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1
Content available remote

Gcd-closed sets and determinants of matrices associated with arithmetical functions

100%
Hong S.
Acta Arithmetica
|
2002
|
tom 101
|
nr 4
321-332
2
Content available remote

Factorization of matrices associated with classes of arithmetical functions

100%
Hong S.
Colloquium Mathematicum
|
2003
|
tom 98
|
nr 1
113-123
EN
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 $x_{i}$ and $x_{i}$ as its i,j-entry divides the matrix $(f[x_{i}, x_{i}])$ having f evaluated at the least common multiple $[x_{i}, x_{i}]$ of $x_{i}$ and $x_{i}$ as its i,j-entry in the ring Mₙ(ℤ) of n × n matrices over the integers. But such a factorization is no longer true if f is multiplicative.
3
Content available remote

Notes on power LCM matrices

88%
Hong S.
Acta Arithmetica
|
2004
|
tom 111
|
nr 2
165-177
4
Content available remote

Infinite divisibility of Smith matrices

75%
Hong S.
Acta Arithmetica
|
2008
|
tom 134
|
nr 4
381-386
5
Content available remote

Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

51%
Hong S., Hu S., Hong S.
Open Mathematics
|
2016
|
tom 14
|
nr 1
146-155
EN
Let f be an arithmetic function and S = {x1, …, xn} be a set of n distinct positive integers. By (f(xi, xj)) (resp. (f[xi, xj])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) (resp. the least common multiple [xi, xj]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi, xj)) and (f[xi, xj]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S1, …, Sk with k ≥ 1 being an integer and S1, …, Sk being gcd-closed sets such that (lcm(Si), lcm(Sj)) = 1 for all 1 ≤ i ≠ j ≤ k). This extends the Bourque-Ligh, Hong’s and the Hong-Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith’s determinant.
6
Content available remote

Divisibility properties of Smith matrices

51%
Hong S., Zhao J., Yin Y.
Acta Arithmetica
|
2008
|
tom 132
|
nr 2
161-175
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