Let X be a reflexive Banach space such that for any x 6= 0 the set {x∗ ∈ X∗ : ‖x∗‖ = 1 and x∗(x) = ‖x‖} is compact. We prove that any unrestricted product of of a finite number of (W ) contractions on X converges weakly. Recall a (bounded) linear operator on a Banach space X is said to be contraction if ‖Tx‖ ≤ ‖x‖ for all x ∈ X . T is said to satisfy condition (W ) if {xn} is bounded and ‖xn‖ −...

Let Pn,r(x) be the generalized weighted power means. We consider bounds for the differences of means in the following form: max { Cu,v,β x 1 , Cu,v,β x n } σn,w′,β ≥ P n,u − P n,v α ≥ min { Cu,v,β x 1 , Cu,v,β x n } σn,w,β . Here β 6= 0, σn,t,β( x) = ∑n i=1 ωi[x β i −P β n,t(x)] andCu,v,β = u−v 2β2 . Some similar inequalities are also considered. The results are applied to inequalities of Ky Fa...

Let X = {Xij : 1 ≤ i, j ≤ n} be an n× n array of independent random variables where n ≥ 2. Let π be a uniform random permutation of {1,2, . . . , n}, independent of X, and let W =∑ni=1 Xiπ(i). Suppose X is standardized so that EW = 0,Var(W)= 1. We prove that the Kolmogorov distance between the distribution of W and the standard normal distribution is bounded by 451 ∑n i,j=1 E|Xij |3/n. Our appr...

Let X $X$ be a compact Kähler manifold of dimension n $n$ and ω $\omega$ form on . We consider the complex Monge–Ampère equation ( d c u + ) = μ $({dd^c}u+\omega )^n=\mu$ , where $\mu$ is given positive measure suitable mass $u$ an -plurisubharmonic function. show that admits Hölder continuous solution if only seen as functional Sobolev space W ∗ $W^*(X)$ continuous. A similar result also obtai...

The main result of the present paper is the sharpening of rationality w x w x w x theorems of Manning 23 , Fried 18 , and Gromov 19 , with an application to hyperbolic groups. The basic concept we shall use is that of Nrationality. A series is called N-rational if it is obtained from polynomials over N in Ž the variable t by applying the following operations: sum, product, star by definition th...

Proof: The proof of this lemma uses a technique called as Padding which has has its roots in complexity theory. Let fk : {0, 1} → {0, 1} be one-way. From the property of one-way functions (efficient evaluation) there exists a specific polynomial p(·) such that for all k ∈ N, fk can be evaluated in time p(n(k)). We now define a function gk : {0, 1} → {0, 1}m(k)+p(n(k))−n(k) such that gk(x||w) = ...

We establish the long time existence of complete non-compact weakly convex and smooth hypersurfaces $$\Sigma _t$$ evolving by $$Q_k$$ -flow. show that maximum T depends on dimension $$d_W$$ vector space $$W{:}{=}\{w \in \mathbb {R}^{n+1}: \sup _{X\in \Sigma _0} |\langle X,w\rangle | = +\infty \}$$ which contains each direction in our initial data _0$$ is infinite. If $$d_W=\text {dim}(W) \ge n-...

Grzegorczyk’s modal logic (Grz ) corresponds to the class of upwards well-founded partially ordered Kripke frames, however all known proofs of this fact utilize some form of the Axiom of Choice; G. Boolos asked in [1], whether it is provable in plain ZF . We answer his question negatively: Grz corresponds (in ZF ) to a class of frames, which does not provably coincide with upwards well-founded ...

Let w be a word in the free group of rank n ∈ N and let V(w) be the variety of groups defined by the law w = 1. Define V(w∗) to be the class of all groups G in which for any infinite subsets X1, . . . , Xn there exist xi ∈ Xi, 1 ≤ i ≤ n, such that w(x1, . . . , xn) = 1. Clearly, V(w) ∪ F ⊆ V(w ∗); F being the class of finite groups. In this paper, we investigate some words w and some certain cl...