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In this paper, we give the four equivalent characterizations for the weighted local hardy spaces on Lipschitz domains. Also, we give their application for the harmonic function defined in bounded Lipschitz domains.

The effect of persistence of physical exercise on the psychological and emotional aspects of primary school students is studied to improve the comprehensive quality of current Chinese primary school students and explore the effect of physical exercise on students’ emotions under the science, technology, engineering, art, mathematics (STEAM) education concept. First, students in a primary school in Nanchang are taken as the survey participants. Second, by formulating a physical exercise scale and a psychological and emotional scale, the current situation of physical exercise of primary school students is investigated by means of mathematical statistics. Finally, the current situation of physical exercise and the overall situation of positive psychological emotions of primary school students are analyzed, and the effect of physical exercise on the positive psychological emotions of primary school students is studied. The data show that there are significant differences in the amount of exercise and its three dimensions of intensity, time, and frequency, as well as the scores of positive emotions in the gender dimension, with the boys scoring higher than the girls. In terms of grades, students in grades 1, 2, and 6 are higher than students in grades 3, 4, and 5 on the level of a small amount of exercise, while students in grades 3, 4, and 5 are higher than the other three grades in terms of a moderate amount of exercise. Moreover, in the aspect of positive psychological emotions, the lower-grade students are obviously higher than the upper-grade students, and the second- and third-grade students present marginal significance, p = 0.058. The correlation and regression between physical exercise and positive psychological emotions are calculated and analyzed, and it is found that there is a significant positive correlation between physical exercise indicators and positive psychological emotions, with a correlation coefficient of 0.297. Physical exercise explains 8.8% of positive emotions. This research also makes relevant recommendations for students and schools and has played a role in strengthening the physical exercise and mental health of primary and secondary school students. Greater attention to the physical exercise of primary school students is recommended.

Student engagement is an essential indicator of educational quality and an instability behavior influenced by teachers. However, research on how teacher support influences this behavioral outcome in physical education has started late compared to academic settings. Hence, this systematic review aims to examine the empirical literature regarding the relationship between perceived teacher support and student engagement in physical education. This review complied with the PRISMA statement and ultimately found 11 eligible studies through the literature utilizing several databases, namely, Web of Science, SCOPUS, PsycINFO, ERIC, and SPORTDiscus. The results revealed a significant positive relationship between perceived teacher support and multiple dimensions of student engagement in physical education, especially in behavioral and emotional engagement. Mediating effects were found in five studies, and autonomous motivation and psychological need satisfaction were the crucial mediators. Teacher support plays a vital role in positive student–teacher interactions and students demonstrate enhanced engagement in physical education learning when teachers provide autonomy, competence, and emotional support. This study has great significance for developing instructional strategies to improve the sustainability of student engagement in physical education and also provides insight for researchers exploring options for optimizing motivational teaching styles to promote the development of sustainable teaching practices.

At present, there are many kinds of intelligent training equipment in tennis sports, but they all need human control. If a single tennis player uses the robot to return the ball, it will save some human resources. This study aims to improve the recognition rate of tennis sports robots in the return action and the return strategy. The human-oriented motion recognition of the tennis sports robot is taken as the starting point to recognize and analyze the return action of the tennis sports robot. The OpenPose traversal dataset is used to recognize and extract human motion features of tennis sports robots under different classifications. According to the return characteristics of the tennis sports robot, the method of tennis return strategy based on the support vector machine (SVM) is established, and the SVM algorithm in machine learning is optimized. Finally, the return strategy of tennis sports robots under eight return actions is analyzed and studied. The results reveal that the tennis sports robot based on the SVM-Optimization (SVM-O) algorithm has the highest return recognition rate, and the average return recognition rate is 88.61%. The error rates of the backswing, forward swing, and volatilization are high in the return strategy of tennis sports robots. The preparation action, backswing, and volatilization can achieve more objective results in the analysis of the return strategy, which is more than 90%. With the increase of iteration times, the effect of the model simulation experiment based on SVM-O is the best. It suggests that the algorithm proposed has a reliable accuracy of the return strategy of tennis sports robots, which meets the research requirements. Human motion recognition is integrated with the return motion of tennis sports robots. The application of the SVM-O algorithm to the return action recognition of tennis sports robots has good practicability in the return action recognition of tennis sports robot and solves the problem that the optimization algorithm cannot be applied to the real-time requirements. It has important research significance for the application of an optimized SVM algorithm in sports action recognition.

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The objective of this research was to examine gender differences in entrepreneurial venture interests drawing on goal congruity theory, which posits that people adopt gender-stereotypic goal orientations in response to social pressures to conform to traditional gender roles. Aspiring entrepreneurs (N = 351) first wrote about what they believed made an entrepreneur successful. They then completed measures of agentic and communal goal orientations (i.e., male and female stereotypic orientations, respectively) and indicated their interests in starting ventures in stereotypically feminine (e.g., salon), masculine (e.g., auto-repair) and science, technology, engineering, and mathematics (STEM; e.g., software developer) ventures. Analysis of open-ended responses demonstrated that participants ascribed more agentic and, specifically, more dominance attributes to entrepreneurs than communal attributes (e.g., warmth). Bifactor structural equation modeling indicated that, as expected, agentic goal orientations included dimensions of competence, self-direction, and dominance orientations; communal goal orientations were unidimensional. Further, as expected, dominance and communal orientations partially accounted for gender differences in all three career types. We discuss implications for entrepreneurial education and practice from a goal congruity perspective and the use of bifactor modeling to improve the measurement of goal orientations.