Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube
- 80 Downloads
In a space of dimension 30 we find a pair of parallel hyperplanes, uniquely determined by vertices of a unit cube lying on them, such that strictly between the hyperplanes there are no vertices of the cube, though there are integer points. A similar two-sided example is constructed in dimension 37. We consider possible locations of empty quadrics with respect to vertices of the cube, which is a particular case of a discrete optimization problem for a quadratic polynomial on the set of vertices of the cube. We demonstrate existence of a large number of pairs of parallel hyperplanes such that each pair contains a large number of points of a prescribed set.
KeywordsInformation Transmission Dynamic Programming Algorithm Quadratic Polynomial Integer Point Great Common Divisor
Unable to display preview. Download preview PDF.
- 1.Beresnev, V.L., Diskretnye zadachi razmeshcheniya i polinomy ot bulevykh peremennykh (Discrete Distribution Problems and Polynomials in Boolean Variables), Novosibirsk: Inst. Mat., 2005.Google Scholar
- 3.Ahlatçioğlu, A., Bussieck, M., Esen, M., Guignard, M., Jagla, J.-H., and Meeraus, A., Combining QCR and CHR for Convex Quadratic Pure 0-1 Programming Problems with Linear Constraints, Ann. Oper. Res. (online publ.), September 30, 2011, DOI:10.1007/s10479-011-0969-1.Google Scholar
- 5.Seliverstov, A.V. and Lyubetsky, V.A., On Symmetric Matrices with Indeterminate Leading Diagonals, Probl. Peredachi Inf., 2009, vol. 45, no. 3, pp. 73–78 [Probl. Inf. Trans. (Engl. Transl.), 2009, vol. 45, no. 3, pp. 258–263].Google Scholar
- 8.Shnurnikov, I.N., Cardinality of a Separable Set of Vertices of a Multidimensional Cube, Vestnik Moskov. Univ., Ser. I: Mat. Mekh., 2010, no. 2, pp. 11–17.Google Scholar
- 9.Seliverstov, A.V. and Lyubetsky, V.A., On Forms Vanishing at Every Vertex of a Cube, Inform. Protsessy, 2011, vol. 11, no. 3, pp. 330–335.Google Scholar