The Fermat-Torricelli problem in normed planes and spaces

We investigate the Fermat-Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat-Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat-Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach

Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tilde\beta_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tilde\beta_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.Comment: 29 pages, 8 figure

The feasibility of canine rabies elimination in Africa: dispelling doubts with data

<p><b>Background:</b> Canine rabies causes many thousands of human deaths every year in Africa, and continues to increase throughout much of the continent.</p> <p><b>Methodology/Principal Findings:</b> This paper identifies four common reasons given for the lack of effective canine rabies control in Africa: (a) a low priority given for disease control as a result of lack of awareness of the rabies burden; (b) epidemiological constraints such as uncertainties about the required levels of vaccination coverage and the possibility of sustained cycles of infection in wildlife; (c) operational constraints including accessibility of dogs for vaccination and insufficient knowledge of dog population sizes for planning of vaccination campaigns; and (d) limited resources for implementation of rabies surveillance and control. We address each of these issues in turn, presenting data from field studies and modelling approaches used in Tanzania, including burden of disease evaluations, detailed epidemiological studies, operational data from vaccination campaigns in different demographic and ecological settings, and economic analyses of the cost-effectiveness of dog vaccination for human rabies prevention.</p> <p><b>Conclusions/Significance:</b> We conclude that there are no insurmountable problems to canine rabies control in most of Africa; that elimination of canine rabies is epidemiologically and practically feasible through mass vaccination of domestic dogs; and that domestic dog vaccination provides a cost-effective approach to the prevention and elimination of human rabies deaths.</p&gt

On the multiple Borsuk numbers of sets

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.Comment: 16 pages, 3 figure

The impact of a decentralised orthopaedic service on tertiary referrals in Cape Town, South Africa

Background. In South Africa (SA), district hospitals (DHs) have limited capacity to manage the high burden of traumatic injuries. Scaling up decentralised orthopaedic care could strengthen trauma systems and improve timely access to essential and emergency surgical care (EESC). Khayelitsha township in Cape Town, SA, has the highest trauma burden in the Cape Metro East health district. Objectives. The primary objective of this study was to describe the impact of Khayelitsha District Hospital (KDH) on acute orthopaedic services in the health district, with a focus on the volume and type of orthopaedic services provided without tertiary referral. Methods. This retrospective analysis described acute orthopaedic cases from Khayelitsha and their management between 1 January 2018 and 31 December 2019. Orthopaedic resources and the proportion of cases referred to the tertiary hospital by all DHs in the Cape Metro East health district are described. Results. In 2018 - 2019, KDH performed 2 040 orthopaedic operations, of which 91.3% were urgent or emergencies. KDH had the most orthopaedic resources and the lowest referral ratio (0.18) compared with other DHs (0.92 - 1.35). In Khayelitsha, 2 402 acute orthopaedic cases presented to community health clinics. Trauma (86.1%) was the most common mechanism of injury for acute orthopaedic referrals. Of clinic cases, 2 229 (92.8%) were referred to KDH and 173 (7.2%) directly to the tertiary hospital. The most common reason for direct tertiary referral was condition related (n=157; 90.8%). Conclusion. This study outlines a successful example of a decentralised orthopaedic surgical service that increased EESC accessibility and alleviated the high burden of tertiary referrals compared with other DHs with fewer resources. Further research on the barriers to scaling up orthopaedic DH capacity in SA is needed to improve equitable access to surgical care

Estimating process capability index Cpm using a bootstrap sequential sampling procedure

Construction of a confidence interval for process capability index CPM is often based on a normal approximation with fixed sample size. In this article, we describe a different approach in constructing a fixed-width confidence interval for process capability index CPM with a preassigned accuracy by using a combination of bootstrap and sequential sampling schemes. The optimal sample size required to achieve a preassigned confidence level is obtained using both two-stage and modified two-stage sequential procedures. The procedure developed is also validated using an extensive simulation study.<br /

Propagation of Respiratory Aerosols by the Vuvuzela

Vuvuzelas, the plastic blowing horns used by sports fans, recently achieved international recognition during the FIFA World Cup soccer tournament in South Africa. We hypothesised that vuvuzelas might facilitate the generation and dissemination of respiratory aerosols. To investigate the quantity and size of aerosols emitted when the instrument is played, eight healthy volunteers were asked to blow a vuvuzela. For each individual the concentration of particles in expelled air was measured using a six channel laser particle counter and the duration of blowing and velocity of air leaving the vuvuzela were recorded. To allow comparison with other activities undertaken at sports events each individual was also asked to shout and the measurements were repeated while using a paper cone to confine the exhaled air. Triplicate measurements were taken for each individual. The mean peak particle counts were 658×103 per litre for the vuvuzela and 3.7×103 per litre for shouting, representing a mean log10 difference of 2.20 (95% CI: 2.03,2.36; p<0.001). The majority (>97%) of particles captured from either the vuvuzela or shouting were between 0.5 and 5 microns in diameter. Mean peak airflows recorded for the vuvuzela and shouting were 6.1 and 1.8 litres per second respectively. We conclude that plastic blowing horns (vuvuzelas) have the capacity to propel extremely large numbers of aerosols into the atmosphere of a size able to penetrate the lower lung. Some respiratory pathogens are spread via contaminated aerosols emitted by infected persons. Further investigation is required to assess the potential of the vuvuzela to contribute to the transmission of aerosol borne diseases. We recommend, as a precautionary measure, that people with respiratory infections should be advised not to blow their vuvuzela in enclosed spaces and where there is a risk of infecting others

A multi-stage genome-wide association study of bladder cancer identifies multiple susceptibility loci.

We conducted a multi-stage, genome-wide association study of bladder cancer with a primary scan of 591,637 SNPs in 3,532 affected individuals (cases) and 5,120 controls of European descent from five studies followed by a replication strategy, which included 8,382 cases and 48,275 controls from 16 studies. In a combined analysis, we identified three new regions associated with bladder cancer on chromosomes 22q13.1, 19q12 and 2q37.1: rs1014971, (P = 8 × 10⁻¹²) maps to a non-genic region of chromosome 22q13.1, rs8102137 (P = 2 × 10⁻¹¹) on 19q12 maps to CCNE1 and rs11892031 (P = 1 × 10⁻⁷) maps to the UGT1A cluster on 2q37.1. We confirmed four previously identified genome-wide associations on chromosomes 3q28, 4p16.3, 8q24.21 and 8q24.3, validated previous candidate associations for the GSTM1 deletion (P = 4 × 10⁻¹¹) and a tag SNP for NAT2 acetylation status (P = 4 × 10⁻¹¹), and found interactions with smoking in both regions. Our findings on common variants associated with bladder cancer risk should provide new insights into the mechanisms of carcinogenesis