Sentence examples for hence thanks from inspiring English sources

Hence, thanks to a trustful numerical model, friction factor and heat transfer correlations are obtained.

(A decade or two hence, thanks to the experiments now starting in Colorado and Washington State, it's possible to think that marijuana legalization might also become uncontroversial).

There is even a fantasy trip to Britain 50 years hence thanks to "Doctor Who" and his Tardis (which celebrate 50 years of time travel in 2013).In short, triskaidekaphobiacs may have their reasons to fear a year with the number 13 in it.

Then and hence thanks to Theorem 3.3.

end{aligned} (16) Now, if (t_{1}rightarrow the2}), the right-hand sides of both (15) and (16) tend to 0. Thus, (mathbb{G}_{1}), (mathbb {G}_{2}) are equicontinuous, and therefore (mathbb{G}=( mathbb{G}_{1},mathbb{G}_{2})) is equicontinuous on D. Hence, thanks to the Arzelà-Ascoli theorem, (mathbb{G}(D)) is compact.

Hence, thanks to the contraction mapping principle applied, we obtain that there exists T > 0 such that the integral equation f ( t ) = ∥ u 0 x ∥ L ∞ + c ∫ 0 t [ ε s − r 4 exp ( c ∫ 0 τ f ( s ) d s ) + 1 2 f 2 + 1 ] d τ. has a unique solution f ( t ) ∈ C ( [ 0, T ] ).

So thanks, but no thanks.

From there we are propelled to a world 50 years hence where, thanks to the advances of medical science, people can live for double their current life spans.

Particles interact with the Higgs field to acquire energy and, hence, mass, thanks to Albert Einstein's famous equivalence of the two, encapsulated in the equation E = mc2.

This mind-boggling ROI is only a "proxy" because the even greater returns to billionaire contributors are invisible and hence unmeasurable, thanks to Citizens United and its progeny.

On 28 July, Austria-Hungary attacked Serbia, which, in turn, required the mobilisation of Russia, hence of Germany, thanks to its responsibilities under the Dual Alliance.