Sentence examples for as in the above theorem from inspiring English sources

Quasi (Embox[0,1]) convexity of (u_{I}^{T} g_{I}) at (x^ ) yields bigl x-x^ bigl x-x^abigr(u_{I} ^{T} g_{I} bigl(x^ bigr)bigr) le 0,quad forall xinablabigl u_{roceed as I} the above theorem to prove that (E(f(x^{),lambda_{2} )) is an opT}mal solution in the objective space of problem ((bar{mathrm{P}})).

Corollary Let z be as in the above theorem.

Proof The proof goes the same way as in the above theorem.

Corollary 3.3 Let m 1, m 2 be as in the above theorem.

Corollary 3.4 Let m 1 and m 2 be as in the above theorem.

Finally, the following is of interest: Let m 1 and m 2 be as in the above theorem.

If z ∈ C, then Y ( z ) is an analytic function, and then it has a power series as defined in the above theorem with Equation (22).

Some properties of the function Y ( z ) = [ f ( z, r, λ, k, l ) ] m in (19) can be expressed as follows: If z ∈ C, then Y ( z ) is an analytic function, and then it has a power series as defined in the above theorem with Equation (22).

As pointed in the above, Theorem 3.3 starts from an arbitrary lower and upper bounds of a stabilizable mean m for giving other lower and upper bounds of the mean m, and so we can iterate the same procedure for obtaining an infinity of lower and upper bounds of m.

Suppose G contains a path P′ of length at least c log n (where c is as fixed in the above theorem) that is equivalent to a subgraph H* of G*.

As previously, taking p = 1 in the above theorem we immediately obtain the following result.