Sentence examples for by considering the mapping from inspiring English sources

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.

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.

Further, T z 1 = T z 2 whenever z 1, z 2 ∈ B e s t ( T ) and z 1 S z 2. Proof The result follows from Theorem 2.6 by considering the mapping α as in the proof of Theorem 3.1.

We illustrate the possible problems by considering the mapping for various candidate data elements in the EMR.

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.

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.

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').