On Extremal Set Partitions in Cartesian Product Spaces

On Extremal Set Partitions in Cartesian Product Spaces

On Set Coverings in Cartesian Product Spaces

Appendix: On Set Coverings in Cartesian Product Spaces

Consider (X, E), where X is a finite set and E is a system of subsets whose union equals X. For every natural number n ∈ N define the cartesian products Xn = ∏n 1 X and En = ∏n 1 E . The following problem is investigated: how many sets of En are needed to cover Xn? Let this number be denoted by c(n). It is proved that for all n ∈ N exp{C · n} ≤ c(n) ≤ exp{Cn+ log n+ log log |X|}+ 1. A formula f...

Data dependence theorems for operators on cartesian product spaces

We present some abstract data dependence theorems of the fixed point set for operators f, g : X × Y → X × Y , using the c-Picard operators technique. 2000 Mathematical Subject Classification: 47H10.

Lower Bounds on Same-Set Inner Product in Correlated Spaces

Let P be a probability distribution over a finite alphabet Ω with all ` marginals equal. Let X(1), . . . , X(`), X(j) = (X 1 , . . . , X (j) n ) be random vectors such that for every coordinate i ∈ [n] the tuples (X i , . . . , X (`) i ) are i.i.d. according to P. The question we address is: does there exist a function cP() independent of n such that for every f : Ω → [0, 1] with E[f(X(1))] = μ...