Sentence examples for by using these relations from inspiring English sources

By using these relations, we compute these numbers and polynomials.

By using these relations, there exist words w x in X ∗, w y ∈ Y ∗ and w a, b ∈ { z a, b : a ∈ A, b ∈ B } ∗ such that w = w x w y w a, b in M for w ∈ Z ∗.

By using these relations, we find that aqueous alteration occurs during a period of 210 yr at 273 K; this period is very short compared to the thermal evolution time scale (of the order of 1 Myr).

On the other hand, in [2], Bailey derived relations between contiguous hypergeometric functions of the type ({_{3}F_{2}}(1)), and by using these relations, he obtained another proof of Watson's reduction formula.

Then, from the compound inequalities (6)–(7), we obtain the equality relation of the diameter of G with the eccentric connectivity, connective eccentric, and augmented eccentric connectivity indices as follows: begin{aligned}& d_{G}=frac{xi^{c}(G)}{2m}, & d_{G}=frac{2m}{C^{xi} G }. end{aligned} By using these relations in inequality (26), we get the required results.

By using these recurrence relations, we then provide a new technique to compare the skew energies of a class of oriented graphs which can tackle the quasi-order incomparable problems in Section 3.

By using this relations, the phase is α t ( k ) = ± π / 2 + β r ( k ).

(IV) By using the relations between asymptotically almost periodic function and the almost periodic functions.

Next, we define two types of monotone mappings by using the relations ≤ ( I ) and ≤ ( II ).

Now we can write the definition of integration by parts of fractional order by using the relations given in (2.1 - 2.4 2.1 - 2.4

The computation of (mu_{n}) (respectively, (nu_{n})) can be done by using the relations (59) and the result of computation of (beta_{n}) (respectively, (gamma_{n})) above.