Sentence examples for there exists a maximal from inspiring English sources

Then by Zorn's lemma there exists a maximal linearly independent set of vectors, which by definition must be a basis for V. (It is known that, without the axiom of choice, it is possible for there to be a vector space without a basis).

Then from Lemma 3.4, there exists a maximal element x 0 in ( E, ≤ φ ).

If S is unionly open-valued, then there exists a maximal element x ¯ ∈ E, that is, S ( x ¯ ) = ∅.

If S is transfer open-valued, then there exists a maximal element x ¯ ∈ K, that is, S ( x ¯ ) = ∅.

Furthermore, (Pcap Mthere exists a maximal subgroup (P_{1}) of (P) such that (Pcap Mle P_{1}).

Theorem 3.2 If b l f l > c M e M, then there exists a maximal periodic solution ( u ¯ ( x, t ), v ¯ ( x, t ) ) of problem (1.1 - 1.3 1.1 - 1.3

There existed a maximal MO decoloration efficiency with the change of oxygen velocity.

For data (f, g) vanishing at infinity, we show that there exist a maximal and a minimal g.e.s. to the Cauchy problem and to the associated stationary problem u+divx φ u)=f.

Therefore, for fixed s ∈ ( d + 2, ∞ ), classical results (see [6]) say that for each m, there exist a maximal time T m ∗ ∈ ( 0, ∞ ] and a solution u m to the Euler equations (2) in C ( [ 0, T m ∗ ) ; C s ( R d ) ). In case of dimension 2, it is well-known that T m ∗ = ∞. That is, there is a global solution u m ∈ C ( [ 0, ∞ ) ; C s ( R 2 ) ).

Moreover, if s is transitive, then there exists an s-maximal element of X.

1. G F is a minimal triangulation of G. 2. Let (T, B) be a clique tree of G F. There exists a minimal separator F ∈ F if and only if there exist two adjacent vertices x and y in T where B x)∩ B y)= F. 3. △ H is a maximal set of pairwise parallel minimal separators of G and G △ H = H.