The smallest sets of points not determined by their X-rays

Let $F$ be an $n$-point set in $\mathbb{K}^d$ with $\mathbb{K}\in\{\mathbb{R},\mathbb{Z}\}$ and $d\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\psi_{\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\mathbb{R}^d$) such that two $n$-point sets in $\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\psi_{\mathbb{Z}^d}(m)\leq 2^{m-1}$ has been observed many times in the literature. In this note we show $\psi_{\mathbb{K}^d}(m)=O(m^{d+1+\varepsilon})$ for $\varepsilon>0$. For the cases $\mathbb{K}^d=\mathbb{Z}^d$ and $\mathbb{K}^d=\mathbb{R}^d$, $d>2$, this represents the first upper bound on $\psi_{\mathbb{K}^d}(m)$ that is polynomial in $m$. As a corollary we derive bounds on the sizes of solutions to both the classical and two-dimensional Prouhet-Tarry-Escott problem. Additionally, we establish lower bounds on $\psi_{\mathbb{K}^d}$ that enable us to prove a strengthened version of R\'enyi's theorem for points in $\mathbb{Z}^2$

Almost ellipsoidal sections and projections of convex bodies

In (1) Dvoretsky proved, using very ingenious methods, that every centrally symmetric convex body of sufficiently high dimension contains a central k-dimensional section which is almost spherical. Here we shall extend this result (Corollary to Theorem 2) to k-dimensional sections through an arbitrary interior point of any convex bod

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

Stigma, health and incarceration: Turning the tide for children with a parent in prison

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Chromatic number, clique subdivisions, and the conjectures of Haj\'os and Erd\H{o}s-Fajtlowicz

For a graph $G$, let $\chi(G)$ denote its chromatic number and $\sigma(G)$ denote the order of the largest clique subdivision in $G$. Let H(n) be the maximum of $\chi(G)/\sigma(G)$ over all $n$-vertex graphs $G$. A famous conjecture of Haj\'os from 1961 states that $\sigma(G) \geq \chi(G)$ for every graph $G$. That is, $H(n) \leq 1$ for all positive integers $n$. This conjecture was disproved by Catlin in 1979. Erd\H{o}s and Fajtlowicz further showed by considering a random graph that $H(n) \geq cn^{1/2}/\log n$ for some absolute constant $c>0$. In 1981 they conjectured that this bound is tight up to a constant factor in that there is some absolute constant $C$ such that $\chi(G)/\sigma(G) \leq Cn^{1/2}/\log n$ for all $n$-vertex graphs $G$. In this paper we prove the Erd\H{o}s-Fajtlowicz conjecture. The main ingredient in our proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on $n$ vertices with independence number $\alpha$.Comment: 14 page

An Exercise in Reverse Engineering for Safety-Critical Systems: An Experience for the Classroom

Since the Y2K crisis, reverse engineering has become a major area of work in industrial software application development, but lacks emphasis in US academia. This issue is exemplified by the high demand for software systems in new and expanding software application areas, which has resulted in systems being implemented before the requirements and design phases have been completed. Towards the maintenance of such systems, it is necessary to conducted reverse engineering for the derivation of software documentation for requirements and high-level and low-level design. When this scenario exists in the domain of safety-critical system, particularly in the aviation industry, reverse engineering takes on greater value because such software systems have to undergo development regulations and certification restrictions. This work reports on the pedagogical revelations gained from conducting reverse engineering on a software system that was developed and deployed for use in managing the assignment of commercial aircrafts to airport terminal gates. The software system incorporated genetic algorithms solutions and was implemented on a high-speed multi-processor system. The reverse engineering methodology applied was based on the RTCA DO-178C Software Considerations in Airborne Systems and Equipment Certification specification for onboard avionic software systems

Association of cardiotrophin-1 with myocardial fibrosis in hypertensive patients with heart failure

Cardiotrophin-1 has been shown to be profibrogenic in experimental models. The aim of this study was to analyze whether cardiotrophin-1 is associated with left ventricular end-diastolic stress and myocardial fibrosis in hypertensive patients with heart failure. Endomyocardial biopsies from patients (n=31) and necropsies from 7 control subjects were studied. Myocardial cardiotrophin-1 protein and mRNA and the fraction of myocardial volume occupied by collagen were increased in patients compared with controls ( P <0.001). Cardiotrophin-1 overexpression in patients was localized in cardiomyocytes. Cardiotrophin-1 protein was correlated with collagen type I and III mRNAs ( r =0.653, P <0.001; r =0.541, P <0.01) and proteins ( r =0.588, P <0.001; r =0.556, P <0.005) in all subjects and with left ventricular end-diastolic wall stress ( r =0.450; P <0.05) in patients. Plasma cardiotrophin-1 and N-terminal pro-brain natriuretic peptide and serum biomarkers of myocardial fibrosis (carboxy-terminal propeptide of procollagen type I and amino-terminal propeptide of procollagen type III) were increased ( P <0.001) in patients compared with controls. Plasma cardiotrophin-1 was correlated with N-terminal pro-brain natriuretic peptide ( r =0.386; P <0.005), carboxy- terminal propeptide of procollagen type I ( r =0.550; P <0.001), and amino-terminal propeptide of procollagen type III ( r =0.267; P <0.05) in all subjects. In vitro, cardiotrophin-1 stimulated the differentiation of human cardiac fibroblast to myofibroblasts ( P <0.05) and the expression of procollagen type I ( P <0.05) and III ( P <0.01) mRNAs. These findings show that an excess of cardiotrophin-1 is associated with increased collagen in the myocardium of hypertensive patients with heart failure. It is proposed that exaggerated cardiomyocyte production of cardiotrophin-1 in response to increased left ventricular end-diastolic stress may contribute to fibrosis through stimulation of fibroblasts in heart failure of hypertensive origi

Historical roots of Agile methods: where did “Agile thinking” come from?

The appearance of Agile methods has been the most noticeable change to software process thinking in the last fifteen years [16], but in fact many of the “Agile ideas” have been around since 70’s or even before. Many studies and reviews have been conducted about Agile methods which ascribe their emergence as a reaction against traditional methods. In this paper, we argue that although Agile methods are new as a whole, they have strong roots in the history of software engineering. In addition to the iterative and incremental approaches that have been in use since 1957 [21], people who criticised the traditional methods suggested alternative approaches which were actually Agile ideas such as the response to change, customer involvement, and working software over documentation. The authors of this paper believe that education about the history of Agile thinking will help to develop better understanding as well as promoting the use of Agile methods. We therefore present and discuss the reasons behind the development and introduction of Agile methods, as a reaction to traditional methods, as a result of people's experience, and in particular focusing on reusing ideas from histor

Packing and Hausdorff measures of stable trees

In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum random tree (1991) which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for levels sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from a joint work with J-F Le Gall (2006).Comment: 40 page

Self-similar disk packings as model spatial scale-free networks

The network of contacts in space-filling disk packings, such as the Apollonian packing, are examined. These networks provide an interesting example of spatial scale-free networks, where the topology reflects the broad distribution of disk areas. A wide variety of topological and spatial properties of these systems are characterized. Their potential as models for networks of connected minima on energy landscapes is discussed.Comment: 13 pages, 12 figures; some bugs fixed and further discussion of higher-dimensional packing