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Integer matrices related to Liouville's function

Shea-Ming Oon (2013)

Czechoslovak Mathematical Journal

In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville's function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion...

Lower bounds for the largest eigenvalue of the gcd matrix on { 1 , 2 , ⋯ , n }

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

Consider the n × n matrix with ( i , j ) ’th entry gcd ( i , j ) . Its largest eigenvalue λ n and sum of entries s n satisfy λ n > s n / n . Because s n cannot be expressed algebraically as a function of n , we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that λ n > 6 π - 2 n log n for all n . If n is large enough, this follows from F. Balatoni (1969).

Matrix problems and stable homotopy types of polyhedra

Yuriy Drozd (2004)

Open Mathematics

This is a survey of the results on stable homotopy types of polyhedra of small dimensions, mainly obtained by H.-J. Baues and the author [3, 5, 6]. The proofs are based on the technique of matrix problems (bimodule categories).

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

Siao Hong, Shuangnian Hu, Shaofang Hong (2016)

Open Mathematics

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...

Netted matrices.

Stănică, Pantelimon (2003)

International Journal of Mathematics and Mathematical Sciences

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