In this dissertation we look at different two models of sorting algorithms based on divideand-conquer algorithms. Quicksort algorithm, sort an unsorted array of n distinct elements. Partial Quicksort sorts the l smallest elements in a list of length n. Both stochastic divide-and-conquer algorithms are widely studied. Our algorithm Quicksort on the fly provides online the first smallest, then se...

Let the sample correlation matrix be W = Y Y T , where Y = (yij)p,n with yij = xij/ √∑n j=1 x 2 ij . We assume {xij : 1 ≤ i ≤ p, 1 ≤ j ≤ n} to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any i, we assume xij , 1 ≤ j ≤ n to be identically distributed. We assume 0 < p < n and p/n → y with some y ∈ (0, 1) as p, n → ∞. In this ...

Let X X be a Kähler manifold which is fibered over complex Y"> Y encoding="application/x-tex">Y such ...

We develop a variety of new techniques to treat Diophantine equations the shape x 2 + D = y</mml...