Student Number : 9804484F MSc dissertation - School of Statistics and Acturial Science - Faculty of Science / This dissertation conducts a comprehensive and up to date review of measures of component and module importance within monotone systems, where it is assumed that components work independent of each other. The dissertation traces the development of these important measures from the initial definition of Birnbaum importance right through to the definition of Meng's criticality importance. Furthermore, the dissertation draws a distinction between time independent measures and time dependent measures (such as the Barlow-Proschan measures). The dissertation demonstrates how such measures may be implemented in analysing the importance of components within the monotone systems by evaluating these measures for a well known bridge structure example. This evaluation also reveals how each defined measure can be compared to each other. In conclusion, the dissertation describes how these measures can be extended to non-monotone systems or systems with dependent components.

In the dissertation, a monotone spline-based nonparametric estimation method is proposed for analyzing longitudinal data with mixture distributions. The innovative and efficient algorithm combining the concept of projected Newton-Raphson algorithm with linear mixed model estimation method is developed to obtain the nonparametric estimation of monotone B-spline functions. This algorithm provides an efficient and flexible approach for modeling longitudinal data monotonically. An iterative 'one-step-forward' algorithm based on the K-means clustering is then proposed to classify mixture distributions of longitudinal data. This algorithm is computationally efficient, especially for data with a large number of underlying distributions. To quantify the disparity of underlying distributions of longitudinal data, we also propose an index measure on the basis of the aggregated areas under the curve (AAUC), which makes no distributional assumptions and fits the theme of nonparametric analysis. An extensive simulation study is conducted to assess the empirical performance of our method under different AAUC values, covariance structures, and sample sizes. Finally, we apply the new approach in the PREDICT-HD study, a multi-site observational study of Huntington Disease (HD), to explore and assess clinical markers in motor and cognitive domains for the purpose of distinguishing participants at risk of HD from healthy subjects.

Weber and Stidham (1987) used submodularity to establish transition monotonicity (a service completion at one station cannot reduce the service rate at another station) for Markovian queueing networks that meet certain regularity conditions and are controlled to minimize service and queueing costs. We give an extension of monotonicity to other directions in the state space, such as arrival transitions, and to arrival routing problems. The conditions used to establish monotonicity, which deal with the boundary of the state space, are easily verified for many queueing systems. We also show that, without service costs, transition-monotone controls can be described by simple control regions and switching functions, extending earlier results. The theory is applied to production/inventory systems with holding costs at each stage and finished goods backorder costs.

Weber and Stidham (1987) used submodularity to establish transition monotonicity (a service completion at one station cannot reduce the service rate at another station) for Markovian queueing networks that meet certain regularity conditions and are controlled to minimize service and queueing costs. We give an extension of monotonicity to other directions in the state space, such as arrival transitions, and to arrival routing problems. The conditions used to establish monotonicity, which deal with the boundary of the state space, are easily verified for many queueing systems. We also show that, without service costs, transition-monotone controls can be described by simple control regions and switching functions, extending earlier results. The theory is applied to production/inventory systems with holding costs at each stage and finished goods backorder costs.

In this paper, the monotone method is extended to the initial-boundary value problems of nonlocal PDE system of first order, both quasi-monotone and non-monotone. A comparison principle is established, and a monotone scheme is given.

We give a necessary and sufficient condition for the existence of an increasing coupling of N (N >= 2) synchronous dynamics on S-Zd (PCA). Increasing means the coupling preserves stochastic ordering. We first present our main construction theorem in the case where S is totally ordered; applications to attractive PCAs are given. When S is only partially ordered, we show on two examples that a coupling of more than two synchronous dynamics may not exist. We also prove an extension of our main result for a particular class of partially ordered spaces.

In this thesis we consider the Einstein-Vlasov system, which models a system of particles within the framework of general relativity, and where collisions between the particles are assumed to be sufficiently rare to be neglected. Here the particles are stars, galaxies or even clusters of galaxies, which interact by the gravitational field generated collectively by the particles. The thesis consists of three papers, and the first two are devoted to cylindrically symmetric spacetimes and the third treats the spherically symmetric case. In the first paper the time-dependent Einstein-Vlasov system with cylindrical symmetry is considered. We prove global existence in the so called polarized case under the assumption that the particles never reach a neighborhood of the axis of symmetry. In the more general case of a non-polarized metric we need the additional assumption that the derivatives of certain metric components are bounded in a vicinity of the axis of symmetry to obtain global existence. The second paper of the thesis considers static cylindrical spacetimes. In this case we prove global existence in space and also that the solutions have finite extension in two of the three spatial dimensions. It then follows that it is possible to extend the spacetime by gluing it with a Levi-Civita spacetime, i.e. the most general vacuum solution of the static cylindrically symmetric Einstein equations. In the third and last paper, which is a joint work with C. Uggla and M. Heinzle, the static spherically symmetric Einstein-Vlasov system is studied. We introduce a new method by rewriting the system as an autonomous dynamical system on a state space with compact closure. In this way we are able to improve earlier results and enlarge the class of distribution functions which give rise to steady states with finite mass and finite extension.

Zugl.: Trier, Univ., Diss., 2007

This thesis is an examination of the structure of irreducible continua, with a particular emphasis on local connectedness and monotone maps. A continuum is irreducible if there exist a pair of points such that no proper subcontinuum contains both, with the arc being the most basic example. Being irreducible has a number of interesting implications for a continuum, both locally and globally, and it is these consequences we shall focus on. As mentioned above, the arc is the most straightforward example of an irreducible continuum. Indeed, an intuitive understanding of an irreducible continuum would be that it is structured like an arc, with the points of irreducibility at either end joined by a subspace with no loops or offshoots. In Chapter 2 we will see that for a certain class of continua this intuition is well founded by constructing a monotone map from an irreducible continuum onto an arc. This monotone map will preserve much of the structure of our continuum and as such will provide an insight into that structure. We will next examine a generalisation of irreducibility which considers finite sets of points rather than just pairs. A number of classical results will be re-examined in this light in Chapter 3. While the majority of these theorems will be shown to have close parallels in higher finite and infinite irreducibility there will be several which do not hold without further conditions on the continuum. Such anomalies will be particularly prevalent in continua which have indecomposable subcontinua dominating their structure. In Chapter 4 monotone maps will be constructed for finitely irreducible continua similar to the map to an arc mentioned previously. Chapters 7 and 8 will generalise irreducibility further to the infinite case and we will again construct monotone maps preserving the structure of our continuum. Along with the arc, another highly significant irreducible continuum is the sin 1 x continuum. Chapter 5 will focus on this continuum, which will be the basis for a nested sequence of continua. A number of results concerning continuous images of these continua will be presented before using the sequence of continua to define an indecomposable continuum. This continuum will be investigated, and it will be shown that the union of our nested continua form a composant of the indecomposable continuum. In Chapter 6 we will turn to the question of compactifications. If a space X is connected then any metric compactification of X will be a continuum. This chapter will answer the question of when a compactification is an irreducible continuum, with the remainder of the compactification consisting of all of the irreducible points. A list of properties will given such that a continuum has such a compactification if and only if it has each property on the list. It will also be demonstrated that each of these properties is independent of the others. Finally, in Chapter 9 we will revisit the idea of structure-preserving monotone maps, but this time in continua which are not irreducible. Motivated by the fibres of the maps in previous chapters, we will introduce two categories of subcontinua of a continuum X. The first will be nowhere dense subcontinua which are maximal with this property and the second will be subcontinua about which X is locally connected and which are minimal with this property. Continua in which every point lies in a maximal nowhere dense subcontinuum will be examined, as well as spaces in which every point lies in a unique minimal subcontinuum about which X is locally connected. We will also look at the properties of monotone maps arising from partitions of X into such subcontinua, and will prove that if every point of X lies in a maximal nowhere dense subcontinuum then the resulting quotient space will be one dimensional.

Les sous-variées isotropes maximales en géométries symplectique sont appelées lagrangiennes ; parmi celles-ci on distingue les lagrangiennes monotones. Historiquement leur définition est motivée en partie par la construction de l'homologie de Floer lagrangiennes ; elles présentent ainsi une classe plus rigide, moins étendue, de lagrangiennes. Ce manuscrit établit une contrainte sur le groupe fondamental de certaines lagrangiennes monotones, qui s'applique en particulier lorsque la variété symplectique ambiante est l'espace projectif complexe. Une des conséquences du théorème principal est d'exclure toute une classe d'exemples classiques de lagrangiennes, due à L. Polterovich, du cas monotone. Elle conduit également à une discussion sur les topologies possibles en dimension 3. / This thesis establishes a topological constraint on the fundamental group of some monotone Lagrangien. One useful consequence is to rule out a class of examples of Lagrangians due to L. Polterovich as monotone ones. It also leads to a discussion on the possible topologies en dimension 3.