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In this paper, we introduce the concept of a supporting set for a general system. We study basic properties of supporting sets for a general system, supporting sets for subsytems of a system, homomorphic images of supporting sets for a system, and supporting sets for free sums and direct sums of systems.

THE use of liquid oxygen as an oxidizer for various fuels in liquid rocket propellent systems is not new. Professor Goddard used liquid oxygen in his rocket experiments and the well known German V‐2 rockets used this material as an oxidizer. However, its effect on non‐metallic materials ordinarily used in rocket systems was not investigated until recent years. This investigation was prompted by phenomena which had been experienced by rocket engine and rocket aircraft manufacturers and by suppliers of the material. It was observed that when some organic materials came in intimate contact with liquid oxygen they became prone to detonation when subjected to certain impact energies. This was undoubtedly due to the formation of unstable organo‐peroxide compounds which when impacted released high levels of energy resulting in an explosion. Specifically, when liquid oxygen was accidently spilled on asphalt and inadvertently stepped on, the asphalt would often explode. Also, leather gaskets immersed in liquid oxygen and subjected to surge impact detonated with disastrous effects.

In the course of development of human communities towards industrially advanced societies of today, there have been changes not only in economic conditions in the majority of fields of human activity but also in the ways of providing for food, which went hand in hand with changes in living conditions. Large‐scale industrial production has brought about major changes in the way of life of the population. The development of industrial agglomerations results in a growing number of city dwellers, most of whom have no chance of producing foodstuffs of their own in kind. The distance between residential and industrial areas has been growing, the number of employed women has risen considerably. These are but a few factors affecting the way of boarding, particularly in households.

12,2 Millionen Bundesbürger sind über 60 Jahre alt. Im Jahre 2000 werden es bereits 14 Millionen sein. Kein Wunder also, dass die Senioren für viele Wirtschaftszweige immer interessanter werden. Dies gilt auch für die Tourismusbranche, die mit der wachsenden Reisefreude der älteren Generation rechnen kann. Im vorliegenden Beitrag werden ausgewählte Ergebnisse einer Studie wiedergegeben, die das Institut für Freizeit‐ und Tourismusberatung aus Ettlingen Anfang des Jahres gemacht hatte.

Let p_[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p_[1,r;t](n) is the generalisation of the two-colour partition function p_[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.

Using q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.

In the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.

The partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.

Economic effects are only one aspect of the total impact of tourism. For example, the development of a tourist industry in a particular country has varying repurcussions on the long term socio‐cultural, environmental and ecological development of the local community as a whole. This article examines the main non‐economic aspects of tourism in Greece.

This study explores the immobilisation of enzymes within porous catalysts of various geometries, including spheres, cylinders and flat pellets. The objective is to understand the irreversible Michaelis-Menten kinetic process within immobilised enzymes through advanced mathematical modelling.

Mathematical models were developed based on reaction-diffusion equations incorporating nonlinear variables associated with Michaelis-Menten kinetics. This research introduces fractional derivatives to investigate enzyme reaction kinetics, addressing a significant gap in the existing literature. A novel approximation method, based on the independent polynomials of the complete bipartite graph, is employed to explore solutions for substrate concentration and effectiveness factor across a spectrum of parameter values. The analytical solutions generated through the bipartite polynomial approximation method (BPAM) are rigorously tested against established methods, including the Bernoulli wavelet method (BWM), Taylor series method (TSM), Adomian decomposition method (ADM) and fourth-order Runge-Kutta method (RKM).

The study identifies two main findings. Firstly, the behaviour of dimensionless substrate concentration with distance is analysed for planar, cylindrical and spherical catalysts using both integer and fractional order Michaelis-Menten modelling. Secondly, the research investigates the variability of the dimensionless effectiveness factor with the Thiele modulus.

The study primarily focuses on mathematical modelling and theoretical analysis, with limited experimental validation. Future research should involve more extensive experimental verification to corroborate the findings. Additionally, the study assumes ideal conditions and uniform catalyst properties, which may not fully reflect real-world complexities. Incorporating factors such as mass transfer limitations, non-uniform catalyst structures and enzyme deactivation kinetics could enhance the model’s accuracy and broaden its applicability. Furthermore, extending the analysis to include multi-enzyme systems and complex reaction networks would provide a more comprehensive understanding of biocatalytic processes.

The validated bipartite polynomial approximation method presents a practical tool for optimizing enzyme reactor design and operation in industrial settings. By accurately predicting substrate concentration and effectiveness factor, this approach enables efficient utilization of immobilised enzymes within porous catalysts. Implementation of these findings can lead to enhanced process efficiency, reduced operating costs and improved product yields in various biocatalytic applications such as pharmaceuticals, food processing and biofuel production. Additionally, this research fosters innovation in enzyme immobilisation techniques, offering practical insights for engineers and researchers striving to develop sustainable and economically viable bioprocesses.

The advancement of enzyme immobilisation techniques holds promise for addressing societal challenges such as sustainable production, environmental protection and healthcare. By enabling more efficient biocatalytic processes, this research contributes to reducing industrial waste, minimizing energy consumption and enhancing access to pharmaceuticals and bio-based products. Moreover, the development of eco-friendly manufacturing practices through biocatalysis aligns with global efforts towards sustainability and mitigating climate change. The widespread adoption of these technologies can foster a more environmentally conscious society while stimulating economic growth and innovation in biotechnology and related industries.

This study offers a pioneering approximation method using the independent polynomials of the complete bipartite graph to investigate enzyme reaction kinetics. The comprehensive validation of this method through comparison with established solution techniques ensures its reliability and accuracy. The findings hold promise for advancing the field of biocatalysts and provide valuable insights for designing efficient enzyme reactors.

In order to improve the computation efficiency of the four-point rainflow algorithm, a new fast four-point rainflow cycle counting algorithm (FFRA) using a novel loop iteration mode is proposed.

In this new algorithm, the loop iteration mode is simplified by reducing the number of iterations, tests and deletions. The high efficiency of the new algorithm makes it a preferable candidate in fatigue life online estimation of structural health monitoring systems.

The extensive simulation results show that the extracted cycles by the new FFRA are the same as those by the four-point rainflow cycle counting algorithm (FRA) and the three-point rainflow cycle counting algorithm (TRA). Especially, the simulation results indicate that the computation efficiency of the FFRA has improved an average of 12.4 times compared to the FRA and an average of 8.9 times compared to the TRA. Moreover, the equivalence of cycle extraction results between the FFRA and the FRA is proved mathematically by utilizing some fundamental properties of the rainflow algorithm. Theoretical proof of the efficiency improvement of the FFRA in comparison to the FRA is also given.

This merit makes the FFRA preferable in online monitoring systems of structures where fatigue life estimation needs to be accomplished online based on massive measured data. It is noticeable that the high efficiency of the FFRA attributed to the simple loop iteration, which provides beneficial guidance to improve the efficiency of existing algorithms.

This paper studies the optimal decision-making and coordination problem of a dual-channel fresh agricultural product (FAP) supply chain. The purpose is to analyze the impact of information sharing on optimal decisions and propose a coordination mechanism to encourage supply chain members to share information.

The two-echelon dual-channel FAP supply chain includes a manufacturer and a retailer. By using the Stackelberg game theory and the backward induction method, the optimal decisions are obtained under information symmetry and asymmetry and the coordination contract is designed.

The results show that supply chain members should comprehensively evaluate the specific situation of product attributes, coefficient of freshness-keeping cost and network operating costs to make decisions. Asymmetric information can exacerbate the deviation of optimal decisions among supply chain members and information sharing is always beneficial to manufacturers but not to retailers. The improved revenue-sharing and cost-sharing contract is an effective coordination mechanism.

The conclusions can provide theoretical guidance for supply chain managers to deal with information asymmetry and improve the competitiveness of the supply chain.

This paper combines the three characteristics that are most closely related to the reality of supply chains, including horizontal and vertical competition of different channels, the perishable characteristics of FAPs and the uncertainty generated by asymmetric demand information.

The aim of this study is to examine the influence of consumer preferences for overseas green products and the implementation of blockchain technology on the performance of a supply chain, which comprises an overseas manufacturer and a domestic e-commerce platform. This research endeavors to identify the optimal pricing decisions and strategies for both the manufacturer and the platform in the context of the expanding e-commerce and globalization of the economy.

The authors propose and analyze four distinct models based on the selection of selling contracts by the manufacturer and the adoption strategy of blockchain by the platform, using game theory to obtain the optimal solutions for these models.

The authors show that consumer migration promotes the manufacturer's green inputs, while the expansion of green consumer proportion is not conducive to it. They also show that blockchain technology has the potential to effectively limit manufacturer cannibalization. Interestingly, the study reveals a cascading effect of advantage where the manufacturer's profit variation trend changes only with the integration of pricing power advantage and blockchain technology inputs. This effect suggests that the equilibrium strategy is achievable under the agency contract with blockchain adoption, while Pareto improvement can be obtained with blockchain technology under both selling contracts.

This research could be extended in several possible directions. First, future work could explore outsourcing strategies for overseas manufacturers. Second, more types of consumer heterogeneity and different risk preferences could be considered. Third, this study can be extended by further exploring the design of mechanisms under asymmetric demand information to make the model more realistic.

The authors examine the impact of market segmentation and consumer preferences on green supply chain decisions, and analyze supply chain members' strategic choices for selling contracts and blockchain adoptions. The research also sheds light on the theoretical underpinnings and practical applications of green supply chain development and blockchain applications.