Sentence examples for using the above identities from inspiring English sources

Thus, using the above identities, we can write.

Multiplying the inequality (2.4) by p 1 ( s ), integrating it over s on [ c, c + d 2 ] and using the above identities, we obtain the inequality (3.1).

Using the above identities, one may estimate the total population with y remaining years of life, P y) directly as Brouard (1986) did, and so approximate a population's thanatological age structure according to some mortality assumptions.

Using the above identity, they gave the following result for the Riemann-Liouville fractional integral.

Integrating the above inequality over s on [ c, c + d 2 ], dividing both sides by d − c and using the above identity, we obtain the inequality (2.1).

Integrating the above inequality over s on [ c, c + d 2 ], dividing both sides by d − c and using the above identity, we have H g ( t 1 ) ≤ H g ( t 2 ).

Dividing the above inequality by ( d − c ) 2, integrating it over s on [ c, d ], over u on [ c, c + d 2 ] and using the above identity, we derive the inequality (2.12).

Integrating the above inequality over s on [ c, c + d 2 ], dividing both sides by d − c and using the above identity, we have P g ( t 1 ) ≤ P g ( t 2 ).

Multiplying the inequality (2.7) by p 1 ( s ), integrating it over s on [ c, c + d 2 ] and using the above identity, we obtain W H g ( t 1 ) ≤ W H g ( t 2 ).

Using the above identity and q integral by parts ∫ a b u ( t ) D q [ v ( t ) ] d q t = [ u ( t ) v ( t ) ] a b - ∫ a b v ( qt ) D q [ u ( t ) ] d q t, Open image in new window.

By using the above generating functions, we get some identities and relation.