We study algorithms for solving a problem of constructing text (a long string) from dictionary sequence small strings). The has an application in bioinformatics and connection with the assembly method reconstructing DNA fragments. Our is construction string t length n using strings $$s^1,\dots , s^m$$ possible overlapping. Firstly, we provide classical (randomized) algorithm running time $$O\le...

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...

A log-linear modelling will take quite a long time if the data involves many variables and if we try to deal with all the variables at once. Fienberg and Kim (1999) investigated the relationship between log-linear model and its conditional, and we will show how this relationship is employed to make the log-linear modelling easier. The result of this article applies to all the hierarchical log-l...

Let G(X) denote the size of the largest gap between consecutive primes below X . Answering a question of Erdős, we show that G(X) > f(X) logX log logX log log log logX (log log logX) , where f(X) is a function tending to infinity with X . Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the e...