Many physics and mathematics problems generate linear equation systems, and solving linear equation systems has become an important proposition. Therefore, combined with the characteristics of modern computers, various methods for solutions need to be sought. This article introduces applications of linear equation systems in different fields, as well as representative figures and works with outstanding achievements. It focuses on providing formulas and corresponding Matlab codes for Gauss elimination, Jacobi iteration, and G-S iteration to solve equation systems, and rigorously proves the sufficient and necessary condition for convergence of iterative formula and also proves the convergence of different iteration methods under different types of coefficient matrices. Based on these solving methods, two examples are practiced in Matlab, the running time and iteration times of different methods are comprehensively compared. Thus, the superiority of the G-S iterative method is obtained. Finally, when there are zero elements in the diagonal elements of the coefficient matrix, the article proposes an improved method to solve this problem.

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With the improvement of people’s living standards in the new era, tourism consumption has gradually become a hotspot of popular entertainment. Beijing faces the challenges of high tourist carrying capacity at attractions and uneven distribution of tourism resources. There is a growing need for personalised travel path planning. This study aims to develop a one-stop personalised intelligent recommendation model for tourist attractions in the Beijing area to enhance tourists’ travel experience. By integrating data from mainstream travel websites such as Ctrip, Tongcheng, and Qunar, the paper uses natural language processing (NLP) technology to conduct analyses of online reviews to derive user sentiment and personalisation indicators. The entropy weight method is used to comprehensively consider the user’s personalised travel preferences, combined with the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method to scientifically rank the attractions and select the candidate set. Finally, the path planning algorithm with distance factor is implemented based on a greedy algorithm to optimise the travel path according to the user’s interest and achieve the recommendation of personalised travel routes. The model proposed in this study shows high accuracy and user satisfaction in empirical tests, which strengthens the user information processing support and personalisation needs in the era of big data, and contributes new solutions to the field of travel path recommendation.

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As urbanization accelerates, noise pollution from public activities like square dancing has become a growing concern. This study focuses on the sound system used in square dancing and introduces directional acoustic propagation technology based on acoustic metasurfaces. The goal is to mitigate the noise impact on surrounding communities without compromising the experience of the dancers. Through a combination of theoretical design, acoustic simulation, 3D modeling and printing, and acoustic testing, we have successfully developed and verified both transmission and reflection acoustic metasurface structures. These innovations effectively reduce community noise pollution while maintaining the enjoyment of square dancing participants. The results demonstrate that this technology is cost-effective and compact, making it highly valuable for practical applications.

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As a complex hydrodynamic phenomenon, cavitation has always been a focus of attention due to its harmful effects. However, it is important to note that the collapse of cavitation bubbles produces instantaneous high temperatures and pressures as well as high-speed jet phenomena, which means this process contains tremendous energy. This energy can not only enhance physical and chemical processes but also achieve the goals of energy conservation and efficiency improvement. Currently, there have been no breakthroughs in effectively generating cavitation, adopting highly efficient cavitation devices, or accurately quantifying and analyzing the effects of cavitation. Therefore, research on the evolution process of cavitation and the factors influencing its effects is significant. In recent years, cavitation technology has developed rapidly in the field of wastewater degradation. To improve the efficiency of cavitation degradation, based on the theory of the Venturi effect in fluid mechanics, this project proposes a novel cavitation generator using a narrow slit type Venturi tube through a combination of motion equations and computational simulation. This design aims to increase shear area and enhance the yield of cavitation bubbles. A low-cost, low-energy consumption new cavitation device was optimized and manufactured based on numerical simulation. The new cavitation generator was used to explore the influence of different throat geometric characteristics and expansion angles on cavitation capability. Firstly, the continuity equation and momentum equation were derived based on Rayleigh's cavitation theory and the Venturi effect in fluid mechanics; secondly, a computational fluid dynamics (CFD) grid was established and finite element simulation calculations were conducted for the structural optimization design of the new cavitation device; finally, by combining fluid mechanics theoretical formulas and numerical simulation, a low-cost, low-energy consumption new cavitation device was manufactured, and experimental analysis was conducted to study the influence of different throat geometric characteristics and expansion angles on cavitation capability.

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