V. I. Rabanovich

In this paper, we consider various new inverse spectral problems (ISP) for metric graphs, using maximal eigen values of the adjacency matrix of the graph and its subgraphs as well as the corresponding eigen vectors or some of their components as spectral data. We give examples of spectral data that uniquely determine the metric on the graph. Effective algorithms for solving the considered ISP are given.

For a certain class of finite posets, we prove that all their irreducible orthoscalar representations are finite-dimensional and describe those, for which there exist essential (non-degenerate) irreducible orthoscalar representations.

In this paper we consider decompositions of the identity operator into a linear combination of $k\ge 5$ orthogonal projections with real coefficients. It is shown that if the sum $A$ of the coefficients is closed to an integer number between $2$ and $k-2$ then such a decomposition exists. If the coefficients are almost equal to each other, then the identity can be represented as a linear combination of orthogonal projections for $\frac{k-\sqrt{k^2-4k}}{2} < A < \frac{k+\sqrt{k^2-4k}}{2}$. In the case where some coefficients are sufficiently close to $1$ we find necessary conditions for the existence of the decomposition.