Graham Higman's PORC theorem

Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments

Optimized power converters for electrically augmented on-orbit propulsion systems

Advances in satellite thruster technology have produced the requirement for high power electrical supplies to operate electrically augmented on-orbit propulsion systems. The power on board satellites is greatly limited by the solar panels that collect energy and the batteries that store the energy. In addition, satellites are constantly using power to operated their mission equipment and handle the communication requirements; This thesis investigates the power systems of modern satellites and the addition of an on-orbit electrically augmented propulsion system. The research process involves determining the system specifications, the power available and the thruster requirements. After determining the requirements, the implementation of the electrically augmented on-orbit propulsion system is characterized in terms of efficiency and potential electromagnetic compatibility. Several circuits are evaluated with the aid of PSPICE circuit simulation software and the results of the evaluation criteria for each circuit are presented

Density Waves in Granular Flow: A Kinetic Wave Approach

It was recently observed that sand flowing down a vertical tube sometimes forms a traveling density pattern in which a number of regions with high density are separated from each other by regions of low density. In this work, we consider this behavior from the point of view of kinetic wave theory. Similar density patterns are found in molecular dynamic simulations of the system, and a well defined relationship is observed between local flux and local density -- a strong indicator of the presence of kinetic waves. The equations of motion for this system are also presented, and they allow kinetic wave solutions. Finally, the pattern formation process is investigated using a simple model of interacting kinetic waves.Comment: RevTeX, HLRZ preprint 46/93, 4 figures available upon reques