We've learned how to use built-in `R` functions like `dnorm()` and `pnorm()` to analyze distributions, but sometimes it's going to be more helpful to be able to (A) do the math by hand or (B) code ...

Extracytoplasmic function (ECF) sigma factors are a distinct subset of sigma factors that enable bacteria to respond robustly to a myriad of environmental challenges. By directing RNA polymerase to ...

Complete proof that odd perfect numbers do not exist. Structured by σ-function constraints, logical contradiction, and density elimination. 奇数完全数が存在しないことをσ関数の制約・背理法・密度除去により証明。構成 ...