Sentence examples for using the uniqueness of from inspiring English sources

Using the uniqueness of the limit, (g u)=f u)), and hence we are done.

By using the uniqueness of the limit in (mathcal{D}'), we get (xi=delta_{a} u).

Using the uniqueness of limit, we conclude that (x_{ast}=y_{ast}).

Using the uniqueness of the ρ-limit, we get (T x) = x), i.e., (x in operatorname {Fix}(T)).

Noting that, each term on the right-hand side belongs to D(A), using the uniqueness of v t), we have that u(t) ∈ D(A).

end{aligned} By using the uniqueness of the asymptotic center, (Tx = x), so x is a fixed point of T. Hence, (F(T)) is nonempty.

We now discuss the conditions, which guarantee the identifiability of CFO estimation by using the uniqueness properties of the PARAFAC model.

Under the conditions (q,q^{prime}in operatorname{AC}(mathbb{R}_)), (lim_{xrightarrow infty} vert q(x vert +vert q^{prime}(x) vert =0), (sup_{xin mathbb{R}_ [ e^{varepsilonsqrt{x}} vert q^{primeprime}(x vert ] 0), using the uniqueness theorems of analytic functions, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities.

Using the uniqueness theorem of [25] and [29], we get begin{aligned} w_{n1}=w_{n3} hbox {in} Btimes (t_1+delta,t_2-delta ), end{aligned} (7.18 where (delta >0) is determined by the metric and by the domain B (cf. Fig. 9).

Using the uniqueness up to scalars of these functions, the eigenfunctions are determined by the differential equation and can be computed explicitly.

Remark (1) One can use the uniqueness property of limits to obtain the result of Theorem 2.2 by combining with the relationship between these three functions 2 ϕ ( − q ) − f ( q ) = f ( q ) + 4 ψ ( − q ) = b ( q ).