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...

Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L²-norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.

MSC 2010: 11B83, 05A19, 33C45This paper is dealing with the Hankel determinants of the special number sequences given in an integral form. We show that these sequences satisfy a generalized convolution property and the Hankel determinants have the generalized Somos-4 property. Here, we recognize well known number sequences such as: the Fibonacci, Catalan, Motzkin and SchrÄoder sequences, like special cases.

If conditional independence constraints define a family of positive distributions that is log-convex then this family turns out to be a Markov model over an undirected graph. This is proved for the distributions on products of finite sets and for the regular Gaussian ones. As a consequence, the assertion known as Brook factorization theorem, Hammersley–Clifford theorem or Gibbs–Markov equivalence is obtained.

A set $𝒮=\{x_1,...,x_n\}$ of $n$ distinct positive integers is said to be gcd-closed if $(x_i,x_j)\in 𝒮$ for all $1\le i,j\le n$. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k\left(t\right)$ depending only on $t$, such that if $n\le k\left(t\right)$, then the power LCM matrix $\left({[x_i,x_j]}^t\right)$ defined on any gcd-closed set $𝒮=\{x_1,...,x_n\}$ is nonsingular, but for $n\ge k\left(t\right)+1$, there exists a gcd-closed set $𝒮=\{x_1,...,x_n\}$ such that the power LCM matrix $\left({[x_i,x_j]}^t\right)$ on $𝒮$ is singular. In 1996, Hong proved $k\left(1\right)=7$ and noted $k\left(t\right)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate...

We study integral similitude 3 × 3-matrices and those positive integers which occur as products of their row elements, when matrices are symmetric with the same numbers in each row. It turns out that integers for which nontrivial matrices of this type exist define elliptic curves of nonzero rank and are closely related to generalized cubic Fermat equations.

For p ≡ 1 (mod 4), we prove the formula (conjectured by R. Chapman) for the determinant of the (p+1)/2 × (p+1)/2 matrix $C=\left(C_{ij}\right)$ with $C_{ij}=((j-i)/p)$.

Let $S=\{x_1,\cdots ,x_n\}$ be a set of $n$ distinct positive integers and $e\ge 1$ an integer. Denote the $n\times n$ 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 $\left({(x_i,x_j)}^e\right)$ (resp. $\left({[x_i,x_j]}^e\right))$. We call the set $S$ an odd gcd closed (resp. odd lcm closed) set if every element in $S$ is an odd number and $(x_i,x_j)\in S$ (resp. $[x_i,x_j]\in S$) for all $1\le i,j\le n$. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that...