Characterisation of Spherical Splits

We investigate the properties of collections of linear bipartitions of points embedded into $\R^3$, which we call collections of affine splits. Our main concern is characterising the collections generated when the points are embedded into $S^2$; that is, when the collection of splits is spherical. We find that maximal systems of splits occur for points embedded in general position or general position in $S^2$ for affine and spherical splits, respectively. Furthermore, we explore the connection of such systems with oriented matroids and show that a maximal collection of spherical splits map to the topes of a uniform, acyclic oriented matroid of rank 4, which is a uniform matroid polytope. Additionally, we introduce the graphs associated with collections of splits and show that maximal collections of spherical splits induce maximal planar graphs and, hence, the simplicial 3-polytopes. Finally, we introduce some methodologies for generating either the hyperplanes corresponding to a split system on an arbitrary embedding of points through a linear programming approach or generating the polytope given an abstract system of splits by utilising the properties of matroid polytopes. Establishing a solid theory for understanding spherical split systems provides a basis for not only combinatorial–geometric investigations, but also the development of bioinformatic tools for investigating non-tree-like evolutionary histories in a three-dimensional manner

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Inscribing a symmetric body in an ellipse

We prove that any bounded, centrally symmetric object K in the plane can be inscribed in an ellipse E touching its boundary aK at at least four points. An application to Minkowskian geometry is given

European Journal of Combinatorics Index, Volume 27

BACKGROUND: Diabetes is an inflammatory condition associated with iron abnormalities and increased oxidative damage. We aimed to investigate how diabetes affects the interrelationships between these pathogenic mechanisms. METHODS: Glycaemic control, serum iron, proteins involved in iron homeostasis, global antioxidant capacity and levels of antioxidants and peroxidation products were measured in 39 type 1 and 67 type 2 diabetic patients and 100 control subjects. RESULTS: Although serum iron was lower in diabetes, serum ferritin was elevated in type 2 diabetes (p = 0.02). This increase was not related to inflammation (C-reactive protein) but inversely correlated with soluble transferrin receptors (r = - 0.38, p = 0.002). Haptoglobin was higher in both type 1 and type 2 diabetes (p &lt; 0.001) and haemopexin was higher in type 2 diabetes (p &lt; 0.001). The relation between C-reactive protein and haemopexin was lost in type 2 diabetes (r = 0.15, p = 0.27 vs r = 0.63, p &lt; 0.001 in type 1 diabetes and r = 0.36, p = 0.001 in controls). Haemopexin levels were independently determined by triacylglycerol (R(2) = 0.43) and the diabetic state (R(2) = 0.13). Regarding oxidative stress status, lower antioxidant concentrations were found for retinol and uric acid in type 1 diabetes, alpha-tocopherol and ascorbate in type 2 diabetes and protein thiols in both types. These decreases were partially explained by metabolic-, inflammatory- and iron alterations. An additional independent effect of the diabetic state on the oxidative stress status could be identified (R(2) = 0.5-0.14). CONCLUSIONS: Circulating proteins, body iron stores, inflammation, oxidative stress and their interrelationships are abnormal in patients with diabetes and differ between type 1 and type 2 diabetes</p

Lattices, Polytopes and Tilings

[no abstract available

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their Z-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results