A function (phiin L_{2}(mathbb{R})) is refinable if it satisfies the following refinement equation: phi=2sum_{kinmathbb{Z}}a k phi 2cdot-k), (2.1) where a is a finitely supported sequence on (mathbb{Z}}a k phi 2cdot-kfinement mask of the refinement function ϕ.
A compactly supported function (phi in L_{2}(mathbb{R})) is refinable if it satisfies the refinement equation begin{aligned} phi=2sum_{ kinmathbb{Z}}tau k phi 2cdot-k), end{aligned} (3) where τ, called the refinement mask of ϕ, is a finitely supported sequence.
Define the Fourier series c ˆ of a sequence { c k } ∈ ℓ 2 ( Z ) by c ˆ : = ∑ k ∈ Z c k e − i k ξ, ξ ∈ R. Then the refinement equation (1.2) becomes ϕ ˆ = a ˆ ( ξ / 2 ) ϕ ˆ ( ξ / 2 ), ξ ∈ R. The function a ˆ is called the refinement mask of ϕ.
In order to overcome this drawback, a refinement approach, called the refined LS1, is presented in Part II and applied to angle-ply rectangular composite laminates.
In order to overcome the drawback of the LS1 model in the vicinity of singularities, a refinement approach, called the refined LS1 is proposed in this part.
To call the collection a refinement of his style — splicing, say, horsehair into arcs of wool or creating sound-wave patterns in the sheer midriff of a black dress — would be to minimize his efforts.
The defendant company, Shoketsu Kinzoku Kogyo Kabushiki, which does business in the United States as the SMC Corporation, told the court that what it called the "doctrinal refinement" of the appeals court's ruling was "a natural and well-justified response to the harmful uncertainty that comes from too flexible a case-by-case approach".
We set these methods within a broader biological M&S approach called the iterative refinement protocol (IR Protocol) [ 27].
We call the resulting graph a refinement of G. Figure 2 (c) shows a situation when we have to retain two edges, leading to a cluster with unresolved one-to-one relationship.
The study calls for the refinement of CPTED concepts to increase the discriminative controllability of potential crime attractors/generators and preventers in the neighbourhood.
