The usual vector cross product of the three-dimensional Euclidian space is considered from an algebraic point of view. It is shown that many proofs, known from analytical geometry, can be distinctly simplified by using the matrix oriented approach. Moreover, by using the concept of generalized matrix inverse, we are able to facilitate the analysis of equations involving vector cross products.
We discuss how to find the well-covered dimension of a graph that is the Cartesian product of paths, cycles, complete graphs, and other simple graphs. Also, a bound for the well-covered dimension of Kn × G is found, provided that G has a largest greedy independent decomposition of length c < n. Formulae to find the well-covered dimension of graphs obtained by vertex blowups on a known graph, and to the lexicographic product of two known graphs are also given.
The Wigner Theorem states that the statistical distribution of the eigenvalues of a random Hermitian matrix converges to the semi-circular law as the dimension goes to infinity. It is possible to establish this result by using harmonic analysis on the Heisenberg group. In fact this convergence corresponds to the topology of the set of spherical functions associated to the action of the unitary group on the Heisenberg group.
Standard facts about separating linear functionals will be used to determine how two cones and and their duals and may overlap. When is linear and and are cones, these results will be applied to and , giving a unified treatment of several theorems of the alternate which explain when contains an interior point of . The case when is the space of Hermitian matrices, is the positive semidefinite matrices, and yields new and known results about the existence of block diagonal...
In this note we show that there are no ring anti-isomorphism between row finite matrix rings. As a consequence we show that row finite and column finite matrix rings cannot be either isomorphic or Morita equivalent rings. We also show that antiisomorphisms between endomorphism rings of infinitely generated projective modules may exist.
Suppose has a 2-dimensional expanding subspace , satisfies a regularity condition, called “good star”, and has , where is an oriented compound of . A morphism of the free group on is called a non-abelianization of if it has structure matrix . We show that there is a tiling substitution whose “boundary substitution” is a non-abelianization of . Such a tiling substitution leads to a self-affine tiling of with as its expansion. In the last section we find conditions on so...
We give a classification of linear endomorphisms up to topological conjugacy.