Sentence examples for by applying the generating from inspiring English sources

Some new formulae of products of the Frobenius-Euler polynomials are established by applying the generating function methods and some summation transform techniques.

In this paper, by applying the generating function methods and summation transform techniques, we establish some new convolution identities for the Frobenius-Euler polynomials.

By applying the generating function technique, some important performance measures are derived, which may be useful for network and software system engineers.

Some closed formulae of sums of products of any number of Apostol-Bernoulli and Apostol-Euler polynomials and numbers are established by applying the generating function methods and some summation transform techniques.

As further applications, we obtain some closed formulae of sums of products of any number of Apostol-Bernoulli and Apostol-Euler polynomials and numbers by applying the generating function methods and some summation transform techniques.

Motivated and inspired by the work of the above authors, in this paper we establish some similar convolution identities for the classical Frobenius-Euler polynomials to (1.6) by applying the generating function methods and summation transform techniques developed in [18].

By applying the generated asset index, sample households were classified into five quintiles.

This work addresses this challenge by applying the generated data to relate glucan enzymatic digestibility to lignin and plant cell wall properties, to better characterize and understand the mechanism of AHP pretreatment for improved cell wall deconstruction, and to compare three analytical platforms (HSQC NMR, Py-GC/MS, and thioacidolysis) for the analysis of grass lignins.

We see the world by applying the expectations generated by the statistical lens of our own past experience, and not (mostly) by applying the more delicately rose-nuanced lenses of our political and social aspirations.

While bacteria and dietary fiber addition did not show any negative effects on product sensory and color properties; dietary fiber addition improved some sensorial features significantly i.e. sweetness, firmness and adherence, The validation of the model had been accomplished by applying the conditions generated by the RSM model.

The lower window on the right shows the detailed building facades created by applying the Computer Generated Architecture (CGA) shape grammar rule files.