Photon number variance in isolated cavities

We consider a strictly isolated single-mode optical cavity resonating at angular frequency omega containing atoms whose one-electron level energies are supposed to be: hbar*omega, 2*hbar*omega,...B*hbar\omega, and m photons. If initially the atoms are in their highest energy state and m=0, we find that at equilibrium: variance(m)/mean(m)=(B+1)/6, indicating that the internal field statistics is sub-Poissonian if the number of atomic levels B does not exceed 4. Remarkably, this result does not depend on the number of atoms, nor on the number of electrons that each atom incorporates. Our result has application to the statistics of the light emitted by pulsed lasers and nuclear magnetic resonance. On the mathematical side, the result is based on the restricted partitions of integers.Comment: 4 pages, to be submitted to Journal of Physics

Global geometry of T2 symmetric spacetimes with weak regularity

We define the class of weakly regular spacetimes with T2 symmetry, and investigate their global geometry structure. We formulate the initial value problem for the Einstein vacuum equations with weak regularity, and establish the existence of a global foliation by the level sets of the area R of the orbits of symmetry, so that each leaf can be regarded as an initial hypersurface. Except for the flat Kasner spacetimes which are known explicitly, R takes all positive values. Our weak regularity assumptions only require that the gradient of R is continuous while the metric coefficients belong to the Sobolev space H1 (or have even less regularity).Comment: 5 page

Productivity and R&D at the Firm Level in French Manufacturing

In a companion study to that of Griliches and Mairesse for the United States, we have investigated the relationship between output, labor, and physical and R&D capital during the 1972-1977 period for a sample of 182 R&D performing firms in the French nnufacturing industries. Our results are quite comparable to those obtained for the U.S. The relationship between firm productivity and R&D appears both strong and robust in the cross-sectional dimension of the data; it is less so in the time dimension. However, the within-firm estimates are still significant and of a likely order of magnitude.In this respect, they are more satisfactory than the U.S. ones. We show that this is largely due to a better measurement of the variables: (1) the fact that we can use a value-added measure of output instead of sales (or equivalently that we include materials among the factors of the production function); (2) the fact that we can correct the measures of labor, physical capital and output for the double counting or expensing out of the labor, capital and materials components of R&D expenditures.

On Classical Ideal Gases

The ideal gas laws are derived from the democritian concept of corpuscles moving in vacuum plus a principle of simplicity, namely that these laws are independent of the laws of motion aside from the law of energy conservation. A single corpuscle in contact with a heat bath and submitted to a $z$ and $t$-invariant force $-w$ is considered, in which case corpuscle distinguishability is irrelevant. The non-relativistic approximation is made only in examples. Some of the end results are known but the method appears to be novel. The mathematics being elementary the present paper should facilitate the understanding of the ideal-gas law and more generally of classical thermodynamics. It supplements importantly a previously published paper: The stability of ideal gases is proven from the expressions obtained for the force exerted by the corpuscle on the two end pistons of a cylinder, and the internal energy. We evaluate the entropy increase that occurs when the wall separating two cylinders is removed and show that the entropy remains the same when the separation is restored. The entropy increment may be defined at the ratio of heat entering into the system and temperature when the number of corpuscles (0 or 1) is fixed. In general the entropy is defined as the average value of $\ln(p)$ where $p$ denotes the probability of a given state. Generalization to $z$-dependent weights, or equivalently to arbitrary static potentials, is made.Comment: Generalization of previous versions to questions of stabilit

Comment on: "Sadi Carnot on Carnot's theorem"

Carnot established in 1824 that the efficiency $\eta_{C}$ of reversible engines operating between a hot bath at absolute temperature $T_{hot}$ and a cold bath at temperature $T_{cold}$ is equal to $1-T_{cold}/T_{hot}$. Carnot particularly considered air as a working fluid and small bath-temperature differences. Plugging into Carnot's expression modern experimental values, exact agreement with modern Thermodynamics is found. However, in a recently published paper ["Sadi Carnot on Carnot's theorem", \textit{Am. J. Phys.} \textbf{70}(1), 42-47, 2002], Guemez and others consider a "modified cycle" involving two isobars that they mistakenly attribute to Carnot. They calculate an efficiency considerably lower than $\eta_{C}$ and suggest that Carnot made compensating errors. Our contention is that the Carnot theory is, to the contrary, perfectly accurate.Comment: Submitted to American Journal of Physic