mm-bigness in compatible systems

Taylor-Wiles type lifting theorems allow one to deduce that for $\rho$ a "sufficiently nice" $l$-adic representation of the absolute Galois group of a number field whose semi-simplified reduction modulo $l$, denoted $\overline{\rho}$, comes from an automorphic representation then so does $\rho$. The recent lifting theorems of Barnet-Lamb-Gee-Geraghty-Taylor impose a technical condition, called \emph{$m$-big}, upon the residual representation $\overline{\rho}$. Snowden-Wiles proved that for a sufficiently irreducible compatible system of Galois representations, the residual images are \emph{big} at a set of places of Dirichlet density $1$. We demonstrate the analogous result in the \emph{$m$-big} setting using a mild generalization of their argument

pp-adic Langlands functoriality for the definite unitary group

We formalise a notion of $p$-adic Langlands functoriality for the definite unitary group. This extends the classical notion of Langlands functoriality to the setting of eigenvarieties. We apply some results of Chenevier to obtain some cases of $p$-adic Langlands functoriality by interpolating known cases of classical Langlands functoriality

An example motivated discourse of the chi-squared test of association (2 by 2)

An example application of the chi-square test of association is given through a set of questions and answers. The emphasis is on methodological aspects and what can be legitimately inferred, if anything. It is strongly highlighted that statistical conclusions can be drawn from a study, but these statistical conclusions might not translate into research or scientific conclusions

A primer on validity and design terminology in comparative designs

An overview of terminology (independent and dependent designs, dependent and independent variables) and validity (internal, external, and measurement) is given using six different scenarios. For simplicity of exposition, all examples have factors at two levels

A primer on statistical hypotheses and statistical errors

An abstract overview of some of the logic and statistical considerations in quantitative inference is give

An example motivated discourse of the independent samples t-test and the Welch test

Analyses are given for two example scenarios. Both scenarios comprise two independent samples. One scenario is analysed using the independent samples t-test; the other the Welch test. Commentary on the designs and limitations is given through a question and answer process

That assumption of normality

Many statistical tests are based around an assumption of “normality”. The reasoning for this choice of distribution, whether to test, what to test, and how to test for normality is covered along with practical recommendations

Cohen’s d for two independent samples

An exposition of standardized effect for two independent samples (under an assumption of normality) is given along with an insight into interpretation and reportin

An example motivated discourse of the paired samples t-test

An application of the paired samples t-test is used to discuss the logic underpinning the test and to consider what may be legitimately inferred