Let$H$^{∞}be the algebra of bounded analytic functions in the unit disk$D$. Let$I=I(f$_{1},...,$f$_{N}) be the ideal generated by$f$_{1},...,$f$_{N}∈$H$^{∞}and$J=J(f$_{1},...,$f$_{N}) the ideal of the functions$f∈H$^{∞}for which there exists a constant$C=C(f)$such that |$f(z)|≤C(|f$_{1}$(z)|+$...;$+|f$_{N}$(z)$|),$z$∈$D$. It is clear that $$I \subseteq J$$ , but an example due to J. Bourgain shows that$J$is not, in general, in the norm closure of$I$. Our first result asserts that$J$is included in the norm closure of$I$if$I$contains a Carleson-Newman Blaschke product, or equivalently, if there exists$s$>0 such that $$\mathop {\inf }\limits_{z \in D} \sum\limits_{k = 0}^s {(1 - |z|)^k } \sum\limits_{j = 1}^N {|f_j^{(k)} (z)| > 0.} $$

Let A be a C*-algebra, L a closed left ideal of A and p the closed projection related to L. We show that for an xp in A**p ( A**/L**) if pAxp pAp and px*xp pAp then xp Ap ( A/L). The proof goes by interpreting elements of A**p (resp. Ap) as admissible (resp. continuous admissible) vector sections over the base space F(p) = { A* : 0, (p) = 1} in the notions developed by Diximier and Douady, Fell, and Tomita. We consider that our results complement both Kadison function representation and Takesaki duality theorem.

In this paper, we construct complete constant scalar curvature Khler (cscK) metrics on the complement of the zero section in the total space of O ( - 1 ) 2 over O ( - 1 ) 2 , which is biholomorphic to the smooth part of the cone <i>C</i> ₀ in O ( - 1 ) 2 defined by equation O ( - 1 ) 2 . On its small resolution and its deformation, we also consider complete cscK metrics and find that if the cscK metrics are homogeneous, then they must be Ricci-flat.

We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell's equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic JacobiDavidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed

In this paper, we give an instability criterion for the Prandtl equations in three-dimensional space, which shows that the monotonicity condition on tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations, in contrast to the classical well-posedness theory of the two-dimensional Prandtl equations under the Oleinik monotonicity assumption. Both linear stability and nonlinear stability are considered. This criterion shows that the monotonic shear flow is linearly stable for the three-dimensional Prandtl equations if and only if the tangential velocity field direction is invariant with respect to the normal variable, and this result is an exact complement to our recent work (A well-posedness theory for the Prandtl equations in three space variables. arXiv:1405.5308 , 2014) on the well-posedness theory for the three-dimensional Prandtl