Sentence examples for using the above result from inspiring English sources

Using the above result, we rewrite the average over the delta function in Sect.

Using the above result with (20), we get the required result (19).

We can construct symmetric matrix polynomials of even degree using the above result.

Using the above result, we are now ready to prove the main fixed point result for -nonexpansive-type mapsivalued maps.

Using the above result, we are now able to specify the output of classical ICA algorithms when applied to a mixture of sources which satisfy A2.

Using the above result, we show the Mazur-Ulam theorem on non-Archimedean n-normed spaces over linear ordered non-Archimedean fields.

Next, for each integer n, where 30using the above results.

The direct H2O2 synthesis under practical conditions was simulated using the above results, and several parameters were suggested to be improved.

Using the above results, the yield strengths reported in literature for these alloys were normalized by their shear moduli to reveal the influence of shear modulus on solid solution strengthening.

Now, we prove the main result of this section using the above results.

Hence, (x^) meets (5) and (6). 3. Using the above results, this result can be obtained.