Sentence examples for concentrating solutions from inspiring English sources

Concentrating solutions for fractional problems involving critical or almost critical exponents were considered in [14].

The earlier result concerning the existence of multiple blowing-up and concentrating solutions of the Equation (3) is given by Baraket and Pacard in [24].

In addition, a weaker stable assumption of critical points of the function φ n,m also helps us construct multiple blowing-up and concentrating solutions of the Equation (3).

However, the authors mainly consider the Dirichlet boundary value problem, and little is known for the problem with singular sources satisfying α i ∈ (-1, 0) for some i = 1,..., N. The main purpose of this article is to study how to construct multiple blowing-up and concentrating solutions of the Equation (1) with the mixed boundary conditions and singular sources.

Then through the finite dimensional reduction procedure and the notions of stability of critical points of the asymptotic reductional functional φ n,m, multiple blowing-up and concentrating solutions can be constructed as a small additive perturbation of the initial approximations.

Moreover, u ε remains uniformly bounded on Ω ∪ i = 1 n + m B λ ( p k i ), and sup B λ ( p k i ) u ε → + ∞ for any λ > 0. Let us point out that from the proof of Theorem 1.1 Robin boundary condition can be considered as a perturbation of Dirichlet boundary condition for the problem (3) in using perturbation techniques to construct multiple blowing-up and concentrating solutions.

It is predicted that the pH of the depleting solution decreases and that of the concentrating solution increases during electrodialysis (ED).

Phenol is quite toxic, however, and concentrated solutions cause severe but painless burns of the skin and mucous membranes.

Dilute and concentrate solutions were circulated through the ED cell until the desired product conductivity (≈0.5 mS cm−1) was achieved in the dilute one.

Competition studies were undertaken, in which increasingly concentrated solutions of H3DFO were added to H3CP256 solutions containing [Zr CP256)]+.

An additional advantage of lyophilisation is that it is possible to concentrate solutions by reconstitution of the dried product with a smaller volume (4).