Let V be a set of curves in the plane. The corresponding intersection graph has V as the set of vertices, and two vertices are connected by an edge if and only if the two corresponding curves intersect in the plane. It is shown that the set of intersection graphs of curves in the plane is a proper subset of the set of all undirected graphs. Furthermore, the set of intersection graphs of straigh...

An obstruction theory for representing homotopy classes of surfaces in 4–manifolds by immersions with pairwise disjoint images is developed, using the theory of non-repeating Whitney towers. The accompanying higher-order intersection invariants provide a geometric generalization of Milnor’s link-homotopy invariants, and can give the complete obstruction to pulling apart 2–spheres in certain fam...

We study the convex hull of the intersection of a disjunctive set defined by parallel hyperplanes and the feasible set of a mixed integer second order cone optimization (MISOCO) problem. We extend our prior work on disjunctive conic cuts (DCCs), which has thus far been restricted to the case in which the intersection of the hyperplanes and the feasible set is bounded. Using a similar technique,...

The algebraic intersection type unification problem is an important component in proof search related to several natural decision problems in intersection type systems. It is unknown and remains open whether the algebraic intersection type unification problem is decidable. We give the first nontrivial lower bound for the problem by showing (our main result) that it is exponential time hard. Fur...

This paper investigates the study of -complex fuzzy sets. The set, where  is a completely distributive lattice, is generalization complex set. fundamental set theoretic operations on sets are discussed properly, including complement, union and intersection. New procedures presented to combine novel concepts -norms -conorms look into conditions that lead comparable representation theorem. W...

Discrete Interval Encoding Trees (Diets) are data structures for the representation of fat, i.e. densely populated sets over a discrete linear order. In this paper we introduce algorithms for set-theoretic operations like intersection, union, etc. on sets represented as balanced diets. We empirically analyse their performance and show that these algorithms can outperform previously known algori...

On a rationally connected variety we would like to use rational curves to obtain a similar result. Complete intersection curves are essentially never rational. (For instance, if X ⊂ P is a hypersurface then a general complete intersection with hyperplanes is rational iff X is a hyperplane or a quadric.) Therefore we have to proceed in a quite different way. Let X be a smooth, projective, ration...