Infiniteness of Double Coset Collections in Algebraic Groups

Let $G$ be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups $X$ and $P$, is the double coset collection $X\backslash G/P$ finite or infinite? We limit ourselves to the case where $X$ is maximal rank and reductive and $P$ parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those $X$ which are spherical. Finally, excluding a case in $F_4$, we show that if $X\backslash G/P$ is finite then $X$ is spherical or the Levi factor of $P$ is spherical. This implies that it is rare for $X\backslash G/P$ to be finite. The primary method of proof is to descend to calculations at the finite group level and then to use elementary character theory.Comment: 24 page

Unipotent classes in the classical groups parameterized by subgroups

This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.Comment: 14 pages. Minor changes and rewriting compared to version

Evaluating the Effects of SARS-CoV-2 Spike Mutation D614G on Transmissibility and Pathogenicity.

Global dispersal and increasing frequency of the SARS-CoV-2 spike protein variant D614G are suggestive of a selective advantage but may also be due to a random founder effect. We investigate the hypothesis for positive selection of spike D614G in the United Kingdom using more than 25,000 whole genome SARS-CoV-2 sequences. Despite the availability of a large dataset, well represented by both spike 614 variants, not all approaches showed a conclusive signal of positive selection. Population genetic analysis indicates that 614G increases in frequency relative to 614D in a manner consistent with a selective advantage. We do not find any indication that patients infected with the spike 614G variant have higher COVID-19 mortality or clinical severity, but 614G is associated with higher viral load and younger age of patients. Significant differences in growth and size of 614G phylogenetic clusters indicate a need for continued study of this variant

Evaluating the Effects of SARS-CoV-2 Spike Mutation D614G on Transmissibility and Pathogenicity

Global dispersal and increasing frequency of the SARS-CoV-2 spike protein variant D614G are suggestive of a selective advantage but may also be due to a random founder effect. We investigate the hypothesis for positive selection of spike D614G in the United Kingdom using more than 25,000 whole genome SARS-CoV-2 sequences. Despite the availability of a large dataset, well represented by both spike 614 variants, not all approaches showed a conclusive signal of positive selection. Population genetic analysis indicates that 614G increases in frequency relative to 614D in a manner consistent with a selective advantage. We do not find any indication that patients infected with the spike 614G variant have higher COVID-19 mortality or clinical severity, but 614G is associated with higher viral load and younger age of patients. Significant differences in growth and size of 614G phylogenetic clusters indicate a need for continued study of this variant

UNIQUE FACTORIZATION IN CYCLOTOMIC INTEGERS OF DEGREE SEVEN

Abstract. This paper provides a survey of some basic results in algebraic number theory and applies this material to prove that the cyclotomic integers generated by a seventh root of unity are a unique factorization domain. Part of the proof uses the computer algebra system Maple to find and verify factorizations. The proofs use a combination of historic and modern techniques and some attempt has been made to discuss the history of this material