Otherwise, there exists such that ; then from (2.6) we have for and so (2.7).
Otherwise, assume that such that is not a Cauchy sequence or, if so, it does not converge to zero while satisfying ; for all Then, such that (2.40).
Formally, we define to be a mapping from VG to sets of nodes in VS, where VS+ is the powerset of VS. is the set of children of M(P g)) such that ; otherwise, (3) One more piece of machinery is needed: to determine, we must know the children of M(P g)), but we do not have that information until we visit P g).
All video games are imperfect, even the ones that appear otherwise (such as computerized versions of Chess).
Otherwise the approximate gradient is negative such that power is decreased through updating.
(Otherwise, if there exists for some such that, then we can get a contradiction as in case (3)).
Otherwise, there is a (eta_{0}>0) such that liminf_{trightarrowinfty}u t,omega_{0} ge2eta_{0} quad mbox{for some }omega_{0}in Omega_{0}.
Otherwise, it attempts to find a unique such that and.
Let be a solution of (1.2) such that case of Lemma 2.1 holds for Then, Next we claim that Otherwise, there exists such that for all By the definition of we have that (2.35) holds.
We may select y = ( y 1, y 2, …, y n ) ∈ R n and D SY = ( y i y j + δ i j ε i ) ∈ M n ( R ), where δ i j = 1 if i = j, otherwise 0, such that exactly one of the following holds: (1) y i ≥ 0 for i ∈ 1 ( x p ) and ∑ i ∈ I 1 y i ≤ ∑ i ∈ I 2 y i ; (2) y i ≤ 0 for i ∈ 1 ( x p ) and ∑ i ∈ I 1 y i ≥ ∑ i ∈ I 2 y i .
