To be more precise, (V^{varepsilon }) is the closure, with respect to the (H^{1}( Omega_{1 varepsilon}))-norm, of the set of the functions in (C^{infty}(Omega_{1 varepsilon})) with a compact support contained in Ω.
We consider a continuous function with the support contained in and (3.2).
By setting (v = e_i ) on ([a-2varepsilon, b+2varepsilon ]), (v = 0) outside ([a-3varepsilon, b+3varepsilon ]), we can let (u_{n}) be a smooth approximation to (mathbf {1}_{[a,b]}e_j), with support contained in ([a-varepsilon, b+varepsilon ]).
Let (mathcal{D}(Omega)) be the space of (mathcal{C}^{infty}) functions with compact support contained in Ω.
Let (psiin C^{infty}(mathbb{R}^{n})) be real, radial, have support contained in ({x:|x|le1}), and (int_{mathbb{R}^{n}}psi(x),dx=0).
Theorem 2.1 Assume that m 0 = u 0 − u 0 x x has compact support, contained in the interval [ α m 0, β m 0 ], and that ρ 0 is also compactly supported, with support contained in [ α ρ 0, β ρ 0 ].
Since q x ≥ 0 and ρ 0 is compactly supported, it follows from this relation that ρ ( ⋅, t ) is compactly supported for all times t ∈ [ 0, T ), with support contained in the interval [ q ( α ρ 0, t ), q ( β ρ 0, t ) ]. Setting α = min { α m 0, α ρ 0 }, β = max { β m 0, β ρ 0 }.
Due to d d t m ( q ( x, t ), t ) q x k 1 ( x, t ) = 0, on x ∈ R − [ α ρ 0, β ρ 0 ], it follows that m ( ⋅, t ) is compactly supported, with its support contained in the interval [ q ( α, t ), q ( β, t ) ], for all t ∈ [ 0, T ). □. In order to prove Theorem 2.3, we will use the following result. Lemma 2.2 [9].
So y is compactly supported with its support contained in the interval [ q ( α, t ), q ( β, t ) ]. Therefore the following functions are well defined: E + ( t ) = ∫ R e x y ( x, t ) d x and E − ( t ) = ∫ R e − x y ( x, t ) d x, with E + ( t ) = ∫ R e x y 0 d x = 0 and E − ( t ) = ∫ R e − x y 0 d x = 0. (4.1).
For any set (Ksubsetmathbb{R}), the subspace (M^{gamma}_{K}) of (M^{gamma}) is defined as the set of functions (uin M^{gamma}) with support contained in K: begin{aligned} M^{gamma}_{K}=M^{gamma}_{K}( mathbb{R};X_{0},X_{1})= bigl{ psiin M ^{gamma}, operatorname{supp}psisubseteq K bigr}.
In addition to support contained in the published literature, the cooperative relationships relating to chromatin modifications are indirectly supported by partial correlation analysis based on experimental data.
