Crystallographic classes in a 4-dimensional Minkowski space

On a crystallographic group, a condition of being topologically discrete is imposed which is weaker than is the conventional requirement for an action on space to be discontinuous. Isomorphism classification is given for crystallographic groups in three crystallographic classes in a 4-dimensional Minkowski space, which are defined by unimodular subgroups of the general Lorentz group. In these classes are, respectively, 24, 36, and 68 crystallographic groups. __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 31–53, January–February, 2008.     

Quasicrystallographic groups on Minkowski spaces

We generalize the quasicrystallographic groups in the sense of Novikov and Veselov from Euclidean spaces to pseudo-Euclidean and affine spaces. We prove that the quasicrystallographic groups on Minkowski spaces whose rotation groups satisfy an additional assumption are projections of crystallographic groups on pseudo-Euclidean spaces. An example shows that the assumption cannot be dropped. We prove that each quasicrystallographic group is a projection of a crystallographic group on an affine space.     

The Yang-Mills fields on the Minkowski space

Let the coordinatex=(x ⁰,x ¹,x ²,x ³) of the Minkowski spaceM ⁴ be arranged into a matrix

Representation of projectors involving Minkowski inverse in Minkowski space

A certain class of results about the different representations of Oblique projectors is present in the literature. These results represent Oblique projectors as the functions of orthogonal projectors with given onto and along spaces. But these results are valid under the restriction that the functions of orthogonal projectors involved are invertible. In this paper we extend and generalize these results. The extension lies in making a transition from Euclidean space to Minkowski space M and the generalization is obtain by voiding the invertibility condition and use of the Minkowski inverse. Furthermore, the nobility lies in utilizing the m-projectors instead of the regular orthogonal projectors.     

Products of conjugacy classes in Chevalley groups II. Covering and generation

In this paper we establish a relationship between generating numbers and covering numbers of conjugacy classes in Chevalley groups over algebraically closed fields. The authors gratefully acknowledge EPSRC grant GR58542. The first author acknowledges further a grant SFB 343 “Diskrete Strukturen in der Mathematik”. The second author thanks the Institute for Advanced Studies at The Hebrew University for its hospitality.     

Crystallographic groups of Sol

We classify all the closed 3‐dimensional orbifolds with Sol‐geometry. These are aspherical orbifolds and so their fundamental groups determine the orbifolds completely. Thus we will classify all the crystallographic groups of Sol, together with all the Bieberbach groups, up to isomorphism.     

A metric space of directions in Minkowski space

A new angular measure in a d-dimensional Minkowski space M was introduced recently. It determines the lengths of rectifiable curves in the (d - 1)-dimensional topological sphere S of all directions in M. Thus, a length structure appears on S. This results in the appropriate intrinsic metric in S. The paper deals with some properties of the length structure and the resulting metric space S. In particular, it shows that diam S 2 .     

Crystallographic classes on the minkowski space <Emphasis Type="Italic">R</Emphasis><Subscript>1,2</Subscript>. I. Theorems

Some preliminary statements are proved necessary for calculating the crystallographic groups in six crystallographic classes in the 3-dimensional Minkowski space. Three classes are determined by the unimodular subgroups of the general Lorentz group and three more classes, by subgroups unimodular in a certain isotropic coordinate system.     

Classification of finite groups according to the number of conjugacy classes II

In the following,G denotes a finite group,r(G) the number of conjugacy classes ofG, β(G) the number of minimal normal subgroups ofG andα(G) the number of conjugate classes ofG not contained in the socleS(G). Let Φ _j = {G|β(G) =r(G) −j}. In this paper, the family Φ₁₁ is classified. In addition, from a simple inspection of the groups withr(G) =b conjugate classes that appear in ϒ _j ^=1/11 Φ _j , we obtain all finite groups satisfying one of the following conditions: (1)r(G) = 12; (2)r(G) = 13 andβ(G) > 1; …; (9)r(G) = 20 andβ(G) > 8; (10)r(G) =n andβ(G) =n −a with 1 ≦a ≦ 11, for each integern ≧ 21. Also, we obtain all finite groupsG with 13 ≦r(G) ≦ 20,β(G) ≦r(G) − 12, and satisfying one of the following conditions: (i) 0 ≦α(G) ≦ 4; (ii) 5 ≦α(G) ≦ 10 andS(G) solvable.     

Outer Minkowski content for some classes of closed sets

We find conditions ensuring the existence of the outer Minkowski content for d-dimensional closed sets in , in connection with regularity properties of their boundaries. Moreover, we provide a class of sets (including all sufficiently regular sets) stable under finite unions for which the outer Minkowski content exists. It follows, in particular, that finite unions of sets with Lipschitz boundary and a type of sets with positive reach belong to this class.     

Hyperbolic euler formula in the n-dimensional Minkowski space

In this paper, the properties ofn-dimensional Minkowski space discussed by using Clifford algebra. The hyperbolic Euler formula is given inn-dimensional Minkowski space.     

Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

Estimates on conjectures of Minkowski and woods II

Let ? ⁿ be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < Rwhich contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ? ⁿ of radius \(\sqrt n /2\) contains a point of ∧. This is known to be true for n ≤ 8. Recently we gave estimates on a more general conjecture of Woods for n ≥ 9. This lead to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms. Here we shall refine our method to obtain improved estimates for Woods Conjecture. These give improved estimates of Minkowski’s conjecture for 9 ≤ n ≤ 31.