The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Displaying similar documents to “Optimal sublinear inequalities involving geometric and power means”

Strong tightness as a condition of weak and almost sure convergence

Grzegorz Krupa, Wiesław Zieba (1996)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A sequence of random elements { X j , j ∈ J } is called strongly tight if for an arbitrary ϵ > 0 there exists a compact set K such that P ⋂ j ∈ J [ X j ∈ K ] > 1 - ϵ . For the Polish space valued sequences of random elements we show that almost sure convergence of { X n } as well as weak convergence of randomly indexed sequence { X τ } assure strong tightness of { X n , n ∈ ℕ } . For L 1 bounded Banach space valued asymptotic martingales strong tightness also turns out to the sufficient condition of convergence. A sequence of r.e. { X n , n ∈ ℕ } is said to converge essentially...

On Hong’s conjecture for power LCM matrices

Wei Cao (2007)

Czechoslovak Mathematical Journal

Similarity:

A set 𝒮 = { x 1 , ... , x n } of n distinct positive integers is said to be gcd-closed if ( x i , x j ) ∈ 𝒮 for all 1 ≤ i , j ≤ n . Shaofang Hong conjectured in 2002 that for a given positive integer t there is a positive integer k ( t ) depending only on t , such that if n ≤ k ( t ) , then the power LCM matrix ( [ x i , x j ] t ) defined on any gcd-closed set 𝒮 = { x 1 , ... , x n } is nonsingular, but for n ≥ k ( t ) + 1 , there exists a gcd-closed set 𝒮 = { x 1 , ... , x n } such that the power LCM matrix ( [ x i , x j ] t ) on 𝒮 is singular. In 1996, Hong proved k ( 1 ) = 7 and noted k ( t ) ≥ 7 for all t ≥ 2 . This paper develops Hong’s method and provides a new idea...