Sentence examples for elementary manipulation from inspiring English sources

(ii) Setting (u=f(s)) in (2.1), we obtain (2.3) after an elementary manipulation.

This inequality is equivalent to (after an elementary manipulation) 8z^{3}- z+1)^{3}zleq(z+1)^{3} sqrt{z^{2}-1}.

Making the change of variables (s=operatorname{argch} t) and (s=arccos t) in the two integrals of (6.1), respectively, we obtain the desired result after an elementary manipulation.

Let m ≥ 1, after an elementary manipulation we obtain T a, d, N ( x m ) = a − m ∑ v = 0 m { ( m v ) ∑ k = 0 a − 1 χ a, N ( k ) ( d k ) m − v } x v. and, therefore, (ii) is satisfied.

If in this inequality we replace u by ((h v,u) )^{1/2}uin C_{1}) and v by ((h u,v) )^{1/2}vin C_{2}) and we use the fact that f and g are homogeneous of degree p and (p^), respectively, h being a semi-inner product, we obtain after elementary manipulation biglvert h u,v bigrvert ^{2}leqbiglvert h u,v bigrvert bigl(pf u) bigr)^{1/p} bigl(p^g v) bigr)^{1/p^.

This section shows how DWFs constructed as a cascade chain of waveguide sections, and reflectively terminated, can be transformed via elementary manipulations to well known ladder and lattice digital filters, such as those used in speech modeling [300,367].

Substituting (17) and (26) in (27) and performing some elementary manipulations, can be expressed as. (28).

Introducing a change of variables, τ=u t, and replacing the exponential by its Taylor series expansion, elementary manipulations lead to the following fractional hazard rate: h_{H}(t) = alpha t^{1-H} E_{1,2-H}(beta t).

On the other hand, our proof of Miki's and Faber-Pandharipande-Zagier's identities follow from the polynomial identity (1.10), which in turn follows immediately from the Fourier series expansion of (1.9), together with the elementary manipulations outlined in (1.11 - 1.13).

Some elementary manipulations on (20) yield the equality begin{array}{*{20}l} E left[ left f,pi_{t}^{N}right -left f,pi_{t}right) right] = Eleft[ left(f,pi_{t}^{N}right -left f(mathbf{1},pi_{t}rightleft(mathbf{1},rightt}^{N}right) }{ (mathbf{1},rho_{t}) } right].

On the other hand, our proofs of Miki's and Faber-Pandharipande-Zagier's identities follow from the polynomial identity (1.9), which in turn follows immediately the Fourier series expansion of (gamma_{m}(langle xrangle )) in Theorems 4.1 and 4.2, with (r=s=1), together with the elementary manipulations outlined in (1.9 - 1.12).