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Where the X's and O's become 1's and 0's.
If a second coherent optical field is added to each of these two fields, however, then the resultant fields become (1 + cos x) and (1 + |cos x|), respectively.
The second part of the referendum asked voters if they wanted the island to become (1) a U.S. state, (2) an independent country, or (3) a "sovereign free associated state".
Otherwise, when the expected rate will not improve nor degrade the average historical rate on that block, the ratio will also tend to become 1.
When ψ equals 0, all the weights for bin-wise cost functions become 1, which boils down the update rules of sc-NTF described in Section 2.1 to those used for PARAFAC-NTF.
(3) and (4), both variance reductions become 1 when (O({varvec{r}}_i, t)) equals (C({varvec{r}}_i, t)), i.e., variance reductions of 1 indicate the best matches.
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The output of the SVM2 become +1 at t =0.077 s detecting the external fault.
At correlation level i = (p + 5)/4, the off-peak array values all become −1.
Because of a potentially odd number of changed signs, det VU T ) can become −1.
Assuming the initial boundary condition, q = 0 at t = 0, the above equation on integration become: 1/q_{t} = 1/beta ln (1 + alpha beta t).
The output of the SVM2 become -1 at t =0.077 s indicating that fault detected by SVM1 is an internal fault.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com