Divisibility of matrices associated with multiplicative functions

Let S={x₁,…,x_n} be a set of n distinct positive integers. For x,y∈S and y<x, we say the y is a greatest-type divisor of x in S if y∣x and it can be deduced that z=y from y∣z,z∣x,z<x and z∈S. For x∈S, let G_S(x) denote the set of all greatest-type divisors of x in S. For any arithmetic function f, let (f(x_i,x_j)) denote the n×n matrix having f evaluated at the greatest common divisor (x_i,x_j) of x_i and x_j as its i,j-entry and let (f[x_i,x_j]) denote the n×n matrix having f evaluated at the least common multiple [x_i,x_j] of x_i and x_j as its i,j-entry. In this paper, we assume that S is a gcd-closed set and . We show that if f is a multiplicative function such that (f∗μ)(d)∈Z whenever and f(a)|f(b) whenever a|b and a,b∈S and (f(x_i,x_j)) is nonsingular, then the matrix (f(x_i,x_j)) divides the matrix (f[x_i,x_j]) in the ring M_n(Z) of n×n matrices over the integers. As a consequence, we show that (f(x_i,x_j)) divides (f[x_i,x_j]) in the ring M_n(Z) if (f∗μ)(d)∈Z whenever and f is a completely multiplicative function such that (f(x_i,x_j)) is nonsingular. This confirms a conjecture of Hong raised in 2004.     

Nonsingularity of least common multiple matrices on gcd-closed sets

Let n be a positive integer. Let S={x₁,…,x_n} be a set of n distinct positive integers. The least common multiple (LCM) matrix on S, denoted by [S], is defined to be the n×n matrix whose (i,j)-entry is the least common multiple [x_i,x_j] of x_i and x_j. The set S is said to be gcd-closed if for any x_i,x_j∈S,(x_i,x_j)∈S. For an integer m>1, let ω(m) denote the number of distinct prime factors of m. Define ω(1)=0. In 1997, Qi Sun conjectured that if S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. In this paper, we settle completely Sun's conjecture. We show the following result: (i). If S is a gcd-closed set satisfying max_x∈S{ω(x)}?2, then the LCM matrix [S] is nonsingular. Namely, Sun's conjecture is true; (ii). For each integer r?3, there exists a gcd-closed set S satisfying max_x∈S{ω(x)}=r, such that the LCM matrix [S] is singular.     

On Hong’s conjecture for power LCM matrices

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 to calculate the determinant of the LCM matrix on a gcd-closed set and proves that k(t) ⩾ 8 for all t ⩾ 2. We further prove that k(t) ⩾ 9 iff a special Diophantine equation, which we call the LCM equation, has no t-th power solution and conjecture that k(t) = 8 for all t ⩾ 2, namely, the LCM equation has t-th power solution for all t ⩾ 2.     

Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets

Let e and n be positive integers and S={x₁,…,x_n} a set of n distinct positive integers. For x∈S, define . The n×n matrix whose (i,j)-entry is the eth power (x_i,x_j)^e of the greatest common divisor of x_i and x_j is called the eth power GCD matrix on S, denoted by (S^e). Similarly we can define the eth power LCM matrix [S^e]. Bourque and Ligh showed that (S)∣[S] holds in the ring of n×n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with |S|≤3, (S)∣[S]. Meanwhile Hong proved that there is a gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S)?[S]. In this paper, we introduce a new method to study systematically the divisibility for the case max_x∈S{|G_S(x)|}≤2. We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S^e)|[S^e]. This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.     

Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets

Let S = {x₁, … , x_n} be a set of n distinct positive integers and f be an arithmetical function. Let [f(x_i, x_j)] denote the n × n matrix having f evaluated at the greatest common divisor (x_i, x_j) of x_i and x_j as its i, j-entry and (f[x_i, x_j]) denote the n × n matrix having f evaluated at the least common multiple [x_i, x_j] of x_i and x_j as its i, j-entry. The set S is said to be lcm-closed if [x_i, x_j] ∈ S for all 1 ? i, j ? n. For an integer x > 1, let ω(x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = {x₁, … , x_n} is an lcm-closed set satisfying , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying (resp. f(p) ? p) for any prime p, then the matrix [f(x_i, x_j)] (resp. (f[x_i, x_j])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1-14], we also obtain reduced formulas for det(f(x_i, x_j)) and det(f[x_i, x_j]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices.     

Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains

Let h be a positive integer and S?=?{x ₁,?…?,?x _h } be a set of h distinct positive integers. We say that the set S is a divisor chain if x _σ(1) ∣?…?∣ x _σ(h) for a permutation σ of {1,?…?,?h}. If the set S can be partitioned as S?=?S ₁?∪?S ₂?∪?S ₃, where S ₁, S ₂ and S ₃ are divisor chains and each element of S _i is coprime to each element of S _j for all 1?≤?i?<?j?≤?3, then we say that the set S consists of three coprime divisor chains. The matrix having the ath power (x _i , x _j ) ^a of the greatest common divisor of x _i and x _j as its i, j-entry is called the ath power greatest common divison (GCD) matrix on S, denoted by (S ^?a ). The ath power least common multiple (LCM) matrix [S ^?a ] can be defined similarly. In this article, let a and b be positive integers and let S consist of three coprime divisor chains with 1?∈?S. We show that if a?∣?b, then the ath power GCD matrix (S ^?a ) (resp., the ath power LCM matrix [S ^?a ]) divides the bth power GCD matrix (S ^?b ) (resp., the bth power LCM matrix [S ^?b ]) in the ring M _h (Z) of h?×?h matrices over integers. We also show that the ath power GCD matrix (S ^?a ) divides the bth power LCM matrix [S ^?b ] in the ring M _h (Z) if a?∣?b. However, if a???b, then such factorizations are not true. Our results extend Hong's and Tan's theorems and also provide further evidences to the conjectures of Hong raised in 2008.     

On the divisibility of power LCM matrices by power GCD matrices

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 2004 that for any integer e ⩾ 1, the n × n power GCD matrix ((x _i , x _j ) ^e ) defined on an odd-gcd-closed (resp. odd-lcm-closed) set S divides the n × n power LCM matrix ([x _i , x _j ] ^e ) defined on S in the ring M _n (ℤ) of n × n matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for n ⩽ 3 but they are both not true for n ⩾ 4. Research is partially supported by Program for New Century Excellent Talents in University, by SRF for ROCS, SEM, China and by the Lady Davis Fellowship at the Technion, Israel Research is partially supported by a UGC (HK) grant 2160210 (2003/05).     

A note on certain functional determinants

Summary We consider the problem when a scalar function ofn variables can be represented in the form of a determinant det(f _i (x _j )), the so-called Casorati determinant off ₁,f ₂,,f _n . The result is applied to the solution of some functional equations with unknown functionsH of two variables that involve determinants det(H(x _i ,x _j )).     

Divisibility properties of power GCD matrices and power LCM matrices

Let a,b and n be positive integers and the set S={x₁,…,x_n} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that x_σ(1)|…|x_σ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (S^a) having the ath power (x_i,x_j)^a of the greatest common divisor of x_i and x_j as its i,j-entry divides the bth power GCD matrix (S^b) in the ring M_n(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (S^a) does not divide the bth power GCD matrix (S^b) in the ring M_n(Z). Similar results are also established for the power LCM matrices.     

Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains

Let a, n ? 1 be integers and S = {x₁, … , x_n} be a set of n distinct positive integers. The matrix having the ath power (x_i, x_j)^a of the greatest common divisor of x_i and x_j as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (S^a). Similarly we can define the ath power LCM matrix [S^a]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S₁ ∪ ? ∪ S_k, where k ? 1 is an integer and all S_i (1 ? i ? k) are divisor chains such that (max(S_i), max(S_j)) = gcd(S) for 1 ? i ≠ j ? k. In this paper, we first obtain formulae of determinants of power GCD matrices (S^a) and power LCM matrices [S^a] on the set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, we then show that det(S^a)∣det(S^b), det[S^a]∣det[S^b] and det(S^a)∣det[S^b] if a∣b and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

The Weak Convergence for Functions of Negatively Associated Random Variables

Let {X_n, n1} be a sequence of stationary negatively associated random variables, S_j(l)=∑^l_i=1 X_j+i, S_n=∑ⁿ_i=1 X_i. Suppose that f(x) is a real function. Under some suitable conditions, the central limit theorem and the weak convergence for sums are investigated. Applications to limiting distributions of estimators of Var S_n are also discussed.     

The <Emphasis Type="Italic">f</Emphasis>-Vector of the Descent Polytope

For a positive integer n and a subset S⊆[n−1], the descent polytope DP  _S is the set of points (x ₁,…,x _n ) in the n-dimensional unit cube [0,1] ⁿ such that x _i ≥x _i+1 if i∈S and x _i ≤x _i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F _S (t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.     

Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Let f,g_i,i=1,…,l,h_j,j=1,…,m, be polynomials on Rⁿ and S?{x∈Rⁿ∣g_i(x)=0,i=1,…,l,h_j(x)≥0,j=1,…,m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.     

A determinantal description of GCD-closed sets and k-sets

Let S={x ₁,x ₂,…,x_n } be a naturally ordered set of distinct positive integers. S is called a k-set if k= gcd(x_i ,x_j ) for x_i ≠x_j any in S. In this paper k-sets are characterized by certain conditions on the determinants of some matrices associated with S.     

行列式detVn-(x1,...,xn)的准确值及其应用

本文给出行列式Ｖ_n^-(x₁,...,x_n)的准确值，它是通常的Ｖａｎｄｅｒｍｏｄｅ行列式计算公式的推广，以及它在理论上的一些重要应用．     

Multiplicative isometries on the Smirnov class

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = [`(f<sup>°</sup> [`(j)] )]\overline {f^\circ \bar \varphi } for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z ₁, ..., z _n ) = (l₁ z_i1 ,...,l_n z_in )(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } ) for |λ _j | = 1, 1 ≤ j ≤ n, and (i ₁; ..., i _n )is some permutation of the integers from 1through n in the case of the n-dimensional polydisk.     

A note on the stability of Jordan triple higher ring derivations

In the present paper, we establish the stability and the superstability of a functional inequality corresponding to the functional equation fn（xyx） = ∑i＋j＋k=n fi（x）fj （y）fk（x）. In addition, we take account of the problem of Jacobson radical ranges for such functional inequality.     

A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.