Block partitions: an extended view

Given a sequence $S=(s_1,\dots,s_m) \in [0, 1]^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a partition of $S$ into $n$ blocks $B_1, \dots , B_n$ with $|b_i - b_j| \le 1$ for every $i, j$. In this paper, we consider a generalization of the problem in higher dimensions

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)

A Colored Version of Tverberg\u27s Theorem

The main result of this paper is that given n red, n white, and n green points in the plane, it is possible to form n vertex-disjoint triangles Δ 1 ,…,Δ n in such a way that the Δ i has one one red, one white, and one green vertex for every i = 1,…, n and the intersection of these triangles is nonempty

CDKN2A/p16INK4a expression is associated with vascular progeria in chronic kidney disease

Patients with chronic kidney disease (CKD) display a progeric vascular phenotype linked to apoptosis, cellular senescence and osteogenic transformation. This has proven intractable to modelling appropriately in model organisms. We have therefore investigated this directly in man, using for the first time validated cellular biomarkers of ageing (CDKN2A/p16INK4a, SA-β-Gal) in arterial biopsies from 61 CKD patients undergoing living donor renal transplantation. We demonstrate that in the uremic milieu, increased arterial expression of CDKN2A/p16INK4a associated with vascular progeria in CKD, independently of chronological age. The arterial expression of CDKN2A/p16INK4a was significantly higher in patients with coronary calcification (p=0.01) and associated cardiovascular disease (CVD) (p=0.004). The correlation between CDKN2A/p16INK4a and media calcification was statistically significant (p=0.0003) after correction for chronological age. We further employed correlate expression of matrix Gla protein (MGP) and runt-related transcription factor 2 (RUNX2) as additional pathognomonic markers. Higher expression of CDKN2A/p16INK4a, RUNX2 and MGP were observed in arteries with severe media calcification. The number of p16INK4a and SA-β-Gal positive cells was higher in biopsies with severe media calcification. A strong inverse correlation was observed between CDKN2A/p16INK4a expression and carboxylated osteocalcin levels. Thus, impaired vitamin K mediated carboxylation may contribute to premature vascular senescence

Increased telomere attrition following renal transplantation: impact of anti-metabolite therapy

Background: The uremic milieu exposes chronic kidney disease (CKD) patients to premature ageing processes. The impact of renal replacement therapy (dialysis and renal transplantation [RTx]) or immunosuppressive treatment regimens on ageing biomarkers has scarcely been studied. Methods: In this study telomere length in whole blood cells was measured in 49 dialysis patients and 47 RTx patients close to therapy initiation and again after 12 months. Forty-three non-CKD patients were included as controls. Results: Non-CKD patients had significantly (P <= 0.01) longer telomeres than CKD patients. Telomere attrition after 12 months was significantly greater in RTx patients compared to dialysis patients (P = 0.008). RTx patients receiving mycophenolate mofetil (MMF) had a greater (P = 0.007) degree of telomere attrition compared to those treated with azathioprine. After 12 months, folate was significantly higher in RTx patients than in dialysis patients (P < 0.0001), whereas the opposite was true for homocysteine (P < 0.0001). The azathioprine group had lower levels of folate after 12 months than the MMF group (P = 0.003). Conclusions: The associations between immunosuppressive therapy, telomere attrition, and changes in folate indicate a link between methyl donor potential, immunosuppressive drugs, and biological ageing. The hypothesis that the increased telomere attrition, observed in the MMF group after RTx, is driven by the immunosuppressive treatment, deserves further attention

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

Continuum Surface Energy from a Lattice Model

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal. The result is valid for a large class of domains, including faceted (polygonal) shapes and regions with piecewise smooth boundaries.Comment: V. 1: 10 pages, no fig's. V 2: 23 pages, no figures. Misprints corrected. Section 3 added, (new results). Intro expanded, refs added.V 3: 26 pages. Abstract changed. Section 2 split into 2. Section (4) added material. V 4, 28 pages, Intro rewritten. Changes in Sec.5 (presentation only). Refs added.V 5,intro changed V.6 address reviewer's comment

Tverberg-type theorems for intersecting by rays

In this paper we consider some results on intersection between rays and a given family of convex, compact sets. These results are similar to the center point theorem, and Tverberg's theorem on partitions of a point set

Knaster's problem for (Z2)k(Z_2)^k-symmetric subsets of the sphere S2k−1S^{2^k-1}

We prove a Knaster-type result for orbits of the group $(Z_2)^k$ in $S^{2^k-1}$, calculating the Euler class obstruction. Among the consequences are: a result about inscribing skew crosspolytopes in hypersurfaces in $\mathbb R^{2^k}$, and a result about equipartition of a measures in $\mathbb R^{2^k}$ by $(Z_2)^{k+1}$-symmetric convex fans