Studies of auxin metabolism rarely express their results as a metabolic rate, although the data obtained would often permit such a calculation to be made. We analyze data from 31 previously published papers to quantify the rates of auxin biosynthesis, conjugation, conjugate hydrolysis, and catabolism in seed plants. Most metabolic pathways have rates in the range 10 nM/h–1 μM/h, with the exception of auxin conjugation, which has rates as high as ~100 μM/h. The high rates of conjugation suggest that auxin metabolic sinks may be very small, perhaps as small as a single cell. By contrast, the relatively low rate of auxin biosynthesis requires plants to conserve and recycle auxin during long-distance transport. The consequences for plant development are discussed.

<p style='text-indent:20px;'>The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions extend to this setting:</p><p style='text-indent:20px;'>1. If <inline-formula><tex-math id="M2">\begin{document}$ (a_n) $\end{document}</tex-math></inline-formula> is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.</p><p style='text-indent:20px;'>2. There exists a sequence <inline-formula><tex-math id="M3">\begin{document}$ (r_n) $\end{document}</tex-math></inline-formula> such that every translate is both a rigidity sequence and a set of recurrence.</p><p style='text-indent:20px;'>The first of these results was shown for <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions by Adams [<xref ref-type="bibr" rid="b1">1</xref>], Fayad and Thouvenot [<xref ref-type="bibr" rid="b20">20</xref>], and Badea and Grivaux [<xref ref-type="bibr" rid="b2">2</xref>]. The latter was established in <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula> by Griesmer [<xref ref-type="bibr" rid="b23">23</xref>]. While techniques for handling <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.</p><p style='text-indent:20px;'>As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>, while others exhibit new phenomena.</p>

We prove that the corona product of two graphs has no Laplacian perfect state transfer whenever the first graph has at least two vertices. This complements a result of Coutinho and Liu who showed that no tree of size greater than two has Laplacian perfect state transfer. In contrast, we prove that the corona product of two graphs exhibits Laplacian pretty good state transfer, under some mild conditions. This provides the first known examples of families of graphs with Laplacian pretty good state transfer. Our result extends of the work of Fan and Godsil on double stars to the Laplacian setting. Moreover, we also show that the corona product of any cocktail party graph with a single vertex graph has Laplacian pretty good state transfer, even though odd cocktail party graphs have no perfect state transfer.

We study state transfer in quantum walk on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer.1 maximum degree k which have perfect state transfer. Therefore, the following relaxation of this notion is often more useful to consider. The state transfer between u and v is called "pretty good" (see Godsil [18]) or "almost perfect" (see Vinet and Zhedanov [27]) if the (u, v)-entry of the unitary matrix U(t) can be made arbitrarily close to one.Christandl et al. [9,8] observed that the path P n on n vertices has antipodal perfect state transfer if and only if n = 2, 3. In a striking result, Godsil et al. [21] proved that P n has antipodal pretty good state transfer if and only if n + 1 is a prime, twice a prime, or a power of two. This provides the first family of graphs with pretty good state transfer which correspond to the quantum spin chains originally studied by Bose.Shortly after, Fan and Godsil [12] studied a family of graphs obtained by taking two cones K 1 +K m and then connecting the two conical vertices. They showed that these graphs, which are called double stars, have no perfect state transfer, but have pretty good state transfer between the two conical vertices if and only if 4m + 1 is a perfect square. These graphs provide the second family of graphs known to have pretty good state transfer.In this work, we provide new families of graphs with pretty good state transfer. Our constructions are based on a natural generalization of Fan and Godsil's double stars. The corona product of an n-vertex graph G with another graph H, typically denoted G • H, is obtained by taking n copies of the cone K 1 + H and by connecting the conical vertices according to G. In a corona product G • H, we sometimes call G the base graph and H the pendant graph. This graph product was introduced by Frucht and Harary [15] in their study of automorphism groups of graphs which are obtained by wreath products.We first observe that perfect state transfer on corona products is extremely rare. This is mainly due to the specific forms of the corona eigenvalues (which unsurprisingly resemble the eigenvalues of cones) coupled with the fact that periodicity is a necessary condition for perfect state transfer. Our negative results apply to corona families G • H when H is either the empty or the complete graph, under suitable conditions on G. In a companion work [1], we observed an optimal negative result which holds for all H but in a Laplacian setting.Given that perfect state transfer is rare, our subsequent results mainly focus on pretty good state transfer. We prove that the family of graphs K 2 • K m , which are called barbell graphs (see Ghosh et al. [16]), admit pretty good state transfer for all m. Here, state transfer occurs between the two vertices of K 2 . This is in contrast to the double stars K 2 • K m where pretty good state transfer requires number-theoretic conditions on m.We observe something curious for corona products when the base graph is c...

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