A new formula on the Hermite expansion is presented in an explicit form. An application of the formula is given to a random boundary value problem: a plane wave reflection from a flat plane, of which position is randomly distributed in the normal direction, is presented. Several numerical results are given for a verification of the formula and for a discussion of the exact behavior of the fluctuation part of the reflection power.

I describe a software reliability growth model that yields accurate parameter estimates even with a small amount of input data. The model is based on a proposed discrete analog of a Gompertz equation that has an exact solution. The difference equation tends to a differential equation on which the Gompertz curve model is defined, when the time interval tends to zero. The exact solution also tends to the exact solution of the differential equation when the time interval tends to zero. The discrete model conserves the characteristics of the Gompertz model because the difference equation has an exact solution. Therefore, the proposed model provides accurate parameter estimates, making it possible to predict in the early test phase when software can be released.

Analytical solutions have been obtained for the electromagnetic scattering by a modified Luneberg lens with the permittivity of arbitrary parabolic function. They are expressed by four spherical vector wave functions for radially stratified medium which were introduced for the Luneberg lens by C. T. Tai. They consist of the confluent hypergeometric function and a "generalized" confluent hypergeometric function, in which the parameters for the permittivity of arbitrary parabolic function are involved. The characteristics of the modified Luneberg lens are numerically investigated using exact solutions in comparison with that of the conventional Luneberg lens. The bistatic cross section, the forward cross section and the radar cross section are studied in detail. The near-field distribution is also investigated in order to study the focal properties of the Luneberg lens. The focal shifts defined by the distance between the geometrical focal point and the electromagnetic focal point are obtained for various ka (k is the wave number and a is the radius of the lens). The focal shift normalized to the radius of the sphere becomes larger as ka is smaller. However it drops down rapidly for ka5 when the peak of the electric field amplitude appears on the surface of sphere.

The previous literature on consecutive k-out-of-r-from-n: F systems give recursive equations for the system reliability only for the special case when all component probabilities are equal. This paper deals with the problem of calculating the reliability for a (linear or circular) consecutive 2-out-of-r-from-n: F system with unequal component probabilities. We provide two new algorithms for the linear and circular systems which have time complexity of O(n) and O(nr), respectively. The results of some computational experiments are also described.