卷 216, 编号 1 (2025)

For a continuous map $F$ from a finite-dimensional real space to another such space the question of the solvability of the nonlinear equation of the form $F(x)=y$ is investigated for $y$ close to a fixed value $F(\overline x)$. To do this, the concept of $\lambda$-truncation of the map $F$ in a neighbourhood of the point $\overline x$ is introduced and examined. A theorem on the uniqueness of a $\lambda$-truncation is proved. The regularity condition is introduced for $\lambda$-truncations; it is shown to be sufficient for the solvability of the equation in question. A priori estimates for the solution are obtained. Bibliography: 16 titles.

We study the equidistribution problem of zeros in relation to a sequence of $Z$-asymptotically Chebyshev polynomials on $\mathbb{C}^{m}$. We use certain results obtained in a very recent work by Bayraktar, Bloom and Levenberg and obtain an equidistribution result in a more general probabilistic setting than what the paper of Bayraktar, Bloom and Levenberg considers, even though the basis polynomials they use are more general than $Z$-asymptotically Chebyshev polynomials. Our equidistribution result is based on the expected distribution and the variance estimate of random zero currents corresponding to the zero sets (zero divisors) of polynomials. This equidistribution result of general nature shows that equidistribution turns out to be true without the random coefficients being independent and identically distributed, which also means that there is no need to use any specific probability distribution function for these random coefficients. In § 3, unlike in the $1$-codimensional case, we study the basis of polynomials orthogonal with respect to the $L^{2}$-inner product defined by the weighted asymptotically Bernstein–Markov measures on a given locally regular compact set, and with a probability distribution studied well by Bayraktar and including the (standard) Gaussian and the Fubini–Study probability distributions as special cases we have an equidistribution result for codimensions larger than $1$. Bibliography: 35 titles.

Let $N$ be a sufficiently large number. We show that, with at most $O(N^{3/32+\varepsilon})$ exceptions, all even positive integers not exceeding $N$ can be represented in the form $p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4$, where $p_1, p_2, …, p_6$ are prime numbers. This is an improvement of the result $O(N^{7/18+\varepsilon})$ due to Zhang and Li.Bibliography: 13 titles.

We obtain some necessary and sufficient conditions for the strong continuity of representations of topological groups in reflexive Frechet spaces. In particular, we show that a representation $\pi$ of a topological group $G$ in a reflexive Frechet space is continuous in the strong operator topology if and only if for some number $q$, $0\le q<1$, and some neighbourhood $V$ of the identity element $e\in G$, for any neighbourhood $U$ of the zero element in $E$, its polar $\overset\circ{U}$ in the dual space $E^*$, any vector $\xi$ in $U$ and any element $f\in\overset\circ{U}$ the inequality $|f(\pi(g)\xi-\xi)|\le q$ holds for all $g\in V$. Bibliography: 26 titles.