BackgroundqPCR has established itself as the technique of choice for the quantification of gene expression. Procedures for conducting qPCR have received significant attention; however, more rigorous approaches to the statistical analysis of qPCR data are needed.ResultsHere we develop a mathematical model, termed the Common Base Method, for analysis of qPCR data based on threshold cycle values (C q) and efficiencies of reactions (E). The Common Base Method keeps all calculations in the logscale as long as possible by working with log10(E) ∙ C q, which we call the efficiency-weighted C q value; subsequent statistical analyses are then applied in the logscale. We show how efficiency-weighted C q values may be analyzed using a simple paired or unpaired experimental design and develop blocking methods to help reduce unexplained variation.ConclusionsThe Common Base Method has several advantages. It allows for the incorporation of well-specific efficiencies and multiple reference genes. The method does not necessitate the pairing of samples that must be performed using traditional analysis methods in order to calculate relative expression ratios. Our method is also simple enough to be implemented in any spreadsheet or statistical software without additional scripts or proprietary components.

Abstract. The intent of this paper is to present a set of axioms that are sufficient for a closure operation to generate a balanced big Cohen-Macaulay module B over a complete local domain R. Conversely, we show that if such a B exists over R, then there exists a closure operation that satisfies the given axioms.In equal characteristic, the tight closure operation has been used to present proofs of the existence of balanced big Cohen-Macaulay modules and algebras. In this article, we present a list of seven axioms for a closure operation defined for finitely generated modules over a complete local domain R. After deriving some simpler consequences of the axioms (including colon-capturing), we prove that a closure operation satisfying the axioms implies the existence of a balanced big Cohen-Macaulay module over R. A balanced big Cohen-Macaulay module B over (R, m) is an R-module where every system of parameters in R is a regular sequence on B (i.e., if x 1 , . . . , x k+1 forms part of a system of parameters in R and b ∈ (x 1 , . . . , x k ) : B x k+1 , then b ∈ (x 1 , . . . , x k )B) and mB = B. Our main tools for proving this theorem will be Hochster's method of modifications [Ho1] and the use of analogues of phantom extensions developed by Hochster and Huneke [HH3].We then show that the existence of a balanced big Cohen-Macaulay module over a complete local domain can be used to create a closure operation that satisfies all of the axioms.Finally, we demonstrate that all of the axioms are satisfied by tight closure. We also examine the axioms in relation to several other common closure operations associated with tight closure theory. The closure axioms and consequencesLet R be a complete local domain, and let N ⊆ M be finitely generated modules. An operation satisfying Axioms (1)-(5) below will be called a closure operation and will be denoted by N M for the closure of N within M . Throughout the paper we will

Abstract. In this article, we delve into the properties possessed by algebras, which we have termed seeds, that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke's "weakly functorial" existence result for big Cohen-Macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic. We also show that every seed over a positive characteristic ring (R, m) maps to a balanced big Cohen-Macaulay R-algebra that is an absolutely integrally closed, m-adically separated, quasilocal domain.

Let R be a commutative, local, Noetherian ring. In a past article, the first author developed a theory of R-algebras, termed seeds, that can be mapped to balanced big Cohen-Macaulay R-algebras. In prime characteristic p, seeds can be characterized based on the existence of certain colon-killers, integral extensions of seeds are seeds, tensor products of seeds are seeds, and the seed property is stable under base change between complete, local domains. As a result, there exist directed systems of big Cohen-Macaulay algebras over complete, local domains. In this work, we will show that these properties can be extended to analogous results in equal characteristic zero. The primary tool for the extension will be the notion of ultraproducts for commutative rings as developed by Schoutens and Aschenbrenner.For a local ring (R, m), a big Cohen-Macaulay R-algebra B is an R-algebra such that some system of parameters for R forms a regular sequence on B with the extra property that mB = B to ensure that B is not trivial. If this is true for every system of parameters for R, then B is a balanced big Cohen-Macaulay algebra. Big Cohen-Macaulay algebras were first shown to exist in [HH92] using characteristic p methods related to tight closure theory (see [HH90]). Hochster and Huneke proved that the absolute integral extension R + is a balanced big Cohen-Macaulay R-algebra when R is an excellent, local, domain of prime characteristic. These results were extended in further articles. For example, in [Ho94] Hochster makes explicit use of tight closure to provide an alternative proof of the existence of big Cohen-Macaulay algebras in prime characteristic based on the notion of algebra modifications. In [HH95] Hochster and Huneke proved the existence of balanced big Cohen-Macaulay algebras for rings containing a field of characteristic zero and proved the "weakly functorial" existence of big Cohen-Macaulay algebras, i.e., given complete local domains of equal characteristic R → S, there exists a balanced big Cohen-Macaulay R-algebra B and a balanced big Cohen-Macaulay S-algebra C such that B → C extends the map R → S. The results in equal characteristic zero are based on reductions to prime characteristic that rely on Artin approximation.In [D07], the first author introduced the notion of seed algebras. Given a local, Noetherian ring R, a seed algebra over R is an R-algebra S such that there exists an R-algebra map S → B where B is a balanced big Cohen-Macaulay Ralgebra. Some of the noteworthy results in that paper for rings of prime characteristic are Theorem 4.8 (which characterizes seeds based on durable, colon-killers),