Sentence examples for hence completing from inspiring English sources

from which (43) follows hence completing the proof.

In addition, we obtain the eigen-solutions to SFSLP-I & -II analytically, also as non-polynomial functions, hence completing the whole family of the Jacobi poly-fractonomials.

Next, the concept-evaluation framework, proposed in an earlier work, is used to organize the topologies obtained in an ascending order of complexity, hence completing the conceptual design phase of the subject motion generators.

The proof of Theorem 3.2 is hence completed.

We hence complete the proof of Theorem 1.2.

which contradicts (3.2), and hence completes the proof of the claim.

Experiments were hence completed using two liquid phases: tap water and a 0.5 wt% aqueous ethanol solution.

Since the size of the inclusions is small with respect to ϵ, we derive a kinetic model for fuel conversion in the region occupied by the solid inclusions and hence complete the description of a single-step chemical kinetics.

As soon as the mouse returned to the test compartment and hence completed the sample run, the test run started.

Proof of Theorem 5.1 It suffices to show that This is the same as to show that This is clearly true, hence completes the proof.

Anoxia, another kind of water stress where plant goes under water completely, hence complete absence of oxygen as a result of long-term flooding.