Sentence examples for modular on the from inspiring English sources

In summary, by comparing the topological properties and features of metabolic networks between chloroplasts and photosynthetic bacteria, we showed that the chloroplast metabolic networks are reduced and simplified on one hand, but highly specialized and modular on the other.

Define the Musielak-Orlicz function modular on the space of all Lebesgue measurable functions by ρ ( f ) = 1 e 2 ∫ 0 ∞ | f ( x ) | x + 1 d m ( x ).

Setting ρ ( f ) = ∫ 0 1 { ∫ 0 1 k ( x, y, | f ( y ) | ) d y } d x and using Jensen's inequality, it is easy to show that ρ is a convex function modular on the space of measurable functions defined in [ 0, 1 ], and that ρ ( T ( f ) − T ( g ) ) ≤ ρ ( f − g ), that is, T is nonexpansive with respect to ρ.

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then ρ is modular (convex modular) on X in the sense of (A.1 - A.4) if and only if w is metric modular (convex metric modular, respectively) on X.

Let ρ be a modular on X having the Fatou property.

If ρ is a modular on X, then the set X_{rho}= Bigl{ xin X: lim_{lambdato0} rho(lambda x)=0 Bigr} is called a modular space.

If ω is metric (pseudo) modular on X, then the modular set X ω is a (pseudo) metric space with (pseudo) metric given by d ω ∘ ( x, y ) = inf { λ > 0 : ω λ ( x, y ) ≤ λ }, x, y ∈ X ω.

It follows from [1, 2] that if ω is a modular on X, then the modular space X ω can be equipped with a (nontrivial) metric generated by ω and given by d ω ( x, y ) = inf { λ > 0 : ω λ ( x, y ) ≤ λ }. for any x, y ∈ X ω.

It follows from [1, 2] that if w is a modular on X, then the modular space X w can be equipped with a (nontrivial) metric, generated by w and given by d w ( x, y ) = inf { λ > 0 : w λ ( x, y ) ≤ λ }. for any x, y ∈ X w.

It follows from [13, 14] that if ω is a modular on X, then the modular space (X_{omega}) can be equipped with a (nontrivial) distance, generated by ω and given by d_{omega} x,y)= infbigl{ lambda>0: omega_{ lambda} x,y) leq lambda bigr}, for any (x, y in X_{omega}).