Because the *p*-parts and global series are closely related, the result above follows from a series of local results concerning the *p*-parts. In particular, we give an explicit recurrence relation on the coefficients of the *p*-parts, which allows us to extend the results of Chinta, Friedberg, and Gunnells [9] to all _ and *n*. Additionally, we show that the *p*-parts of Chinta and Gunnells [10] agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg [4, 5] (in the cases when both constructions apply).

It is a common practice to apply some kind of filter to the comparison study data. These filters go from outliers detection and exclusion to exclusion of the entire data from a participant when its measurements are very “different". When the measurements are not so “different" the usual assumption is that the laboratories are unbiased then the simple mean, the weighted mean or the one way random effects model are applied to obtain estimates of the true value.

Instead we explore methods to analyze these data under weaker assumptions and apply them to some of the available data. More specifically we explore estimation of models assessing the laboratories performance and way to use those fitted models in estimating a consensus value for new study material. This is done in various ways starting with models that allow a separate bias for each lab with each compound at each point in time and then considering generalizations of that. This is done first by exploiting models where, for a particular compound, the bias may be shared over labs or over time and then by modeling systematic biases (which depend on the concentration) by combining data from different labs. As seen in the analyses, the latter models may be more realistic.

Due to uncertainty in the certified reference material analyzing systematic biases leads to a measurement error in linear regression problem. This work has two differences from the standard work in this area. First, it allows heterogeneity in the material being delivered to the lab, whether it be control or study material. Secondly, we make use of Fieller's method for estimation which has not been used in the context before, although others have suggested it. One challenge in using Fieller's method is that explicit expressions for the variance and covariance of the sample variance and covariance of independent but non-identically distributed random variables are needed. These are developed.

Simulations are used to compare the performance of moment/Wald, Fieller and bootstrap methods for getting confidence intervals for the slope in the measurement model. These suggest that the Fieller's method performs better than the bootstrap technique. We also explore four estimators for the variance of the error in the equation in this context and determine that the estimator based on the modified squared residuals outperforms the others.

Homogeneity is a desirable property in control and study samples. Special experiments with nested designs must be conducted for homogeneity analysis and assessment purposes. However, simulation shows that heterogeneity has low impact on the performance of the studied estimators. This work shows that a biased but consistent estimator for the heterogeneity variance can be obtained from the current experimental design.

]]>The proof of our main theorem primarily rests on three lemmas. The first lemma uses the reduction-exact sequence of an abelian survace defined over an unramified extension *K *of Q*p* to give a mod *p*^{2 }condition for detecting when *A* has a nontrival *p*-torsion point defined over *K*. The second lemma employs crystalline Dieudonne theory to count the number of isomorphism classes of lifts of abelian surfaces over F*p* to Z/*p ^{p} *that satisfy the conditions from our first lemma. Finally, the third lemma addresses the issue of the assumption in the first lemma that