Sentence examples for by using the properties from inspiring English sources

This Lemma can be easily proven by using the properties of the positive (semi) definite matrices.

By using the properties of a log function, we get immediately (44).

By using the properties of M -matrix, a new delay differential-difference inequality is established.

By using the properties of multivalent prestarlike functions, we obtain various inclusion relationships between defined classes of functions.

By using the properties of the metric projection and m-accretive mappings, we can easily prove the following two lemmas.

Many authors have studied the twisted -Bernoulli polynomials by using the properties of the -adic invariant -integral on (cf. [4]).

By using the properties of the epigraph of the conjugate functions, the optimality conditions and strong duality of are obtained.

With some processing software, an analysis of bottom materials is also possible by using the properties of return signals.

We prove the conclusions of Theorem 1.2 by using the properties of two functions X ( x ) and U ( x ).

The ascent and the algebraic multiplicity of their eigenvalues are determined by using the properties of the eigenvalues and associated eigenvectors.

By using the properties of Green's function together with the Schauder fixed point theorem, it has been proved that this problem has at least one positive solution.