Substituting this relation into Eq. (8) yields (9) where Atg = AQ.
Substituting this relation in (2.31) and using the recurrences of the kernels (1.9) and (1.10), (2.22) holds.
Substituting this relation to Equation (13), one can obtain 〈p〉 = edα / (1 + α), i.e., dipole moment does not depend on temperature.
Substituting the relation V = CU into Eq. (14) yields mathbf{C}leftlangle mathbf{U},{mathbf{U}}^{mathrm{T}}rightrangle {mathbf{C}}^{mathrm{T}}=mathbf{C}{mathbf{R}}_{nn}{mathbf{C}}^{mathrm{T}}={mathbf{D}}^2, (15 which then obtains {mathbf{R}}_{nn}^{-1}={mathbf{C}}^{mathrm{T}}{mathbf{D}}^{-2}mathbf{C}.
Still, the diaries do seem to confirm that Reagan had an ingrained habit of substituting public relations for serious policy.
So too should the criticism that he is substituting public relations gimmickry (the recruitment of Bob Geldof as an adviser on global poverty especially riled Tory traditionalists) for serious policy.
Substituting the relations given in Equation 24 into Equation 19 results in the optimal filter matrix: mathbf{H}_{n} = mathbf{U}_{n} mathbf{B}^{T} left[mathbf{Q}_{n} left(mathbf{Q}_{n}^{T} mathbf{Q}_{n}right)^{-1} mathbf{Q}_{n}^{T} right] mathbf{B}^{-T}.
Hence, substituting these relations in the inequality (9) eventually yields (10). in which { a i 1, a i 2, …, a i m } i = 1 n and { b i 1, b i 2, …, b i k } i = 1 n are two sequences of real numbers, m, k ∈ N and h 1, m ( x 1, x 2, …, x m ) and h 2, k ( y 1, y 2, …, y k ) are two arbitrary functions of m and k variables.
"A plea of guilty was substituted in relation to that and he did accept that perhaps he could have done more to intercept the notice and ensure that the details were completed," he said.
Building on Theorem 4.1, we give a further generalization of Caristi type, by substituting the ordering relation with an oriented graph.
