Detection of norbolethone,an anabolic steroid never marketed,in athletes' urine

Norbolethone (13-ethyl-17-hydroxy-18,19-dinor-17alpha-pregn-4-en-3-one) is a 19-nor anabolic steroid first synthesized in 1966. During the 1960s it was administered to humans in efficacy studies concerned with short stature and underweight conditions. It has never been reported by doping control laboratories. Norbolethone was identified in two urine samples from one athlete by matching the mass spectra and chromatographic retention times with those of a reference standard. The samples also contained at least one likely metabolite. The samples were also unusual because the concentrations of endogenous steroids were exceptionally low. Since norbolethone is not known to be marketed by any pharmaceutical company, a clandestine source of norbolethone may exist.     

A bound on the chromatic number of a graph

We give an upper bound on the chromatic number of a graph in terms of its maximum degree and the size of the largest complete subgraph. Our results extends a theorem due to Brooks.     

Graph family operations

In previous papers, Catlin introduced four functions, denoted , , , and , between sets of finite graphs. These functions proved to be very useful in establishing properties of several classes of graphs, including supereulerian graphs and graphs with nowhere zero k-flows for a fixed integer k3. Unfortunately, a subtle error caused several theorems previously published in Catlin (Discrete Math. 160 (1996) 67–80) to be incorrect. In this paper we correct those errors and further explore the relations between these functions, showing that there is a sort of duality between them and that they act as inverses of one another on certain sets of graphs.     

Extension of C R Structures on three dimensional compact pseudoconvex C R manifolds

Let M be a smooth compact orientable pseudoconvex C R manifold of real dimension three and assume that there is a smooth function λ which is strictly subharmonic in any direction where the Levi-form vanishes on M. Then we extend the given C R structure on M to an integrable almost complex structure on the concave side of M. As an application, if M is a non-compact pseudoconvex C R manifold of real dimension three, we prove that the given C R structure on M can be locally extended to an integrable almost complex structure on the concave side of M. Partially supported by Korea research foundation Grant (KRF-2001-015-DP0016) and by R14-2002-044-01000-0 from KRF     

Sufficient conditions for the extension ofC R structures

LetM be a smoothC R manifold of dimension 2n ? 1 such that at each point, either the Levi form has at least 3 positive eigenvalues or it hasn ? 1 negative eigenvalues. IfD is a smoothly bounded subdomain ofM, then there is a smoothly bounded integrable almost complex manifoldX of dimension 2n such thatM is contained in the boundary ofX and such that theC R structure thatM inherits as a subset ofX coincides with the original structure ofM.     

The reduction of graph families closed under contraction

Let be a family of graphs. Suppose there is a nontrivial graph H such that for any supergraph G of H, G is in if and only if the contraction G/H is in . Examples of such an : graphs with a spanning closed trail; graphs with at least k edge-disjoint spanning trees; and k-edge-connected graphs (k fixed). We give a reduction method using contractions to find when a given graph is in and to study its structure if it is not in . This reduction method generalizes known special cases.     

Two problems in metric diophantine approximation,I

There is no known necessary and sufficient condition on a sequence {α_n} of nonnegative real numbers such that for almost all x (in the sense of Lebesgue measure) there are infinitely many fractions $\text{p}\text{q}$ satisfying $\text{| x ?}\text{p}\text{q}\text{| <}\text{α}{}_{\mathrm{q}}\text{q}$. Nor is any condition known when $\text{p}\text{q}$ is required to be reduced. We shall consider the relation of these problems to each other, and we shall discuss conjectured answers for these questions.