Constructing monotone homotopies and sweepouts

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.Comment: 16 pages, 6 figure

Pollen, biomarker and stable isotope evidence of late Quaternary environmental change at Lake McKenzie, southeast Queensland

Unravelling links between climate change and vegetation response during the Quaternary is important if the climate–environment interactions of modern systems are to be fully understood. Using a sediment core from Lake McKenzie, Fraser Island, we reconstruct changes in the lake ecosystem and surrounding vegetation over the last ca. 36.9 cal kyr. Evidence is drawn from multiple sources, including pollen, micro-charcoal, biomarker and stable isotope (C and N) analyses, and is used to gain a better understanding of the nature and timing of past ecological changes that have occurred at the site. The glacial period of the record, from ca. 36.9 to 18.3 cal kyr BP, is characterised by an increased abundance of plants of the aquatic and littoral zone, indicating lower lake water levels. High abundance of biomarkers and microfossils of the colonial green alga Botryococcus occurred at this time and included large variation in individual botryococcene d13C values. A slowing or ceasing of sediment accumulation occurred during the time period from ca. 18.3 to 14.0 cal kyr BP. By around 14.0 cal kyr BP fire activity in the area was reduced, as was abundance of littoral plants and terrestrial herbs, suggesting wetter conditions from that time. The Lake McKenzie pollen record conforms to existing records from Fraser Island by containing evidence of a period of reduced effective precipitation that commenced in the mid-Holocene

Diastereoselective synthesis of protected syn 1,3-Diols: Preparation of the C16-C24 portion of dolabelides

International audience(Equation Presented) We have designed a new method to make synthons encompassing a protected syn 1,3-diol motif and an aldehyde a to the 1,3-dioxane ring. An additional stereocenter was also created, potentially leading to stereochemically defined 1,2,4-triols. This method was successfully applied to the synthesis of the C16-C24 portion of Dolabelides

Monotone contractions of the boundary of the disc

In this paper, we study contractions of the boundary of a Riemannian 2-disc where the maximal length of the intermediate curves is minimized. We prove that with an arbitrarily small overhead in the lengths of the intermediate curves, there exists such an optimal contraction that is monotone, i.e., where the intermediate curves are simple closed curves which are pairwise disjoint. This proves a conjecture of Chambers and Rotman

Monotone contractions of the boundary of the disc

In this paper, we study contractions of the boundary of a Riemannian 2-disc where the maximal length of the intermediate curves is minimized. We prove that with an arbitrarily small overhead in the lengths of the intermediate curves, there exists such an optimal contraction that is monotone, i.e., where the intermediate curves are simple closed curves which are pairwise disjoint. This proves a conjecture of Chambers and Rotman