On the perimeters of simple polygons contained in a disk

A simple $n$-gon is a polygon with $n$ edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass asked the following question: For $n \geq 5$ odd, what is the maximum perimeter of a simple $n$-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.Comment: 6 pages, 2 figure

On the perimeters of simple polygons contained in a plane convex body

A simple n-gon is a polygon with n edges such that each vertex belongs to exactly two edges and every other point belongs to at most one edge. Brass, Moser and Pach asked the following question: For n > 3 odd, what is the maximum perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet, Hansen and Messine answered this question, and showed that the supremum is the perimeter of an isosceles triangle inscribed in the disk, with an edge of multiplicity n-2. L\'angi generalized their result for polygons contained in a hyperbolic disk. In this note we find the supremum of the perimeters of simple n-gons contained in an arbitrary plane convex body in the Euclidean or in the hyperbolic plane.Comment: 7 pages, 7 figure

Determining All Universal Tilers

A universal tiler is a convex polyhedron whose every cross-section tiles the plane. In this paper, we introduce a certain slight-rotating operation for cross-sections of pentahedra. Based on a selected initial cross-section and by applying the slight-rotating operation suitably, we prove that a convex polyhedron is a universal tiler if and only if it is a tetrahedron or a triangular prism.Comment: 18 pages, 12 figure

Point-of-Care Manufacture: Regulatory Opportunities and Challenges for Advanced Biotherapeutics

On 29 June 2021, UCL’s Future Targeted Healthcare Manufacturing Hub (FTHMH) held an online workshop to discuss the concepts and rationale of a new point-of-care (POC) manufacturing regulatory framework in development by the UK’s Medicines and Healthcare products Regulatory Agency (MHRA). The proposal, which seeks to address the unique challenges of manufacturing healthcare products at, (or close to), the POC, is anticipated for publication and public consultation in summer 2021

A Bichromatic Incidence Bound and an Application

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = \Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal and Aronov. We use this incidence bound to prove that a set of n points, no more than n-k of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also provide an infinite family of counterexamples to a conjecture of Purdy's on the number of hyperplanes spanned by a set of points in dimensions higher than 3, and present new conjectures not subject to the counterexample.Comment: 12 page

The UK’s emerging regulatory framework for point-of-care manufacture: insights from a workshop on advanced therapies

Point-of-care (POC) manufacture can be defined as the production of therapies in clinical settings or units close to hospitals and patients. This approach is becoming increasingly via- ble due to the emergence of flexible manufacturing technologies. Expecting an increase in this kind of production, the UK’s regulatory agency, the Medicines and Healthcare products Regulatory Agency (MHRA) is proposing a regulatory framework specifically designed for POC manufacture. To discuss the challenges of POC manufacture and the MHRA’s pro- posal, the EPSRC Future Targeted Healthcare Manufacturing Hub (FTHMH) organized a workshop drawing insights from specialists in cell and gene therapy manufacture. Through presentations and discussion roundtables, the workshop highlighted the challenges for the UK and other countries implementing POC manufacture. The workshop attendees stressed four main issues: quality control; standardization and equipment use; availability of qualified personnel; and the challenges to be met by hospitals participating in POC manufacture systems. This commentary provides a summary of the points discussed in this workshop

Impaired muscle oxygen use at onset of exercise in peripheral arterial disease

ObjectivesIn patients with peripheral arterial disease (PAD), abnormal muscle metabolism and impaired oxygen delivery distal to the arterial occlusions may contribute to the exercise limitation observed in this population. Muscle tissue hemoglobin saturation (StO2), measured with near-infrared spectroscopy, reflects the relative contributions of oxygen delivery and oxygen use. Thus differences in the kinetics of StO2 in response to exercise may yield important insight into the potential mechanisms associated with the PAD exercise impairment. The purposes of this study were to characterize the muscle oxygenation responses in patients with PAD and in healthy control subjects at the onset of exercise, and to compare the kinetics of StO2 desaturation. We hypothesized that at the onset of exercise the kinetics of StO2 desaturation would be slowed in PAD compared with control responses.Material and methodsSix patients with PAD and 6 healthy control subjects from a university center were examined in a prospective cross-sectional analysis that evaluated the desaturation kinetics of StO2 at the onset of walking exercise. On separate visits subjects performed graded treadmill exercise and 3 constant work rate treadmill tests equivalent to ∼60% (low), ∼80% (medium), and 100% (peak) of their peak exercise work rate. Gastrocnemious muscle StO2 response profiles (InSpectra tissue spectrometer) were measured at rest and across the rest to exercise transition. Muscle StO2 responses were characterized by an exponential mathematical model. The end point value was taken as the time constant of StO2 desaturation after onset of exercise (ie, equivalent to time to reach approximately 63% of StO2 decrease).ResultsThe patients with PAD and the control subjects were of similar age and activity level. The qualitative patterns of StO2 responses at onset of exercise were also similar between patients and control subjects at all work rates. However, the kinetic time constants of StO2 desaturation were prolonged in patients with PAD versus control subjects (averaged time constant across all work rates, 21.9 ± 9.4 seconds vs 4.9 ± 2.2 seconds; P < .01).ConclusionsThe slowed muscle StO2 kinetics in PAD are consistent with an impairment in muscle oxygen use at the onset of walking exercise. Impaired muscle metabolism may contribute to the altered physiologic responses to exercise and to exercise impairment in patients with PAD

Disjoint edges in topological graphs and the tangled-thrackle conjecture

It is shown that for a constant $t\in \mathbb{N}$, every simple topological graph on $n$ vertices has $O(n)$ edges if it has no two sets of $t$ edges such that every edge in one set is disjoint from all edges of the other set (i.e., the complement of the intersection graph of the edges is $K_{t,t}$-free). As an application, we settle the \emph{tangled-thrackle} conjecture formulated by Pach, Radoi\v{c}i\'c, and T\'oth: Every $n$-vertex graph drawn in the plane such that every pair of edges have precisely one point in common, where this point is either a common endpoint, a crossing, or a point of tangency, has at most $O(n)$ edges

Simultaneous Embeddings with Few Bends and Crossings

A simultaneous embedding with fixed edges (SEFE) of two planar graphs $R$ and $B$ is a pair of plane drawings of $R$ and $B$ that coincide when restricted to the common vertices and edges of $R$ and $B$. We show that whenever $R$ and $B$ admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if $R$ and $B$ are trees then one bend per edge and four crossings per edge pair suffice (and one bend per edge is sometimes necessary), (2) if $R$ is a planar graph and $B$ is a tree then six bends per edge and eight crossings per edge pair suffice, and (3) if $R$ and $B$ are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. Our results improve on a paper by Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge pair suffice.Comment: Full version of the paper "Simultaneous Embeddings with Few Bends and Crossings" accepted at GD '1