Sentence examples for fact a solution from inspiring English sources

In fact, a solution of (1.1) is constructed by the common solution of (2.3) and (2.4).

In fact, a solution to the integral equations in (12) is a continuous function satisfying the conditions in (12).

Moreover, in our work proving that the solution of the modified problem is in fact a solution of the original one involves different comparison functions since our penalization is slightly different.

Although we have proven that any solution found using the unconstrained Lagrangian form is in fact a solution to the original constrained optimization problem, we have not proven that we can find a solution corresponding to a particular value for.

The same arguments as at the end of the proof of Theorem 4.1 apply, which shows that is in fact a solution of problem (1.1) belonging to the interval.

Now we shall show that this solution u satisfies the inequalities (3.1) and (3.2) which in view of the definitions of (F_{3}), (F_{2}), (F_{1}), and (F_{0}) will imply that u is in fact a solution of (1.1 - 1.2 1.1 - 1.2

The scale-shift approach takes advantage of the fact that a solution to a stage-wise separation problem is more easily obtained the fewer the number of stages.

He does, in fact, provide a solution that would, in its ironic appreciation of the principal actors, have brought a smile to the face of Father Brown, G. K. Chesterton's ecclesiastical sleuth.

Then by Corollary 2.8, every solution of (3.3) is either oscillatory or satisfies In fact, is a solution of (3.3).

Hence, has a fixed point in [ 0, 1 ] by Theorem 2.5, which in fact is a solution of problem (1.1).

Hence, N has a fixed point in [ 0, 1 ] by Theorem 2.4, which in fact is a solution of the problem (1.1).