Sentence examples for by multiplying the first from inspiring English sources

This follows directly by multiplying the first equation in (1) by (w_{t}) (the second by (( 3s-psi ) _{t}) and the third by (s_{t})) and integrating over ((0,1)).

By multiplying the first and second equation of (1.1) by u t and v t, respectively, integrating over Ω and summing them up, we obtain d d t E ( t ) = − ( ∥ u t ∥ m + 1 m + 1 + ∥ v t ∥ r + 1 r + 1 ) + 1 2 ( g ′ ∘ ∇ u + h ′ ∘ ∇ v ) − 1 2 ( g ( t ) ∥ ∇ u ∥ 2 + h ( t ) ∥ ∇ v ∥ 2 ).

Then, by multiplying the first equation in (2) by (x_{n-1}) and the second one by (y_{n-1}), and using the following changes of variables begin{aligned} u_{n}=frac{1}{x_{n}x_{n-1}},qquad v_{n}= frac{1}{y_{n}y_{n-1}}, end{aligned} (4) (nge-1), system (2) is transformed in the following one: begin{aligned} u_{n}=a_{n}v_{n-1}+b_{n}, qquad v_{n}=alpha _{n}u_{n-1}+beta _{n},quad n in mathbb {N}_{0}.

Proof of Theorem 1.2 We introduce the notation M = sup t ∈ [ 0, T ] ∥ u ( t ) ∥ H s. The first step we will give estimates on ∥ u ( x, t ) ∥ L ∞. Integrating the both sides with respect to x variable by multiplying the first equation of (2.1) by u 2 p − 1 with p ∈ Z +, we get ∫ R u 2 p − 1 u t d x + ∫ R u 2 p − 1 ( u n + 1 u x ) d x + ∫ R u 2 p − 1 ( P ∗ E ( u ) ) d x = 0. (2.2).

By multiplying the first 2 fractions by the total population, we obtained the number of patients with ILI who had influenza A in a country.

In this method, the second-order bending moment is calculated by multiplying the first-order one by a magnification factor k that depends on the flexural stiffness EI and the equivalent moment distribution.

The rates of synthesis of lanosterol in the presence or absence of sterols were calculated by multiplying the first-order rate constant by the concentration of lanosterol.

The disposition index (DI), a measure of β-cell function and adjusted for insulin sensitivity, was calculated by multiplying the first-phase incremental C-peptide secretion with the M value (AIRgluc*M) (12).

By definition, dividing one number by another is the same as multiplying the first number by the inverse of the second.

The equation Ri = k[CV]i was used to determine the initial rate (Ri) by multiplying the apparent first-order rate constant k with the initial concentration of CV, [CV]i.

The equation Ri = k[PS]i was used to determine the initial rate (Ri) by multiplying the apparent first-order rate constant k with the initial concentration of PS, [PS]i.