منابع مشابه
Covering Spheres with Spheres
Given a sphere of radius r > 1 in the n-dimensional Euclidean space, we study the coverings of this sphere with unit spheres. Our goal is to design a covering of the lowest covering density, which defines the average number of unit spheres covering a point in a bigger sphere. For growing n, we obtain the covering density of (n lnn)/2. This new upper bound is half the order n lnn established in ...
متن کاملCovering and Packing with Spheres
We address the problem of covering R with congruent balls, while minimizing the number of balls that contain an average point. Considering the 1-parameter family of lattices defined by stretching or compressing the integer grid in diagonal direction, we give a closed formula for the covering density that depends on the distortion parameter. We observe that our family contains the thinnest latti...
متن کاملCovering space with convex bodies
1. A few years ago Rogers [1] showed that, if K is any convex body in n-dimensional Euclidian space, there is a covering of the whole space by translates of K with density less than nlogn+nloglogn+5n, provided n > 3. However the fact that the covering density is reasonably small does not imply that the maximum multiplicity is also small. In the natural covering of space by closed cubes, the den...
متن کاملA SUFFICIENT CONDITION FOR AN EXTREME COVERING OF n-SPACE BY SPHERES
The problem of finding the most economical coverings of M-dimensional Euclidean space by equal spheres whose centres form a lattice, which is equivalent to a problem concerning the inhomogeneous minima of positive definite quadratic forms, has been discussed recently by Barnes and Dickson [1]. The reader is referred to [1] for a complete background on the problem. Terms and notations used will ...
متن کاملTiling Hamming Space with Few Spheres
Recently there has been some interest in the combinatorics of the geometry of the Hamming space, e.g., [10], and in particular, in tilings of this space [8]. Here, we investigate partitions of the Hamming space into spheres with possibly different radii. Such a partition is sometimes called a generalized perfect code, see e.g. [1, 3, 6, 13, 15]. Generalized spherepacking bounds can be found in ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1972
ISSN: 0022-314X
DOI: 10.1016/0022-314x(72)90061-3