By Wolfgang Arendt, Wolfgang P. Schleich

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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. Note added in proof: in a recent unpublished paper [59] it is claimed to present a proof of Hardy’s conjecture.

103–104 and p. ” He then states, without proof: “For high frequencies ν there are approximately V 8πν2 dν/c3 modes of oscillation in the frequency interval ν, ν + dν. ” In a footnote he adds: “This result is easily obtained by elementary methods for a rectangular parallelepiped. For the general proof see H. ” It is clear that the limit V → ∞ is an idealization which can never be realized in a physical experiment. Rather the “assumption must always hold that the linear dimensions of all cavities considered and also the curvature of the radii of all surfaces considered must be large compared with the wavelengths of the radiation.

For his contributions to the solution of the Horn conjecture, the Clay Research Award was assigned to Terence Tao in 2003. He was also awarded the Fields Medal in 2006 for his results in the arithmetic of prime numbers. It is remarkable that the once abstract graph theory has also found useful applications in algorithmic computer science, especially in data clustering (Chapters 8 and 9), sequencing (Chapter 5), classification (Chapter 11), and sorting (Chapter 17). Δ Δ Δ In spite of its limitations in large data analysis, the most efficient system for information processing and computing is still a natural one – the brain.

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