Although the analysis via four different methods of estimation is consistent and valid, the existence of ethnic Turks in Bulgaria obstructs the analysis and induces an incorrect proposition.
For valid spines, existence of a neck is the first decision point in the scheme.
The conclusion of Theorem 3.1 is still valid if the existence of a w-monotone lower solution for problem (3.1) is replaced by the existence of a w-monotone upper solution for problem (3.1).
"That warrant remains valid and in existence.
If (1.12) is not valid, then the existence of both an escape solution and a homoclinic solution is proved in [12], provided that satisfies moreover.
It is worth remarking that if (3.13) is not valid, then the existence of a lower solution α and an upper solution β with (alpha (x leqbeta(x)) in ([0,1]) cannot guarantee the existence of solutions in the order interval ([alpha(x), beta(x)]).
To measure people's knowledge, the existence of valid and accurate tool such as questionnaires is very important.
As with some later feminist criticisms of Freud, Horney attempted to retrieve female sexuality, and by extension a valid form of feminine existence, by appealing to a genuinely independent nature and holding culture culpable for women's subordinate status.
A similar procedure is valid to show the existence of a global minimum for Problem 2. However, for the sake of completeness, we describe it as given in this section.
While each of these three descriptions casts a different light on why the inference to God's existence is valid, Bonaventure here seems to prefer "the highest truth" because the definition of truth as "indivision" is the basis for re-conceiving the ontological argument as establishing an "indivision" between the middle term and the predicate "exists".
