On the root mean square weighted L2 discrepancy of scrambled nets

On the mean square weighted L2 discrepancy of randomized digital nets in prime base

We study the mean square weighted L2 discrepancy of randomized digital (t,m, s)nets over Zp. The randomization method considered here is a digital shift of depth m, i.e., for each coordinate the first m digits of each point are shifted by the same shift whereas the remaining digits in each coordinate are shifted independently for each point. We also consider a simplified version of this shift. ...

Formulas for the Computation of the Weighted L2 Discrepancy

For a d-dimensional quasi-Monte Carlo method with n points, calculating the weighted L 2 discrepancy using a formula obtained directly from its deenition requires O(2 d?1 dn 2) operations. Here we give an alternative formula requiring O(dn 2) operations. We also present the rst numerical calculations of the weighted L 2 discrepancy. These results give supporting evidence for thèlimiting discrep...

On Weighted Pcm and Mean Square Deviation

On the square root of quadratic matrices

Here we present a new approach to calculating the square root of a quadratic matrix. Actually, the purpose of this article is to show how the Cayley-Hamilton theorem may be used to determine an explicit formula for all the square roots of $2times 2$ matrices.

ON l2 NORMS OF SOME WEIGHTED MEAN MATRICES

k=1 |ak|, in which C = (cj,k) and the parameter p are assumed fixed (p > 1), and the estimate is to hold for all complex sequences a. The lp operator norm of C is then defined as the p-th root of the smallest value of the constant U : ||C||p,p = U 1 p . Hardy’s inequality thus asserts that the Cesáro matrix operator C, given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤ ...