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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2002, Volume 239, Pages 52–62
(Mi tm358)
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A Relative Ryshkov Perfect Polyhedron As a Generatrix of a $W$-Tiling
R. G. Barykinskii M. V. Lomonosov Moscow State University
Abstract:
For an arbitrary positive definite quadratic form $f$ in $n$ variables ($n$-PQF) and any positive number $\rho$, the notion of $(f,\rho)$-perfect $(n+m)$-PQF is introduced. The problem of finding all such forms for any given $n$-PQF $f$ and $\rho>0$ is studied. Two representations of all $(f,\rho)$-perfect $(n+1)$-PQFs are obtained: one in the form of the vertices of a tiling of the Euclidean $n$-space (we call this tiling a $W$-tiling corresponding to the $n$-PQF $f$ and the number $\rho$), and the other in the form of the vertices of an $n$-dimensional polyhedral surface $\mu _f(\rho )$ (we call it a relative Ryshkov perfect polyhedron corresponding to the $n$-PQF $f$ and the number $\rho$). It is proved that the polyhedron $\mu _f(\rho )$ is a generatrix of the $W$-tiling corresponding to the $n$-PQF $f$ and the number $\rho$.
Received in April 2002
Citation:
R. G. Barykinskii, “A Relative Ryshkov Perfect Polyhedron As a Generatrix of a $W$-Tiling”, Discrete geometry and geometry of numbers, Collected papers. Dedicated to the 70th birthday of professor Sergei Sergeevich Ryshkov, Trudy Mat. Inst. Steklova, 239, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 52–62; Proc. Steklov Inst. Math., 239 (2002), 45–54
Linking options:
https://www.mathnet.ru/eng/tm358 https://www.mathnet.ru/eng/tm/v239/p52
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Abstract page: | 204 | Full-text PDF : | 74 | References: | 44 |
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