Displaying similar documents to “Determinants of matrices associated with incidence functions on posets”

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...

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...

M -estimation in nonlinear regression for longitudinal data

Martina Orsáková (2007)

Kybernetika

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The longitudinal regression model Z i j = m ( θ 0 , 𝕏 i ( T i j ) ) + ε i j , where Z i j is the j th measurement of the i th subject at random time T i j , m is the regression function, 𝕏 i ( T i j ) is a predictable covariate process observed at time T i j and ε i j is a noise, is studied in marked point process framework. In this paper we introduce the assumptions which guarantee the consistency and asymptotic normality of smooth M -estimator of unknown parameter θ 0 .

Inequality-sum : a global constraint capturing the objective function

Jean-Charles Régin, Michel Rueher (2005)

RAIRO - Operations Research - Recherche Opérationnelle

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This paper introduces a new method to prune the domains of the variables in constrained optimization problems where the objective function is defined by a sum y = Σ x i , and where the integer variables x i are subject to difference constraints of the form x j - x i ≤ c . An important application area where such problems occur is deterministic scheduling with the mean flow time as optimality criteria. This new constraint is also more general than a sum constraint defined on a set of ordered variables. Classical...