Sentence examples for derived with the help from inspiring English sources
The results are derived with the help of generalization (6) of Kummer's formula (4).
Unit water output coefficients of various departments can be derived with the help of their individual water consumption volumes.
The result is derived with the help of extension of classical Saalschütz's summation theorem recently added in the literature.
Next, the explicit algebraic expressions for the dynamic stiffness elements are derived with the help of symbolic computation.
The existence, stability and convergence of the proposed numerical schemes are rigorously derived with the help of functional analysis.
The governing equation and boundary conditions are derived with the help of strain gradient elasticity theory and Hamilton principle.
Thus, using in (18) the PDF transformation for two random variables, related as (here ), we can derive with the help of (B.6) the PDF of the SNR as a sum of weighted gamma PDFs, that is, (19).
Then, we can derive with the help of the matrix inversion lemma that the the solution of the problem is given by θ i * = s i 1 - σ 1 C ( γ R 1 ) + σ 1 C ( γ R 2 ) + ∑ j = 1 N σ j C ( γ j R ) (29). with s i = σ i C ( γ i R ) if 1 ≤ i ≤ N σ 1 C ( γ R 2 ) if i = N + 1 σ i - N C ( γ R 1 ) if N + 2 ≤ i ≤ 2 N - 1 (30).
A sufficient condition for synchronization is derived analytically with the help of Lyapunov stability theory.
Moreover, in [15] the maximal L p - L q regularity in a domain is derived automatically with the help of the Weis' operator valued Fourier multiplier theorem, so that a local in time unique existence theorem is proved by using the usual contraction mapping principle based on the maximal L p - L q regularity.
It is important to underline that the well known for this case expressions for the energies (22) and functions (23a) (23b) (23can(23d) can be derived also with the help of the asymptotic properties of the Airy functions [ 29] as the limit of the corresponding dependencies from Eqs. (18) and (14), respectively.
