Notes about the Caratheodory number

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carath\'{e}odory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carath\'{e}odory theorem and give a Tverberg type theorem for families of convex compacta

Development and characterisation of injection moulded, all-polypropylene composites

In this work, all-polypropylene composites (all-PP composites) were manufactured by injection moulding. Prior to injection moulding, pre-impregnated pellets were prepared by a three-step process (filament winding, compression moulding and pelletizing). A highly oriented polypropylene multifilament was used as the reinforcement material, and a random polypropylene copolymer (with ethylene) was used as the matrix material. Plaque specimens were injection moulded from the pellets with either a film gate or a fan gate. The compression moulded sheets and injection moulding plaques were characterised by shrinkage tests, static tensile tests, dynamic mechanical analysis and falling weight impact tests; the fibre distribution and fibre/matrix adhesion were analysed with light microscopy and scanning electron microscopy. The results showed that with increasing fibre content, both the yield stress and the perforation energy significantly increased. Of the two types of gates used, the fan gate caused the mechanical properties of the plaque specimens to become more homogeneous (i.e., the differences in behaviour parallel and perpendicular to the flow direction became negligible)

Nash equilibria in random games

We consider Nash equilibria in 2-player random games and analyze a simple Las Vegas algorithm for finding an equilibrium. The algorithm is combinatorial and always finds a Nash equilibrium; on m × n payoff matrices, it runs in time O(m2n log log n + n2m log lo gm) with high probability. Our result follows from showing that a 2-player random game has a Nash equilibrium with supports of size two with high probability, at least 1 − O(1 / log n). Our main tool is a polytop

On the toughness of thermoplastic polymer nanocomposites as assessed by the essential work of fracture (EWF) approach

The essential work of fracture (EWF) approach is widely used to determine the plane stress fracture toughness of highly ductile polymers and related systems. To shed light on how the toughness is affected by nanofillers EWF-suited model polymers, viz. amorphous copolyester and polypropylene block copolymer were modified by multiwall carbon nanotube (MWCNT), graphene (GR), boehmite alumina (BA), and organoclay (MMT) in 1 wt% each. EWF tests were performed on deeply double-edge notched tensile-loaded specimens under quasistatic loading conditions. Data reduction occurred by energy partitioning between yielding and necking/tearing. The EWF prerequisites were not met with the nanocomposites containing MWCNT and GR by contrast to those with MMT and BA. Accordingly, the toughness of nanocomposites with homogeneously dispersed and low aspect ratio fillers may be properly determined using the EWF. Results indicated that incorporation of nanofillers may result in an adverse effect between the specific essential and non-essential EWF parameters

A Tverberg type theorem for matroids

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq \sqrt{b(M)}/4 disjoint independent sets \sigma_1,\ldots,\sigma_t \in M such that \bigcap_{i=1}^t f(\sigma_i) \neq \emptyset.Comment: This article is due to be published in the collection of papers "A Journey through Discrete Mathematics. A Tribute to Jiri Matousek" edited by Martin Loebl, Jaroslav Nesetril and Robin Thomas, due to be published by Springe

Rotation sets of billiards with one obstacle

We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets

Acknowledgement of priority - A fractional Helly theorem for boxes

In our recent paper [1] we prove a fractional Helly type theorem for boxes in Rd. This short note is to acknowledge priority: in 1980 Meir Katchalski [4] proved exactly the same result and in 1988 Jürgen Eckhoff [2] proved the same result in much more generality. In fact, Eckhoff established an upper bound theorem for the f -vectors of finite families of boxes in Rd from which his result is derived. Besides apologies for our ignorance we would like to mention that Eckhoff extended his results further in a more recent paper [3]

Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

We introduce a new structure for a set of points in the plane and an angle $\alpha$, which is similar in flavor to a bounded-degree MST. We name this structure $\alpha$-MST. Let $P$ be a set of points in the plane and let $0 < \alpha \le 2\pi$ be an angle. An $\alpha$-ST of $P$ is a spanning tree of the complete Euclidean graph induced by $P$, with the additional property that for each point $p \in P$, the smallest angle around $p$ containing all the edges adjacent to $p$ is at most $\alpha$. An $\alpha$-MST of $P$ is then an $\alpha$-ST of $P$ of minimum weight. For $\alpha < \pi/3$, an $\alpha$-ST does not always exist, and, for $\alpha \ge \pi/3$, it always exists. In this paper, we study the problem of computing an $\alpha$-MST for several common values of $\alpha$. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point $p \in P$, we associate a wedge $W_p$ of angle $\alpha$ and apex $p$. The goal is to assign an orientation and a radius $r_p$ to each wedge $W_p$, such that the resulting graph is connected and its MST is an $\alpha$-MST. (We draw an edge between $p$ and $q$ if $p \in W_q$, $q \in W_p$, and $|pq| \le r_p, r_q$.) Unsurprisingly, the problem of computing an $\alpha$-MST is NP-hard, at least for $\alpha=\pi$ and $\alpha=2\pi/3$. We present constant-factor approximation algorithms for $\alpha = \pi/2, 2\pi/3, \pi$. One of our major results is a surprising theorem for $\alpha = 2\pi/3$, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set $P$ of $3n$ points in the plane and any partitioning of the points into $n$ triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by $P$ is connected. We apply the theorem to the {\em antenna conversion} problem

Характеристика сили нервових процесів у плавців

OBJECTIVE: Higher levels of the novel inflammatory marker pentraxin 3 (PTX3) predict cardiovascular mortality in patients with chronic kidney disease (CKD). Yet, whether PTX3 predicts worsening of kidney function has been less well studied. We therefore investigated the associations between PTX3 levels, kidney disease measures and CKD incidence. METHODS: Cross-sectional associations between serum PTX3 levels, urinary albumin/creatinine ratio (ACR) and cystatin C-estimated glomerular filtration rate (GFR) were assessed in two independent community-based cohorts of elderly subjects: the Prospective Investigation of the Vasculature in Uppsala Seniors (PIVUS, n = 768, 51% women, mean age 75 years) and the Uppsala Longitudinal Study of Adult Men (ULSAM, n = 651, mean age 77 years). The longitudinal association between PTX3 level at baseline and incident CKD (GFR &lt;60 mL( ) min(-1)  1.73 m(-) ²) was also analysed (number of events/number at risk: PIVUS 229/746, ULSAM 206/315). RESULTS: PTX3 levels were inversely associated with GFR [PIVUS: B-coefficient per 1 SD increase -0.16, 95% confidence interval (CI) -0.23 to -0.10, P &lt; 0.001; ULSAM: B-coefficient per 1 SD increase -0.09, 95% CI -0.16 to -0.01, P &lt; 0.05], but not ACR, after adjusting for age, gender, C-reactive protein and prevalent cardiovascular disease in cross-sectional analyses. In longitudinal analyses, PTX3 levels predicted incident CKD after 5 years in both cohorts [PIVUS: multivariable odds ratio (OR) 1.21, 95% CI 1.01-1.45, P &lt; 0.05; ULSAM: multivariable OR 1.37, 95% CI 1.07-1.77, P &lt; 0.05]. CONCLUSIONS: Higher PTX3 levels are associated with lower GFR and independently predict incident CKD in elderly men and women. Our data confirm and extend previous evidence suggesting that inflammatory processes are activated in the early stages of CKD and drive impairment of kidney function. Circulating PTX3 appears to be a promising biomarker of kidney disease

Algorithms for Colourful Simplicial Depth and Medians in the Plane

The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure