2019
Том 71
№ 11

### All Issues

Articles: 2
Brief Communications (English)

### Local Maxima of the Potential Energy on Spheres

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1427–1429

Let S d be a unit sphere in ℝ d+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N S d , we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ ij ||x i x j ||α . The following theorem is proved: for αd − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima.

Article (Ukrainian)

### Representations of Algebras Defined by a Multiplicative Relation and Corresponding to the Extended Dynkin Graphs $\tilde{D}_4, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$

Ukr. Mat. Zh. - 2012. - 64, № 12. - pp. 1654-1675

We describe, up to unitary equivalence, all $k$-tuples $(A_1, A_2,..., A_k)$ of unitary operators such that $A^{n_i}_i = I$ for $i = \overline{1, k}$ and $A_1 A_2 ... A_k = \lambda I$, where the parameters $(n_1,... ,n_k)$ correspond to one of the extended Dynkin diagrams $\tilde{D}_4, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8$, and $\lambda \in \mathbb{C}$ is a fixed root of unity.