95 research outputs found

    Inertia, positive definiteness and â„“p\ell_p norm of GCD and LCM matrices and their unitary analogs

    Get PDF
    Let S={x1,x2,…,xn}S=\{x_1,x_2,\dots,x_n\} be a set of distinct positive integers, and let ff be an arithmetical function. The GCD matrix (S)f(S)_f on SS associated with ff is defined as the n×nn\times n matrix having ff evaluated at the greatest common divisor of xix_i and xjx_j as its ijij entry. The LCM matrix [S]f[S]_f is defined similarly. We consider inertia, positive definiteness and ℓp\ell_p norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions

    Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

    Get PDF
    Let m,n≥2m,n\ge 2, m≤nm\le n. It is well-known that the number of (two-dimensional) threshold functions on an m×nm\times n rectangular grid is {eqnarray*} t(m,n)=\frac{6}{\pi^2}(mn)^2+O(m^2n\log{n})+O(mn^2\log{\log{n}})= \frac{6}{\pi^2}(mn)^2+O(mn^2\log{m}). {eqnarray*} We improve the error term by showing that t(m,n)=\frac{6}{\pi^2}(mn)^2+O(mn^2). $
    • …
    corecore