Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of m le...

The purpose of this paper is to define the notion of “real” intersection between paths drawn on the 3d digital boundary of a connected object. We consider two kinds of paths for different adjacencies, and define the algebraic number of oriented intersections between these two paths. We show that this intersection number is invariant under any homotopic transformation we apply on the two paths. ...

In this paper we consider the problem of privately computing the intersection of sets (setintersection), as well as several variations on this problem: cardinality set-intersection, threshold set-intersection, and over-threshold set-intersection. Cardinality set-intersection is the problem of determining the size of the intersection set, without revealing the actual threshold set. In threshold ...

We give a simpler proof of the result of Boros and Füredi that for any finite set of points in the plane in general position there is a point lying in 2/9 of all the triangles determined by these points. Introduction Every set P of n points in R in general position determines ( n d+1 ) d-simplices. Let p be another point in R. Let C(P, p) be the number of the simplices containing p. Boros and F...

Four different vector fields on (x,y)-plane A vector field on a sphere ♣ Curves. A smooth curve is a continuous curve (no gaps or discontinuities) with no corners (no abrupt changes in slope at a point such as you would get from the intersection of two lines). A continuous and differentiable produces a smooth function. A path C is called closed if its initial and final points are the same point...

If φ : L → L′ is a bijection from the set of lines of a linear space (P,L) onto the set of lines of a linear space (P ′,L′) (dim (P,L), dim (P ′,L′) ≥ 3), such that intersecting lines go over to intersecting lines in both directions, then φ is arising from a collineation of (P,L) onto (P ′,L′) or a collineation of (P,L) onto the dual linear space of (P ′,L′). However, the second possibility can...

A method of representing transformations of a finite set pictorially is described. These pictures of a function are used to count certain idempotent transformations, and interesting formulae for the Fibonacci numbers are obtained. No part of any string extends beyond the region ° : : ; z ::; 1, and if two strings intersect then they join together, from the point of intersection, to meet the sam...

We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that Dominating Set on such intersection graphs is polynomially solvable whenever Q contains at least one interval,...

Ray tracing requires many ray-obect intersection tests. A way of reducing the number of ray-object intersection tests is to subdivide the space occupied by objects into many nonoverlapping subregions, called voxels, and to construct an octree for the subdivided space. In this paper, we propose the Octree-R, an octree-variant data structure for efficient ray tracing. The algorithm for constructi...

Uncontrolled intersections are the intersections where there are no external signs or signals to control the movement of vehicles. In mixed traffic conditions priority rules are often violated by the road users. In All-way-stop-controlled intersections (AWSC), the vehicle should stop themselves before they enter the intersection and should check whether any vehicles are present in the other app...