when the degree of saturation at intersection approaches one, webster’s optimum cycle length equation becomes inapplicable, because the cycle length will becomes very big when the degree of saturation approaches one and will be fully unrealistic when the degree of saturation becomes greater than one. this is not a problem for hcm2000 method. but optimum cycle length calculation in this method h...

υ(X,L) = lim sup k→∞ n! kn h(X,OX(kL)). If L is ample, or more generally nef and big, υ(L) = (L), the self-intersection number of L. The volume of a general big line bundle has been studied in [F] and [DEL]; in particular, υ(L) has been given the following geometric interpretation (Proposition 3.6 of [DEL]): Let (kL) [n] be the moving selfintersection number of kL, that is, the number of inters...

We address the problem of computing bounds for the self-intersection number (the minimum number of self-intersection points) of members of a free homotopy class of curves in the doubly-punctured plane as a function of their combinatorial length L; this is the number of letters required for a minimal description of the class in terms of a set of standard generators of the fundamental group, and ...

When the degree of saturation at intersection approaches one, Webster’s optimum cycle length equation becomes inapplicable, because the cycle length will becomes very big when the degree of saturation approaches one and will be fully unrealistic when the degree of saturation becomes greater than one. This is not a problem for HCM2000 method. But optimum cycle length calculation in this method h...

This paper focuses on the relationship between an $L$-subset and the system of level-elements induced by it, where the underlying lattice $L$ is a complete residuated lattice and the domain set of $L$-subset is an $L$-partially ordered set $(X,P)$. Firstly, we obtain the sufficient and necessary condition that an $L$-subset is represented by its system of level-elements. Then, a new representat...

Let V be a finite-dimensional vector space over a field k. A hyperplane arrangement in V is a collection A = (H1, . . . , Hn) of codimension one affine subspaces of V . The arrangement A is called central if the intersection ⋂ Hi is nonempty; without loss of generality the intersection contains the origin. We will always denote by n the number of hyperplanes in the arrangement, and by l the dim...

The intersection dimension of a bipartite graph with respect to a type L is the smallest number t for which it is possible to assign sets Ax ⊆ {1, . . . , t} of labels to vertices x so that any two vertices x and y from different parts are adjacent if and only if |Ax ∩Ay| ∈ L. The weight of such a representation is the sum ∑x |Ax| over all vertices x. We exhibit explicit bipartite n×n graphs wh...