326 research outputs found

    Random structures for partially ordered sets

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    This thesis is presented in two parts. In the first part, we study a family of models of random partial orders, called classical sequential growth models, introduced by Rideout and Sorkin as possible models of discrete space-time. We analyse a particular model, called a random binary growth model, and show that the random partial order produced by this model almost surely has infinite dimension. We also give estimates on the size of the largest vertex incomparable to a particular element of the partial order. We show that there is some positive probability that the random partial order does not contain a particular subposet. This contrasts with other existing models of partial orders. We also study "continuum limits" of sequences of classical sequential growth models. We prove results on the structure of these limits when they exist, highlighting a deficiency of these models as models of space-time. In the second part of the thesis, we prove some correlation inequalities for mappings of rooted trees into complete trees. For T a rooted tree we can define the proportion of the total number of embeddings of T into a complete binary tree that map the root of T to the root of the complete binary tree. A theorem of Kubicki, Lehel and Morayne states that, for two binary trees with one a subposet of the other, this proportion is larger for the larger tree. They conjecture that the same is true for two arbitrary trees with one a subposet of the other. We disprove this conjecture by analysing the asymptotics of this proportion for large complete binary trees. We show that the theorem of Kubicki, Lehel and Morayne can be thought of as a correlation inequality which enables us to generalise their result in other directions

    Modular Decomposition and the Reconstruction Conjecture

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    We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs.Comment: 9 pages, 2 figure

    Design Wind Speeds In Tropical Cyclone-prone Regions

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    Simulation methods have recently emerged as the most reliable, and in some cases, the only means available for predicting design wind speeds in tropical cyclone-prone regions of the world. This thesis presents a refined simulation procedure. The windfield model is considerably more complex than those used in previous simulations. Wind speeds and directions are defined at three levels, (i) at the 700 mb height, where gradient balance is assumed to apply, (ii) at the 500 metre height, where surface friction and translation of the storm system itself combine to modify the circulation in a manner which, in the current simulation, is computed using a recently developed numerical hurricane boundary layer model, and (iii) at the surface or 10 metre height. Within the windfield model, the boundary layer variation of wind speed and direction and the influences on the three circulation levels when landfall is made are defined using data gathered in recently occurring tropical cyclones. The simulation also contains a new filling model taking into account the geographical variation of this phenomenon along the U.S. coastline.;The simulation windfield model is evaluated by comparing actual wind records obtained in recently occurring tropical cyclones with wind speeds and directions predicted by the model itself. The storms chosen for these comparisons include North Atlantic, Northwest Pacific and Australian tropical cyclones. The statistical representation of the tropical cyclone characteristic parameters required in the simulation is reviewed and in some cases new distributions have been proposed. The description of these climatological parameters has application in fields other than the prediction of design wind speeds.;The simulation procedure was applied to the Gulf and Atlantic Coasts of the United States to yield extreme wind speed estimates. The results show significant variation along the coastline and compare reasonably with two recent studies of U.S. hurricane design wind speeds.;Finally, the simulation procedure is extended to handle the increased risk sustained by line-like structures, such as transmission lines, and to be compatible with the methodology currently used in the calculation of structural response exceedances, basic to the prediction of wind-induced building response

    Non-homogeneous random walks on a semi-infinite strip

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    We study the asymptotic behaviour of Markov chains (Xn,ηn) on Z+×S, where Z+ is the non-negative integers and S is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of Xn, and that, roughly speaking, ηn is close to being Markov when Xn is large. This departure from much of the literature, which assumes that ηn is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for Xn given ηn. We give a recurrence classification in terms of increment moment parameters for Xn and the stationary distribution for the large- X limit of ηn. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between Xn (rescaled) and ηn. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on Z+ (the case where S is a singleton). Motivation arises from modulated queues or processes with hidden variables where ηn tracks an internal state of the system

    Deposition, diffusion, and nucleation on an interval

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    Motivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model studied in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparently, no simple closed form. We show that the same splitting density governs the fixed deposition rate, large time asymptotics of the normalized gap distribution, so these asymptotics are independent of deposition rate. The splitting density is derived by solving an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson

    A virtual earth model of the dementias in China

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    © 2017 International Medical Informatics Association (IMIA) and IOS Press. This developmental project was undertaken to explore how applying spatial science analysis and visualisation methods might inform societies undergoing significant structural and demographic change. China is rapidly transitioning to an aged society. It already exceeds all other countries in its population aged 65 years and over. Dementia is closely correlated with ageing and intersects with a variety of physical and cognitive disabilities. Information dashboards are a growing part of health and social policy data environments. These visual data applications increasingly include mapping capabilities. In this paper, we explore the utility of a geographic modelling approach to exploring the complex nature of population ageing and the dementias in China

    A comprehensive overview of social network measures for older adults : a systematic review

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    Objectives: The size and type of older adults' social networks is associated with health, mental and social outcomes. Investigators within many disciplines are now measuring social networks, but it is not always clear what they are assessing, or which measures may best meet their objectives. To undertake a systematic review to identify (i) social network measures used for older adults, (ii) variety of social network dimensions and (iii) how measures have developed over time. Materials and Methods: The MEDLINE, EMBASE, CINAHL, PsycInfo and Cochrane Library databases were systematically searched to identify social network instruments, followed by categorization of the domains into quantitative, qualitative and alter domains. Results: A total of 229 studies and 21 social network measures were included, with 11 quantitative dimensions (e. g., size, frequency), 5 qualitative dimensions (e.g., support satisfaction, emotional bond) and 7 alter members (e. g., family, neighbours) of social networks identified. Measures commonly clustered on quantifiable network size (n = 19), availability of supportive networks (n = 14) and presence of family ties (n = 21). The period between 1985 and 1995 produced the greatest number of newly developed social network measures (n = 10) with a stronger focus on qualitative features. Discussion and Implications: This review provides researchers with an organized summary of measures and dimensions for consideration when appraising social connections in older adults. This can enable better study design through providing information that makes explicit inevitable trade-offs between survey length, comprehensiveness of dimension coverage, and utilization of the measure for researchers

    Modular decomposition and the Reconstruction Conjecture

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    Abstract We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs