• Journal of the Korean Mathematical Society
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• v.36 no.5
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• pp.923-943
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• 1999

In this paper we consider functionals of the empirical age distribution of supercritical Bellman-Harris processes. Let f : R+ longrightarrow R be a measurable function that integrates to zero with respect to the stable age distribution in a supercritical Bellman-Harris process with no extinction. We present sufficient conditions for the asymptotic normality of the mean of f with respect to the empirical age distribution at time t.

• Proceedings of the KIEE Conference
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• 1996.11a
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• pp.39-41
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• 1996

While the z-transform method is a basic mathematical tool to relate the signals only at the sampling instants in analyzing and designing sampled-data control systems, the modified z-transform which is a variation of the z-transform is widely used to represent the details of continuous signals between the sampling instants. Regarding the modified z-transform method, some properties were established to relate the modified z-transform to the regular z-transform. This paper will show that these properties, in their current forms, cause some analytic problems, when they are applied to the signals with discontinuities at the sampling instants, which accordingly limit their applications significantly. In this paper, those analytic problems will be investigated, and the theorems of the modified z-transform will be revised by adopting a new notation so that those can be correctly interpreted and used without any analytic problems in the analysis of sampled data systems. Also some useful schemes of applying the modified z-transform will be developed.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.51 no.12
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• pp.549-555
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• 2002

While the z-transform method is a basic mathematical tool to relate the imput/output signals only at the sampling instants in analyzing and designing sampled-data control systems, the modified z-transform which is a variation of the z-transform is widely used to represent the details of continuous signals between the sampling instants. To relate the modified z-transform to the corresponding regular z-transform, some properties were established regarding the modified z-transform method. This paper will show that these properties, in their current forms, cause come analytic problems, when they are applied to the signals with discontinuities at the sampling instants, which accordingly limit their applications significantly. In this paper, those analytic problems will be investigated, and the theorems of the modified z-transform will be revised by adopting new notations on the z-transform so that those can be correctly interpreted and used without any analytic problems. Also some additional useful schemes of applying the modified z-transform will be developed.