Sentence examples for be the modulus of from inspiring English sources

Let be the modulus of concavity of.

Let be the modulus of smoothness of defined by (2.6).

Let X be a quasi-β-Banach space with the quasi-β-norm ∥ ⋅ ∥ X and K be the modulus of concavity of ∥ ⋅ ∥ X.

[17]Let E be a uniformly convex and uniformly smooth Banach space, δ E be the modulus of convexity of E, and ρ E (t) be the modulus of smoothness of E; then the inequalities 8 d 2 δ E ( | | x - ξ | | ∕ 4 d ) ≤ ϕ ( x, ξ ) ≤ 4 d 2 ρ E ( 4 | | x - ξ | | ∕ d ).

Then a function ρ E : R + → R + is said to be the modulus of smoothness of E if ρ E ( t ) = sup { ∥ x + y ∥ + ∥ x − y ∥ 2 − 1 : ∥ x ∥ = 1, ∥ y ∥ = t }.

Then a function (delta_{E} : [0, 2]rightarrow[0,1]) is said to be the modulus of convexity of E if delta_{E} epsilon)=infbiggl{ 1-frac{Vert x+yVert }{2}:Vert xVert leq1, Vert yVert leq 1, Vert x-yVert geqepsilonbiggr}.

B is the modulus of the field.

where δ is the modulus of convexity of the norm.

Suppose K is the modulus of concavity of || · ||.

where is the modulus of smoothness of, defined by.

Case 3. Let, where is the modulus of and.