Criteria for the existence of systems of subspaces related to a certain class of unicyclic graphs
UDC 512.552.4We study the configurations of subspaces of a Hilbert space associated with a unicyclic graph, which is a cycle of length $m\geqslant 3$ and has, at each vertex of the cycle, a chains of length $s\geqslant 1$ glued to the vertex. There is a one-to-one correspondence between the vertices...
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| Date: | 2021 |
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| Main Authors: | , , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2021
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/6354 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 512.552.4We study the configurations of subspaces of a Hilbert space associated with a unicyclic graph, which is a cycle of length $m\geqslant 3$ and has, at each vertex of the cycle, a chains of length $s\geqslant 1$ glued to the vertex. There is a one-to-one correspondence between the vertices and subspaces. If an edge connects two vertices, then the angle between subspaces is equal to $\psi\in(0;\pi/2),$ otherwise the subspaces are orthogonal. Applying the theorem on reduction of unicyclic graph, we prove that nonzero configurations exist if and only if $\cos\psi\in(0;\tau_{m,s}].$ We obtain formulas for $\tau_{m,s}$ and show that~$\bigcap\limits_{m,s}(0;\tau_{m,s}] = (0;2/5].$ |
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| DOI: | 10.37863/umzh.v73i4.6354 |