Sentence examples for and hence the second from inspiring English sources

Next, the unknown of the third row is found by division, and hence the second and the first unknowns are found as well.

Moreover, all the remaining phrases are false, and hence the second one is true, contradiction.

Thus, we know that B < 0, and hence, the second term in right hand side of Eq. 5 is a positive number.

It is interesting to note that (Q_{i}^approx Q_{i}) when the auxiliary W approximates X, and hence the second term in the expectation of Σ 2 approximates to (1−ρ)Q i.

As an immediate consequence of Theorem 4.1, in the case T 1 = T 2 = T, we find that the integral equation (11) has a solution in C, and hence the second order nonlinear boundary value problem (10) has a solution.

The lack of fit was insignificant and, hence, the second order regression surface model was satisfactorily significant.

The Anglo-Norman family of Bruce, which had come to Scotland in the early 12th century, was related by marriage to the Scottish royal family, and hence the sixth Robert de Bruce (died 1295), grandfather of the future king, claimed the throne when it was left vacant in 1290.

On Monday morning, chez Millard will be nothing short of blissful, since it is the beginning of the school holidays and hence the first morning for quite a long time in which no packed lunch will have to be prepared for anyone.

However, such a protocol is not IC, and hence, the first-best is not achievable.

and, hence, the first relation in Eq. (19), where k is substituted by k + 1, is proved.

However, current regulation does not allow to run local peer-to-peer energy markets in most countries and, hence, the seventh component cannot be satisfied yet.