Precoloring extension on unit interval graphs

In the precoloring extension problem a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper k-coloring of the graph. Answering an open question of Hujter and Tuza [Precoloring extension. III. Classes of perfect graphs, Combin. Probab. Comput. 5 (1) (1996) 35-56], we show that the precoloring extension problem is NP-complete on unit interval graphs.     

Complexity results on restricted instances of a paint shop problem for words

We study the following problem: an instance is a word with every letter occurring twice. A solution is a 2-coloring of its letters such that the two occurrences of every letter are colored with different colors. The goal is to minimize the number of color changes between adjacent letters.This is a special case of the paint shop problem for words, which was previously shown to be NP-complete. We show that this special case is also NP-complete and even APX-hard. Furthermore, derive lower bounds for this problem and discuss a transformation into matroid theory enabling us to solve some specific instances within polynomial time.     

The proportional hazards regression model with staggered entries: A strong martingale approach

The proportional hazards regression model, when subjects enter the study in a staggered fashion, is studied. A strong martingale approach is used to model the two-time parameter counting processes. It is shown that well-known univariate results such as weak convergence and martingale inequalities can be extended to this two-dimensional model. Strong martingale theory is also used to prove weight convergence of a general weighted goodness-of-fit process and its weighted bootstrap counterpart.     

The constrained degree and fixed-point index theory for set-valued maps

In the first part of the paper we give a construction of a topological degree theory for set-valued tangent vector fields with convex and nonconvex values defined on nonsmooth closed subsets of a Banach space. The obtained homotopy invariant is an extension of the classical degree for vector fields on manifolds. In the second part we propose a fixed-point index for inward maps on arbitrary closed convex sets.     

A general multi-valued hybrid fixed point theorem and perturbed differential inclusions

Combining three basic multi-valued versions of Banach, Schauder and Tarski fixed point theorems, a general hybrid fixed point theorem for multi-valued mappings in Banach spaces is proved via measure of noncompactness and it is further applied to perturbed differential inclusions for proving the existence results under mixed Lipschitz, compactness and monotone conditions.     

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x^″+a(t)x−b(t)x²+c(t)x³=0, where a(t),b(t),c(t) are T-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T-periodic solutions for this equation, using coincidence degree theories.