Separate continuity, joint continuity and the Lindelöf property

In this paper we prove a theorem more general than the following. Suppose that is Lindelöf and -favourable and is Lindelöf and Cech-complete. Then for each separately continuous function there exists a residual set in such that is jointly continuous at each point of .

     

The effective communication of system dynamics to improve insight and learning in management education

Experienced exponents of system dynamics in management education appreciate that feedback, non-linearities and delays are ubiquitous, and create difficulties for making intuitive judgements about the dynamic behaviour of business systems. They have applied much effort to solving this problem, offering simple, high-level causal mapping techniques to conceptualise dynamic issues, formal simulation modelling, and simulation-based learning environments. However, the use of these methods is still not extensive, and it is often disconnected from other management development pedagogies.System dynamics is ideally suited to operationalising certain concepts central to the management field, notably the resource-based and competence-based view of strategic management. However, three developments must be made if this potential is to be exploited. Firstly, system dynamics must connect with those established concepts and frameworks in the management field. Secondly, the barriers for managers to appreciate the power of system dynamics must be lowered, to make the method an integral part of the managerial mind-set. Thirdly, managers need help to climb the learning curve of understanding regarding the dynamic behaviour of the business systems they endeavour to manage, through a comprehensive set of resource-mapping and gaming simulation tools.This paper describes how these developments can be implemented, reports on experience of using the resulting learning devices and comments on possible future directions. There is an exciting opportunity for system dynamics to make a major contribution to a new strategy paradigm, based on a dynamic resource-system view of the firm, a perspective that can be extended to other fields in management and to non-business contexts.     

Stiff Oscillatory Systems, Delta Jumps and White Noise

Two model problems for stiff oscillatory systems are introduced. Both comprise a linear superposition of harmonic oscillators used as a forcing term for a scalar ODE. In the first case the initial conditions are chosen so that the forcing term approximates a delta function as and in the second case so that it approximates white noise. In both cases the fastest natural frequency of the oscillators is <e6>OM</e6>(N). The model problems are integrated numerically in the stiff regime where the time-step satisfies The convergence of the algorithms is studied in this case in the limit and For the white noise problem both strong and weak convergence are considered. Order reduction phenomena are observed numerically and proved theoretically. August 25, 1999. Final version received: May 3, 2000.     

Optimal Control of Non-convex Measure-driven Differential Inclusions

Necessary conditions for optimality in control problems with differential-inclusion dynamics have recently been developed in the non-convex case by Clarke, Vinter, and others. Using appropriate reparametrizations of the time variable, we extend these results to systems whose dynamics involve a differential inclusion where a vector-valued measure appears. An auxiliary result central to our proof is an extension of existing free end-time necessary conditions to Clarke’s stratified framework.     

Wall shear rate distribution for flow in random sphere packings

The wall shear rate distribution P(gamma) is investigated for pressure-driven Stokes flow through random arrangements of spheres at packing fractions 0.1< or =varphi< or =0.64. For dense packings, P(gamma) is monotonic and approximately exponential. As varphi-->0.1, P(gamma) picks up additional structure which corresponds to the flow around isolated spheres, for which an exact result can be obtained. A simple expression for the mean wall shear rate is presented, based on a force-balance argument.     

Functional magnetic resonance imaging with intermolecular multiple-quantum coherences

For the first time, we demonstrate here functional magnetic resonance imaging (fMRI) using intermolecular multiple-quantum coherences (iMQCs). iMQCs are normally not observed in liquid-state NMR because dipolar interactions between spins average to zero. If the magnetic isotropy of the sample is broken through the use of magnetic field gradients, dipolar couplings can reappear, and hence iMQCs can be observed. Conventional (BOLD) fMRI measures susceptibility variations averaged over each voxel. In the experiment performed here, the sensitivity of iMQCs to frequency variations over mesoscopic and well-defined distances is exploited. We show that iMQC contrast is qualitatively and quantitatively different from BOLD contrast in a visual stimulation task. While the number of activated pixels is smaller in iMQC contrast, the intensity change in some pixels exceeds that of BOLD contrast severalfold.     

Explicit gradient estimates for minimal Lagrangian surfaces of dimension two

We derive explicit, uniform, a priori interior Hessian and gradient estimates for special Lagrangian equations of all phases in dimension two. Y. Yuan is partially supported by an NSF grant.     

Mechanisms of sulfanyl (RS) migrations: synthesis of heterocycles

Thiiranium (episulfonium) ions had been acknowledged as reaction intermediates for many years, but it was not until 1977 that Nicolaou demonstrated systematically that these reactive heterocyclic cations could be trapped by carboxylic acids to give lactones. In the years that followed this report, extensive research greatly extended the scope of this reaction, particularly with regard to the methods for generating thiiranium ions, the types of nucleophiles that are compatible with this reaction, and the selectivity involved in the cyclization reactions. For many years we have been using thiiranium ions for the synthesis of saturated heterocycles. Whereas Nicolaou's method relied on electrophilic sulfenylation of alkenes, we have generated thiiranium ions by displacement of a leaving group with neighboring-group participation by a sulfanyl group. Many of the examples we have reported are of cyclizations that are reversible and so where two (and in some cases more) products can result, the outcome of the reactions provides fundamental information about the relative stability of different heterocyclic ring systems. This Review will begin with a brief introduction to sulfanyl participation as a method for generating thiiranium (and thiolanium) ions, and will go on to explore the idea of using sulfanyl migrations in synthesis. Initially, emphasis will be placed on mechanisms of [1,2] sulfanyl migrations: we will look specifically at [1,2] sulfanyl migrations (usually PhS) with elimination, substitution, and cyclization. Emphasis will then shift to the factors that affect the outcome of cyclization reactions. In particular, we will cover cyclizations with hydroxy nucleophiles and examine situations in which there are more than one hydroxy nucleophile present. We will also examine cyclizations with other nucleophiles, namely amines and sulfides. After our discussion of [1,2] sulfanyl migrations, we will look very briefly at the scope of [1,4] sulfanyl participation, before finally drawing up some guidelines that (we hope) will help other organic chemists take advantage of the rearrangement reactions that the sulfanyl group has to offer.