# Plakhotnyk M. V.

### Topological conjugate piecewise linear unimodal mappings of an interval into itself

Kirichenko V. V., Plakhotnyk M. V.

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 217-226

Let $f, g : [0, 1] \rightarrow [0, 1]$ be a pair of continuous piecewise linear unimodal mappings and let $f$ be defined as follows: $f(x) = 2x$ for $x \leq 1/2$ and $f(x) = 2 - 2x$ for $x > 1/2$. Also let $h : [0, 1] \rightarrow [0, 1]$ be a piecewise differentiable homeomorphism such that $fh = hg$. Then $h$ is piecewise linear and the mapping $f$ completely determines $g$ and $h$, together with the ascending or descending monotone parts of $g$.

### Simple strongly connected quivers and their eigenvectors

Dudchenko I. V., Kirichenko V. V., Plakhotnyk M. V.

Ukr. Mat. Zh. - 2012. - 64, № 3. - pp. 291-306

We study the relationship between the isomorphism of quivers and properties of their spectra. It is proved that two simple strongly connected quivers with at most four vertices are isomorphic to one another if and only if their characteristic polynomials coincide and their left and right normalized positive eigenvectors that correspond to the index can be obtained from one another by the permutation of their coordinates. An example showing that this statement is not true for quivers with five vertices is given.