Displaying similar documents to “On Hong’s conjecture for power LCM matrices”

On the divisibility of power LCM matrices by power GCD matrices

Jian Rong Zhao, Shaofang Hong, Qunying Liao, Kar-Ping Shum (2007)

Czechoslovak Mathematical Journal

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Let S = { x 1 , , x n } be a set of n distinct positive integers and e 1 an integer. Denote the n × n power GCD (resp. power LCM) matrix on S having the e -th power of the greatest common divisor ( x i , x j ) (resp. the e -th power of the least common multiple [ x i , x j ] ) as the ( i , j ) -entry of the matrix by ( ( x i , x j ) e ) (resp. ( [ x i , x j ] e ) ) . We call the set S an odd gcd closed (resp. odd lcm closed) set if every element in S is an odd number and ( x i , x j ) S (resp. [ x i , x j ] S ) for all 1 i , j n . In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in...

An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices

Assaf Goldberger, Neumann, Michael (2004)

Czechoslovak Mathematical Journal

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Suppose that A is an n × n nonnegative matrix whose eigenvalues are λ = ρ ( A ) , λ 2 , ... , λ n . Fiedler and others have shown that det ( λ I - A ) λ n - ρ n , for all λ > ρ , with equality for any such λ if and only if A is the simple cycle matrix. Let a i be the signed sum of the determinants of the principal submatrices of A of order i × i , i = 1 , ... , n - 1 . We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: det ( λ I - A ) + i = 1 n - 1 ρ n - 2 i | a i | ( λ - ρ ) i λ n - ρ n , for all λ ρ . We use this inequality to derive the inequality that: 2 n ( ρ - λ i ) ρ n - 2 i = 2 n ( ρ - λ i ) . In the spirit of a celebrated conjecture...

Comparison of order statistics in a random sequence to the same statistics with I.I.D. variables

Jean-Louis Bon, Eugen Păltănea (2006)

ESAIM: Probability and Statistics

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The paper is motivated by the stochastic comparison of the reliability of non-repairable k -out-of- n systems. The lifetime of such a system with nonidentical components is compared with the lifetime of a system with identical components. Formally the problem is as follows. Let U i , i = 1 , . . . , n , be positive independent random variables with common distribution F . For λ i > 0 and μ > 0 , let consider X i = U i / λ i and Y i = U i / μ , i = 1 , . . . , n . Remark that this is no more than a change of scale for each term. For k { 1 , 2 , . . . , n } , let us define X k : n to be the k th order...

Factorization of matrices associated with classes of arithmetical functions

Shaofang Hong (2003)

Colloquium Mathematicae

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Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix ( f ( x i , x i ) ) having f evaluated at the greatest common divisor ( x i , x i ) of...

Generalized induced norms

S. Hejazian, M. Mirzavaziri, Mohammad Sal Moslehian (2007)

Czechoslovak Mathematical Journal

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Let · be a norm on the algebra n of all n × n matrices over . An interesting problem in matrix theory is that “Are there two norms · 1 and · 2 on n such that A = max { A x 2 x 1 = 1 } for all A n ?” We will investigate this problem and its various aspects and will discuss some conditions under which · 1 = · 2 .