Sentence examples for preceding equation from inspiring English sources

With U regarded as a function U S, V), its differential dU is      (40) A comparison with the preceding equation shows immediately that the partial derivatives are     (41) Furthermore, the cross partial derivatives,     (42) must be equal because the order of differentiation in calculating the second derivatives of U does not matter.

We use the identity m ( m − 1 i − 1 ) = i ( m i ). in the third line of the preceding equation.

Unfortunately, the preceding equation cannot be solved independently because the surface heat flux is unknown and coupled to transient conduction of the rock mass surrounding the fracture.

The preceding equation means that it is possible to predict the financial situation of firm i at time t using its situation at time t-1 and some exogenous variables z it.

Furthermore, from the preceding equation, we have f ′ ( x n ) = c 1 + 2 c 2 e n + 3 c 3 e n 2 + 4 c 4 e n 3 + O e n 4. Open image in new window (9).

Neglecting the higher order terms, the preceding equation reduces to: φ u T Δ K φ u = Λ d - Λ u - φ u T K u Δ φ - Δ φ T K u φ u (18).

To generate uniform [0,1] random vectors X ± = X 1 ±, X 2 ±, X 3 ± with correlation matrices ρ±, one uses two times the preceding equations.

Accordingly, there must be a sudden change in the amount of,, and, from zero to a certain positive value which is determined by the preceding equations: (21).

In order for the later use, we express the preceding equations in vector format that x ̄ n = [ x n ( t 1 ), x n ( t 2 ), …, x n ( t I ) ] T and ψ ̄ n = [ e j 2 Π n I t 1, e j 2 Π 2 n I t 2, …, e j 2 Π nI I t I ] T = [ ψ 1 n, ψ 2 n, …, ψ In ] T.

The addition of feedback changes only the functional form of x in the preceding equations, as detailed below.

So the terms p ^ u, i in the preceding equations represent the probabilities that the mate of the focal parent is contributing an i allele.