We derive in this paper closed formulae for the joint probability generating function of the number of customers in the two FIFO queues of a generalized processor-sharing (GPS) system with two classes of customers arriving according to Poisson processes and requiring exponential service times. In contrast to previous studies published on the GPS system, we show that it is possible to establish explicit expressions for the generating functions of the number of customers in each queue without calling for the formulation of a Riemann–Hilbert problem. We specifically prove that the problem of determining the unknown functions due to the reflecting conditions on the boundaries of the positive quarter plane can be reduced to a Poisson equation. The explicit formulae are then used to derive some characteristics of the GPS system (in particular the tails of the probability distributions of the numbers of customers in each queue).

This paper presents a theoretical analysis of oversampled complex modulated transmultiplexers. The perfect reconstruction (PR) conditions are established in the polyphase domain for a pair of biorthogonal prototype filters. A decomposition theorem is proposed that allows it to split the initial system of PR equations, that can be huge, into small independent subsystems of equations. In the orthogonal case, it is shown that these subsystems can be solved thanks to an appropriate angular parametrization. This parametrization is efficiently exploited afterwards, using the compact representation we recently introduced for critically decimated modulated filter banks. Two design criteria, the out-of-band energy minimization and the time-frequency localization maximization, are examined. It is shown, with various design examples, that this approach allows the design of oversampled modulated transmultiplexers, or filter banks with a thousand carriers, or subbands, for rational oversampling ratios corresponding to low redundancies. Some simulation results, obtained for a transmission over a flat fading channel, also show that, compared to the conventional OFDM, these designs may reduce the mean square error.

The angular factors of atomic orbitals are real spherical harmonics. This is independent of the choice of basis function. In the course of molecular electronic structure calculations, numerous rotations of real spherical harmonics are required in a suitably defined space-fixed coordinate system. The origin and axes are space-fixed and rotation matrices defined on a basis of spherical harmonics. In this work, a highly compact expression and efficient evaluation of the rotation matrices are given for a real spherical harmonic basis. Relations to Gaunt coefficients are shown explicitly as are recurrence formulae for rotation matrices. This leads to extremely rapid and precise rotation algorithms. The Wigner rotation matrices which are still often used in orbital rotations are shown to be completely surpassed by this approach. The present work is related to a method described by Kautz in the field of image processing but significant improvements have been made, especially in the study of structure and storage of the rotation matrices. After complete testing using computer algebra, a numerical program was written in C. Numerical tests are cited in the closing sections of this work.

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