Sentence examples for a partial generalization to from inspiring English sources

Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom (AGH), save for one key difference.

This then provided a partial generalization of the results of Johnson and Korányi to the stated context.

Thus Corollary 3.3 is a partial generalization of Theorem 3.1 and can be situationally useful.

Also, it is a partial generalization of Theorem 2.1 in [19].

Theorem 2.10 is a partial generalization of the main results of [16, Theorem 2.2].

Remark Corollary 3.5 is Theorem 3.2 in [8] which is a partial generalization of Lemma 2 in [3].

By setting, and in Theorem 3.15, one can deduce the following result for certain iterates of mappings which is a partial generalization of Theorem 3.9.

In the present work we introduced a new concept of fuzzy mappings in the Hausdorfffuzzy metric space on compact sets, which is a partial generalization of fuzzycontractive mappings in the sense of George and Veeramani.

Then A, B, S and T have a unique common fixed point provided A S = S A and B T = T B. Remark 3.7 Corollary 3.4 is a slight but partial generalization of Theorem 3.1 as the commutativity requirements (that is, A S = S A and B T = T B ) in this corollary are relatively stronger as compared to weak compatibility in Theorem 3.1.

Then, A, B, S, and T have a unique common fixed point provided that A S = S A and B T = T B. Notice that Corollary 2.4 is a slight but partial generalization of Theorem 2.1 as the commutativity requirements (that is, A S = S A and B T = T B) in this corollary are relatively stronger as compared to weak compatibility in Theorem 2.1.

Corollary 3.11 is a slight but partial generalization of Theorem 3.3 as the commutativity requirements (i.e., and ) in this corollary are stronger as compared to weak compatibility in Theorem 3.3.