Sentence examples for first corollary from inspiring English sources

We can now prove our first corollary of Proposition 8.4.

The first corollary is on the commonly used logarithmic transformation, and is applied to a geometric auto-regressive (AR) process, as well as to a positive moving-average (MA) process.

As the first corollary, we shall now prove a generalization of Wiman's theorem for the case of quasiregular mappings where is a warped Riemannian product.

For our first corollary, we take the function ({ F X,Y) = int _0^infty frac{1}{lambda +Y}mathrm{d}lambda +Y}mathrm{d}lambda }), which evidently satisfies the conditions of Theorem 1.1 with (q = -1).

The first corollary is a generalization of Geraghty's fixed point theorem [22] and it is obtained by taking in Theorem 3.1 as R-function (eta t,s) = psi (s) s -t) for all (t,s in [0,+infty[), where ψ is endowed with a suitable property. Let ((Z, d)) be a complete metric space and (h : Z rightarrow Z) be a mapping.

Remark: For the first corollary, we can note that if N1≤N s and N2≤N d, then H sr T ⊗ H rd is full column-rank, which ensures that Ω 1 ∈ C N D N s K × N 2 is full column-rank due to its Khatri-Rao product structure [24].

There is no issue for (n=3) in the second corollary.

The second corollary is on the tan−1 transformation which will turn possibly unstable series into stable ones.

The same reasoning is valid for the third corollary, which is analogous to the second one.

Regarding the second corollary, it corresponds to a special case of our system model where the first relay tier reduces to a single-antenna relay.

A second corollary of our hypothesis is that a β adrenergic receptor agonist should alleviate the SMase C-caused suppression of CFTR current by boosting intracellular cAMP.