Generalization of GCD matrices
- PDF / 1,620,105 Bytes
- 7 Pages / 595.276 x 790.866 pts Page_size
- 93 Downloads / 217 Views
SPECIAL ISSUE
Generalization of GCD matrices Haiqing Han1 · Qin Li2 · Yi Wen1 · Shuang Wen1 · Jie Li1 Received: 11 November 2019 / Revised: 7 June 2020 / Accepted: 27 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Special matrices are widely used in information society. The gcd-matrices have be conducted to study over Descartes direct-product of some finite positive integer sets. If Descartes direct-product S = S1 × S2 × ⋯ × Sn{ with n finite}positive integer sets as direct product terms, then S is finite too. Without loss of generality, set S = d1 , d2 , … , dt , and ∏n ∀a = (a1 , a2 , … , an ), b = (b1 , b2 , … , bn ) ∈ S , the general greatest common factor is defined as gcd(a, b) = i=1 gcd(ai , bi ) . And create a square matrix ⟨S⟩ = (sij )t×t = (gcd(di , dj ))t×t possessed the general greatest common factors gcd(di , dj ) as arrays sij = gcd(di , dj ) . We have researched upper bound and lower bound of the determinant det ⟨S⟩ of the t × t gcd-matrix ⟨S⟩ , and compute the determinant’s value under special or specific conditions in the article. At last, some well results about the gcd-matrix has been extend from Descartes direct-product of some finite positive integer sets to general direct product of the posets. Keywords Greatest common divisor matrix · Descartes direct-product · Meet semilattice · Meet matrices · Generalized Euler’s φ-function · Mobius inversion
1 Introduction The gcd-matrix ⟨S⟩ standing for the greatest common divisor matrix is a special type of matrix that the}arrays come { from a positive integers set S = a1 , a2 , … , an . The nature of its determinants det ⟨S⟩ has increasingly become a hot topics of scientific research in many application fields. The authors such as S. Beslin, S. Ligh etc. have taken the lead in giving a definition of the greatest common factor matrix [4, 18, 19, 22, 23] (that { is gcd-matrix } ⟨S⟩ ) over the finite natural number set S = a1 , a2 , … , an . Set the square matrix ⟨S⟩ = (sij )n×n with the arrays sij = gcd(ai , aj ) , the array sij = gcd(ai , aj ) is the greatest common factor between ai and aj . The greatest common factor (or greatest common divisor)is abbreviated to GCD. And that naturally gives rise to the concept of a gcd-matrix ⟨S⟩ . Z. Li has pointed * Qin Li [email protected] Haiqing Han [email protected] 1
School of Mathematics and Physics, Hubei Polytechnic Univ, Huangshi 435003, Hubei, China
Normal Department, Hubei Polytechnic Univ, Huangshi 435003, Hubei, China
2
out det ⟨S⟩ = 𝜙(a1 )𝜙(a2 ) ⋯ 𝜙(an ) in [14, 26], where S is the FC set and 𝜙 is the Euler’s totient function. While the set is said a FC set, if the all positive integer factors of a ∈ S are contained in the set S . Namely, if ∀a ∈ S and d|a then d ∈ S . The matrix ⟨S⟩g has been defined as ⟨S⟩g = (sij )n×n ,sij = g((ai , aj )) (i, j = 1, 2, … , n) and g((ai , aj )) refers to the value of the arithmetic function g on the greatest common factor gcd(ai , aj ) [1, 2, 7, 8, 12, 13, 21]. The following conclusions are proved by H.J.S. S
Data Loading...
