On the L₂-discrepancy for anchored boxes

For a family of subsets S of a finite set V, a coloring @g:V->{-1,1}, and S"j@?S, let @g(S"j)=@?"v"@?"S"""j@g(v). We consider the problem to minimize @?"j@g(S"j)^2 and we call the problem L"2-discrepancy problem. We show that the problem is NP-complete, ...

One measure of the "goodness" of a quadrature rule Q for approximating integrals over the unit hypercube is the L₂ discrepancy D(Q). For number-theoretic rules of prime order n we derive an expression for the average of D²(Q). Having this average makes ...

Let I be a protoset of d-dimensional polyominoes. Which boxes (rectangular parallelepipeds) can be tiled by I? A nice result of Klarner and Göbel asserts that the answer to this question can always be given in a particularly simple form, namely, by ...