The elliptic elements of M, each with two conjugate complex fixed points, are precisely the conjugates of nontrivial powers of A and B. The parabolic elements, each with a single real fixed point, are precisely the conjugates of nontrivial powers of C = AB: z ~ z + 1. The remaining nontrivial elements of M are hyperbolic, each with two real fixed points. A subgroup S of M is torsionfree (and th...

The maximal representation of a shape is defined and algorithms for shape arithmetic are developed. Introduction A shape is any arrangement of spatial elements from among points, or lines, planes, volumes, or higher dimensional hyperplanes of limited but nonzero measure. According to Stiny (1991), a shape is in algebra U, or simply in U, whenever it is made up of elements from U. Thus, a shape ...

A symmetric scaling of a square matrix A #0 is a matrix of the form XAX-' where X is a nonnegative, nonsingular, diagonal matrix having the same dimension of A. An asymmetric scaling of a rectangular matrix B # 0 is a matrix of the form XBY-' where X and Y"are nonnegative, nonsingular, diagonal matrices having appropriate dimensions. We consider two objectives in selecting a symmetric scaling o...

2 Stabilisers of Parabolic Centralisers 6 2.1 Stabiliser Subgroups . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Generators for the Stabiliser of L . . . . . . . . . . . . . . . . 7 2.3 Ordering L and the stabiliser of L-maximal elements . . . . . 9 2.4 Ordering X and closures of elements . . . . . . . . . . . . . . 10 2.5 A matrix representation of St(L) . . . . . . . . . . . . . . . . 1...

We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for se...

Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or re-cursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identiication of elements in a pca. We establish several facts concerning nal collapses (maximal identiication of elements). `En passan...

Filters of a lattice, maximal filters (ultrafilters), operation to create a filter generating by an element or by a nonempty set of elements of the lattice are discussed. Besides, there are introduced implicative lattices such that for every two elements there is an element being pseudo-complement of them. Some facts concerning these concepts are presented too, i.e. for any proper filter there ...