Dimension estimates for sets of uniformly badly approximable systems of linear forms

The set of badly approximable m × n matrices is known to have Hausdorff dimension mn. Each such matrix comes with its own approximation constant c, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed c. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions. The lower bounds are established via the technique of Schmidt games, while for the upper bound we use homogeneous dynamics methods, namely exponential mixing of flows on the space of lattices.

APPROXIMATION PROPERTIES FOR CROSSED PRODUCTS BY ACTIONS AND COACTIONS

We investigate approximation properties for C*-algebras and their crossed products by actions and coactions by locally compact groups. We show that Haagerup's approximation constant is preserved for crossed products by arbitrary amenable groups, and we show why this is not always true in the non-amenable case. We also examine similar questions for other forms of the approximation property.

Are Novae Standard Candles?

Novae in outburst can be used as distance indicators. The distance measurement process is considerably simplified if novae could be regarded as standard candles. The term standard candles in the context of this paper means that the absolute magnitude at maximum for different novae is, to a good approximation, constant. The t3 – Mb diagram (Schmidet 1957) implies that this is the case for M31 and LMC novae (Della Valle & Livio 1995). However, this conclusion does not necessarily apply to non Local Group novae (Livio 1992; Della Valle & Livio 1995); for example the Virgo novae are not similar to Local Group novae (Della Valle & Livio 1995). Here we focus on a question of principle, namely: Does the fact that novae have similar spectra imply that they have similar luminosities?

Effect of Pressurization on the Vibration Isolation Capability of Squeeze Film Bearings

This paper investigates theoretically the effect of pressurization on the vibration isolation capability of centrally preloaded squeeze film bearings supporting a rigid rotor which in turn is mounted in rolling element bearings. Assuming the short bearing approximation, constant lubricant properties, and that steady state conditions have been reached with the journal center describing synchronous circular orbits about the bearing center, the theory is developed for the general case of arbitrary pressurization at either end of the bearing. The design data are for bearings pressurized at one end only as in circumferentially grooved bearings and conservatively assume that the saturation vapor pressure of the lubricant is atmospheric. These design curves show the effect of the relevant system parameters on the possibility of undesirable operation modes, on the unbalance force transmissibility and on rotor vibration amplitudes. Hence, the influence of lubricant viscosity, lubricant supply pressure, bearing dimensions, rotor speed, rotor mass, rotor unbalance and support flexibility may be readily determined, allowing for optimal system design. It is shown that significant unbalance force isolation is a practical possibility with consequent decrease in the vibration level of the rotor mounts and increase in rolling element bearing life, while maintaining rotor excursion amplitudes at an acceptable level, even with relatively high unbalance loading. In particular, with increased pressurization, the likelihood of bistable operation can be considerably reduced. The data suggest that by varying the supply pressure and/or the lubricant viscosity, the rotor bearing system may be gainfully controlled to run at minimum vibration level and/or with minimum unbalance transmissibility.

The observation by means of a string electrometer of fluctuations in the ionisation produced by γ -rays

A γ -ray is a projected entity, according to the theories of J. J. Thomson, of Bragg, and of J. Stark, and the effect of it will be localised in space in the sense that the effect of a bullet is. Any one of these theories requires the ionisation produced by γ -radiation in a gas at any instant to fluctuate both in space and time. In this respect the effect of γ -rays would be similar to that of α -particles; as is well known, the spinthariscope exhibits for the latter rays both fluctuations in time and discontinuity in space. Similarly, the effects predictable for β -rays have been observed, but, when the experiments recorded below were begun, no such fluctuation had been observed for γ -rays. In 1908 one of us described how they might be sought for experimentally: "At the same distance from a source of γ rays, two identical and adjacent ionisation chambers would be placed to catch the γ -rays. With one chamber at + v volts and the other at — v , the electrode of one receives positive ions, the other negative; connecting the electrodes, the electroscope would receive the small excess of ions of one sign. If the γ -radiation is a spherical wave surface, then the ionisations in the chambers will have a constant ratio. This is true .... even when the energy varies continuously over the wave front; the excess current to the electroscope is then to a first approximation constant. If, on the other hand, the γ -rays are particles emitted in random directions as α -rays are, then the number entering each chamber in the given time will fluctuate. In the extreme case, during a short time a few γ -particles might enter one chamber, and none the other. In the more general case, when some large number of particles enter both chambers, the current to the electroscope will fluctuate.