An Axiomatic Line Geometry

Abstract

In their classic treatment (5) Veblen and Young build n-dimensional projective geometry from points and lines. Naturally, each line becomes identified with the set of points with which it is incident, and many treatments build from points alone, postulating the existence of certain distinguished subsets of the set of points. From either point of view, some labour is required, even in the two-dimensional case, to establish duality; hence a considerable interest attaches to self-dual systems of axioms; cf. (2; 3).

Possible primitive notions for geometry of spine spaces

“…[19,4]; comp. also [1,22]). Assume that 1 = k = dim(V) − 1, since otherwise A is simply an affine space.…”

“…Thus the sets of the form S(a) are distinguished within C. 2For at least 3-dimensional affine spaces and for at least 4-dimensional projective spaces the result is well known as it is a result of a search for an adequate system of primitive notions for line geometry (cf [8,13,12,9,22]…”

Swimming Together Upstream: How to Align MLP Services with U.S. Healthcare Delivery

“…Sources for MLP financing typically include legal fellowships, federal, and state legal services dollars, project funding through Medicaid waivers, HRSA funding of Federally Qualified Health Centers (FQHCs), hospital community benefit dollars, administrative line items, and, in the academic space, universities. 58 A 2019 national environmental scan found that the median MLP budget is a paltry $100,000 annually, typically cobbled together from multiple sources, which supports a median of 1.0 FTE of attorney time and 0.2 FTE of support staff. 59 Because of unreliable funding, MLPs may operate with a skeleton legal staff, which disrupts programmatic integrity in the same way that other aspects of philanthropy-driven social services frequently are forced to survive on a grant-togrant, competitive basis.…”

On the axiomatics of projective and affine geometry in terms of line intersection

“…Pieri [15] first noted that projective three-dimensional space can be axiomatized in terms of lines and line-intersections. A simplified axiom system was presented in [7], and two new ones in [17] and [10], by authors apparently unaware of [15] and [7]. Another axiom system was presented in [16,Ch.…”