Generalizations of Sperner\u27s Theorem: Packing Posets, Families Forbidding Posets, and Supersaturation

Sperner\u27s Theorem is a well known theorem in extremal set theory that gives the size of the largest antichain in the poset that is the Boolean lattice. This is equivalent to finding the largest family of subsets of an $n$-set, $[n]:=\{1,2,\dots,n\}$, such that the family is constructed from pairwise unrelated copies of the single element poset. For a poset $P$, we are interested in maximizing the size of a family $\mathcal{F}$ of subsets of $[n]$, where each maximally connected component of $\mathcal{F}$ is a copy of $P$, and finding the extreme configurations that achieve this value. For instance, Sperner showed that when $P$ is one element, $\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$ is the maximum number of copies of $P$ and that this is only achieved by taking subsets of a middle size. Griggs, Stahl, and Trotter have shown that when $P$ is a chain on $k$ elements, $\dfrac{1}{2^{k-1}}\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$ is asymptotically the maximum number of copies of $P$. We find the extreme families for a packing of chains, answering a conjecture of Griggs, Stahl, and Trotter, as well as finding the extreme packings of certain other posets. For the general poset $P$, we prove that the maximum number of unrelated copies of $P$ is asymptotic to a constant times $\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$. Moreover, the constant has the form $\dfrac{1}{c(P)}$, where $c(P)$ is the size of the smallest convex closure over all embeddings of $P$ into the Boolean lattice. Sperner\u27s Theorem has been generalized by looking for $\operatorname{La}(n,P)$, the size of a largest family of subsets of an $n$-set that does not contain a general poset $P$ in the family. We look at this generalization, exploring different techniques for finding an upper bound on $\operatorname{La}(n,P)$, where $P$ is the diamond. We also find all the families that achieve $\operatorname{La}(n,\{\mathcal{V},\Lambda\})$, the size of the largest family of subsets that do not contain either of the posets $\mathcal{V}$ or $\Lambda$. We also consider another generalization of Sperner\u27s theorem, supersaturation, where we find how many copies of $P$ are in a family of a fixed size larger than $\operatorname{La}(n,P)$. We seek families of subsets of an $n$-set of given size that contain the fewest $k$-chains. Erd\H{o}s showed that a largest $k$-chain-free family in the Boolean lattice is formed by taking all subsets of the $(k-1)$ middle sizes. Our result implies that by taking this family together with $x$ subsets of the $k$-th middle size, we obtain a family with the minimum number of $k$-chains, over all families of this size. We prove our result using the symmetric chain decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951)

Diving into the vertical dimension of elasmobranch movement ecology

Knowledge of the three-dimensional movement patterns of elasmobranchs is vital to understand their ecological roles and exposure to anthropogenic pressures. To date, comparative studies among species at global scales have mostly focused on horizontal movements. Our study addresses the knowledge gap of vertical movements by compiling the first global synthesis of vertical habitat use by elasmobranchs from data obtained by deployment of 989 biotelemetry tags on 38 elasmobranch species. Elasmobranchs displayed high intra- and interspecific variability in vertical movement patterns. Substantial vertical overlap was observed for many epipelagic elasmobranchs, indicating an increased likelihood to display spatial overlap, biologically interact, and share similar risk to anthropogenic threats that vary on a vertical gradient. We highlight the critical next steps toward incorporating vertical movement into global management and monitoring strategies for elasmobranchs, emphasizing the need to address geographic and taxonomic biases in deployments and to concurrently consider both horizontal and vertical movements

Diving into the vertical dimension of elasmobranch movement ecology

Knowledge of the three-dimensional movement patterns of elasmobranchs is vital to understand their ecological roles and exposure to anthropogenic pressures. To date, comparative studies among species at global scales have mostly focused on horizontal movements. Our study addresses the knowledge gap of vertical movements by compiling the first global synthesis of vertical habitat use by elasmobranchs from data obtained by deployment of 989 biotelemetry tags on 38 elasmobranch species. Elasmobranchs displayed high intra- and interspecific variability in vertical movement patterns. Substantial vertical overlap was observed for many epipelagic elasmobranchs, indicating an increased likelihood to display spatial overlap, biologically interact, and share similar risk to anthropogenic threats that vary on a vertical gradient. We highlight the critical next steps toward incorporating vertical movement into global management and monitoring strategies for elasmobranchs, emphasizing the need to address geographic and taxonomic biases in deployments and to concurrently consider both horizontal and vertical movements

Diving into the vertical dimension of elasmobranch movement ecology

Knowledge of the three-dimensional movement patterns of elasmobranchs is vital to understand their ecological roles and exposure to anthropogenic pressures. To date, comparative studies among species at global scales have mostly focused on horizontal movements. Our study addresses the knowledge gap of vertical movements by compiling the first global synthesis of vertical habitat use by elasmobranchs from data obtained by deployment of 989 biotelemetry tags on 38 elasmobranch species. Elasmobranchs displayed high intra- and interspecific variability in vertical movement patterns. Substantial vertical overlap was observed for many epipelagic elasmobranchs, indicating an increased likelihood to display spatial overlap, biologically interact, and share similar risk to anthropogenic threats that vary on a vertical gradient. We highlight the critical next steps toward incorporating vertical movement into global management and monitoring strategies for elasmobranchs, emphasizing the need to address geographic and taxonomic biases in deployments and to concurrently consider both horizontal and vertical movements

Dual Catalysis for Selective Ring-Opening Polymerization of Lactones: Evolution toward Simplicity

Much work has been directed to the design of complex single-site catalysts for ring-opening polymerization (ROP) to enhance both activity and selectivity. More simply, however, cooperative effects between Lewis acids and organocatalytic nucleophiles/Lewis bases provide a powerful alternative. In this study we demonstrate that the combination of <i>N</i>-heterocyclic carbenes, 1,8-diazabicycloundec-7-ene (DBU) and 4-dimethylaminopyridine (DMAP) with simple Lewis acids enables the ROP of the macrolactone pentadecalactone in a rapid and efficient manner. Remarkably, regardless of the nature of the nucleophile, the order of activity was observed to be MgX₂ ≫ YCl₃ ≫ AlCl₃ and MgI₂ > MgBr₂ > MgCl₂ in every case. The minimal influence of the organobase on polymerization activity allows for the use of simple and inexpensive precursors. Furthermore, extension of the study to other cyclic (di)­ester monomers reveals the choice of Lewis acid to lead to monomer selective ROP activity and hence control over copolymer composition by choice of Lewis acid. This approach could lead to the realization of complex polymer structures with tunable physical properties from simple catalyst combinations

Diving into the vertical dimension of elasmobranch movement ecology.

Knowledge of the three-dimensional movement patterns of elasmobranchs is vital to understand their ecological roles and exposure to anthropogenic pressures. To date, comparative studies among species at global scales have mostly focused on horizontal movements. Our study addresses the knowledge gap of vertical movements by compiling the first global synthesis of vertical habitat use by elasmobranchs from data obtained by deployment of 989 biotelemetry tags on 38 elasmobranch species. Elasmobranchs displayed high intra- and interspecific variability in vertical movement patterns. Substantial vertical overlap was observed for many epipelagic elasmobranchs, indicating an increased likelihood to display spatial overlap, biologically interact, and share similar risk to anthropogenic threats that vary on a vertical gradient. We highlight the critical next steps toward incorporating vertical movement into global management and monitoring strategies for elasmobranchs, emphasizing the need to address geographic and taxonomic biases in deployments and to concurrently consider both horizontal and vertical movements

Genomic reconstruction of the SARS-CoV-2 epidemic in England

AbstractThe evolution of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus leads to new variants that warrant timely epidemiological characterization. Here we use the dense genomic surveillance data generated by the COVID-19 Genomics UK Consortium to reconstruct the dynamics of 71 different lineages in each of 315 English local authorities between September 2020 and June 2021. This analysis reveals a series of subepidemics that peaked in early autumn 2020, followed by a jump in transmissibility of the B.1.1.7/Alpha lineage. The Alpha variant grew when other lineages declined during the second national lockdown and regionally tiered restrictions between November and December 2020. A third more stringent national lockdown suppressed the Alpha variant and eliminated nearly all other lineages in early 2021. Yet a series of variants (most of which contained the spike E484K mutation) defied these trends and persisted at moderately increasing proportions. However, by accounting for sustained introductions, we found that the transmissibility of these variants is unlikely to have exceeded the transmissibility of the Alpha variant. Finally, B.1.617.2/Delta was repeatedly introduced in England and grew rapidly in early summer 2021, constituting approximately 98% of sampled SARS-CoV-2 genomes on 26 June 2021.</jats:p