Sentence examples for equivalent provided that from inspiring English sources

We show that the ac-parts ˜Aac and Aac0 of A˜="A˜⁎∈ExtA and A0 are unitarily equivalent provided that the resolvent difference KA˜:="(A˜−i)−1−(A0−is−1 is compact and the Weyl function M of the pair {A,A0} admits weak boundary limits M t):= w-limy→+0M(t+iy) for a.e.

The two problems (19) and (20) are equivalent provided that the underlying graph is strongly connected [20].

The optimization theory asserts that problems (BP ε ) and (QP λ ) are equivalent provided that the parameters ε and λ satisfy certain relationship [17].

Let T / T s M = N where N is an integer, then the block diagrams depicted in Figure 2 a,b are equivalent, provided that H ˜ M w satisfies H ˜ M w = 1 T s M H M w T s M, w ≤ π b. (13) Figure 2 Interpolation identity for bandlimited signals in the LCT domain.

Then, the block diagrams illustrated in Figure 1 a,b are equivalent, provided that H ˜ M w satisfies H ˜ M w = H M w T s M, w ≤ π b. (4) Figure 1 Sampling identity for bandlimited signals in the LCT domain.

This implicates a detection range from 100 to 5000 leptospires per ml or tissue equivalent, provided that the DNA extraction is efficient.

Then (f(Z t ))t≥0 and (Z t )t≥0 are equivalent in law provided that f(Z0)=Z0.

[8] The two non-negative sequences x = (x k ) and y = (y k ) are said to be asymptotically equivalent of multiple L provided that lim k → ∞ x k y k = L, Open image in new window.

Definition 3.3 Let θ be a lacunary sequence; two number sequences x = ( x k ) and y = ( y k ) are strong ℐ-asymptotically lacunary equivalent of multiple L provided that { r ∈ N : 1 h r ∑ k ∈ I r | x k y k − L | ≥ ε } ∈ I. (denoted by x ∼ N θ L ( I ) y ) and strong simply ℐ-asymptotically lacunary equivalent if L = 1.

Definition 3.10 Two non-negative double sequences ( x k, l ) and ( y k, l ) are said to be strongly asymptotically lacunary equivalent of multiple L provided that P − lim r, s 1 h r, s ∑ ( k, l ) ∈ J r, s | x k, l y k, l − L | = 0. denoted by ( x k, l ) ∼ [ N θ ¯ ] L ( y k, l ) and simply strongly asymptotically lacunary equivalent if L = 1.

Arbitrarily-branching systems are inherently difficult to solve, a fact recognised early by Rall, who proposed a method of mapping the branching structure onto an equivalent cylinder provided that certain geometrical restrictions were satisfied [12].