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

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

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

In search of the origins and enduring impact of agile software development

The Agile Manifesto is a philosophical touchpoint for all agile software development (ASD) methods. We examine the manifesto and some of its associated agile methods in an effort to identify the major impacts of ASD. We have encountered some difficulty in delineating agile and non-agile software processes, which is partially the result of terminological confusion. It is clear from the volume of published research that ASD has made a significant contribution, and we have identified two lasting and important impacts. Firstly, the reduction in iteration durations and secondly, the push for reduced levels of documentation (especially in relation to software requirements). Other aspects of the Agile Manifesto may not have exerted a significant impact; for example, the use of tooling to automate processes has become central to continuous software engineering (CSE) and may not be wholly congruent with the manifesto. Furthermore, many organisations may still rely on business contracts despite calls in the manifesto for greater levels of informal customer collaboration

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

Some Findings Concerning Requirements in Agile Methodologies

gile methods have appeared as an attractive alternative to conventional methodologies. These methods try to reduce the time to market and, indirectly, the cost of the product through flexible development and deep customer involvement. The processes related to requirements have been extensively studied in literature, in most cases in the frame of conventional methods. However, conclusions of conventional methodologies could not be necessarily valid for Agile; in some issues, conventional and Agile processes are radically different. As recent surveys report, inadequate project requirements is one of the most conflictive issues in agile approaches and better understanding about this is needed. This paper describes some findings concerning requirements activities in a project developed under an agile methodology. The project intended to evolve an existing product and, therefore, some background information was available. The major difficulties encountered were related to non-functional needs and management of requirements dependencies

Human Missions to Near-Earth Asteroids: An Update on NASA's Current Status and Proposed Activities for Small Body Exploration

Introduction: Over the past several years, much attention has been focused on the human exploration of near-Earth asteroids (NEAs). Two independent NASA studies examined the feasibility of sending piloted missions to NEAs, and in 2009, the Augustine Commission identified NEAs as high profile destinations for human exploration missions beyond the Earth-Moon system as part of the Flexible Path. More recently the current U.S. presidential administration directed NASA to include NEAs as destinations for future human exploration with the goal of sending astronauts to a NEA in the mid to late 2020s. This directive became part of the official National Space Policy of the United States of America as of June 28, 2010. Dynamical Assessment: The current near-term NASA human spaceflight capability is in the process of being defined while the Multi-Purpose Crew Vehicle (MPCV) and Space Launch System (SLS) are still in development. Hence, those NEAs in more accessible heliocentric orbits relative to a minimal interplanetary exploration capability will be considered for the first missions. If total mission durations for the first voyages to NEAs are to be kept to less than one year, with minimal velocity changes, then NEA rendezvous missions ideally will take place within 0.1 AU of Earth (approx about 5 million km or 37 lunar distances). Human Exploration Considerations: These missions would be the first human expeditions to inter-planetary bodies beyond the Earth-Moon system and would prove useful for testing technologies required for human missions to Mars, Phobos and Deimos, and other Solar System destinations. Missions to NEAs would undoubtedly provide a great deal of technical and engineering data on spacecraft operations for future human space exploration while conducting detailed scientific investigations of these primitive objects. Current analyses of operational concepts suggest that stay times of 15 to 30 days may be possible at these destinations. In addition, the resulting scientific investigations would refine designs for future extraterrestrial In Situ Resource Utilization (ISRU), and assist in the development of hazard mitigation techniques for planetary defense. Conclusions: The scientific and hazard mitigation benefits, along with the programmatic and operational benefits of a human venture beyond the Earth-Moon system, make a piloted mission to a NEA using NASA's proposed human exploration systems a compelling endeavo

Developing Ontologies withing Decentralized Settings

This chapter addresses two research questions: “How should a well-engineered methodology facilitate the development of ontologies within communities of practice?” and “What methodology should be used?” If ontologies are to be developed by communities then the ontology development life cycle should be better understood within this context. This chapter presents the Melting Point (MP), a proposed new methodology for developing ontologies within decentralised settings. It describes how MP was developed by taking best practices from other methodologies, provides details on recommended steps and recommended processes, and compares MP with alternatives. The methodology presented here is the product of direct first-hand experience and observation of biological communities of practice in which some of the authors have been involved. The Melting Point is a methodology engineered for decentralised communities of practice for which the designers of technology and the users may be the same group. As such, MP provides a potential foundation for the establishment of standard practices for ontology engineering