Sentence examples for obtain propositions from inspiring English sources

Using a functional (1.5), we can only obtain propositions concerning the stability.

Since these automorphisms generate P(G), we obtain: Proposition 1.

From (2.18) and (2.21), we can obtain Proposition 2.5 below.

After the change of variable, we obtain: Proposition 3.

For (gamma < 1), contrasting the principal's surplus attainable in both settings, we obtain Proposition 1.

To reduce the computation complexity, we further analyze the probability in (19) and obtain Proposition 1. Proposition 1.

By the convexity (resp. concavity) and constant preserving condition, one can easily obtain Proposition 2.1, which shows that (S^{mathrm{cv}}) (resp. (S^{mathrm{conca}})) has minimal members (resp. maximal members).

To reduce the computation complexity, we further analyze the probability in (25) and obtain Proposition 2. Proposition 2. The energy availability of the nth,∀n ∈ {1,2,…,N} relay node can be expressed approximately as follows: Pr A n A 0 A 1 … A n - 1 ≈ Γ M n ′ + 1, Λ t, (30).

Indeed, if we put s = 1, t = 0 in Theorem 1.7, then we obtain Proposition 1.4 restricted to 1 ≤ p. On the other hand, being different from Theorem 1.3, even if all parameters are positive, Theorem 1.7 does not show that the range 1 ≤ p, 1 ≤ s, 0 ≤ t ≤ 1, t ≤ r. cannot be expanded anymore for the grand Furuta inequality to be valid.

The following proposition is obtained: Proposition 7 Suppose r = g.

By Theorems 2.2, 2.3 and 2.4, we obtained Proposition 2.5 immediately.