Sentence examples for modular equivalents of from inspiring English sources

Let us recall the modular equivalents of uniform convexity introduced in [3].

The importance of applications of nonexpansive mappings in modular function spaces lies in the richness of structure of modular function spaces, that is, besides being Banach spaces (or F-spaces in a more general setting --are equipped with modular equivalentsetting --aremetric notionsetting --arerequippeded with almodularequivalentsnvergence and cofvergenormin submeasure.

The importance of applications of nonexpansive mappings in modular function spaces lies in the richness of structure of modular function spaces that besides being Banach spaces (or F-spaces in a more general settings) are equipped with modular equivalents of norm or metric notions, and also are equipped with almost everywhere convergence and convergence in submeasure.

We say that ρ has the Δ 2 -property if sup n ρ ( 2 f n, D k ) → 0. whenever D k ↓ ∅ and sup n ρ ( f n, D k ) → 0. The modular equivalents of uniform convexity were introduced in [11].

The importance for applications of mappings defined within modular function spaces consists in the richness of the structure of modular function spaces, which, besides being Banach spaces, are equipped with modular equivalents of norm or metric notions and also are equipped with almost everywhere convergence and convergence in sub-measure.

The importance for applications of nonexpansive mappings in modular function spaces consists in the richness of structure of modular function spaces, that-besides being Banach spaces (or F-spaces in a more general settings -are equipped with modular equivalentsettings -aremequippedtions, and also are equipped with almodularequivalentsnvergence and cofvergenormin submeasure.

We will need the following result, being a modular equivalent of a norm property inuniformly convex Banach spaces; see e.g.[26].[26]

For instance, it can be proved that in Orlicz spaces over a finite atomless measure [37] or in sequence Orlicz spaces [11] the uniform continuity of the Orlicz modular is equivalent to the Δ 2 -type condition.

For instance, it can be proved that in Orlicz spaces over a finite atomless measure [25] or in sequence Orlicz spaces [26] the uniform continuity of the Orlicz modular is equivalent to the (Delta_{2}) property.

For instance, it can be proved that in Orlicz spaces over a finiteatomless measure [44] or in Orlicz sequence spaces [45] the uniform continuity of the Orlicz modular is equivalent to the -type condition.

Analysis of an eight-gene model constructed from available experimental data reveals that it has a modular structure equivalent to the successful two-node case.