We study how the typical gradient and height of a random surface are modified by addition quenched disorder in form independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, following cases. It is shown that for real-valued random-field surfaces $$\nabla \phi $$ type with uniformly convex interaction potential: (i) delocalizes dimensions $$1\l...

in this paper, stochastic generalizations of some fixed point for operators satisfying random contractively generalized hybrid and some other contractive condition have been proved. we discuss also the existence of a solution to a nonlinear random integral equation in banah spaces.

In a recent paper, Oliver Riordan shows that for and p up to slightly larger than the threshold Kr-factor, hypergraph formed by copies of Kr in G(n, p) contains copy binomial random with . For r = 3, he gives weaker result where density is reduced constant factor. Recently, Jeff Kahn announced an asymptotically sharp bound Shamir's matching problem all With Riordan's result, this immediately im...