On the Maximum Crossing Number

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.Comment: 16 pages, 5 figure

Six topics on inscribable polytopes

Inscribability of polytopes is a classic subject but also a lively research area nowadays. We illustrate this with a selection of well-known results and recent developments on six particular topics related to inscribable polytopes. Along the way we collect a list of (new and old) open questions.Comment: 11 page

Theory of commensurable magnetic structures in holmium

The tendency for the period of the helically ordered moments in holmium to lock into values which are commensurable with the lattice is studied theoretically as a function of temperature and magnetic field. The commensurable effects are derived in the mean-field approximation from numerical calculations of the free energy of various commensurable structures, and the results are compared with the extensive experimental evidence collected during the last ten years on the magnetic structures in holmium. In general the stability of the different commensurable structures is found to be in accord with the experiments, except for the tau=5/18 structure observed a few degrees below T_N in a b-axis field. The trigonal coupling recently detected in holmium is found to be the interaction required to explain the increased stability of the tau=1/5 structure around 42 K, and of the tau=1/4 structure around 96 K, when a field is applied along the c-axis.Comment: REVTEX, 31 pages, 7 postscript figure

Small grid embeddings of 3-polytopes

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

Impact of family structure on long-term survivors of osteosarcoma.

GOALS OF WORK: Long-term outcomes of osteosarcoma have dramatically improved with the use of modern combination therapies. Such aggressive treatments, however, entail chronic complications. In the present study, we assessed the functional, psychological, and familial status of long-term survivors of osteosarcoma treated at our institution. MATERIALS AND METHODS: Fifteen long-term survivors of osteosarcoma were evaluated for functional and psychological sequelae. Functional assessment was based on a method described by Enneking et al. Psychological assessment was based on General Health Questionnaire 28, Inventory Scale for Traumatic Neurosis, and Family System Test. MAIN RESULTS: Ten patients showed mild functional impairments; only five patients were handicapped more seriously. Depressive symptoms were diagnosed in four patients. A total of six patients revealed unbalanced family structures, including three of the four patients with depressive symptoms, all four patients with symptoms of posttraumatic stress disorder, and five of seven patients who showed poor emotional acceptance. CONCLUSIONS: Osteosarcoma survivors will generally recover good functional performance. Only a minority of them remain seriously impaired. One third of the patients present depressive symptoms and posttraumatic stress disorder. Poor coping is closely associated with unbalanced family structures. Therefore, the psychological and familial situation of patients with newly diagnosed osteosarcoma should be carefully assessed

Pancreatic enzyme replacement therapy in patients with pancreatic cancer: A national prospective study

Objective: UK national guidelines recommend pancreatic enzyme replacement therapy (PERT) in pancreatic cancer. Over 80% of pancreatic cancers are unresectable and managed in non-surgical units. The aim was to assess variation in PERT prescribing, determine factors associated with its use and identify potential actions to improve prescription rates. Design: RICOCHET was a national prospective audit of malignant pancreatic, peri-ampullary lesions or malignant biliary obstruction between April and August 2018. This analysis focuses on pancreatic cancer patients and is reported to STROBE guidelines. Multivariable regression analysis was undertaken to assess factors associated with PERT prescribing. Results: Rates of PERT prescribing varied among the 1350 patients included. 74.4% of patients with potentially resectable disease were prescribed PERT compared to 45.3% with unresectable disease. PERT prescription varied across surgical hospitals but high prescribing rates did not disseminate out to the respective referring network. PERT prescription appeared to be related to the treatment aim for the patient and the amount of clinician contact a patient has. PERT prescription in potentially resectable patients was positively associated with dietitian referral (p = 0.001) and management at hepaticopancreaticobiliary (p = 0.049) or pancreatic unit (p = 0.009). Prescription in unresectable patients also had a negative association with Charlson comorbidity score 5–7 (p = 0.045) or >7 (p = 0.010) and a positive association with clinical nurse specialist review (p = 0.028). Conclusion: Despite national guidance, wide variation and under-treatment with PERT exists. Given that most patients with pancreatic cancer have unresectable disease and are treated in non-surgical hospitals, where prescribing is lowest, strategies to disseminate best practice and overcome barriers to prescribing are urgently required

Genetic and ecological outcomes of Inga vera subsp. affinis (leguminosae) tree plantations in a fragmented tropical landscape

Planting of native trees for habitat restoration is a widespread practice, but the consequences for the retention and transmission of genetic diversity in planted and natural populations are unclear. Using Inga vera subsp. affinis as a model species, we genotyped five natural and five planted populations in the Atlantic forest of northeastern Brazil at polymorphic microsatellite loci. We studied the breeding system and population structure to test how much genetic diversity is retained in planted relative to natural populations. We then genotyped seedlings from these populations to test whether genetic diversity in planted populations is restored by outcrossing to natural populations of I. vera. The breeding system of natural I. vera populations was confirmed to be highly outcrossing (t = 0.92; FIS = -0.061, P = 0.04), with populations showing weak population substructure (FST = 0.028). Genetic diversity in planted populations was 50% less than that of natural populations (planted: AR = 14.9, HO = 0.865 and natural: AR = 30.8, HO = 0.655). However, seedlings from planted populations showed a 30% higher allelic richness relative to their parents (seedlings AR = 10.5, parents AR = 7.6). Understanding the processes and interactions that shape this system are necessary to provide ecologically sensible goals and successfully restore hyper-fragmented habitats. Future restoration plans for I. vera must consider the genetic diversity of planted populations and the potential for gene flow between natural populations in the landscape, in order to preserve ecological interactions (i.e. pollination), and promote opportunities for outcrossing