Abstract. In the present paper, we consider a wave system that is fixed at one end and a boundary control input possessing a partial time delay of weight (1 − µ) is applied over the other end. Using a simple boundary velocity feedback law, we show that the closed loop system generates a C0 group of linear operators. After a spectral analysis, we show that the closed loop system is a Riesz one, that is, there is a sequence of eigenvectors and generalized eigenvectors that forms a Riesz basis for the state Hilbert space. Furthermore, we show that when the weight µ > 1 2 , for any time delay, we can choose a suitable feedback gain so that the closed loop system is exponentially stable. When µ = 1 2 , we show that the system is at most asymptotically stable. When µ < 1 2 , the system is always unstable.Mathematics Subject Classification. 34H05, 49J25, 49K25, 93D15.

Over the past decade, research studies on benefits, critical success factors, difficulties, process, conceptual and theoretical models of construction partnering have been ubiquitous in the construction management discipline. In fact, there is adequate evidence that an increasing number of client organizations are adopting a partnering approach to undertake their building and construction projects both locally and worldwide during the last decade. With the perceived benefits that partnering brings about, research into Key Performance Indicators (KPIs) to evaluate the success of partnering projects in construction becomes vital as it can help set a benchmark for measuring the performance level of partnering projects. However, although there are some related studies and papers on this research area, few, if any, comprehensive and systematic studies focus on developing a comprehensive, objective, reliable and practical performance evaluation model for partnering projects. A model has been developed using the Delphi survey technique to objectively measure the performance of partnering projects in Hong Kong based on a consolidated KPIs' conceptual framework previously developed for partnering projects. Four rounds of Delphi questionnaire survey were conducted with 31 construction experts in Hong Kong. The results reveal that the top seven weighted KPIs to evaluate the success of partnering projects in Hong Kong were: (1) time performance; (2) cost performance; (3) top management commitment; (4) trust and respect; (5) quality performance; (6) effective communications; and (7) innovation and improvement. A statistically significant consensus on the top seven weighted KPIs was also obtained. Finally, a composite Partnering Performance Index (PPI) for partnering projects in Hong Kong was derived to provide an all-round assessment of partnering performance. Different partnering projects can now be assessed on the same basis for benchmarking purposes. Construction senior executives and project managers can thus use the Index to measure, evaluate and improve the performance of their partnering projects to strive for construction excellence. Although the PPI was developed locally in Hong Kong, the research method could be replicated in other parts of the world to produce similar indices for international comparison. Such an extension would aid the understanding of managing partnering projects across different geographic locations.Key Performance Indicators (KPIs), Partnering Performance Index (PPI), Delphi method, Delphi study, Hong Kong,

This paper develops an algorithm to calculate the Brownian multivariate normal probability subject to any preset error tolerance criteria. The algorithm is founded upon the computational simplicity of the tridiagonal structure of the inverse of the Brownian correlation matrix. Compared with existing pricing technologies without the "barrier too close" problem, our calculation method can produce a more accurate and efficient analytic evaluation of barrier options monitored at discrete instants with well- or ill-behaved barrier levels, or discrete hindsight options, for a reasonably large number of monitorings.Brownian Motion, Multivariate Normal Probability Evaluation Technique, Discrete Barrier Options, Discrete Lookback Options

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This article presents preliminary research on the general problem of reducing the number of neurons needed in a neural network so that the network can perform a specific recognition task. We consider a singlehidden-layer feedforward network in which only McCulloch-Pitts units are employed in the hidden layer. We show that if only interconnections between adjacent layers are allowed, the minimum size of the hidden layer required to solve the n-bit parity problem is n when n ≤ 4.