Sentence examples for proved uniform from inspiring English sources

By properties of the Schur-convex function, Schur-concavity for a class of symmetric functions is simply proved uniform.

In particular, given that a special persistency of excitation condition is satisfied, we prove uniform global asymptotic stability and semi-global exponential stability of the origin of the state and parameter estimation error, and give explicit lower bounds on the convergence rate of both the state and parameter estimation error dynamics.

We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T, and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N. Finally, we prove uniform pointwise boundedness and power-law decay for ψ up to and including the event horizon H+.

To prove uniform asymptotic stability, let ε 0 : = ε = 1, choose δ 0 : = δ ( 1 ) > 0 corresponding to uniform stability.

McCord [6] provided methods of proving uniform integrability of such functions of (Y_{n}) for the case (r=1).

In this paper, we prove uniform optimal-order error estimates for characteristics-mixed finite element methods for two-dimensional convection-dominated diffusion equations.

In the third section, we consider an approximate problem and get an approximate solution for each n ∈ N. In the forth section, we prove uniform a priori estimates for the approximate solutions.

By virtue of Kahane's contraction principle, additional and product properties of R-bounded operators, see, e.g., Lemma 3.5, Proposition 3.4 in [5], and in view of (3.7), it is sufficient to prove uniform R-boundedness of the following set B = { Q ; ξ ∈ R n ∖ { 0 } }, Q = ∑ | α | ≤ l | λ | 1 − | α | l a α ξ α D.

Here we have set u k : = u | Ω k for k = 1, 2. The aim of the paper is to prove uniform a priori estimates for the solutions of (1.1) and (1.3) under suitable ellipticity and smoothness assumptions on A and C, see Section 2 below for the precise formulations.

They proved the uniform energy decay rates of the solutions, by utilizing the multiplier method.

In particular they proved that uniform convexity implies the property (R).