching.

The novelty of the current investigation is the analysis of NF ND–Cu/H₂O along with an exponentially stretching velocity in a system with gyrotactic microorganisms. The investigation of fluid flow at an exponentially stretching velocity using NFs is still relatively unexplored.

This study aims to establish a centralized decision-making game model and manufacturer-led Stackelberg game model based on factors of risk aversion of supply chain members and product greenness. The research aims to study whether the introduction of the “cost + risk sharing” contract affects coordination of this type of green supply by calculating the optimal decision of each mode.

This research designs a supply chain model under centralized and decentralized decision-making. This model uses the Stackelberg game to calculate the optimal decision under decentralized decision-making to evaluate the effect of a green supply chain and then analyze the “cost + risk sharing” contract and the degree of coordination of the supply chain. A sensitivity analysis is conducted on the centralized mode for the impact of variables on the supply chain.

This research finds a double marginalization effect in decentralized decision-making, and the risk aversion coefficient plays a decisive role in the utility of supply chain members. The specific range of risk- and cost-sharing factors allows supply chain members to achieve Pareto improvements and provides decision-making based on the corresponding management strategies according to each other’s risk preference degree.

The influence of each variable on the green supply chain in the centralized mode is studied by MATLAB numerical simulation. It provides reference for green supply chain members to formulate corresponding management strategies according to each other's risk preference degree.

This research innovatively considers manufacturers and retailers to explore the market demand for product greenness. It introduces a novel “cost + risk sharing” contract to coordinate the green supply chain.

The main purpose of this paper is to introduce the gradient discretisation method (GDM) to a system of reaction diffusion equations subject to non-homogeneous Dirichlet boundary conditions. Then, the authors show that the GDM provides a comprehensive convergence analysis of several numerical methods for the considered model. The convergence is established without non-physical regularity assumptions on the solutions.

In this paper, the authors use the GDM to discretise a system of reaction diffusion equations with non-homogeneous Dirichlet boundary conditions.

The authors provide a generic convergence analysis of a system of reaction diffusion equations. The authors introduce a specific example of numerical scheme that fits in the gradient discretisation method. The authors conduct a numerical test to measure the efficiency of the proposed method.

This work provides a unified convergence analysis of several numerical methods for a system of reaction diffusion equations. The generic convergence is proved under the classical assumptions on the solutions.