Modern education attaches great importance to the innovation of teaching concepts, and teachers should be guided by it to provide students with high-quality educational resources and learning environment. Teachers should conduct in-depth research on auditing course materials, set certain training goals for students, and optimize their teaching ideas to conduct diversified evaluations of students. Teachers should create an environment for students to learn auditing and choose corresponding teaching methods based on their learning situation. Teachers should also guide students to master the courses and basic theories of auditing, so that they have certain operational skills and can apply relevant theories to analyze and develop problems encountered in the management profession.

The scientificity and truthfulness of Marxist theory are crucial aspects of our research. To gain a comprehensive understanding, it is essential to examine Marxism within a reference frame and comparative perspective. In this regard, exploring Marx's critique of Proudhonism becomes significant, as it provides insights into the ideological transformations he achieved. Proudhonism, a small bourgeoisie socialism and anarchism that emerged in France in the 1840s and spread across Western Europe in the 1850s-1860s, had intricate connections with Marxism. This analysis not only broadens our research horizon but also holds practical implications for today's society. Thus, our research holds both theoretical and practical significance.

With the popularization of the Internet and the rapid development of computer network technology, human beings have entered a brand new era - the information age. This kind of network technology beyond space not only brings well-being to people, but also subtly affects the ideas and behaviors of teenagers. It not only changes their lifestyle and values, but also quietly makes them mentally ill, resulting in an endless series of problems of juvenile cybercrimes. For the purpose of promoting the governance of Internet crimes among young people effectively and avoiding crimes among special groups of young people, this paper plans to base on the concept of Internet crimes of teenagers, summarize the characteristics of youth crimes in our country, analyze its influence factors and propose the measures to deal with it.

The cultivation of English majors in applied universities aims to cultivate versatile foreign language talents that meet the needs of social development. The construction and development of the English major need to keep up with the pace of changes in the times, and meeting the needs of current social development is the main criterion for measuring the value of this major. The satisfaction of English major students with their major is an important factor affecting the construction and reform of the major. This article mainly uses the questionnaire survey method to survey English majors in several applied universities and analyze their major satisfaction.

In the current context of information technology development, many vocational colleges are reforming and developing their education majors in the direction of information technology. Now that we have entered the era of informatization, the direction of reform and innovation in the education industry can be considered to use the development methods of informatization to improve the comprehensive efficiency of teaching. In the context of informatization, the teaching of pharmacy majors in vocational colleges needs to seize the development advantages of informatization, reform classroom teaching methods, and improve teaching efficiency. This article mainly discusses how to integrate the teaching reform of pharmaceutical majors in vocational colleges with the development trend of informatization, achieve better reform and development efficiency, and cultivate pharmaceutical professionals.

It is well known that determining the exact values of crossing number for circulant graphs is very difficult. Even so, some important results in this field are still proved. D.J. Ma was proved that the crossing number of C(2m + 2, m) is m + 1[8]. Then such problem for C(n, 3) was further solved [7]. Pak Tung Ho and X. Lin obtained accurate values for the crossover numbers of C (3m, m) and C (3m + 1, m)[4][5]. In this paper, as a complement, we show that the edges from the principal cycle of C(9, 3) do not cross each other in an optimal drawing.