On the punctured torus the number of essential self-intersections of a homotopy class of closed curves is bounded (sharply) by a quadratic function of its combinatorial length (the number of letters required for its minimal description in terms of the two generators of the fundamental group and their inverses). We show that if a homotopy class has combinatorial length L, then its number of esse...

number of k-subsets of an n-set so that the size of the intersection of any two among them is in L. It is proven that for every rational number r 2 1 there is a choice of k and L so that cn'<m(n, k, L) idn', where c, d depend on k and L but not on n. 0 1986 Academic Press. Inc

We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet A. To this purpose, we consider automata that recognize such submonoids and we study the product automata recognizing their intersection. By using automata methods we obtain a new proof of a result of Karhumäki on the characterization of the intersection of two submonoids of rank two, in...

pointing towards the sensor and a i is a binary number depending on the intersection of l and Q i , and The reason behind taking a weighted sum of the intersection points is that the sensor measurement of the position of a surface point is less reliable if the local surface is facing away from the sensor. The weight w i is a reflection of this heuristic. In general, the sensor direction informa...