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5 Black Body Radiation: Rayleigh’s Law In June 1900, that is several months before Planck’s revolutionary discovery, Lord Rayleigh made another proposal [26] which for the first time introduces into the black body radiation problem the density of states D(ν), that is the density of the vibrational modes of a black body cavity. 3). 18) “viewed from the theoretical side [. . ] appears to me little more than a conjecture . . ”, and “. . 18) lim ρW (ν, T ) = αν3 . ” [26] Without further explanation, Rayleigh assumes, first of all, that the equilibrium distribution ρ is proportional to the density of the vibrational modes of the cavity per unit volume, that is ρ(ν, T ) ~ D(ν)/V, where D(ν) := dN with N(ν) := N (2π/cν)2 dν and N(λ) denotes the leading asymptotic term of the counting function expressed in terms of the frequency ν.

In 1900, Rayleigh realized that the same formula can be applied to a physically completely different, but mathematically equivalent problem: the heat radiation from a body in thermal equilibrium with radiation, the importance of which had been pointed out already in 1859 by Kirchhoff. The amount of energy emitted by a body is determined by the high-frequency spectrum of standing electromagnetic waves and that spectrum should be essentially the same as for the high overtones of an organ pipe, say.

E. P(x) = O x1/4+ε for every ε > 0 [53, 54]. Actually, Hardy proved α0 v 1/4. 330 . . 324 . . 318 . . (Iwaniec and Mozzochi 1988 [57]). 315 . . is still far away from 1/4! ) Since P(x) is a wildly fluctuating function, it might be that some very rare spikes exceeding the conjectured x1/4 -behavior make it extremely difficult to improve the best existing bound. In order to “tame” these spikes, one can consider moments of P(x) and hope that the spikes are being washed out. 9 making use of the trace formula for ν(x) which we shall now derive.

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10 Lies About The Atkins Diet by Tom Venuto


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