Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to canonical symplectic form. We consider an arbitrary C?-smooth hypersurface ? ? ?n+1 that is either a global strictly convex closed hypersurface, or germ hypersurface. deal pseudogroup generated compositional ratios reflections and its small deformations. In case when we show C?-closure latter...

In this letter, we tackle the problem of planning an optimal coverage path for a robot operating indoors. Many existing approaches attempt to discourage turns in by covering environment along least number coverage lines , i.e., straight-line paths. This is because turning not only slows down bu...

We are interested in embedding trees T with ∆(T) ≤ 4 a rectangular grid, such that the vertices of correspond to grid points, while edges non-intersecting straight segments lines. Such embeddings called models. While each edge is represented by segment, path model union corresponding its edges, which may consist having several bends. The aim determine given tree minimizing maximum number bends ...

We give a linear time algorithm which, given a simply connected gure of the plane divided into cells, whose boundary is crossed by some colored inputs and outputs, produces non-intersecting directed ow lines which match inputs and outputs according to the colors, in such a way that each edge of any cell is crossed by at most one line. The main tool is the notion of height function, previously i...

A blocking set in a projective plane is point intersecting each line. The smallest sets are lines. second minimal Baer subplanes (subplanes of order q). Our aim to study the stability PG(2,q). If we delete q+1−k points from subplane, then resulting has size q+k and (q+1−k)(q−q) 0-secants. have somewhat more 0-secants, our main theorem says that this can be obtained subplane or line by deleting ...

An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that every pair is either disjoint or intersects in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. Kang and Müller showed tha...

Determining the common zeroes of a set of polynomials is further complicated over nonalgebraically closed fields such as the real numbers. A special case is whether a problem of enumerative geometry can have all its solutions be real. We call such a problem fully real. Little is known about enumerative geometry from this perspective. A standard proof of Bézout’s Theorem shows the problem of int...

This paper investigates t-intersecting families of injections, where two injections a, b from [k] to [n] t-intersect if there exists X ⊆ [k] with |X| ≥ t such that a(x) = b(x) for all x ∈ X. We prove that if F is a 1-intersecting injection family of maximal size then all elements of F have a fixed image point in common. We show that when n is large in terms of k and t, the set of injections whi...