Sentence examples for denotes the completion of from inspiring English sources

We shall prove that for every second countable (not precompact) group G, we have Zt LUC(G ∗)="M(Gˆ), where ˆG denotes the completion of G with respect to its right uniform structure (if G is precompact, then Zt LUC G ∗)= LUC(G ∗, of course).

(mathbb{Z}_{p}) denotes the ring of integers, (mathbb{Q}_{p}) denotes the field of p-adic numbers, and (mathbb{C}_{p}) denotes the completion of the algebraic closure of (mathbb{Q}_{p}).

Throughout this paper, we make use of the following notations: (mathbb{Z} _{p}) denotes the ring of p-adic rational integers, ℚ denotes the field of rational numbers, (mathbb{Q} _{p}) denotes the field of p-adic rational numbers and (mathbb{C} _{p}) denotes the completion of algebraic closure of (mathbb{Q} _{p}).

Throughout this work, we use the following notations, where Z p denotes the ring of p-adic rational integers, ℚ denotes the field of rational numbers, Q p denotes the field of p-adic rational numbers, and C p denotes the completion of algebraic closure of Q p. Let ℕ be the set of natural numbers and N ∗ = N ∪ { 0 }.

By Z p, we denote the ring of p -adic rational integers, Q denotes the field of rational numbers, Q p denotes the field of p -adic rational numbers, and C p denotes the completion of algebraic closure of Q p. Let N be the set of natural numbers and N ∗ = N ∪ 0. The normalized p -adic absolute value is defined by p p = 1 p. In this paper, we assume q - 1 p < 1 as an indeterminate.

Let D 1, 2 ( R + n ) denote the completion of C c ∞ ( R ¯ + n ) with respect to the Dirichlet norm.

Assume that: (i) the multifunction F is (mathcal{T}_{mu}otimesmathcal{B}(X -measurable ((mathcal {T}_{mu}) denoting the completion of the Borel σ-algebra (mathcal{B}(T)) of T with respect to the measure μ);   (ii) for a.e.

end{aligned}  . the multifunction F is (mathcal{T}_{mu}otimesmathcal{B}(X -measurable ((mathcal {T}_{mu}) denoting the completion of the Borel σ-algebra (mathcal{B}(T)) of T with respect to the measure μ); for a.e.

We use (H^1_0(Omega )) to denote the completion of (C_0^{infty }(Omega )) with respect to the norm: begin{aligned} ||z||_{H^1_0(Omega )}=left( int _{Omega }left( |nabla z|^2-sigma frac{ z^2}{|x|^2}right) mathrm{d}xright) ^{frac{1}{2}}.

Let (overline{L(Gamma ) }) be the completion of the topological algebra (L(Gamma ).) Let (overline{L(Gamma ) }_i) denote the completion of the homogeneous component (L(Gamma )_i) of degree i in the algebra (overline{L(Gamma ) }.) The main focus of this paper will be on the completion begin{aligned} {widehat{L}}(Gamma )=sum limits _{iin {mathbb {Z}}} overline{L(Gamma )_i}.

Let (mathscr{D}^{2, 2}(mathbb{R}^{N})) denote the completion of (mathscr{C}_{0}^{infty}(mathbb{R}^{N})) under the norm ((int_{mathbb{R}^{N}} vert Delta u vert ^{2},dx)^{1/2}), associated with the inner product given by (langle u, varphirangle=int_{mathbb{R}^{N}}Delta uDeltavarphi,dx).