Furthermore, the reverse implications are valid.
From the above definition, it is easy to see that the following implications are valid: E is uniformly convex ⇒ E is locally uniformly convex ⇒ E is strictly convex.
In other words, the implications are valid regarding the knowledge formalised in the Boolean network and can also be checked by supplementary human expert knowledge or further literature research, e.g., for co-regulation of genes or possible or forbidden resulting states.
It is clear that the following diagram, where arrows stand for implications, is valid: contraction → quasicontraction ↓ ↓ ordered contraction → ordered quasicontraction.
then the following implication is valid: if (p=p^{r}_{0}(x)), then mathbb{P} left widetilde{V}_{T} (x,p,phi,mathcal{C})=x right)=1.
Moreover, if condition (i) holds, then condition (ii) is equivalent to the following condition if p is such that there exists ((x,p,phi,mathcal {C} in Psi ^{0,x} (p,mathcal {C})) satisfying mathbb{P} left widetilde{V}_{T} (x,p,phi,mathcal{C}) geq x right)=1, (53) then the following implication is valid: if (p=p^{r}_{0}(x)), then mathbb{P} left widetilde{V}_{T} (x,p,phi,mathcal{C})=x right)=1.
Rules 3 and 4 show that the following implications among formulas (a)–(f) are valid: The implications holding among the negations of (a)–(f) follow from these by the law of transposition; e.g., since (a) ⊃ (b) is valid, so is ∼(b) ⊃ ∼(a).
Those discussions have focused on whether the findings are valid, the potential implications, and the best way to move forward.
There are valid criticisms.
These suspicions are valid".
All are valid explanations.
