f is called: (1) totally convex at x ∈ int ( dom f ) if its modulus of total convexity at x, that is, the function ν f : int ( dom f ) × [ 0, + ∞ ) → [ 0, + ∞ ) defined by ν f ( x, t ) : = inf { D f ( y, x ) : y ∈ dom f, ∥ y − x ∥ = t }, .
Let f : E → (-∞, +∞] be a convex and Gâteaux differentiable function. The function f is said to be totally convex at x ∈ int dom f if its modulus of total convexity at x, that is, the function ν f : int dom f × [0, +∞) → [0, +∞] defined by. is positive, whenever t > 0. The function f is said to be totally convex when it is totally convex at every point x ∈ int dom f.
With a view to achieve optimized gradation, there are various aggregate gradation tools or methods, such as, gradation curve, individual percent retained (IPR) curve, "8-18" band and combined fineness modulus (fineness modulus of total aggregate) etc.
This rate Ψ depends - in addition to the data on which Φ depends (see Remark 2.3 iii)) - only on a modulus of total boundedness for K.
and then the modulus of total convexity is the function ν f : D ∘ × [ 0, ∞ ) → [ 0, ∞ ] defined by ν f ( x, t ) = inf { D f ( y, x ) : y ∈ D, | | y - x | | = t }.
The modulus of total convexity of f at x ∈ intdom f is the function v f ( x, t ) : [ 0, + ∞ ) → [ 0, + ∞ ) defined by v f ( x, t ) : = inf { D f ( y, x ) : y ∈ dom f, ∥ y − x ∥ = t }.
The influence of both the increasing incorporation of PUR foam waste and the presence of superplasticizer on the workability, bulk density, mass loss, drying shrinkage, compressive strength, dynamic modulus of elasticity, total porosity, gas permeability and chloride diffusion coefficient of the different concretes, has been investigated and analyzed.
The modulus of the total convexity of f at x ∈ int dom f is the function v f ( x, t ) : [ 0, + ∞ ) → [ 0, + ∞ ) defined by v f ( x, t ) : = inf { D f ( y, x ) : y ∈ dom f, ∥ y − x ∥ = t }.
Let f : E → ( − ∞, + ∞ ] be a Gâteaux differentiable function. The modulus of the total convexity of f at x ∈ dom f is the function ν f ( x, ⋅ ) : [ 0, + ∞ ) → [ 0, + ∞ ] defined by ν f ( x, t ) : = inf { D f ( y, x ) : y ∈ dom f, ∥ y − x ∥ = t }.
It was also found that the drying shrinkage property in concrete can be affected by the static modulus of elasticity and total amount of water content, in which the water absorption ratio, unit mass of aggregate and unit water content were considered.
Thirdly, water production modulus presents the ratio of total water resource to the total area of the region.
