Sentence examples for proved the characterization of from inspiring English sources

In 1991, Xu [11] proved the characterization of uniform convexity as follows.

We proved the characterization of a vector solution of the vector matrix game in [6].

Moreover, Kakavandi also proved the characterization of △-convergence for (operatorname {CAT}(0)) spaces satisfying the ((mathbb{S})) property as follows.

We have proved that the characterization of expressivity is a key for gestures understanding.

It can be observed in Figure 1a and 1b that the obtained material Co-SiO2 presented as a highly dispersed, uniform-sized nanosphere, which was further proved by the characterization of SEM (Figure 1c).

Kowalski and Vanhecke[16, 17] proved the following characterization of geodesic vectors.

For all tested matrices, the p35S and the tNOS bidirectional DNA walking methods were successfully applied as the GMO presence was proven via the characterization of the junction between the transgenic cassette and the plant genome as well as of a part of the transgenic cassette.

Recently, Du [28] first proved the following characterizations of MT -functions which are quite useful for proving our main results.

A function φ : [ 0, ∞ ) → [ 0, 1 ) is said to be an MT -function (or ℛ-function) [2 5] if lim sup s → t + φ ( s ) < 1 for all t ∈ [ 0, ∞ ). It is evident that if φ : [ 0, ∞ ) → [ 0, 1 ) is a nondecreasing function or a nonincreasing function, then φ is a MT -function. So the set of MT -functions is a rich class. Recently, Du [5] first proved the following characterizations of MT -functions.

(i) sup n ∑ k = 1 ∞ h n + k < ∞, (ii) for each p ∈ ℕ, there exists lim n ∑ k = p ∞ h n + k = h p. Moreover, if (i) and (ii) hold and ξ k → s as k → ∞, then lim n ∑ k = 1 ∞ h n + k ξ k = s h 1 + ∑ k = 1 ∞ h k - h k + 1 ξ k - s. First, we prove the following characterization of a Hankel matrix transform convergence of a trigonometric sequence.

Pre-fractionation by SDS PAGE prior to MS has proved useful in the characterization of samples that are not amenable to separation by 2-DE.