Sentence examples for there exist infinite from inspiring English sources

If (varphi(a)=varphi(b)) for some (a,b inmathbb{D}) with (a neq b), then by using the idea similar to that used in [2, Lemma] there exist infinite sets ({a_{n}}) and ({b_{n}}) in (mathbb{D}) which are disjoint and such that (varphi(a_{n})= varphi(b_{n})).

This axiom eliminates from the models of the first nine axioms those in which there exist infinite descending ∊-chains (i.e., sequences x1, x2, x3, … such that x2 ∊ x1, x3 ∊ x2, …), a phenomenon that does not appear in the model based on an iterative hierarchy described above.

If there exist infinite (n_{i}) s.t.

If (I) is infinite, there exist infinite subsets (I_1 ) and (I_2) of (I) such that (I = I_1 cup I_2).

One fact is that for a given transfer function matrix, there exist infinite number of state space representations related by a similarity transform.

Proposition 4 implies that there exist infinite ways to reach the NE in game as long as the sequential best-response path is admissible.

Does there exist an infinite, residually finite, locally finite group G such that (C^*(G)) is just-infinite?

Does there exist an infinite, residually finite, locally finite group G satisfying a nontrivial identity such that (C^*(G)) is just-infinite?

Under a fixed, there exist an infinite number of threshold pairs achieving a fixed.

For any arbitrary value of, there exist an infinite number of pairs satisfying (A.6); however, to guarantee an intersection between the sensing range of the sensor and, should fall within where (A19)In fact, and are the desired integral bounds.

Haven't there always existed infinite Egyptian jokes to this effect?