On the correlation function of the characteristic polynomials of the hermitian Wigner ensemble

We consider the asymptotics of the correlation functions of the characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$. We show that for the correlation function of any even order the asymptotic coincides with this for the GUE up to a factor, depending only on the forth moment of the common probability law $Q$ of entries $\Im W_{jk}$, $\Re W_{jk}$, i.e. that the higher moments of $Q$ do not contribute to the above limit.Comment: 20

The formation of primary school teachers’ professional readiness for teaching Russian as a foreign language using the opportunities of cross-border education

The article is devoted to the problem of using the opportunities of cross-border education as a factor of professional training of primary school teachers for tuition Russian as a second language. The authors note the relevance of this problem both in theoretical aspects and at the practical level. The article describes an effective way to form the readiness of primary school teachers to implement cross-border education. In this study, the authors reveal new opportunities for professional development of primary school teachers. Based on the content analysis of the regulatory database, it is shown that this problem is related to an upward trend in the Russian higher education system. The article reveals the relationship between teachers' professional training in the field of tuition Russian language to younger students, for whom Russian is a second language, and the existing problems with using Online learning. The article substantiates the need for training primary school teachers, which contributes to the development of their professional competencies both in the Offline mode of training and the Online training. A set of pedagogical measures related to the professional training of primary school teachers to teach Russian as a non-native language in cross-border education can become one of the foundations for the development of the Russian Federation education system. The article is addressed to the teachers of higher and secondary vocational training institutions, as well as future and certified teachers of primary General education