We consider the vorticity-stream formulation of axisymmetric incompressible ows and its equivalence with the primitive formulation. It is shown that, to characterize the regularity of a divergence free axisymmetric vector eld in terms of the swirling components, an extra set of pole condition is necessary to give a full description of the regularity. In addition, smooth solutions up to the axis of rotation gives rise to smooth solutions of primitive formulation in the case of Navier-Stokes equations, but not the Euler equations. We also establish proper weak formulations and show its equivalence to Leray's solutions.

We prove that for a large class of functions P and Q, the discrete bilinear operator T P, Q (f, g)(n)= m Z{0} f (n P (m)) g (n Q (m)) 1 m is bounded from l 2 l 2 into l 1+ , for any (0, 1].

Let <i>X, Y</i> be compact Hausdorff spaces and <i>E, F</i> be Banach spaces. A linear map <i>T: C(X, E) C(Y, F)</i> is separating if <i>Tf, Tg</i> have disjoint cozeroes whenever <i>f, g</i> have disjoint cozeroes. We prove that a biseparating linear bijection <i>T</i> (that is, <i>T</i> and <i>T</i>^-1 are separating) is a weighted composition operator <i>Tf = h f</i> o . Here, <i>h</i> is a function from <i>Y</i> into the set of invertible linear operators from <i>E</i> onto <i>F</i>, and , is a homeomorphism from <i>Y</i> onto <i>X</i>. We also show that <i>T</i> is bounded if and only if <i>h(y)</i> is a bounded operator from <i>E</i> onto <i>F</i> for all <i>y</i> in <i>Y</i>. In this case, <i>h</i> is continuous with respect to the strong operator topology.

In this paper, we obtain several new continuous selection theorems for multi-valued mappings on completely regular spaces and fixed point theorems for multi-valued maps on nonmetrizable spaces. They, in particular, provide a partial solution of a conjecture of X. Wu.

We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization problem can be reformulated as a lattice point problem where one wishes to maximize the number of points of (N−12)×(N−12) inside triangles with vertices (0,0),(0,λβ−−√) and (λ/β−−√,0) with respect to β>0, for fixed λ≥0. This lattice point formulation of the problem naturally leads to a family of generalized problems where one instead considers the shifted lattice (N+σ)×(N+τ), for σ,τ>−1. We show that the nature of these problems are rather different depending on the shift parameters, and in particular that the problem corresponding to harmonic oscillators, σ=τ=−12, is a critical case.