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...

Global behavior of a third order rational difference equation

Raafat Abo-Zeid (2014)

Mathematica Bohemica

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In this paper, we determine the forbidden set and give an explicit formula for the solutions of the difference equation x n + 1 = a x n x n - 1 - b x n + c x n - 2 , n ∈ ℕ 0 where a , b , c are positive real numbers and the initial conditions x - 2 , x - 1 , x 0 are real numbers. We show that every admissible solution of that equation converges to zero if either a < c or a > c with ( a - c ) / b < 1 . When a > c with ( a - c ) / b > 1 , we prove that every admissible solution is unbounded. Finally, when a = c , we prove that every admissible solution converges to zero.

A note on k-c-semistratifiable spaces and strong β -spaces

Li-Xia Wang, Liang-Xue Peng (2011)

Mathematica Bohemica

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Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are G δ -sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T 2 -space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x ∈ X , { x } = ⋂ { g ( x , n ) : n ∈ ℕ } and g ( x , n + 1 ) ⊆ g ( x , n ) for each...