By L. Boi, D. Flament, Jean-Michel Salanskis

Within the first 1/2 the nineteenth century geometry replaced considerably, and withina century it helped to revolutionize either arithmetic and physics. It additionally positioned the epistemology and the philosophy of technology on a brand new footing. In this quantity a legitimate assessment of this improvement is given through prime mathematicians, physicists, philosophers, and historians of technology. This interdisciplinary strategy offers this assortment a distinct personality. it may be utilized by scientists and scholars, however it additionally addresses a basic readership.

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**Example text**

Num Den 0 0 0 0 0 0 0 1! 0 0 0 0 0 0 Den"(1! )(1! k I>. and it gives the generating function for the number of states of the four-dimensional isotropic harmonic oscillator. Generating functions for numbers of states of de"nite symmetry within N polyads follows P immediately from Table 17 taking into account the resonance relation between frequencies. e. for states of k"0, g,# type): F 1# g (k"0, g,#; )" . (73) (1! )(1! )(1! )(1! )(1! ) Power series expansion gives, for example g (k"0, g,#; )"1#2 # #4 #3 #7 #7 #12 #2 .

1! I. Zhilinskin& / Physics Reports 341 (2001) 85}171 135 Fig. 19. erent symmetry types as a function of the total polyad quantum number N . Numbers of states are given for four one-dimensional representations of D group. k I> , k"1, 2,2 , (1! )(1! )(1! )(1! )(1! k I>) , k"1, 2,2, g (2k#1, , )" (1! )(1! )(1! )(1! )(1! ) "u, g . (79) (80) Fig. erent symmetry types as a function of N . erent symmetry types scale in fact by a constant factor in the high energy region (Quack, 1977).

In each case we consider K vibrational modes with frequencies , i"1,2, K, and suppose a resonance G condition : : 2 : +n : n : 2 : n . All n should be taken as positive integers; they can be ) ) G large in order to reproduce the ratio of the actual frequencies with desired accuracy. erent meaning, are possible. To label the vibrational polyads we introduce the polyad quantum number N. The physical meaning of the polyad quantum number N can be understood in several ways. erent modes N , N ,2, N in accordance with the ) resonance condition.