Communications in Contemporary Mathematics

Smooth K-functors are introduced and the smooth K-theory of locally convex algebras is developed. It is proved that the algebraic and smooth K-functors are isomorphic on the category of quasi -stable real (or complex) Fréchet algebras.

We associate what we call vertex ℂ((z))-algebras and their modules in a certain category with elliptic affine Lie algebras. To a certain extent, this association is similar to that of vertex algebras and their modules with affine Lie algebras. While the notion of vertex ℂ((z))-algebra is a special case of that of quantum vertex ℂ((z))-algebra, which was introduced and studied by one of us (Li), here we use those results on quantum vertex ℂ(z))-algebras in an essential way. In the course of this work, we also construct and exploit two families of Lie algebras which are closely related to elliptic affine Lie algebras.

In this paper, we deal with positive solutions for singular quasilinear problems whose model is

The paper is devoted to local derivations on the algebra of τ-measurable operators affiliated with a von Neumann algebra and a faithful normal semi-finite trace τ. We prove that every local derivation on which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra for the algebra to admit local derivations which are not derivations.

This paper is concerned with the mean field equation of the equilibrium turbulence on a compact Riemannian surface. The solution existence is shown in the super-critical case without any assumption on the topology and the geometry of the surface.

We give a description of singularity formation in terms of energy quanta for 2-dimensional radially symmetric equivariant harmonic map heat flows. Adapting Struwe's energy method we first establish a finite bubble tree result with a discrete multiple of energy quanta disappearing in the singularity. We then use intersection-comparison arguments to show that the bubble tree consists of a single bubble only and that there is a well defined scale R_BHK(t) ↓ 0 in which the solution converges to the standard harmonic map.

We study existence of solutions with singular limits for a 2-dimensional semilinear elliptic problem with exponential dominated nonlinearity and a singular source term given by Dirac masses, imposing Dirichlet boundary condition. This paper extends previous results obtained in [3, 8]. We mainly use the method of domain decomposition.