Several risk factors associated with colorectal cancer (CRC) have been identified including β-catenin/CTNNB1 hotspot mutations. The levels of β-catenin within a cell are regulated via phosphorylation of the N terminus of β-catenin by GSK-3β. Thus far three serines (S33, 37, 45) and one threonine (T41) are considered to be the substrates for GSK-3β phosphorylation. In the present investigation an attempt was made to study the role of β-catenin mutations in exon-3 in 60 colorectal cancer patients from Kingdom of Saudi Arabia (KSA).

We consider weighted path lengths to the extremal leaves in a random binary search tree. When linearly scaled, the weighted path length to the minimal label has Dickman's infinitely divisible distribution as a limit. By contrast, the weighted path length to the maximal label needs to be centered and scaled to converge to a standard normal variate in distribution. The exercise shows that path lengths associated with different ranks exhibit different behaviors depending on the rank.

Two processes of random fragmentation of an interval are investigated. For each of them, there is a splitting probability at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. More precisely, we consider two models. In the first model, the fragmentation stops which a probability p witch can not depend on the fragment size. The number of stable fragments with sizes less than a given t≥ 0, denoted by K(t), is introduced and studied. In the second one the probability to split a fragment of size x is p(x)=1-e-x.

In this paper we consider a non balanced generalized Polya urn model. At each step, m balls are drawn at random from an urn containing white and black balls. Balls are added according the inspected colors. We consider two different models: the self reinforcement model and the opposite reinforcement model. We use the Stochastic approximation to obtain the almost sure convergence of the proportion of white balls in the urn, then we give the asymptotic of the expectation and the variance.