1. Introduction. To determine cohomology groups of holomorphic vector bundles or more general coherent analytic sheaves on complex manifolds is very important in several complex variables and complex geometry. For example, Cartan-Serre's theorem B and Kodaira's vanishing theorem are fundamental respectively in the studies of two classes of complex manifolds: Stein manifolds and projective algeb...

In this paper, the kinetics of a class delayed reaction-diffusion mussel-algae system under Neumann boundary conditions with half-saturation constant is studied. The global existence and priori bounds as well positive equilibrium are obtained. affects stability may result in Turing instability. When exceeds certain critical value, globally asymptotically stable which means that larger forces mu...

Abstract. Given any simple biorientable graph it is shown that there exists a weak *-Hopf algebra constructed on the vector space of graded endomorphisms of essential paths on the graph. This construction is based on a direct sum decomposition of the space of paths into orthogonal subspaces one of which is the space of essential paths. Two simple examples are worked out with certain detail, the...

William M. Singer’s theory of extensions of connected Hopf algebras is used to give a complete list of the cocommutative connected Hopf algebras over a field of positive characteristic p which have vector space dimension less than or equal to p3. The theory shows that there are exactly two noncommutative non-primitively generated Hopf algebras on the list, one of which is the Hopf algebra corre...

We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions which was recently studied as a vector space by Rosas and Sagan [12]. The bases for this algebra are indexed by set partitions. We show that there exist a natural inclusion of the Hopf algebra of noncommutative symmetric functions studied in [17] in this larger space. We also consider this algebra a...

We consider the regularity of Leray-Hopf solutions to Navier-Stokes equations on critical case u ∈ L 2 w (0, T ; L ∞ (R 3)). By a new embedding inequality in Lorentz space we proved that if u L 2 w (0,T ;L ∞ (R 3)) is small then the Leray-Hopf solutions are regular. Particularly, an open problem proposed in [7] was solved.

The Hopf term in the 2+ 1 dimensional O(3) nonlinear sigma model, which is known to be responsible for fractional spin and statistics, is re-examined from the viewpoint of quantization ambiguity. It is confirmed that the Hopf term can be understood as a term induced quantum mechanically, in precisely the same manner as the θ-term in QCD. We present a detailed analysis of the topological aspect ...