Sentence examples for in the two expressions from inspiring English sources

We prove mathematical 'propositions'trueue'correctect') by 'seeing' that two expressions have the same meaning, which "must be manifest in the two expressions themselves" (6.23), and by substituting one expression for another with the same meaning.

Therefore, we obtain the following power series for h ( t ) by symmetry: h ( t ) = ∑ n = 0 ∞ ( ∑ k = 0 n ( n k ) E k ( a x ; λ b ) λ − a b k a n − k ∑ p = 0 n − k ( n − k p ) ( − 1 ) n − k − p T n − k ( b ; λ a ) ) t n n !. Equating the coefficient of t n n ! in the two expressions for h ( t ) gives us the desired result.

In a similar manner, we have k ( t ) = ∑ n = 0 ∞ { ∑ s = 0 n ( n s ) E n − s ( α + 1 ) ( a y ; λ b ) λ ( − a α ) b n − s × ∑ k = 0 s ( s k ) ∑ p = 0 k ( k p ) k − p T k ( b ; λ a ) E s − k ( b x ; λ b ) a s } t n n !. Equating the coefficient of t n n ! in the two expressions for k ( t ) gives us the desired result.

It would actually be sufficient to restrict contraction by forbidding only those cases in which the two expressions being contracted have been introduced in semantically different ways.

It is difficult to see any important respect in which the two expressions divide their reference differently.

FTIR analysis of PDC1 protein expressed in the two expression systems showed that expression in P. pastoris gave a product with more β-sheets and less random unordered structure than when it was expressed in E. coli.

Firstly, the effects of the 9' LT mutation are different in the two expression systems and in oocytes they depend on α∶β cRNA ratios.

The significantly up- or down- regulated genes in cancer samples in the two expression microarray data sets described above assuming a paired sample design was analyzed further.

In this example, the two expressions given match the start and the end of a global definition for a variable or structure in the Clive Go compiler.

We notice that the two expressions in (46) and (47) are dual to each other.

Author response For each probe the variance was calculated with the two expressions in the two libraries.