Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

Loading next page...
 
/lp/springer_journal/geometric-relationship-between-parallel-hyperplanes-quadrics-and-hOure0khRl
Publisher
SP MAIK Nauka/Interperiodica
Copyright
Copyright © 2012 by Pleiades Publishing, Ltd.
Subject
Engineering; Information Storage and Retrieval; Systems Theory, Control; Electrical Engineering; Communications Engineering, Networks
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946012020081
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 13, 2012

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve Freelancer

DeepDyve Pro

Price
FREE
$49/month

$360/year
Save searches from
Google Scholar,
PubMed
Create lists to
organize your research
Export lists, citations
Read DeepDyve articles
Abstract access only
Unlimited access to over
18 million full-text articles
Print
20 pages/month
PDF Discount
20% off