A note on Faber operator

For a rectifable Jordan curve Γ with complementary domainsD and D,Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces Bp(1 p ∞) of analytic functions in the unit disk and in the inner domain D,respectively.We point out that the conjecture is not true in the special case p=2,and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.     

A Remark on Pal Type Interpolation on Non-Uniformly Distributed Nodes on the Unit Circle

In this paper we study the problem of explicit representation and convergence of Pal type （0;1） interpolation and its converse, with some additional conditions, on the non-uniformly distributed nodes on the unit circle obtaIned by projecting the interlaced zeros of Pn （x） and Pn′ （x） on the unit circle. The motivation to this problem can be traced to the recent studies on the regularity of Birkhoff interpolation and Pal type interpolations on non-uniformly distributed zeros on the unit circle.     

A note on the blow-up criterion of smooth solutions to the 3D incompressible MHD equations

In this note, we will give a new proof of the blow-up criterion of smooth solutions to the 3D incompressible magneto-hydrodynamic equations by a simple application of Gagliardo-Nirenberg?? s inequality.     

A note on economic equilibrium and financial networks

Nagurney （1999） used variational inequalities to study economic equilibrium and financial networks and applied the modified projection method to solve the problem. In this paper, we formulate the problem as a nonlinear complementarity problem. The complementarity model is just the KKT condition for the model of Nagurney （1999）. It is a simpler model than that of Nagurney （1999）. We also establish sufficient conditions for existence and uniqueness of the equilibrium pattern, which are weaker than those in Nagurney （]999）. Finally, we apply a smoothing Newton-type algorithm to solve the problem and report some numerical results.     

A note on equilibrium of eventually strongly monotone skew-product semiflows

In this paper, we show that for an eventually strongly monotone skew-product semiflow τ, the strict ordering on Ec （the set consisting of continuous equilibria of τ） implies the strong one.     

A note on the existence of positive solutions for a fourth-order semilinear elliptic system

In this note, we prove the existence of positive solutions to a class of biharmonical systems. Our main ingredients are proving a Liouville type result for biharmonical system in R＋^N via the method of moving plane combined with integral inequality, and establishing a prior estimates for positive solutions of the system via the blowing-up method.     

A Note on Wigner Functions and ＊-Genvalue Equation

In deformation quantization, static Wigner functions obey functional ,-genvalue equation, which is equivalent to time-independent Schrodinger equation in Hilbert space operator formalism of quantum mechanics. This equivalence is proved mostly for Hamiltonian with form H^ = （1/2）p^2 ＋ V（x^） [D. Fairlie, Proc. Camb. Phil. Soc. 60 （1964） 581]. In this note we generalize this proof to a very general Hamiltonian H^（x^,p^） and give examples to support it.     

Remarks on one-dimensional Compressible Navier-Stokes equations with density-dependent viscosity and vacuum

The Navier-Stokes system for one-dimensional compressible fluids with density-dependent viscosities when the initial density connects to vacuum continuously is considered in the present paper. When the viscosity coefficient u is proportional to pθ with 0 〈 θ 〈 1, the global existence and the uniqueness of weak solutions are proved which improves the previous results in [Vong, S. W., Yang, T., Zhu, C. J.： Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum II. J. Differential Equations, 192（2）, 475-501 （2003）]. Here p is the density. Moreover, a stabilization rate estimate for the density as t → ＋∞ for any θ 〉 0 is also given.