let and be a sequence with non-negative entries. if , denote by the infimum of those satisfying the following inequality: whenever . the purpose of this paper is to give an upper bound for the norm of operator t on weighted sequence spaces d(w,p) and lp(w) and also e(w,?). we considered this problem for certain matrix operators such as norlund, weighted mean, ceasaro and copson matrices. th...

We prove that for the mean curvature flow of closed embedded hypersurfaces, intrinsic diameter stays uniformly bounded as approaches first singular time, provided all singularities are neck or conical type. In particular, assuming Ilmanen’s multiplicity one conjecture and no cylinder conjecture, we conclude in two-dimensional case always bounded. also obtain sharp $$L^{n-1}$$ bound curvature. T...

In this paper we study the average distance in weighted graphs. More precisely, we consider assignments of families of non-negative weights to the edges. The aim is to maximise (minimise, respectively) the average distance in the resulting weighted graph. Two variants of the problem are considered depending on whether the collection of weights is fixed or not. The main results of this paper are...