We prove the Faber intersection number conjecture and other more general results by using a recursion formula of n-point functions for intersection numbers on moduli spaces of curves. We also present several conjectural properties of Gromov-Witten invariants generalizing results on intersection numbers.

Let the braid group Bn act as (isotopy classes of) diffeomorphisms of an n-punctured disc Dn. Then there is an action of Bn on a polynomial algebra R = C [a1 , . . . , aN ] and a way of representing simple closed curves on Dn as elements of R. Fix k ∈ 2N. Using this approach we show that the image in Aut(R) of each Dehn twist τ about a simple closed γ in Dn satisfies a kind of characteristic eq...

The problem of finding all intersections between two surfaces has many applications in computational geometry and computer-aided geometric design. We propose an algorithm based on Newton’s method and subdivision for this problem. Our algorithm uses a test based on the Kantorovich theorem to prevent the divergence or slow convergence problems normally associated with using unsuitable starting po...

Many problems in the verification of concurrent software systems reduce to checking the non-emptiness of the intersection of contextfree languages, an undecidable problem. We propose a decidable underapproximation, and a semi-algorithm based on the under-approximation, for this problem through bounded languages. Bounded languages are context-free subsets of regular languages of the form w∗ 1w ∗...

We investigate edge-intersection graphs of paths in the plane grid regarding a parameter called the bend-number. The bend-number is related to the interval-number and the track-number of a graph. We provide new upper and lower bounds of the bend-number of any given simple graph in terms of the coloring number, edge clique covers and the maximum degree. We show that the bend-number of an outerpl...

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

The intersection number of a graph G is the minimum size of a ground set S such that G is an intersection graph of some family of subsets F ⊆ 2 . The overlap number of G is defined similarly, except that G is required to be an overlap graph of F . In this paper we show two algorithmic aspects concerning both these graph invariants. On the one hand, we show that the corresponding optimization pr...

Modifying the notion of numerically trivial foliation of a pseudoeffective line bundle L introduced by the author in [Eck04a] (see also math.AG/0304312) it can be shown that the leaves of this foliation have codimension ≥ the numerical dimension of L as defined by Boucksom, Demailly, Paun and Peternell, math.AG/0405285. Furthermore, if the Kodaira dimension of L equals its numerical dimension t...

The intersection problem for additive (extended and non-extended) perfect codes, i.e. which are the possibilities for the number of codewords in the intersection of two additive codes C1 and C2 of the same length, is investigated. Lower and upper bounds for the intersection number are computed and, for any value between these bounds, codes which have this given intersection value are constructe...