Estimation of the distance between two bodies inside an nn-dimensional ball of unit volume

We consider the problem of estimating the distance between two bodies of volume $\varepsilon$ located inside a $n$-dimensional ball $U$ of unit volume for $n\to\infty$. Let $A$ be a closed set with a smooth boundary of the volume $\varepsilon$ ($0 \leq \varepsilon \leq 1/2$) inside a $n$-dimensional ball $U$ of unit volume that implements among all the sets of volume $\varepsilon$ is a set with the smallest possible free surface area, lying in one half-space with respect to a certain hyperplane that passes through the center of the ball. Then $A$ has the same free surface area as the set representing the intersection of a ball perpendicular to $U$ and the ball $U$ itself.Comment: 6 pages, in Russian, The work was carried out with the help of the Russian Science Foundation Grant N 17-11-01377, to appear in Math. Note

On Shirshov bases of graded algebras

We prove that if the neutral component in a finitely-generated associative algebra graded by a finite group has a Shirshov base, then so does the whole algebra.Comment: 4 pages; v2: minor corrections in English; to appear in Israel J. Mat

Small cancelation rings

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gr\"obner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. This is a short version of paper arXiv:2010.02836Comment: arXiv admin note: substantial text overlap with arXiv:2010.0283

Describing the set of words generated by interval exchange transformation

Let $W$ be an infinite word over finite alphabet $A$. We get combinatorial criteria of existence of interval exchange transformations that generate the word W.Comment: 17 pages, this paper was submitted at scientific council of MSU, date: September 21, 200