Notes on various things

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It may be useful to have a “dump document” or "junk file" where you put your discards that you cut out of the paper you are working on. Keeping old draft is a separate good thing to do. Name your working document 12a-piesplitting.txt, and your dump document 00dump-piesplitting.txt, and your notes file 00piesplitting.txt. says:

Although the symbols {\displaystyle \mathbb {R} _{+}}\mathbb {R} _{+} and {\displaystyle \mathbb {R} ^{+}}{\mathbb {R}}^Template:+ are ambiguously used for either of these, the notation {\displaystyle \mathbb {R} _{+}}\mathbb {R} _{+} or {\displaystyle \mathbb {R} ^{+}}{\mathbb {R}}^Template:+ for {\displaystyle \left\{x\in \mathbb {R} \mid x\geq 0\right\}}{\displaystyle \left\{x\in \mathbb {R} \mid x\geq 0\right\}} and {\displaystyle \mathbb {R} _{+}^{*}}{\displaystyle \mathbb {R} _{+}^{*}} or {\displaystyle \mathbb {R} _{*}^{+}}{\displaystyle \mathbb {R} _{*}^{+}} for {\displaystyle \left\{x\in \mathbb {R} \mid x>0\right\}}{\displaystyle \left\{x\in \mathbb {R} \mid x>0\right\}} has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians.

Robin Hanson in You Don't Understand Our Culture says

What are the key differences between GMU econ blogger culture and mainstream intellectual culture?

D. Bnonn Tennant says on Twitter:

When women hold power in a church, officially or unofficially, two things happen: (1) They strive to include anyone agreeable, regardless of error; (2) They strive to exclude anyone disagreeable, regardless of orthodoxy.

This is how women are designed, and it is good—in its proper context. This is why the proper context for feminine influence is under masculine rulership. False teachers are experts at being agreeable; men mimicking Jesus and the prophets are highly proficient at the other thing.

In an ecclesial context, therefore, women's social instincts are roughly inversely proportional to ensuring orthodoxy. Women are likely to approve and endorse flatterers, hirelings, soft men; likely to disapprove and ostracize truth-tellers, shepherds, tough men.