Pentti Haukkanen

On Smith's Determinant

Abstract We give a brief review of papers relating to Smiths determinant and point out a common structure that can be found in many extensions and analogues of Smiths determinant. We present the common structure in the language of posets. We also make an investigation on a conjecture of Beslin and Ligh on greatest common divisor (GCD) matrices in the sense of meet matrices and give characterizations of the posets satisfying the conjecture. Further, we give a counterexample for the conjecture of Bourque and Ligh that the least common multiple matrix on any GCD-closed set is invertible.

On meet and join matrices associated with incidence functions

We study recently meet matrices on meet-semilattices as an abstract generalization of greatest common divisor (GCD) matrices. Analogously, in this paper we consider join matrices on lattices as an abstract generalization of least common multiple (LCM) matrices. A formula for the determinant of join matrices on join-closed sets, bounds for the determinant of join matrices on all sets and a formula for the inverse of join matrices on join-closed sets are given. The concept of a semi-multiplicative function gives us formulae for meet matrices on join-closed sets and join matrices on meet-closed sets. Finally, we show what new the study of meet and join matrices contributes to the usual GCD and LCM matrices.

On meet matrices on posets

Abstract We consider meet matrices on posets as an abstract generalization of greatest common divisor (GCD) matrices. Some of the most important properties of GCD matrices are presented in terms of meet matrices.

Higher-dimensional gcd matrices

Abstract We evaluate the higher-dimensional determinants of the greatest-common-divisor matrix defined on a gcd-closed set and on an arbitrary set of distinct positive integers.

Some analogues of smith's determinant

We calculate the determinants of the greatest common divisor (GCD) and the least common multiple (LCM) matrices associated with an arithmetical function on gcd-closed and lcm-closed sets. We also consider some unitary analogues of these determinants.

Bounds for determinants of meet matrices associated with incidence functions

Abstract We consider meet matrices on meet-semilattices as an abstract generalization of greatest common divisor (gcd) matrices. Some new bounds for the determinant of meet matrices and a formula for the inverse of meet matrices are given.

Some characterizations of totients

An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Eulers phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we present several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.

On the positive definiteness and eigenvalues of meet and join matrices

In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set

Notes on the divisibility of GCD and LCM Matrices

S_n

Some characterizations of specially multiplicative functions

is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix