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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With the economic development, environmental regulating equipment such as air conditioners have improved indoor temperature while also bringing about problems such as dry air and deteriorated air quality. This paper takes a rectangular room as the research object, based on the Navier-Stokes equation and Fourier's law of heat conduction, designs the optimal shape and size, and simulates the temperature changes in summer and winter under different influencing factors. Through random search algorithm and finite difference method for solution, the optimal shape and size of the air conditioner are determined. Then, using the Lagrangian CFD method, the influence of the position, quantity, wind direction, angle, wind speed and air volume of the air conditioner on temperature changes is deeply studied. The results show that the summer regulation effect of the air conditioner is better than that of winter. The research in this paper provides theoretical basis and optimization schemes for air conditioner design, and helps to improve the energy efficiency and user experience of air conditioners.

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Galaxy mergers are a crucial pathway for the growth and evolution of galaxies at the cosmic center. Supermassive black holes (SMBHs) are commonly present at the centers of galaxies. During galaxy mergers, the central SMBHs may form binary black hole systems and, in some cases, merge. Detecting and studying supermassive binary black hole systems within galaxies is a significant aspect of galaxy evolution research. Active black holes produce prominent broad emission lines, which serve as effective probes of such activity. The presence of two distinct sets of broad emission lines with velocity differences has traditionally been considered a key indicator for identifying binary black holes. However, the probability of both black holes in a binary system being active is extremely low. More commonly, binary systems consist of one active and one quiescent black hole. The spectral signature of such systems is characterized by a significant velocity offset between the broad emission lines and the system's narrow emission lines. This velocity difference has also led to the discovery of "recoiling" black holes, driven by gravitational wave radiation from closely bound binary systems. Recoiling black holes are crucial observational targets for studying black hole mergers, binary orbital evolution, and galaxy mergers.We have identified a class of recoiling black holes with partially obscured nuclear regions. Dust in the nuclear region attenuates the intense radiation from the active black hole, allowing the host galaxy's emission to become visible alongside the nuclear radiation and broad-line features. This discovery provides a unique perspective for exploring the physical connection between the evolution of binary black hole systems and galaxy evolution. Further studies using multi-wavelength photometry, high-resolution spectroscopic analysis, and long-term spectral monitoring of partially obscured supermassive binary black hole systems are expected to reveal the physical processes underlying black hole mergers and galaxy mergers, as well as their significant role in galaxy evolution.

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