Sentence examples for by considering the map from inspiring English sources

Proof The proof follows directly from the previous theorem by considering the map J : α ↦ { j }.

The rigid solid is obtained by considering the map X ∗ = I, where I denotes the identity map.

Similarly by considering the map (p_2) from (M = P' oplus C) to (P') we have (B cong C cap B oplus J) for some submodule (J) of (P').

In [4], Rhoades and Temir established the weak convergence of the sequence of the Mann iterates to a common fixed point of T and I by considering the map T to be I-nonexpansive.

Then a left-module isomorphism R R → R R 2 is easy to establish, by considering the map that associates with any row-finite matrix M the pair of row-finite matrices ( M 1, M 2 ), where M 1 is built from the odd-indexed columns of M, and M 2 from the even-indexed columns.

By considering the mapping in Remark 3.5, we note that the converse of Proposition 3.7 is not true (see [4]).

Proof This result follows from Theorem 2.9 by considering the mapping α given in the proof of Theorem 3.1.

Ilić and Rakočević [9] determined some common fixed point theorems by considering the maps on cone metric spaces.

In [14], Ilić and Rakočević determined some common fixed point theorems by considering the maps on cone metric spaces.

Proof The result follows from Theorem 17 by considering the mapping α given by (4.4) and by observing that condition implies condition (iii′).

The main feature of the sub inner decoder (with a concatenated doping RA decoder) can be roughly estimated by considering the mapping alone since the doping rate is very low.