Given a vector space V and a non-singular quadratic form Q on V , the orthogonal group O(Q) is the subgroup of GL(V ) that preserves Q. The special orthogonal group SO(Q) is given by O(Q) ∩ SL(V ). If Q is the standard inner product on R then SO(Q) is denoted SO(n). This is a connected, compact Lie group. For n > 2, the group is semi-simple. In the case that n is odd, SO(n) corresponds to the D...

1. When a system of sets covers a space, the star number of the covering is the supremum over the sets of the system of the cardinals of the numbers of sets of the system meeting a set of the system . The standard Lebesgue 'brick-laying' construction provides an example, for each positive integer n, of a lattice covering of R" by closed rectangular parallelepipeds with star number 2"T'-1 . In v...

It is shown that every plane compact convex set /f with an interior point admits a covering of the plane with density smaller than or equal to 8(2\/3 — 3)/3 = 1.2376043 For comparison, the thinnest covering of the plane with congruent circles is of density 2n/\Z21 = 1.209199576... (see R. Kershner [3]), which shows that the covering density bound obtained here is close to the best possible. It ...

To cite: Cowart D, Cowart KA, Hougeir FG. BMJ Case Rep Published online: [please include Day Month Year] doi:10.1136/ bcr-2013-201434 DESCRIPTION A 39-year-old African-American woman presented with a 12-year history of ‘itchy dark spots all over my body,’ excluding the face and acral surfaces. The lesions are slightly pruritic at baseline. Flares occur frequently, however, mostly with friction ...

In this paper, we propose a design method of wholebody tactile interface for elevating human-robot coordination. The proposed surface cover sensor as the tactile interface enables robots to detect accurate 3-D force vector applied on various parts of the body during physical interaction with

In this article, we introduce the concept of co-covering which is the dual of covering concept. Then, we prove several theorems being similar to the theorems that have been de-veloped for the covering concept. For example, we provide the lifting criterion for co-coverings of a topological space X, which helps us to classify them by subgroups of the group of all homeomorphisms of X.

A classical theorem of Rogers states that for any convex body K in n-dimensional Euclidean space there exists a covering of the space by translates of K with density not exceeding n log n+n log log n+5n. Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of n the same bound can be attained by a covering which is the union of ...

For a finite connected graph let be the spectral radius of its universal cover. We prove that for any graph of average degree and derive from it the following generalization of the Alon Boppana bound. If the average degree of the graph after deleting any radius ball is at least , then its second largest eigenvalue in absolute value is at least for some absolute constant . This result is tight i...