What Is the Philosophy of Mathematics,and What Should It Be?

This article was begun while the author was on sabbatical at the Université Catholique de Louvain and was partially supported by Lilly Faculty Open Fellowship grant number 910043. A preliminary version was presented in Belgium to the logic seminar, and the author thanks the participants in that seminar, especially Thierry Lucas, as well as Jan Denef for their suggestions.     

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (noncommutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Noncommutative Gr?bner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment?Cangle complex. We apply our result to the loop-space homology of this space.     

Millimeter-wave gyroklystron amplifier experiment using arelativistic electron beam

A fundamental-mode TE₁₁₁° two-cavity intense-beam gyroklystron amplifier experiment, operating at an accelerating voltage of 1 MV, is reported. The two cavities that were tested are designed to serve as bunching cavities for a high-power output cavity. The two-cavity amplifier has demonstrated a linear gain of 15 dB and an unsaturated output power of ~40 kW, with the intracavity gain and power ~4 dB higher. The frequency of the second cavity has been found to track the frequency of the driven cavity over a range of 300 MHz around a center frequency of 35 GHz. Stable amplifier operation was achieved with beam currents as large as 150 A and a velocity pitch ratio of 0.36. The stable operating range was limited by spurious oscillation in the TE₁₁₂° mode. Theoretical calculations indicate that higher gains might be attainable if this mode could be suppressed     

Generalized poisson distributions

Poisson distributions evidently can be treated in a unified manner by dealing with the G.P. which contains a characteristic parameterq. The properties of the PEBL, long recognized as significant in a variety of statistical applications like telephone trunking, queueing, etc., may be deduced by passage to the limitq = 1.     

Barrier penetration effects for electrons in quantum wells: screening,mobility, and shallow impurity states

A two-dimensional electron gas in a quantum well confined by finite barriers is considered. We present analytical expressions for the finite confinement effects of a square-well potential and calculate the electron-electron interaction potential, the electron-impurity interaction potential, the interface-roughness scattering potential and the alloy-disorder scattering potential. The dielectric function of the interacting electron gas, the mobility (for charged-impurity scattering, for interface-roughness scattering, and for alloy-disorder scattering), and the binding energy of hydrogenic impurities (screened and unscreened) are discussed.     

Simulation of transient behavior in a pulse-line-driven gyrotronoscillator

Results are reported for a set of slow-time-scale single-moded and fast-time-scale single-mode and multimode simulations of the transient-mode excitation phenomena in a short-pulse high-peak-power Ka-band gyrotron oscillator experiment. Both the slow- and fast-time-scale single-mode simulations are generally in good agreement with each other, and, within experimental uncertainties, with the experimental observations of the time dependence and magnetic-field dependence of 35-GHz emission in the TE₆₂ mode. However, the multimode simulations suggest the presence of mode suppression, mode beating, and other nonlinear multimode phenomena that could not easily by observed in the experiment, and generally agree less well with the experimental measurements than the single-mode simulations. The multimode simulations also suggest that steady-state behavior may not be obtainable with the highly time-dependent voltage waveform employed in the experiment, and indicate the importance of carrying out future high-voltage gyrotron experiments with less highly transient voltage waveforms     

A modulated lamp system for solar UV-B (280--320 nm). Supplementation studies in the field

Precise control and modulation of fluorescent sunlamps are necessary for ecologically valid simulation of solar UV-B (280-320 nm) radiation enhancement which would correspond to ozone reduction. A control system is described which allows lamp emittance to be modulated over a large dynamic range (50:1), permits stable lamp operation and starting at low temperatures, and provides a sensitive feedback loop to compensate for both atmospheric conditions, such as cloud cover, as well as changes in radiant emittance from the lamps resulting from factors such as temperature and lamp age.