Pretentiously detecting power cancellation

Granville and Soundararajan have recently introduced the notion of pretentiousness in the study of multiplicative functions of modulus bounded by 1, essentially the idea that two functions which are similar in a precise sense should exhibit similar behavior. It turns out, somewhat surprisingly, that this does not directly extend to detecting power cancellation - there are multiplicative functions which exhibit as much cancellation as possible in their partial sums that, modified slightly, give rise to functions which exhibit almost as little as possible. We develop two new notions of pretentiousness under which power cancellation can be detected, one of which applies to a much broader class of multiplicative functions

The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point

In recent work, Bhargava and Shankar have shown that the average size of the $2$-Selmer group of an elliptic curve over $\mathbb{Q}$ is exactly $3$, and Bhargava and Ho have shown that the average size of the $2$-Selmer group in the family of elliptic curves with a marked point is exactly $6$. In contrast to these results, we show that the average size of the $2$-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous curves is the Tamagawa ratio, which measures the relative sizes of the Selmer groups associated to the isogeny and its dual. Building on previous work in which we considered the Tamagawa ratio in quadratic twist families, we show that, in the family of all elliptic curves with a two-torsion point, the Tamagawa ratio is essentially governed by a normal distribution with mean zero and growing variance

Effect of Tryptophan Analogs on Derepression of the \u3cem\u3eEscherichia coli\u3c/em\u3e Tryptophan Operon by Indole-3-Propionic Acid

The abilities of 14 tryptophan analogs to repress the tryptophan (trp) operon have been studied in Escherichia coli cells derepressed by incubation with 0.25 mM indole-3-propionic acid (IPA). trp operon expression was monitored by measuring the specific activities of anthranilate synthase (EC 4.1.3.27) and the tryptophan synthase (EC 4.2.1.20) β subunit. Analogs characterized by modification or removal of the α-amino group or the α-carboxyl group did not repress the trp operon. The only analogs among this group that appeared to interact with the trp aporepressor were IPA, which derepressed the trp operon, and d-tryptophan. Analogs with modifications of the indole ring repressed the trp operon to various degrees. 7-Methyl-tryptophan inhibited anthranilate synthase activity and consequently derepressed the trp operon. Additionally, 7-methyltryptophan prevented IPA-mediated derepression but, unlike tryptophan, did so in a non-coordinate manner, with the later enzymes of the operon being relatively more repressed than the early enzymes. The effect of 7-methyltryptophan on IPA-mediated derepression was likely not due to the interaction of IPA with the allosteric site of anthranilate synthase, even though feedback-resistant mutants of anthranilate synthase were partially resistant to derepression by IPA. The effect of 7-methyltryptophan on derepression by IPA was probably due to the effect of the analog-aporepressor complex on trp operon expression