Unbounded Components in the Solution Sets of Strictly Quasiconcave Vector Maximization Problems

Let (P) denote the vector maximization problem

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives.First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, …Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincaré compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincaré disc for the separable quadratic polynomial differential systems.     

Compactification équivariante d’un tore (d’après Brylinski et Künnemann)

We give a combinatorial, self-contained proof of the existence of a smooth equivariant compactification for an algebraic torus defined over an arbitrary field.     

Geometry of quadratic differential systems in the neighborhood of infinity

In this article we give a complete global classification of the class QS_ess of planar, essentially quadratic differential systems (i.e. defined by relatively prime polynomials and whose points at infinity are not all singular), according to their topological behavior in the vicinity of infinity. This class depends on 12 parameters but due to the action of the affine group and re-scaling of time, the family actually depends on five parameters. Our classification theorem (Theorem 7.1) gives us a complete dictionary connecting very simple integer-valued invariants which encode the geometry of the systems in the vicinity of infinity, with algebraic invariants and comitants which are a powerful tool for computer algebra computations helpful in the route to obtain the full topological classification of the class QS of all quadratic differential systems.     

Smoothness on bubble tree compactified instanton moduli spaces

The bubble tree compactified instanton moduli space -Mκ （X） is introduced. Its singularity set Singκ（X） is described. By the standard gluing theory, one can show that- Mκ（X） - Singκ（X） is a topological orbifold. In this paper, we give an argument to construct smooth structures on it.