Conditional expectation values in quantum mechanics

The general question of defining the expectation value of an operator for a fixed value of another noncommuting observable is considered and explicit expressions are derived. Due to the noncommutivity of operators a unique definition is not possible, and we consider different possible expressions. Special cases which have previously been considered in the literature are shown to be derivable from the methods presented.In honor of Ilya Prigogine on the occasion of his 70th birthday.     

Single-vendor multi-buyer inventory coordination under private information

This paper considers the problem of coordinating a single-vendor multi-buyer inventory system when there are privacy restrictions in the information required to solve the problem. The objective function and cost parameters of each facility are regarded as private information that no other facilities in the system have access to. Moreover, each facility is responsible to specify its own replenishment policy. The objective is to minimize the total average setup/ordering and inventory-related cost. Solution methodologies under private and global information are developed to find two types of nested power-of-two stationary policies. The first policy assumes all the buyers must replenish simultaneously. The second policy is a more general case where the common replenishment assumption is relaxed. A simple form of information exchange is uncovered that allows the solution methodologies for private and global information yield the same results. The experimental results suggest that the performance of the proposed heuristics is comparable or better than an existing method.     

On the risk-based steam generator lifetime optimisation

A probabilistic fracture mechanics model is employed to estimate the failure probability of axially cracked steam generator tubes. The model estimates the failure probability from the random changes of the influencing parameters such as tube and crack geometry, material properties and non-destructive examination results, reliability and sizing accuracy and stable crack propagation. The performance of the model is illustrated by a numerical example. A steam generator tubing severely affected by the stress corrosion cracking is studied during most unfavourable accidental conditions. Two different plugging approaches are analyzed and the quality is compared, showing the superior performance of crack length oriented approach over tube wall thickness reduction both in terms of SG failure probability and extent of plugging. Thus, apart from setting the acceptable SG failure probability, all elements for the risk-based SG lifetime optimisation are provided on the example of stress corrosion cracking in the tube expansion transition zone.     

Light at the interface: the potential of attenuated total reflection infrared spectroscopy for understanding heterogeneous catalysis in water

IR spectroscopy has been an important tool for studying detailed interactions of reactants and reaction-intermediates with catalyst surfaces. Studying reactions in water is, however, far from trivial, due to the excessive absorption of infrared light by water. One way to deal with this is the use of Attenuated Total Reflection spectroscopy (ATR-IR) minimizing the path length of infrared light through the water. Moreover, ATR-IR allows for a direct comparison of reactions in gas and water on the same sample, which bridges the gap between separate catalyst investigations in gas and liquid phase. This tutorial review describes recent progress in using ATR-IR for studying heterogeneous catalysts in water. An overview is given of the important aspects to be taken into account when using ATR-IR to study heterogeneous catalysts in liquid phase, like the procedure to prepare stable catalyst layers on the internal reflection element. As a case study, CO adsorption and oxidation on noble metal catalysts is investigated with ATR-IR in gas and water. The results show a large effect of water and pH on the adsorption and oxidation of CO on Pt/Al(2)O(3) and Pd/Al(2)O(3). From the results it is concluded that water affects the metal particle potential as well as the adsorbed CO molecule directly, resulting in higher oxidation rates in water compared to gas phase. Moreover, also pH influences the metal particle potential with a clear effect on the observed oxidation rates. Finally, the future outlook illustrates that ATR-IR spectroscopy holds great promise in the field of liquid phase heterogeneous catalysis.     

Morse�CSmale theory for flows on covering spaces of compact manifolds

For a dynamical system X on a compact differentiable manifold M and for the dynamical system X(ρ) induced from X by a covering map r :  [(M)tilde] ?  M{rho , : , widetilde{M}, rightarrow , M}, we develop algebraic topology methods for estimating the lower bounds on the number of codimension-1 surfaces (i.e., on the number of index-1 equilibria of flows and their stable manifolds) on the boundary of regions of stability on [(M)tilde]{widetilde{M}}. We also develop methods for estimating the number of equilibria on the boundaries of stability regions of noncompact manifolds with very general assumptions. Our methods allow us to obtain results for noncompact manifolds in cases when Morse–Smale approach does not work.