Notice that with the introduction of the μ operator for the first time we encounter a recursive function that might fail to be defined for some arguments.
2When the BER is over 20%, we define that the watermark is failed to be recovered.
The conditional probabilities between most pairs of sentences fail to be objectively defined in a way that suits likelihoodists.
In such situations, the iterative schema for the approximation of fixed points of T may fail to be well defined.
(3.22) With these restrictions, the sum in (3.21) contains only a finite number of terms, and a simple estimate shows this number is bounded by ((2M_{m+1})^{ 2L_{m+1})}). To prevent that the sum in (3.21) fails to be well defined, we exclude R^{m}equivbigl{ xiinPi_{m}:exists kinmathbb {X}^{m+1} text{ s.t.
However, if the domain of T, D ( T ), is a proper subset of E (and this is the case in several applications) and T maps D ( T ) into E, then the iteration processes of Mann type and Ishikawa type have been studied by the authors mentioned above, their modifications introduced may fail to be well defined.
If, however, the domain of T, D(T), is a proper subset of X (and this is the case in several applications), and T maps D(T) into X, then the iterative sequence {x n } may fail to be well defined.
Therefore, it is necessary to consider the following representations for those proteins that fail to be meaningfully defined in the GO space.
Therefore, it is necessary to consider the following representation modes for those proteins which fail to be meaningfully defined in the GO space.
However, if the gradient ∇g fails to be strongly monotone, the operator W defined by (1.5) would fail to be contractive; consequently, the sequence { x n } generated by the algorithm (1.3) may fail to converge strongly (see [19]).
