Sentence examples for we considered certain from inspiring English sources

In the present work, we considered certain power series with algebraic coefficients from a certain algebraic number field K of degree m and showed that under certain conditions these series take values belonging to either the algebraic number field K or ⋃ i = 1 m U i in Mahler's classification of the complex numbers for some Liouville number arguments.

We consider certain fundamental values to be essential to international relations in the 21st century.

In this paper we consider certain dual integral equations involving trigonometrical functions whose closed form solutions are obtained.

In this section, we consider certain boundary value problems which are related to q-Lidstone polynomials, and then we define these polynomials by using Green's function.

In this paper, we consider certain families of functions to prove log-convexity and exponential convexity of functionals due to the differences in inequalities given in Theorems 1.4-1.13.

As the illustrative identities in Section 1, we consider certain interesting identities about finite-series involving binomial coefficients, harmonic numbers and generalized harmonic numbers.

In particular, the charge of incompleteness seems most plausible when we consider certain "problems of extension", that is, topics where it is unclear whether public reason can be extended to provide reasonable answers (Rawls 1996, 244 246).

If we consider certain major components of parenting to be mentoring, then it has been the prevailing methodology by which the continuous components of civilization and culture are passed on from one generation to another.

In the present work, we consider certain power series with rational coefficients and show that these series take values of either the set of Liouville numbers or rational numbers under certain conditions.

In the study, we will expand this problem for matrix operators from (l_{p}(w)) into (c_{p} (w)) and matrix operators from (c_{p}(w)) into (l_{p} (w)), and we consider certain matrix operators such as Cesàro, Nörlund and weighted mean.

We consider certain fundamental values to be essential to international relations in the twenty-first century.