Sentence examples for a fixed natural from inspiring English sources

Let k≥2 be a fixed natural number.

Let k be a fixed natural number.

Let (ngeq1) be a fixed natural number and (varepsilon>0).

For a fixed natural number (ngeq1) let (N={0,1,ldots,n}).

In this section is a fixed natural number, will be a fixed complete metric space, and all the GIFSs are of order and have the form, where is a natural number.

Theorem 3.5 Let n 0 be a fixed natural number, f : A → B a mapping with f ( 0 ) = 0 and let ϕ : A 5 → [ 0, ∞ ) be a function such that (22).

Then, there exists a constant such that for any fixed natural number, (2.8).

Then there exists a constant, independent of, such that for any fixed natural number, (2.9).

Let M : R → 2 R be a set-valued mapping defined by M ( x ) = { x, x 2, x 3, …, x n }, for any fixed natural number  n.

For (2le qle infty ) and any fixed natural number n, assume (C_{1}2^{m}le nle C_{1}^{2}2^{m}) with (C_{1}>0) to be specified later.

end{aligned} Therefore, Delta_{tau}^{m}y(t)+bar{q}(t)sum _{i=0}^{n}p^{i}ybigl(g(t -irbigr)< 0 holds for any fixed nat -irbigrber n and for all large enough t.