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

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

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

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

A cost-effectiveness analysis of a preventive exercise program for patients with advanced head and neck cancer treated with concomitant chemo-radiotherapy

In recent years, concomitant chemo-radiotherapy (CCRT) has become an indispensable organ preserving treatment modality for advanced head and neck cancer, improving local control and overall survival in several anatomical sites [1]. Unfortunately, CCRT can have a detrimental effect on many functions of the upper respiratory and digestive system. Sequellae such as pain, oedema, xerostomia and fibrosis negatively affect mouth opening (trismus), chewing, swallowing and speech [1]. Several studies investigating long-term effects of CCRT have concluded that swallowing and nutritional dysfunction tend to be persistent and can be severe [2-4]. Not surprisingly, therefore, CCRT can have a negative effect on patients‟ quality of life (QoL) [2]. Moreover, even before onset of treatment patients may already present with pain, impaired swallowing, trismus, aspiration, dietary restrictions and tube dependency, and loss of body weight, because the tumour may disrupt the normal anatomy and thus interfere with normal function [1]. Many studies refer to the importance of rehabilitation after, and even during treatment, in order to support and improve those functions [2]. However, as yet, few studies have investigated the effects of (preventive) rehabilitation exercises on the predictable and inevitable swallowing and mouth opening problems for this patient group. In addition, little is known about the costs and benefits of such exercise programs for head and neck cancer. As the clinical effectiveness is established [4], it is now relevant to embark on cost-effectiveness as a contribution to decision making on coverage. The aim of this study was to analyze the incremental cost-effectiveness for a preventive exercise program (PREP) versus usual care (UC) for patients with advanced head and neck cancer treated at the Netherlands Cancer Institute - Antoni van Leeuwenhoek Hospital (NKI-AVL)

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]