Sentence examples for principle u from inspiring English sources

By the comparison principle, u ≤ u ¯ ≤ M 1.

This makes it analogous to the moral principle (U), which specifies (D) for moral discourse.

The moral principle "U" specifies that the rule of argumentation for moral discourse is univeralizability (Habermas 1990, 65 66).

By the comparison principle, u ˜ ≤ u ¯ ≤ M 2, then v ˜ must blow up.

By the strong maximum principle, u is a positive solution of problem (1.1), so Theorem 1.1 holds.

By the comparison principle, ( u ̲, v ̲ ) is a sub-solution of (1.1), thus the solutions of (1.1) also blow up.

Hence, the comparison principle gives ((u, v leq(bar{u}, bar{v})), which implies ((u, v)) exists globally.

Then we know from the assumption and comparison principle that u is radially decreasing in r with max Ω u ( x, t ) = u ( 0, t ).

Moreover, it follows from the comparison principle that u 1 ( x, t ) ≥ δ 1 w ∗ ( x, t ) and u 2 ( x, t ) ≥ δ 2 w ∗ ( x, t ) in Ω × [ 0, T 0 ), where T 0 is the maximal existent time of u 1, u 2, and w ∗. Hence ( u ̲ 1, u ̲ 2 ) = ( q 1 ( δ 1 w ∗ ), q 2 ( δ 2 w ∗ ) ) is a lower solution of (1.1).

Hence, it follows from the sub-solution comparison principle that U ( x, t ) > 0 in Ω × ( 0, ∞ ).

Hence the maximum principle yields u n + 1 ≫ 0, i.e., α n ≪ α n + 1 in [ 0, 1 ].