Let A and B be complex matrices of same dimension. Given their eigen-values and singular values, we survey and further develop simple inequalities for eigenvalues and singular values of A + B, AB, and A • B. Here • denotes the Hadamard product. As corollaries, we find inequalities for additive and multi-plicative spreads of these matrices.
The incorporation of nonnegativity constraints in image reconstruction problems is known to have a stabilizing effect on solution methods. In this paper, we both demonstrate and provide an explanation of this phenomena when the image reconstruction problem of interest has least squares form. The benefits of using this natural constraint suggest the… (More)
Let l(n) be the number of lines through at least two points of an n × n rectangular grid. We prove recursive and asymptotic formulas for it using respectively combinatorial and number theoretic methods. We also study the ratio l(n)/l(n−1). All this originates from Mustonen's experimental results.
Let m, n ≥ 2, m ≤ n. It is well-known that the number of (two-dimensional) threshold functions on an m × n rectangular grid is t(m, n) = 6 π 2 (mn) 2 + O(m 2 n log n) + O(mn 2 log log n) = 6 π 2 (mn) 2 + O(mn 2 log m). We improve the error term by showing that t(m, n) = 6 π 2 (mn) 2 + O(mn 2).
We present an asymptotic formula for the number of line segments connecting q + 1 points of an n × n square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case q = 2 is so generalized.
Given n ∈ Z, its arithmetic derivative n ′ is defined as follows: (i) 0 ′ = 1 ′ = (−1) ′ = 0. (ii) If n = up 1 · · · p k , where u = ±1 and p 1 ,. .. , p k are primes (some of them possibly equal), then n ′ = n k j=1 1 p j = u k j=1 p 1 · · · p j−1 p j+1 · · · p k. An analogous definition can be given in any unique factorization domain. What about the… (More)
Let A be a nonnegative n × n matrix with row sums r