Sentence examples for modules and classical from inspiring English sources

Making use of Theorem 1.4 and the relations between random normed modules and classical normed spaces, we give the proof of Theorem 1.5.

We do not obtain a sharp estimate as the classical result, since random normed modules have a complicated stratification structure, which is the essential difference between random normed modules and classical normed spaces.

For Theorem 1.4, it is easy to see that it has the same shape as the classical Mazur-Ulam theorem, but it is not trivial since we must make full use of the relations between random normed modules and classical normed spaces in the process of the proof.

Remark 3.4 In Theorem 1.5, we do not obtain a sharp estimate as the classical result, namely Theorem 1.2, since random normed modules have a complicated stratification structure, which is the essential difference between random normed modules and classical normed spaces.

For Theorem 1.5, we do not get a sharp estimate as the classical result, namely Theorem 1.2, since the complicated stratification structure in the random setting needs to be considered, which is the essential difference between random normed modules and classical normed spaces.

Making use of this theorem and the relations between random normed modules and classical normed spaces, we show that if f : E 1 → E 2 is a surjective random ε-isometry with f ( 0 ) = 0 and has the local property, where ε ∈ L 0 ( F, R ) and ε ≥ 0, then there is a surjective L 0 -linear random isometry U : E 1 → E 2 such that ∥ f ( x ) − U ( x ) ∥ 2 ≤ 4 ε, for all x ∈ E 1.

The overall mathematical model integrates the membrane modules and the classical solution-diffusion transport models for the depiction of the permeate flux and the rejection coefficients.

It should be pointed out that further development of random smoothness under Definition 3.5 confronted some obstacles, one of which is how to establish the connection between random smoothness of an RN module E and classical smoothness of the abstract (L^{p}) space derived from E. It is an interesting problem and attracts some attention in different literature [19, 24].

The software provides a variety of modules for quantum mechanical and classical mechanical simulation.

This emerging EpoR signaling pathway could explain the puzzling finding that EpoR mutants devoid of docking sites for class Ia PI3K p85 SH2 domains and classical Ras activator modules (like Shc/Grb2/SoS) are still able to transmit Epo signals and apparently do not compromise the viability of the mutant mice [ 22- 25].

We mapped all strongly-conserved residues found in the pPIWI-RE family onto available structures of classical PIWI modules and compared those positions to those required for RNase activity or nucleic acid binding in the latter modules (see Figures  1B-C, 2A).