102 TEMPORAL ORGANIZATION IN CELLS 



Consider an oscillatory function F(t). According to a result of M. Kac 

 (Kerner, 1959), the mean frequency of zeros of^ F{t), call it aj(F(0), is 



co(F(t))=^-j \F'(t)\8(F(t))dt 



where 8 signifies the delta-function and F'(t) is the first derivative of F(t). This 

 result follows from the observation that the integral 



T 



jS(F{t))dt 







will give a value 



near a zero t = to ofF, but is zero elsewhere. Now using the canonical ensemble 

 to obtain phase averages in place of time averages, we have for the mean 

 frequency of zeros of a variable, say >',, about its steady state (i.e. the mean 

 frequency with which j, takes its steady state value 0) 



Since yi is dependent only on Xj, this integral reduces to 



<o(yd = ^ J §(>;,■) e-^^^.^j, J |j>,k-^^Wx,- 



-Tj —pi 



00 





-Pi 



CO 



^Pi^Qi J 



(50) 



-Pi 



Similarly for the variable x,- we have 



a>(x,) = J \xi\S(xde-^''dvlJe-^'^dv 



00 



= ^ r \x,\e-P'''"dy^ (51) 



because .v, is dependent only upon j^,. 



The formula can be extended slightly to the mean frequency of zeros of a 



