Thysanoptera diversity associated with Mexican lemon (Citrus aurantifolia Christm.)

Objective: To calculate the monthly relative abundance of Thysanoptera species, according to the Margalef, Simpson and Shannon-Wiener diversity indices. Design / Methodology / Approach: The work was carried out in three geographic units with conventional management, during January-December, 2019 in the Reserva de la Biosfera Zicuirán-Infiernillo (Biosphere Reserve), Michoacán, Mexico. In each unit, 10 trees were selected through simple random sampling. Thrips counts were performed on ten shoots per tree every 15 d, for a total of 7200 shoots in the three geographic units. Thysanoptera individuals were placed in entomological jars. The variables were: number of thrips collected per shoot in sampled tree and geographic unit (orchard). To estimate the specific richness and structure of species, the program "calculation of diversity indices DIVERS" was used. Results: In the three geographical units studied, the recorded presence of Thysanoptera accounted for 12 to 17 species. For Nueva Italia 12 recorded species, two were permanent (16.66%), five abundant (41.66%), one scarce (8.3%) and four rare (33.33%). In Zicuirán, three species were permanent (17.64%), six abundant (35.29%), two scarce (11.76%) and six rare (35.29%). In Los Hoyos, four species were permanent (26.66%), four abundant (26.66%) and seven rare (46.66%). The abundance of species was represented by the genus Frankliniella and the species Scolothrips sexmaculatus and Scirtothrips citri. The highest species richness and abundance was found from January to May. In October and November, the value of the calculated indices was zero, which shows less richness and abundance of individuals. The best species uniformity was recorded during January and December, which meant a more stable and homogeneous relation. Study limitations/Implications. Pest resurgence, presence of Candidatus Liberibacter spp. and its vector Diaphorina citri. Findings / Conclusions: in Nueva Italia, 12 species were taxonomically determined; in Los Hoyos 15, and in Zicuirán 17 species, which are reported for the first time in the state of Michoacán, Mexico. At the geographic unit "Los Hoyos" diversity was higher, uniform and stable.

On properties of the lower central series of associative algebras

We give an accessible introduction into the theory of lower central series of associative algebras, exhibiting the interplay between algebra, geometry and representation theory that is characteristic for this subject, and discuss some open questions. In particular, we provide shorter and clearer proofs of the main results of this theory. We also discuss some new theoretical and computational results and conjectures on the lower central series of the free algebra in two generators modulo a generic homogeneous relation.

The structure of Rényi entropic inequalities

We investigate the universal inequalities relating the α -Rényi entropies of the marginals of a multi-partite quantum state. This is in analogy to the same question for the Shannon and von Neumann entropies ( α =1), which are known to satisfy several non-trivial inequalities such as strong subadditivity. Somewhat surprisingly, we find for 0< α <1 that the only inequality is non-negativity: in other words, any collection of non-negative numbers assigned to the non-empty subsets of n parties can be arbitrarily well approximated by the α -entropies of the 2 n −1 marginals of a quantum state. For α >1, we show analogously that there are no non-trivial homogeneous (in particular, no linear) inequalities. On the other hand, it is known that there are further, nonlinear and indeed non-homogeneous, inequalities delimiting the α -entropies of a general quantum state. Finally, we also treat the case of Rényi entropies restricted to classical states (i.e. probability distributions), which, in addition to non-negativity, are also subject to monotonicity. For α ≠0,1, we show that this is the only other homogeneous relation.