Fracture of Brittle Solids. IV. Two-Dimensional Distribution Function for Fragment Size in Single Fracture (Theoretical)

1962 ◽  
Vol 33 (11) ◽  
pp. 3214-3217 ◽  
Author(s):  
J. J. Gilvarry
Keyword(s):  
Distribution Function ◽  
Fragment Size ◽  
Single Fracture ◽  
Two Dimensional ◽  
Brittle Solids ◽  
Dimensional Distribution

scholarly journals Sur les processus arithmétiques liés aux diviseurs

2016 ◽  
Vol 48 (A) ◽  
pp. 63-76 ◽  
Author(s):  
R. de la Bretèche ◽  
G. Tenenbaum
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Mean Value ◽  
Two Dimensional ◽  
Limit Laws ◽  
Uniform Probability ◽  
Dimensional Distribution ◽  
The Mean ◽  
Generalized Moments

AbstractFor natural integer n, let Dn denote the random variable taking the values log d for d dividing n with uniform probability 1/τ(n). Then t↦ℙ(Dn≤nt) (0≤t≤1) is an arithmetic process with respect to the uniform probability over the first N integers. It is known from previous works that this process converges to a limit law and that the same holds for various extensions. We investigate the generalized moments of arbitrary orders for the limit laws. We also evaluate the mean value of the two-dimensional distribution function ℙ(Dn≤nu, D{n/Dn}≤nv).

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