Sentence examples for by straightforward calculation from inspiring English sources

Moreover, ({ m_{n} t, V t)) }_{n in mathbb {N}_{0}}) are (mathcal {F}_{t} -martingales, t∈ [ 0,∞). By straightF}_{t} -martingales we can check (2.9). We can prove t∈at ({ m_{n}(t, V(t)) }_{n in mathbb {N}_{0}}) are (mathcal {F}_{t})-martingales from the fact that G α (t,x) is an (mathcal {F}_{t})-martingale for all (alpha in mathbb {R}). Suppose (N in mathbb {N}).

Proof The identity is obtained by straightforward calculation.

By straightforward calculation, one can verify assumption (e) of Theorem 1.

Proof In view of Lemma 4.2, the identity is obtained by straightforward calculation.

Let D k denote the determinant of the submatrix of Discr ( f ˜ ), formed by the first 2k rows and the first 2k columns, for k = 1, 2, 3, 4. So, by straightforward calculation one can see that D 1 = 4, D 2 = 4 Δ 1, D 3 = 4 c 2 Δ 2, and D 4 = c 4 Δ 3. The rest of the proof follows in view of [[44], Theorem 1].

By straightforward calculations with covariant derivatives defined by the coefficients Equations 86 and 87, we can verify that one holds true to all conditions of the theorem.

By straightforward calculations with covariant derivatives defined by the coefficients (Equation 122), we can verify that one holds true all conditions of the theorem.

Although this follows by straightforward calculations, we describe all the details, since this was the reason why we had to consider the space (H^{2}_{mathrm{rad}}(mathbb{R}^{N})) rather than (H^{2}(mathbb{R}^{N})).

By straightforward calculations with Equation 53, we can verify that the conditions (Equation 52) are satisfied, and that the d-torsions are really subjected to the conditions T ^ b ' c ' a ' = 0 Open image in new window and T ^ bc a = 0 Open image in new window (see Torsions and curvatures on Lie N-algebroids section).

Once equations are obtained, AUC values can be estimated by straightforward calculations that can even be done manually.

giving by a straightforward calculation (4.23).