Sentence examples for tool in the proof of from inspiring English sources

Next, we give some lemmas which are an essential tool in the proof of main results.

The following lemma is a primarily used tool in the proof of our main results.

The pullback method is a powerful tool in the proof of uniformly asymptotic stability.

Moreover, we present a new generalized Gronwall inequality with singularity, which is an important tool in the proof of solvability.

The main tool in the proof of the results in this context combines the ideas in the contraction principle with those in the monotone iterative technique [18].

Now we present a Rosenthal-type inequality for maximum partial sums of m-NA random variables, which is the crucial tool in the proof of our main results.

The main tools in the proof of the Sobolev inequalities for Riemannian manifolds are the Hadamard parametrix (cf. L. Hörmander, Acta Math.88 1968, 341 370, and "The Analysis of Linear Partial Differential Equations," Vol. III, Springer-Verlag, New York 1985) and oscillatory integral theorems of L. Carleson and P. Sjölin (Studia Math.44 1972, 287 299) and Stein Annn. Math. Stud.112 1986, 307 357).

Siu himself used it as one of two key tools in the proof of the deformation invariance of plurigenera, the second tool being Skoda's ideal membership theorem.

To prove the positivity of function (82), the following formula and inequality are used as basic tools in the proof of [[34], Theorem 1]. 1.

Our main tools in the proof of Theorems 1.1-1.2 1.1-1.2ologicareargumentopological the globargumentsation theorem for mappings which are not necessary smooth [15, 14].

The main tools in the proof of Theorem 1.3 are the perturbation theory of operators in Hilbert spaces and the spectral theory of (mathcal{P}mathcal{T} -symmetric differenT} -symmetrics.