Sentence examples for mathematical manipulation the from inspiring English sources

Based on this equation, after mathematical manipulation, the calculations of the volume, thickness and width of the annular MR fluid within the cylindrical MR fluids brake are yielded.

After some mathematical manipulation, the maximum permissible transmit power P s of secondary source under the outage constraint ϵ of primary receivers is given by P S = X P P P ω θ P S ω β S (16).

After a straightforward mathematical manipulation, the following expression is obtained: δ T MVDR ; ℋ 1 = ( h 0 H R xx − 1 h ) 2 h 0 H R xx − 1 h 0, (25).

After some mathematical manipulation, the approximate expectation of the relative standard error estimator (AERSEE) turns out to be mathrm{AERSEE}=mathrm{R}mathrm{S}mathrm{E}{left(1+mathrm{R}mathrm{B}right)}^{1/2}.

By the same mathematical manipulation, the stability boundaries are found as σ n = d 1 2 ω n α − 1 cos ( π 2 α ) ± d 3 2 ω n 2 α − 2 a n 2 − d 1 2 4 ω n 2 α − 2 sin 2 ( π 2 α ).

Using (23) and (50) and after some mathematical manipulation, the PDF of γ κ can be setup as begin{array}rcl@ f_{gamma_{kappa}}left(gamma_{kappa}right) & = & frac{gamma_{kappa}^{N-1}left(frac{N}{overline{gamma_{kappa}}}right)^{N}expleft -frac{Ngamma_{kappa}}{overliN}expleft -frac{Ngamma_{kappa}(N-1right)!}{overline{gamma_{kappa}

Finally, through some mathematical manipulations, the six ordinary differential equations governing the beam structural behaviour are derived.

Through a series of mathematical manipulations, the problem is reduced into a group of singular integral equations that can be solved by numerical methods.

Now, setting the partial derivatives of J2 to zero and after some mathematical manipulations, the powers p b,k can be shown to be given by (16).

Replacing all the unknowns with their estimates, and after some mathematical manipulations, the final expression for the unstructured GLRT in (9) turns out to be given by: Λ UG X = | Σ ̂ x | 1 K Tr ( Σ ̂ x ) K ≷ ℋ 1 ℋ 0 γ.

Next, α in (18) can be plugged into φ in (15) and after some mathematical manipulations the following: text{Var}left({hat{mathbf{X}}_{n + 1}-alphamathbf{X}_{n}}right) = sigma_{hat{mathbf{X}}_{n + 1}}^{2}left 1-rho_{hat{mathbf{X}}_{n + 1}mathbf{X}_{n}}^{2}left 1-rho_{hat{mathbf{X}