Sentence examples for hence substitute from inspiring English sources

Third, carers might reach a point where they no longer felt able to manage and hence, substitute care arrangements had to be made.

Lady was rushed all spring & didn't have time to make new figures, hence substitutes.

Hence, substituting (3.8) in (3.3) and by condition (iv), we obtain (3.9).

Hence substituting the expressions for, and into the above equation yields (4.10).

Hence, substituting (7) into (16) gives the following ASEP expression for the considered dual-branch SSC receiver: (17).

Hence, substituting the bounds in (62) and (63) back into (61) and using Part of Lemma 5, we obtain.

Upon examination, many theses were outside the target years or from other institutions and were, hence, substituted.

Hence, substituting z n = z ^ n | n into Equation (26), we have  F n = I N A 0 H ^ n | n A ^ n | n (28).

Hence, substituting (71), (73), and (75) into the SVD of and, we can have the desired result (68), which decomposes the MIMO relay channel into parallel channels.

(30) Hence, substituting (29) and (30) into (26) and considering the definition of (V_{i}) defined by (25), one can complete the proof.

Hence, substituting 1 − y for y and μ − 1 for λμ in (4.27), in view of the complementary addition theorems of the Apostol-Bernoulli and Apostol-Euler polynomials, the desired result follows immediately.