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

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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

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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...