Sentence examples for From relation from inspiring English sources

If x ∗ is not a weakly efficient solution for problem (FP), then there exists x ∈ X ∘ such that f i ( x ) g i ( x ) < f i ( x ∗ ) g i ( x ∗ ) = v i ∗ for  i ∈ K. <span class="lh">From relation (6) and α i 1 ( x, x ∗ ) > 0, we have ∑ i = 1 k y i ∗ A i ( x ) α i 1 ( x, x ∗ ) < ∑ i = 1 k y i ∗ A i ( x ∗ ) α i 1 ( x, x ∗ ), which contradicts (7).

Then there is a feasible solution x ∈ X ∘ such that f i ( x ) g i ( x ) < f i ( x ∗ ) g i ( x ∗ ) = v i ∗ for  i ∈ K. From the above inequality, we have f i ( x ) − v i ∗ g i ( x ) < f i ( x ∗ ) − v i ∗ g i ( x ∗ ) for  i ∈ K. <span class="lh">From relation (6) and α i 1 ( x, x ∗ ) > 0, for i ∈ K, we obtain ∑ i = 1 k α i 1 ( x, x ∗ ) y i ∗ A i ( x ) < ∑ i = 1 k α i 1 ( x, x ∗ ) y i ∗ A i ( x ∗ ).

From relation (3.38) we have that.

From relation (3.42) it follows that, for all.

From relation (3.3) it is obtained that the sequence ({ a_{n}/b_{n}}_{ngeq1}) is increasing.

From relation (28) to the linear change of variable (13), we can get a periodic solution ((x theta),y(theta ),z(theta))) of system (2).

Recall that the following-from relation is a causal relation.

Recall that the modal transfer principle states that necessity transfers down the following-from relation.

Hence necessity will track the following-from relation because both are grounded in the same relations of conceptual involvement.

To see why, notice that the difference between [iii] and [iv] is not a function of the presence or absence of the following-from relation itself.

Of course, we need not and should not interpret Spinoza's following-from relation as the strict logical entailment of contemporary modal logic.