Efficient, optimal stochastic-action selection when limited by an action budget

The problem that we consider here is a basic operations research problem, but it also a special case of the Stochastic Shortest Path with Recourse Problem and the Canadian Travellers Problem in the probabilistic path planning literature, and it is also a special case of maximizing a submodular set function subject to a matroid constraint. Specifically, suppose an agent has a task and suppose that there is a set of actions, any of which the agent might perform, with respective probabilities of the actions successfully accomplishing the task and respective rewards for the agent if the actions are successful; the agent is to select a sequence of some of these actions that will be performed sequentially, until the task is accomplished or the selected actions are exhausted, but there is a budget on the number of actions that can be performed. We provide an efficient algorithm that chooses a sequence of actions that, under the budget, maximize the agent’s expected reward. An example illustrates how, when conditioning on partial selection of actions, there can be changes to the order of the remaining actions’ adjusted utilities. However, we prove and exploit a nesting result involving solutions.     

Spectral flow and Dixmier traces

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd -summable Breuer-Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential operators along the orbits of an ergodic action on a compact space X. Finally, we give a short proof of an index theorem of Lesch for generalised Toeplitz operators.     

Bis-functionalized fullerene-dibenzo[18]crown-6 conjugate: synthesis and cation-complexation dependent redox behavior

A one-step synthesis of bis-pyrrolidine functionalized fullerene-dibenzo[18]crown-6 conjugate and its metal cation complexation to the crown ether entity dependent redox behavior is reported.     

Hydrophosphination of boron–boron multiple bonds

Five compounds containing boron–boron multiple bonds are shown to undergo hydrophosphination reactions with diphenylphosphine in the absence of a catalyst. With diborenes, the products obtained are highly dependent on the substitution pattern at the boron atoms, with both 1,1- and 1,2-hydrophosphinations observed. With a symmetrical diboryne, 1,2-hydrophosphination yields a hydro(phosphino)diborene. The different mechanistic pathways for the hydrophosphination of diborenes are rationalised with the aid of density functional theory calculations.

Compounds containing boron–boron double and triple bonds are shown to undergo uncatalysed hydrophosphination reactions with diphenylphosphine.     

An integrality theorem for subnormal operators

In a recent paper we conjectured that the principal function of a cyclic subnormal operator T is a.e. equal to the negative of a characteristic function. We showed that this was true in a variety of cases - including the general arc length Swiss Cheese.Now we prove stronger results. The conjecture is a consequence of:The principal function of a subnormal operator with trace class self-commutator assumes a.e. nonpositive integer values.It is an interesting fact that this integrality is a basic geometric property of subnormal operators and is not associated with any smoothness or "thinness" of the essential spectrum of T.This result is actually a simple corollary of a much more basic fact:The mosaic of a subnormal operator with trace class self-commutator is projection valued a.e.We have long known that the mosaic is a complete unitary invariant for T. Thus, this theorem establishes a map z Range B(z) which associates a subspace of Hilbert space with almost every point of the plane; and this generalized bundle completely characterizes the subnormal operator T. If T is cyclic then its mosaic B(·) is a.e. either the zero operator or a rank one projection.     

Induced representations,reproducing kernels and the conformal group

The properties of induced (or multiplier) representations of groups which act in Hilbert spaces with a reproducing kernel are investigated. A resumé of earlier work is followed by a discussion of criteria for the irreducibility of such representations. The notions of reproducing kernel and positive definite spherical function are found to overlap. As a result, functional equations (analogous to those of Godement for spherical functions) are found for the reproducing kernel. The abstract theory is illustrated by certain discrete series representations of the conformal group and by their limit points. In particular the so-called ladder representations (which give rise to the conformal symmetry of zero mass particles) are analysed from this viewpoint.