Continuous Blooming of Convex Polyhedra

We construct the first two continuous bloomings of all convex polyhedra. First, the source unfolding can be continuously bloomed. Second, any unfolding of a convex polyhedron can be refined (further cut, by a linear number of cuts) to have a continuous blooming.Comment: 13 pages, 6 figure

Locked and Unlocked Chains of Planar Shapes

We extend linkage unfolding results from the well-studied case of polygonal linkages to the more general case of linkages of polygons. More precisely, we consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are hinged together sequentially at rotatable joints. Our goal is to characterize the families of planar shapes that admit locked chains, where some configurations cannot be reached by continuous reconfiguration without self-intersection, and which families of planar shapes guarantee universal foldability, where every chain is guaranteed to have a connected configuration space. Previously, only obtuse triangles were known to admit locked shapes, and only line segments were known to guarantee universal foldability. We show that a surprisingly general family of planar shapes, called slender adornments, guarantees universal foldability: roughly, the distance from each edge along the path along the boundary of the slender adornment to each hinge should be monotone. In contrast, we show that isosceles triangles with any desired apex angle less than 90 degrees admit locked chains, which is precisely the threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof details. (Fixed crash-induced bugs in the abstract.

Dynamic ham-sandwich cuts in the plane

We design efficient data structures for dynamically maintaining a ham-sandwich cut of two point sets in the plane subject to insertions and deletions of points in either set. A ham-sandwich cut is a line that simultaneously bisects the cardinality of both point sets. For general point sets, our first data structure supports each operation in O(n1/3+ε) amortized time and O(n4/3+ε) space. Our second data structure performs faster when each point set decomposes into a small number k of subsets in convex position: it supports insertions and deletions in O(logn) time and ham-sandwich queries in O(klog4n) time. In addition, if each point set has convex peeling depth k , then we can maintain the decomposition automatically using O(klogn) time per insertion and deletion. Alternatively, we can view each convex point set as a convex polygon, and we show how to find a ham-sandwich cut that bisects the total areas or total perimeters of these polygons in O(klog4n) time plus the O((kb)polylog(kb)) time required to approximate the root of a polynomial of degree O(k) up to b bits of precision. We also show how to maintain a partition of the plane by two lines into four regions each containing a quarter of the total point count, area, or perimeter in polylogarithmic time.Engineering and Applied Science

Bichromatic compatible matchings

Abstract For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BRmatchings are compatible if their union is also non-crossing. We prove that, for any two distinct BRmatchings M and M , there exists a sequence of BR-matchings M = M 1 , . . . , M k = M such that M i−1 is compatible with M i . This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [6]

Bichromatic compatible matchings

ABSTRACT For a set R of n red points and a set B of n blue points, a BR-matching is a non-crossing geometric perfect matching where each segment has one endpoint in B and one in R. Two BR-matchings are compatible if their union is also noncrossing. We prove that, for any two distinct BR-matchings M and M , there exists a sequence of BR-matchings M = M1, . . . , M k = M such that Mi−1 is compatible with Mi. This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [5]

Cauchy's Arm Lemma on a Growing Sphere

We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases. The main focus of this work is a comparison of three alternate proofs, to show the links between Toponogov's Comparison Theorem, Legendre's Theorem and Cauchy's Arm Lemma

Do salivary bypass tubes lower the incidence of pharyngocutaneous fistula following total laryngectomy? A retrospective analysis of predictive factors using multivariate analysis

Salivary bypass tubes (SBT) are increasingly used to prevent pharyngocutaneous fistula (PCF) following laryngectomy and pharyngolaryngectomy. There is minimal evidence as to their efficacy and literature is limited. The aim of the study was to determine if SBT prevent PCF. The study was a multicentre retrospective case control series (level of evidence 3b). Patients who underwent laryngectomy or pharyngolaryngectomy for cancer or following cancer treatment between 2011 and 2014 were included in the study. The primary outcome was development of a PCF. Other variables recorded were age, sex, prior radiotherapy or chemoradiotherapy, prior tracheostomy, type of procedure, concurrent neck dissection, use of flap reconstruction, use of prophylactic antibiotics, the suture material used for the anastomosis, tumour T stage, histological margins, day one post-operative haemoglobin and whether a salivary bypass tube was used. Univariate and multivariate analysis were performed. A total of 199 patients were included and 24 received salivary bypass tubes. Fistula rates were 8.3% in the SBT group (2/24) and 24.6% in the control group (43/175). This was not statistically significant on univariate (p value 0.115) or multivariate analysis (p value 0.076). In addition, no other co-variables were found to be significant. No group has proven a benefit of salivary bypass tubes on multivariate analysis. The study was limited by a small case group, variations in tube duration and subjects given a tube may have been identified as high risk of fistula. Further prospective studies are warranted prior to recommendation of salivary bypass tubes following laryngectomy

Natural archives of long-range transported contamination at the remote lake Letšeng-la Letsie, Maloti Mountains, Lesotho

Naturally accumulating archives, such as lake sediments and wetland peats, in remote areas may be used to identify the scale and rates of atmospherically deposited pollutant inputs to natural ecosystems. Co-located lake sediment and wetland cores were collected from Letšeng-la Letsie, a remote lake in the Maloti Mountains of southern Lesotho. The cores were radiometrically dated and analysed for a suite of contaminants including trace metals and metalloids (Hg, Pb, Cu, Ni, Zn, As), fly-ash particles, stable nitrogen isotopes, polycyclic aromatic hydrocarbons (PAHs) and persistent organic pollutants such as polychlorinated biphenyls (PCBs), polybrominated flame retardants (PBDEs) and hexachlorobenzene (HCB). While most trace metals showed no recent enrichment, mercury, fly-ash particles, high molecular weight PAHs and total PCBs showed low but increasing levels of contamination since c.1970, likely the result of long-range transport from coal combustion and other industrial sources in the Highveld region of South Africa. However, back-trajectory analysis revealed that atmospheric transport from this region to southern Lesotho is infrequent and the scale of contamination is low. To our knowledge, these data represent the first palaeolimnological records and the first trace contaminant data for Lesotho, and one of the first multi-pollutant historical records for southern Africa. They therefore provide a baseline for future regional assessments in the context of continued coal combustion in South Africa through to the mid-21st century

H3K36 methylation maintains cell identity by regulating opposing lineage programmes

The epigenetic mechanisms that maintain differentiated cell states remainincompletely understood. Here we employed histone mutants to uncovera crucial role for H3K36 methylation in the maintenance of cell identitiesacross diverse developmental contexts. Focusing on the experimentalinduction of pluripotency, we show that H3K36M-mediated depletionof H3K36 methylation endows fibroblasts with a plastic state poisedto acquire pluripotency in nearly all cells. At a cellular level, H3K36Mfacilitates epithelial plasticity by rendering fibroblasts insensitive to TGFβsignals. At a molecular level, H3K36M enables the decommissioning ofmesenchymal enhancers and the parallel activation of epithelial/stemcell enhancers. This enhancer rewiring is Tet dependent and redirectsSox2 from promiscuous somatic to pluripotency targets. Our findingsreveal a previously unappreciated dual role for H3K36 methylation inthe maintenance of cell identity by integrating a crucial developmentalpathway into sustained expression of cell-type-specific programmes, andby opposing the expression of alternative lineage programmes throughenhancer methylation