Miscellaneous notes on Thai Gramineae

7 Notes for Miscellaneous Lectures

Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious. 1 Nemirovski Estimate of Mean of Arbitrary Distributions with Bounded Variance The popular Chernoff bounds assume severe restrictions on distribution: it must be...

Notes on Early Land Plants Today. 26. Miscellaneous synonyms in liverworts (Marchantiophyta)

Aneura bipinnata (≡ Riccardia bipinnata, described from Jamaica as Jungermannia bipinnata (“2-pinnata“) by Olof Swartz (1788, 1806), is one of the oldest names in Aneuraceae and one of the poorest known. Swartz' original description and the brief diagnosis in Synopsis Hepaticarum (Gottsche et al. 1846) are the only available descriptions for the species. Stephani (1899) was unable to locate mat...

Miscellaneous preliminaries on arithmetic geometry

One definition of a hyperelliptic curve is a curve C over an algebraically closed field k whose function field K is a degree 2 extension of a purely transcendental extension of k. 1. (a) Show that every hyperelliptic curve is birational to a curve of the form y 2 = f (x) where f ∈ k[x] is a monic squarefree polynomial. (b) Conversely, show that every squarefree f ∈ k[x] gives rise to a hyperell...

Miscellaneous HOL

Miscellaneous HOL