Let Z3 be the affine space of dimension d on the Z3 field. The main goal of this paper is the geometric representation of the maximum linefree set E in Z3. Therefore the complement T of E is the minimum size set in Z3 intersecting all the affine lines in Z3. We describe, from an affine point of view, the geometric structure for d = 2, 3 of set E producing the maximal cardinality. The case d = 4...

We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of non-singular bilinear maps, and the immersion problem for real projec...

We investigate the monopole and vortex content of a meron pair by calculating the points at which the transformation to the Laplacian Center Gauge is ill-defined and by studying the behavior of Wilson loops. These techniques reveal complementary aspects of the vortex and monopole structure, including the presence of closed monopole lines and closed vortex surfaces joining the two merons, and ev...

In a Steiner triple system of order v, STS(v), a set of three lines intersecting pairwise in three distinct points is called a triangle. A set of lines containing no triangle is called triangle-free. The minimum number of triangle-free sets required to partition the lines of a Steiner triple system S, is called the triangle chromatic index of S. We prove that for all admissible v, there exists ...

We confirm a conjecture of J. Marklof regarding the limiting distribution of certain sparse collections of points on expanding horospheres. These collections are obtained by intersecting the expanded horosphere with a certain manifold of complementary dimension and turns out to be of arithmetic nature. This result is then used along the lines suggested by J. Marklof to give an analogue of a res...

According to a particular view, chance events are not uncaused but they are simply the result of intersecting causal lines. More precisely, the intersections between different processes that belong to independent causal chains are the origin of accidental events, called absolute coincidences. This paper provides a new account devoted to showing the strong relation between absolute coincidences ...

Detecting intersecting surfaces and computing their intersection curves is one of the fundamental problems in solid modeling algebra. In this paper, we introduce a new strategy to classify surfaces against their intersection status. We replace exact geometric entities by fuzzy geometric ones. Intersecting surfaces are then replaced by fuzzy intersecting faces and their fuzzy intersection algori...