Sentence examples for by interchanging the order from inspiring English sources
By interchanging the order of the summation we get (5.53).
Then, by interchanging the order of summation, we obtain the following (2.4).
By interchanging the order of integration and some rearrangement of (2.22), we obtain (2.23).
Weng avoids that problem by interchanging the order of histogram modification and IWT.
By interchanging the order of integration, we have int_{T_{1}}^{eta}frac{q(t)}{r(t+tau-sigma)} biggl int_{t+tau-sigma }^{t}lambiggl int_{t+tau-sigma int _{T_{1}}^{t}lambdasigma}lambda(t) biggl(int _{t}^{t-tau+sigma}frac{q(s)}{r(s+tau-sigma)},ds biggr),dtgeq
By interchanging the order of integration, we have int_{T}^{xi}q(t) biggl int _{t+tau-sigma}^{t}psi(s),ds biggr),dt geqint _{T}^{xi+tau-sigma}psi(t) biggl( int_{t}^{t+sigma-tau }q(s),ds biggr),dt.
Proof The proof can be achieved as in Theorem 2.1 [12], by expressing the left-hand side of (60), interchanging the order of summation and using the power formula (59).
Thus, interchanging the order of summation has no effect.
Dividing both sides of (2.19) by and summing up over from 1 to first, then summing up over from 1 to, using again inverse Hölder's inequality, and then interchanging the order of summation, we obtain (2.20).
Interchanging the order of integration, which is permissible under the conditions given in Theorem 2.1, we find that (2.2).
Dividing both sides of (2.9) by summing up over from 1 to first, then summing up over from 1 to, using again Hölder's inequality, then interchanging the order of summation, we obtain (2.10).
