Sentence examples for exists a uniformly from inspiring English sources

then there exists a uniformly bounded random variable such that.

(2) There exists a uniformly bounded set (mathcal{X} subset mathcal{C}) such that forallxiinXimboxquad mathbb{P}(xinotinmathcal{X}) < alpha.  .

There exists a uniformly bounded set (mathcal{X} subset mathcal{C}) such that forallxiinXimboxquad mathbb{P}(xinotinmathcal{X}) < alpha.

By Lemma 4.1 there exists a uniformly bounded set (mathcal{X} subsetmathcal{C}) such that mu_{mathrm{H},infty}(mathcal{X}) < epsilon/2 (13) and forallxiinXimbox quad mathbb{P} (xinotinmathcal{X} ) < alpha.

(2) For each (epsilon> 0), there exists a uniformly bounded set (mathcal{X} subsetmathcal{C}) such that (a) (mu_{mathrm{H},infty}(mathcal{X}) < epsilon),   (b) (forallxiinXi): (mathbb{P}(xinotinmathcal{X}) < alpha).    .

For each (epsilon> 0), there exists a uniformly bounded set (mathcal{X} subsetmathcal{C}) such that (a) (mu_{mathrm{H},infty}(mathcal{X}) < epsilon),   (b) (forallxiinXi): (mathbb{P}(xinotinmathcal{X}) < alpha).  .

In particular, if f ( n, x ) is periodic of period ω, then there exists a unique uniformly asymptotically stable periodic solution of system (8) of period ω.

It follows that there exists a unique uniformly asymptotically stable almost periodic solution X = ( x ( n ), u ( n ) ) of system (4) which satisfies x ∗ ≤ x ( n ) ≤ x ∗ and u ∗ ≤ u ( n ) ≤ u ∗ for all n ∈ Z +. □.

Then, there exists a unique uniformly asymptotically stable almost periodic solution X = ( x ( n ), u ( n ) ) of system (4) which satisfies x ∗ ≤ x ( n ) ≤ x ∗ and u ∗ ≤ u ( n ) ≤ u ∗ for all n ∈ Z +.

Thus, we conclude that there exists a unique uniformly asymptotically stable almost periodic solution X = ( p ( n ), u ( n ) ) of system (13) which satisfies ln x ∗ ≤ p ( n ) ≤ ln x ∗ and u ∗ ≤ u ( n ) ≤ u ∗ for all n ∈ Z +.

Based on Lemma 2.4 and Remark 2.5, there exists a unique uniformly asymptotically stable almost periodic solution ( p 1 ( n ), p 2 ( n ) ) of system (B.1), that is, there is a unique uniformly asymptotically stable positive almost periodic solution ( x 1 ∗ ( n ), x 2 ∗ ( n ) ) of system (1.1) which satisfies m i ≤ x i ∗ ( n ) ≤ M i, i = 1, 2, for all n ∈ K.