Displaying similar documents to “On total incomparability of mixed Tsirelson spaces”

On the classes of hereditarily ℓ p Banach spaces

Parviz Azimi, A. A. Ledari (2006)

Czechoslovak Mathematical Journal

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Let X denote a specific space of the class of X α , p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ℓ p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of ℓ p . It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of ℓ q where 1 p + 1 q = 1 . For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of...

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...