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Let $$f:C \rightarrow \mathbb {P}^1$$ be a degree k genus g cover. The stratification of line bundles $$L \in {{\,\mathrm{Pic}\,}}^d(C)$$ by the splitting type of $$f_*L$$ is a refinement of the stratification by Brill–Noether loci $$W^r_d(C)$$ . We prove that for general degree k covers, these strata are smooth of the expected dimension. In particular, this determines the dimensions of all irreducible components of $$W^r_d(C)$$ for a general k-gonal curve (there are often components of different dimensions), extending results of Pflueger (Adv Math 312:46–63, 2017) and Jensen and Ranganathan (Brill–Noether theory for curves of a fixed gonality, arXiv:1701.06579 , 2017). The results here apply over any algebraically closed field.

Poyang Lake is China's largest freshwater lake, and it has been an internationally important wintering ground for migratory waterbirds. Based on waterbird censuses from 2001 to 2016, community structure and abundance trends of waterbirds were analyzed, as well as the potential correlations with the water level of Poyang Lake. The results showed that the annual average number of waterbirds in Poyang Lake was 426,707 ± 150,170, with 111 species from 17 families. Waterfowl was the most abundant family accounting for 74.4 ± 8.8% of all individuals, followed by shorebirds (14.8 ± 8.5%), wading birds (6.5 ± 1.8%) and open-water/waterbirds (4.4 ± 1.9%). Although waterbird abundance fluctuated dramatically, there were no significant trends in the abundance of most guilds or in total waterbird abundance; only geese significantly increased among the eight groups. Analysis of trends of 37 relatively abundant or regularly occurring species indicated that population trends appeared to be species-specific. As for the correlations between water level and waterbird abundance, only shorebirds showed significant correlations with average July water level, average water level, and high water level duration in wet season among four guilds, i.e. waterfowl, wading bird, shorebird and, open-water and waterbird. At the group level, abundances of swans, geese, and ducks were significantly correlated with monthly average water level during the wet season, and wader abundance was significantly correlated with average water levels and high water level duration during the wet season. Correlations between population abundance and monthly water level also exhibited species-specific patterns.

Recent work of Kass--Wickelgren gives an enriched count of the $27$ lines on a smooth cubic surface over arbitrary fields. Their approach using $\mathbb{A}^1$-enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related problem of the $28$ bitangents to a smooth plane quartic. However, it turns out the relevant vector bundle is not relatively orientable and new ideas are needed to produce enriched counts. We introduce a fixed "line at infinity," which leads to enriched counts of bitangents that depend on their geometry relative to the quartic and this distinguished line., 17 pages, comments welcome!

Biological systems are typically dependent on transportation networks for the efficient distribution of resources and information. Revealing the decentralized mechanisms underlying the generative process of these networks is key in our global understanding of their functions and is of interest to design, manage and improve human transport systems. Ants are a particularly interesting taxon to address these issues because some species build multi-sink multi-source transport networks analogous to human ones. Here, by combining empirical field data and modelling at several scales of description, we show that pre-existing mechanisms of recruitment with positive feedback involved in foraging can account for the structure of complex ant transport networks. Specifically, we find that emergent group-level properties of these empirical networks, such as robustness, efficiency and cost, can arise from models built on simple individual-level behaviour addressing a quality-distance trade-off by the means of pheromone trails. Our work represents a first step in developing a theory for the generation of effective multi-source multi-sink transport networks based on combining exploration and positive reinforcement of best sources.

Given an arithmetic function $$a: \mathbb {N}\rightarrow \mathbb {R}$$ , one can associate a naturally defined, doubly infinite family of Jensen polynomials. Recent work of Griffin et al. shows that for certain families of functions $$a: \mathbb {N}\rightarrow \mathbb {R}$$ , the associated Jensen polynomials are eventually hyperbolic (i.e., eventually all of their roots are real). This work proves Chen et al. conjecture that the partition Jensen polynomials are eventually hyperbolic as a special case. Here, we make this result explicit. Let N(d) be the minimal number such that for all $$n \ge N(d)$$ , the partition Jensen polynomial of degree d and shift n is hyperbolic. We prove that $$N(3)=94$$ , $$N(4)=206$$ , and $$N(5)=381$$ , and in general, that $$N(d) \le (3d)^{24d} (50d)^{3d^{2}}$$ .

The partition function p(n), which counts the number of partitions of a positive integer n, is widely studied. Here, we study partition functions pS(n) that count partitions of n into distinct parts satisfying certain congruence conditions. A shifted partition identity is an identity of the form $${{p_{S_1}} (n - H) = {p_{S_2}} (n)}$$ for all n in some arithmetic progression. Several identities of this type have been discovered, including two infinite families found by Alladi. In this paper, we use the theory of modular functions to determine the necessary and sufficient conditions for such an identity to exist. In addition, for two specific cases, we extend Alladi’s theorem to other arithmetic progressions.