Indeed, pools showed rather heterogeneous expression levels under these induction conditions, most likely due to integration site effects with individual loci following their inherent partial induction schemes.
Statistical methods were employed to compare these expression profiles to search for specific gene expression signatures that might aid in understanding the molecular mechanisms at work in each subtype of B cell malignancy and the differences in tumor latent periods resulting from variation in the PCT induction schemes.
We introduce explicit induction, a new induction scheme targeted to temporal properties, and to interactive development of inductive proofs.
Moreover, Σ01-induction can be shown (Paris and Kirby 1978) to be equivalent to the induction scheme restricted to (roughly) purely universal formulas (Π01-formulas).
Suppose furthermore that A and B regard the collection of predicates for which mathematical induction is permissible as open-ended, and are both willing to accept the other's induction scheme as true.
However, essentially the same system is obtained if one takes just the axioms of Q and the induction scheme restricted to, roughly, purely existential formulas (in technical terms, Σ01-formulas; see below) (this was first showed by Parsons 1970).
Note that unlike Q, PA contains infinitely many axioms, because all (infinitely many) instances of the induction scheme, one corresponding to every formula φ(x) (with at least one free variable) of the language, are taken as axioms.
The innovative idea in explicit induction is to make the induction scheme an explicit part of the specification, where it can be easily controlled, using a highly expressive language like ForSpec.
It contains not just the above axioms of Q governing successor, addition and multiplication, but also defining axioms for all primitive recursive functions (see the entry on recursive functions), and the application of the induction scheme is restricted to quantifier-free formulas (i.e., φ(x) is not allowed to contain any (unbounded) quantifiers).
In contrast to the complex induction scheme of clusters representing upregulated genes, clusters containing genes that decrease upon differentiation form fewer clusters that fall mainly in two categories, of genes suppressed early, at the onset of differentiation (clusters 3 and 18), and of genes downregulated in more gradual fashion (clusters 1, 8 and 13; Figure 3B).
After an induction scheme with intravenous infliximab infusions given at weeks 0, 2, and 6, infliximab is typically administered at a dosage of 3 mg/kg every 8 weeks in combination with MTX.
