Sentence examples for By propositions from inspiring English sources
By Propositions 2.1 and 2.3, we readily have the following: □.
By Propositions 4.2 and 4.4, we get the following strong duality theorem straightforwardly.
By Propositions 3.1 and 3.2 we can obtain some new inequalities for trigonometric functions.
By Propositions 3.4 and 3.10, we obtain the conclusion of Theorem 1.1.
By Propositions 3.1 and 3.2, we have: Theorem 3.1 Assume that (H1 - H7) hold, then system (1.2) is uniformly persistent.
By Propositions 2.6 and 2.10, we see that the sequence { T n p } n ∈ N is bounded and ∥ p − T n p ∥ → 0 as n → ∞.
By Propositions 2.4 and 2.8, we see that the sequence ({T_{n}p_{n}}) is bounded and (|p-T_{n}p|rightarrow0) as (nrightarrowinfty).
By Propositions 4.1 4.12 and the intermediate value theorem, the inequalities in (3.1)–(3.2) can been illustrated with the graph of (see Figures 1 3).
By Propositions 2.3 and 2.8, ∇ f ∗ is uniformly continuous on bounded subsets of E ∗ and thus ∥ T n j x n j − x ¯ ∥ → 0 as j → ∞.
By Proposition 3.4, whenever.
