| 1. |
- Almanasreh, Hasan, 1981
(författare)
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The Dirac Operator; From Numerics to the Theory of G-convergence
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We consider two main issues concerning the Dirac operator, the first is widely known as the appearance of spurious eigenvalues within the spectrum. The second is the study of the asymptotic behavior of the eigenvalues for a family of Dirac operators with oscillatory potential added to the Coulomb-Dirac Hamiltonian. In the first problem a stable Finite element scheme is used to treat the problem of spuriousity of the radial Dirac operator with a Coulomb potential. The numerical accuracy depends strongly on the derivation of a fine-intrinsic stability parameter. In the second problem we consider the Coulomb-Dirac operator with addition to an abstract oscillating potential. Using the spectral measure of operators we project into the positive part of the perturbed Hamiltonian. By using G-convergence theory of positive self-adjoint operators in Hilbert spaces and ¡- convergence of the associated quadratic functionals we prove G-compactness for a family of positive Dirac operators under certain assumptions on the potentials.
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| 2. |
- Cronie, Ottmar, 1979
(författare)
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Different Aspects of Inference for Spatio-Temporal Point Processes
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis deals with inference problems related to the Renshaw-Särkkä growth interaction model (RS-model). It is a continuous time spatio-temporal point process with time dependent interacting marks, in which the immigrationdeath process (a continuous time Markov chain) controls the arrivals of new marked points as well as their potential life-times. The data considered are marked point patterns sampled at fixed time points. First we propose three edge correction methods for discretely sampled (marked) spatio-temporal point processes. These are all based on the idea of placing an approximated expected behaviour of our process at hand (based on simulated realisations) outside the study region, which in turn interacts with the data during the estimation. We study the methods and evaluate them numerically in the context of the RS-model. The parameters related to the development of the marks are estimated using the least-squares approach. Secondly, we propose (approximate) maximum likelihood (ML) estimators for the two parameters of the immigration-death process; the arrival intensity and the death rate. The arrival intensity is assumed to be constant and the death rate is assumed to be proportional to a function of the current mark size of a point. The arrival intensity estimator is constructed to compensate for the (unobserved) individuals arriving and dying between two sampled time points. When assumed that the death rate is constant we can derive the transition probabilities of the immigration-death process. These in turn give us the exact likelihood of its parameter pair. We are able to reduce the likelihood maximisation problem from two dimensions to one dimension. Furthermore, under the condition that the parameter pair lies in some compact subset of the positive part of the real plane, we manage to show the consistency and the asymptotic normality of its ML-estimator under an equidistant sampling scheme. These results are also evaluated numerically.
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| 3. |
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| 4. |
- Eriksson, Frank, 1980
(författare)
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Semiparametric survival models for routine register data
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Routine registers offer researchers opportunities to carry out studies of covariate effects on lifetimes of rare diseases otherwise infeasible because of the large cohorts required. Familial relationships necessary for analysis of environmental or genetic factors can be identified by record linking. The vast amount of data and clustering of related individuals pose statistical challenges. As most statistical information is associated with the cases, an estimator based on a sample where cases are overrepresented can drastically reduce the sample size with only a minor loss of efficiency. This thesis concerns regression of clustered cohort sampled survival data within the broad class of semiparametric transformation models. This class includes the proportional hazards and proportional odds models as special cases. Correlations within clusters are modeled by random effects. We derive consistency and asymptotic normality of a weighted maximum likelihood estimator and provide a consistent estimator of its asymptotic variance. A likelihood ratio test for regression coefficients is also proposed. The method is shown to perform well on simulated data and is illustrated by application to a study on cardiovascular diseases among Swedish men.
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| 5. |
- Freij, Ragnar, 1984
(författare)
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Enumeration on words, complexes and polytopes
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis presents four papers, studying enumerative problems on combinatorial structures. The first paper studies Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. The main motivation is the case where some group acts transitively on the vertex set of the complex, and G is some large subgroup of this group. In particular we are interested in complexes of graph properties. As an application, we determine the (C_2 x S_{n-2})-homotopy type of the complex of non-connected graphs on n nodes. The motivation behind the second paper is Gil Kalai's conjecture from 1989, that a centrally symmetric d-polytope must have at least 3^d non-empty faces. Looking for examples that are close to achieving the lower bound, we study the centrally symmetric Hansen polytopes, associated to perfect graphs. In particular, we study Hansen polytopes of split graphs. Among them, we find an infinite family of polytopes with 3^d+16 faces. We also prove that a Hansen polytope of a split graph has at least 3^d non-empty faces. The third paper studies the problem of packing a pattern as densely as possible into compositions. We are able to find the packing density for some classes of generalized patterns and all the three letter binary patterns. In the fourth paper, we enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point lambda-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given. We also prove that except in a trivial special case, the event that pi has descents in a set S of positions is positively correlated with the event that pi is a derangement, if pi is chosen uniformly in S_n.
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| 6. |
- Hammar, Oscar, 1977
(författare)
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Inference in a Partially Observed Percolation Process
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this licentiate thesis, inference in a partially oberved percolation process living on a graph, is considered. Each edge of the graph is declared open with probability $\theta$ and closed with probability $1-\theta$ independently of the states of all other edges. The inference problem is to draw inference about $\theta$ based on the information on whether or not particular pairs of vertices are connected by open paths. Consistency results under certain conditions on the graph are given for both a Bayesian and a frequentist approach to the inference problem. Moreover, a simulation study is presented which in addition to illustrating the consistency results, also indicates that the consistency results might hold for percolation processes on more general graphs.
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| 7. |
- Lärkäng, Richard, 1985
(författare)
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Residue currents on analytic spaces
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns residue currents on analytic spaces. In the first paper, we construct Coleff-Herrera products and Bochner-Martinelli type currents associated with a weakly holomorphic mapping, and show that these currents satisfy well-known properties from the strongly holomorphic case. This includes the transformation law, the Poincaré-Lelong formula and the equivalence of the Coleff-Herrera product and the Bochner-Martinelli current associated with a complete intersection of weakly holomorphic functions. In the second paper, we discuss the duality theorem on singular varieties. In the case of a complex manifold, the duality theorem, proven by Dickenstein-Sessa and Passare, says that the annihilator of the Coleff-Herrera product associated with a complete intersection $f$ equals the ideal generated by $f$. We give sufficient and in many cases necessary conditions in terms of certain singularity subvarieties of the sheaf $\mathcal{O}_Z$ for when the duality theorem holds on a singular variety $Z$.
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| 8. |
- Sznajdman, Jacob, 1983
(författare)
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Some Analytic generalizations of the Briancon-Skoda Theorem
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The Brian\c con-Skoda theorem appears in many variations in recent literature. The common denominator is that the theorem gives a sufficient condition that implies a membership $\phi\in \ideala^l$, where $\ideala$ is an ideal of some ring $R$. In the analytic interpretation $R$ is the local ring of an analytic space $Z$, and the condition is that $|\phi|\leq C|\ideala|^{N+l}$ holds on the space $Z$. The theorem thus relates the rate of vanishing of $\phi$ along the locus of $\ideala$ to actual membership of (powers of) the ideal. The smallest integer $N$ that works for all $\ideala \subset R$ and all $l\geq 1$ simultaneously will be called the Brian\c con-Skoda number of the ring $R$. The thesis contains three papers. The first one gives an elementary proof of the original Brian\c con-Skoda theorem. This case is simply $Z=\C^n$. The second paper contains an analytic proof of a generalization by Huneke. The result is also sharper when $\ideala$ has few generators if the geometry is not to complicated in a certain sense. Moreover, the method can give upper bounds for the Brian\c con-Skoda number for some varieties such as for example the cusp $z^p = w^q$. In the third paper non-reduced analytic spaces are considered. In this setting Huneke's generalization must be modified to remain valid. More precisely, $\phi$ belongs to $\ideala^l$ if one requires that $|L \phi| \leq C |\ideala|^{N+l}$ holds on $Z$ for a given family of holomorphic differential operators on $Z$. We impose the assumption that the local ring $\O_Z$ is Cohen-Macaulay for technical reasons.
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| 9. |
- Säfström, Ida, 1984
(författare)
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Tensor Products of Highest Weight Representations and Skew-Symmetric Matrix Equations A+B+C=0
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The question of characterizing the eigenvalues for the sum of two Hermitian matrices, was solved in 1999, after almost a century of efforts. The saturation conjecture for GL_C(n) was proven by Knutson and Tao, filling in the last gap in Horn’s conjecture. Under certain conditions, this problem is equivalent to decomposing the tensor product of two finite dimensional irreducible highest weight representations of GL_C(n). In the first part of this thesis we use the methods of moment maps and coadjoint orbits to find equivalence between the eigenvalue problem for skew-symmetric matrices and the decomposition of tensor products of irreducible highest weight representations of SO_C(2k). We characterize the eigenvalues in the cases k = 2,3, where we can take advantage of Lie algebra isomorphisms. In the second part, we consider irreducible, infinite dimensional, unitary highest weight representations of GL_C(n + 1) as representations on spaces of vector valued polynomials, and we find irreducible factors in the tensor product of two such representations.
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| 10. |
- Wojciechowski, Adam, 1982
(författare)
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On the optimization of opportunistic maintenance activities
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Maintenance is a source of large costs; in the EU the maintenance costs amountto between 4% and 8% of the total sales turnover. Opportunistic maintenance is anattempt to lower the maintenance cost by considering the failure of onecomponent as an opportunity to replace yet non-failed components in order toprevent future failures. At the time of failure of one component, a decision is to be made on which additional components toreplace in order to minimize the expected maintenance cost over a planningperiod.This thesis continues the work of Dickman et. al. (1991) and Andreasson (2004) on the opportunistic replacement problem. In Paper I, we show thatthe problem with time-dependent costs is NP-hard and present a mixed integerlinear programming model for the problem. We apply the model to problems withdeterministic and stochastic component lives with data originating from theaviation and wind power industry. The model is applied in a stochastic settingby employing the expected values of component lives. In Paper II, a first steptowards a stochastic programming model that considers components with uncertain lives istaken by extending the problem to allow non-identical lives for component individuals.This problem is shown to be NP-hard even with time-independent costs. We present amixed integer linear programming model of the problem. The solution time of the model is substantially reduced compared to the model presented in Andreasson (2004). In Paper III, we then study the opportunisticreplacement problem with uncertain component lives and present a two-stagestochastic programming approach. We present a deterministic equivalent model anddevelop a decomposition method. Numerical studies on the same data as in Paper I from the aviation and wind power industry show that the stochastic programming approachproduces maintenance decisions that are on average less costly than decisions obtained from simple maintenance policies and the approach used in Paper I. The decomposition method requires less CPU-time than solving the deterministic equivalent on three out of four problems.
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