This property may be used to characterize realizers of finite partial orders : A nonempty set of linear extensions is a realizer if and only if it reverses every critical pair.
22.
Note that one formulation of AC is that the Cartesian product of a family of nonempty sets is nonempty; but since the empty set is most certainly compact, the proof cannot proceed along such straightforward lines.
23.
Let \ mathcal { X } be a nonempty set and k a positive-definite real-valued kernel on \ mathcal { X } \ times \ mathcal { X } with corresponding reproducing kernel Hilbert space H _ k.
24.
Let A = \ cup _ i A _ i and B = \ cup _ j B _ j be partitions of disjoint nonempty sets A and B, where all A _ i and B _ j share a common size.
25.
Then TT is a topology on M . It is called the manifold topology induced by AA . A manifold is a nonempty set with an atlas so that the manifold topology is separated, i . e . distinct points have disjoint neighborhoods.
26.
It seems to me that if K = { " B " } where " B " has 2 elements, then PH ( K ) contains only sets whose sizes are powers of 2, whereas HP ( K ) contains all nonempty sets.
27.
If we had a black box that solved FIND-SUBSET-SUM in unit time, then it would be easy to solve SUBSET-SUM . Simply ask the black box to find the subset that sums to zero, then check whether it returned a nonempty set.
28.
They observed ( as did some other contemporary authors ) that the lattice of stable matchings was reminiscent of the conclusion of Tarski s fixed point theorem, which states that an increasing function from a complete lattice to itself has a nonempty set of fixed points that form a complete lattice.
29.
It states that for every indexed family ( S _ i ) _ { i \ in I } of nonempty sets there exists an indexed family ( x _ i ) _ { i \ in I } of elements such that x _ i \ in S _ i for every i \ in I.
30.
Let \ mathcal { X } be a nonempty set, k a positive-definite real-valued kernel on \ mathcal { X } \ times \ mathcal { X } with corresponding reproducing kernel Hilbert space H _ k, and let R \ colon H _ k \ to \ R be a differentiable regularization function.
