We have constructed a concrete example of a non-factorable positive operator. As a result, for the well-known problems (Ringrose, Kadison and Singer problems) we replace existence theorems by concrete examples.
There exists a surface of a convex polyhedron P and a partition L of P into geodesic convex polygons such that there are no connected edge unfoldings of P without self-intersections (whose spanning tree is a subset of the edge skeleton of L).
In [B1, Theorem 2.36] we proved the equivalence of six conditions on a continuous function f on an interval. These conditions define a subset of the set of operator convex functions, whose elements are called strongly operator convex. Two of the six conditions involve operator-algebraic semicontinuity theory, as given by C. Akemann and G. Pedersen in [AP], and the other four conditions do not involve operator algebras at all. Two of these conditions are operator inequalities, one is a global condition on f, and the fourth is an integral representation of f stronger than the usual integral representation for operator convex functions. The purpose of this paper is to make the equivalence of these four conditions accessible to people who do not know operator algebra theory as well as to operator algebraists who do not know the semicontinuity theory. We also provide a similar treatment of one theorem from [B1] concerning (usual) operator convex functions. And in two final sections we give a somewhat tentative treatment of some other operator inequalities for strongly operator convex functions, and we give a differential criterion for strong operator convexity.
We construct a family of map which is shown to be positive when imposing certain condition on the parameters. Then we show that the constructed map can never be completely positive. After tuning the parameters, we found that the map still remain positive but it is not completely positive. We then use the positive but not completely positive map in the detection of bound entangled state and negative partial transpose entangled states.
We prove a Fatou-type theorem and its converse for certain positive eigenfunctions of the Laplace-Beltrami operator $mathcal{L}$ on a Harmonic $NA$ group. We show that a positive eigenfunction $u$ of $mathcal{L}$ with eigenvalue $beta^2-rho^2$, $betain (0,infty)$, has an admissible limit in the sense of Koranyi, precisely at those boundary points where the strong derivative of the boundary measure of $u$ exists. Moreover, the admissible limit and the strong derivative are the same. This extends a result of Ramey and Ullrich regarding nontangential convergence of positive harmonic functions on the Euclidean upper half space.
Is it possible to form an image using light produced by stimulated emission? Here we study light scatter off an assembly of excited chromophores. Due to the Optical Theorem, stimulated emission is necessarily accompanied by excited state Rayleigh scattering. Both processes can be used to form images, though they have different dependencies on scattering direction, wavelength and chromophore configuration. Our results suggest several new approaches to optical imaging using fluorophore excited states.