Sentence examples for there exist minimal from inspiring English sources

Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length.

There exist minimal period-two solutions ( p, q ) ≤ se ( x ¯, y ¯ ) ≤ se ( r, s ) such that ( p, q ) and ( r, s ) lie at the north-west and south-east corners of B ∗, respectively, and ( x ¯, y ¯ ) lies in its interior.

For example, we prove that the difference of any two functions in R c, c ≥ 0, retains its sign on [ 0, T ), and that there exist minimal and maximal solutions v c, min, v c, max ∈ R c for each c ≥ 0, cf. Theorem 5.

(iii) There exist minimal period-two solutions ( p, q ) ≤ se ( x ¯, y ¯ ) ≤ se ( r, s ) such that ( p, q ) and ( r, s ) lie at the north-west and south-east corners of B ∗, respectively, and ( x ¯, y ¯ ) lies in its interior.  .

Our main result is that for some subspaces there exist minimal-volume shadows that are far from parallelepipeds with respect to the Banach Mazur distance.

We conclude that: (1) the epidemiologic evidence for an increased risk of low birth weight (<2500 g) is insufficient, although there exists limited evidence for birth weight decreases; (2) the evidence for increased preterm delivery is insufficient; and, (3) there exists minimal evidence for decreased birth size.

Since B and C are finite sets and (varnothing ne )B subseteq )C, there exist the minimal sets (R_{mathrm{min},S'} equiv ) Minimal(B)(ne varnothing ), (R_{mathrm{min}} equiv )Minimal(C)(ne varnothing ).

In the previous section, we have proved that there exist coupled minimal and maximal solutions for the differential equation equation (15).

If (1.4) is nonoscillatory, that is, (2.1) possesses a solution which exists on some interval, among all such solutions of (2.1), there exists the minimal one, minimal in the sense that any other solution of (2.1) which exists on some interval satisfies in this interval, see [9, 10] for details.

Then there exist maximal and minimal solutions Y max, Y min ∈ [ Y L, Y U ] for equation (3.2).

Then there exist maximal and minimal solutions X max, X min ∈ [ X L, X U ] for equation (3.1).