Sentence examples for the proof utilizes from inspiring English sources

The proof utilizes the kinetic formulation and the averaging lemma.

Yet, since the proof utilizes ordinal numbers, there are notions of constructiveness that the proof would not satisfy.

Our proof of the CFL/n-pseudorandomness of the generator is quite elementary and, in particular, one part of the proof utilizes a special feature of the behaviors of nondeterministic pushdown automata, called a swapping property, which is interesting in its own right, generalizing the swapping lemma for context-free languages.

In order to illustrate to the reader the method of proof utilized by Dahal and Goodrich, we provide the proof in full.

Bell's proof and the Kochen-Specker proof utilize similar constructions in 3-dimensional Hilbert space, though they differ in their details.

In the proof, we utilize the notion of ⋆-positive definiteness, proposed in [Y.G.

With the help of (2.4), we can complete the proof by utilizing Banach's fixed-point theorem and an argument used frequently to prove the well-posedness of Cauchy problems for ordinary differential equations.

Proof Utilizing the condition λ n ( α n + L ) < 1, ∀ n ≥ 1, and repeating the same arguments as in Remark 3.1, we can see that { y n }, { t n } and { z n } are defined well.

The proof of this theorem utilizes the Steinitz lemma and may be adapted to provide a nonstandard proof of this type of theorem for various other probabilistic categories.

In the above proof we utilized the property that (LG_x) acts transitively on the unit sphere in (T_x(X)) for all (xin X).

For the proof of (A3), we utilize (A1) and (A2) in the following calculations.