Synthesis of Benzazepine Analogues of Noscapine

The synthesis of benzazepine analogues of the opium alkaloid noscapine ( 1 ) is described. The benzazepines 2 and 3 were prepared starting from nornarceine ethyl ester ( 4 ; readily available from 1 ) in several steps. X-Ray analysis of compound 2 revealed that it is not a diastereosisomer mixture but a racemate of the threo-form and thus has the same configuration as 1 .     

Level of nodes in increasing trees revisited

Simply generated families of trees are described by the equation T(z) = ϕ(T(z)) for their generating function. If a tree has n nodes, we say that it is increasing if each node has a label ∈ { 1,…,n}, no label occurs twice, and whenever we proceed from the root to a leaf, the labels are increasing. This leads to the concept of simple families of increasing trees. Three such families are especially important: recursive trees, heap ordered trees, and binary increasing trees. They belong to the subclass of very simple families of increasing trees, which can be characterized in 3 different ways. This paper contains results about these families as well as about polynomial families (the function ϕ(u) is just a polynomial). The random variable of interest is the level of the node (labelled) j, in random trees of size n ≥ j. For very simple families, this is independent of n, and the limiting distribution is Gaussian. For polynomial families, we can prove this as well for j,n → ∞ such that n − j is fixed. Additional results are also given. These results follow from the study of certain trivariate generating functions and Hwang's quasi power theorem. They unify and extend earlier results by Devroye, Mahmoud, and others. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

An analytic approach for the analysis of rotations in fringe-balanced binary search trees

This paper presents an analytic approach to the construction cost of fringe-balanced binary search trees. In [7], Mahmoud used a bottom-up approach and an urn model of Pólya. The present method is top-down and uses differential equations and Hwang's quasi-power theorem to derive the asymptotic normality of the number of rotations needed to construct such afringe balanced search tree. We also obtain the exact expectation and variance with this method. Although Pólya's urn model is no longer needed, we also present an elegant analysis of it based on an operator calculus as in [4].This research was supported by the Austrian Research Society (FWF) under the project number P12599-MAT.     

Monte Carlo Hamiltonian: Linear Potentials

We further study the validity of the Monte Carlo Hamiltonian method. The advantage of the method,in comparison with the standard Monte Carlo Lagrangian approach, is its capability to study the excited states. Weconsider two quantum mechanical models: a symmetric one V(x) = |x|/2; and an asymmetric one V(x) = ∞, forx ＜ 0 and V(x) = x, for x ≥ 0. The results for the spectrum, wave functions and thermodynamical observables are inagreement with the analytical or Runge-Kutta calculations.