Sentence examples for by applying the theorem from inspiring English sources

Now, think of EQI[ck | hi/hj | b] as generated by applying the theorem in successive steps: So each such division of (oku∨oku+1∨…∨okm) into two separate tatements, oku and (oku+1∨…∨okm), adds a strictly positive contribution to the size of EQI[ck | hi/hj | b]  just when  P[oku | hi·b·ck] / P[oku | hj·b·c k]   ≠   P[ oku+1∨…∨okm) | hi·b·ck] / P[ oku+1∨…∨okm) | hj·b·c k].

whenever n ≥ N. Let f n ( t ) = ( t 2 − α u n ( t ) ) ′ and f ( t ) = ( t 2 − α u ( t ) ) ′. Then (2.4) means that f n → f uniformly on ( 0.1 ], and so lim t → 0 + f ( t ) = lim n → ∞ lim t → 0 + f n ( t ) exists . by applying the theorem to the limit convergence of function sequences again.

The proof of the above theorems is completed by applying the theorem of Arzella-Ascoli on each interval ( t n, t n + 1 ], n = 0, 1, 2, … and constitutes a diagonal line sequence. Let C be a nonempty subset of X. Set. We now have the following lemma. Lemma 3 ([17]).

By applying the theorem of Riesz, it is easy to see that the operators (mathcal{J}_{epsilon}) and N̂ differ by a relatively compact perturbation in (mathcal{L}^{p}_{s}), for (pin 3,4)) and (s>3 1-frac{1}{p})).

Because lim t → 0 + t 2 − α u n ( t ) exists for all n ∈ N, by applying the theorem to the limit convergence of function sequences, we know that lim n → ∞ lim t → 0 + t 2 − α u n ( t )  exists, and lim t → 0 + t 2 − α u ( t ) = lim t → 0 + lim n → ∞ t 2 − α u n ( t ) = lim n → ∞ lim t → 0 + t 2 − α u n ( t ).

The predictive values for the laboratory were calculated by applying the Theorem of Bayes assuming a prevalence of 54.82% for the year 2005 with the same software.

Therefore, by applying the Theorems 2.1 and 2.2, the result follows.

By applying the bifurcation theorem of López-Gómez [[7], Theorem 6.4.3], we shall establish the following: Theorem 1.1.

Now we give the proof of Theorems 2.1 and 2.2 by applying the fountain theorem and the principle of symmetric criticality.

The estimated parameters μ, σ and f (Table 1) adequately fit the data (Fig. 2b) and were used to compute the probability that each event belongs to the low inter-arrival time populations (either L1 or L2) or to the high inter-arrival times population H by applying the Bayes theorem: Pr(C|x) = (Pr(x|C) Pr(C))/Pr(x).

By applying the Coase theorem, emphasising transaction costs and property rights, this paper argues that strong public support is needed to create private incentives for exploring economic and environmental win win innovations.