Spaces with a covariant constant linear geometric object

The Geometric Interior in Real Linear Spaces

For simplicity, we adopt the following rules: x denotes a set, r, s denote real numbers, n denotes a natural number, V denotes a real linear space, v, u, w, p denote vectors of V , A, B denote subsets of V , A1 denotes a finite subset of V , I denotes an affinely-independent subset of V , I1 denotes a finite affinelyindependent subset of V , F denotes a family of subsets of V , and L1, L2 denot...

On Linear Spaces of Skew-symmetric Matrices of Constant Rank

Linear sections of the Grassmannians G(1, n) of lines in P appear naturally in several different situations. In complex projective algebraic geometry, 3-dimensional linear sections of G(1, 4) appear in the classification of Fano threefolds, 2-dimensional linear sections of G(1, 5) define one of the smooth scrolls of P. Linear sections of dimension n − 1 of the Grassmannian of lines of P are cla...

Geometric spaces with no points

Some models of set theory are given which contain sets that have some of the important characteristics of being geometric, or spatial, yet do not have any points, in various ways. What’s geometrical is that there are functions to these spaces defined on the ambient spaces which act much like distance functions, and they carry normable Riesz spaces which act like the Riesz spaces of real-valued ...

Light Orthogonal Networks with Constant Geometric Dilation

An orthogonal network for a given set of n points in the plane is an axis-aligned planar straight line graph that connects all input points. We show that for any set of n points in the plane, there is an orthogonal network that (i) is short having a total edge length of O(jT j), where jT j denotes the length of a minimum Euclidean spanning tree for the point set; (ii) is small having O(n) verti...

Piecewise Constant Controlled Linear Fuzzy Systems