Chemical pesticides have long been used to control insect populations on field crops, and their widespread availability and ease of manipulation present an ideal system for the exploration of the relationship between vital rates and population dynamics. Research has shown that models of population effects of insecticides are more accurate when time-varying parameters are used, but the scientific community has been slow to accept such models, which may involve complicated mathematics such as partial differential equations. Additionally, many of the pesticides now in use have both acute and sublethal effects, and these differing modes of action, along with concerns for the scientific community's understanding of the models, motivate a new approach to mathematical modeling of the effects of insecticides on population dynamics. We first consider an ordinary differential equation model, which, while simple, provides highly accurate fits to population data, though it accounts for only one generation at a time without explicitly considering reproductive effects. An evaluation of the effects of insecticides on insect fecundity is considered separately. We then develop, implement, and analyze a coupled time-delay differential equation model, which incorporates information for multiple generations and is an intermediate model in terms of ease of understanding for non-mathematicians. Sensitivity results for the time delay kernel are also presented.

Free and/or moving boundary problems occur in a wide range of applications. These boundaries can obey either local or global conditions. In this dissertation, new numerical techniques for solving some of these problems are developed, analyzed, implemented and tested. The new techniques for free and moving boundary problems are 1) a second order method for solving moving boundary problems and 2) a hybrid level set/boundary element method for solving some free boundary problems. The main tool used in both is the Fast Marching method, a fast algorithm for solving the eikonal equation. An application using Fast Marching to solve a model for sand pile formation in domains with obstacles is shown. A new, second order Fast Marching scheme for domains with obstacles is introduced. We look at the stability and accuracy of discretizations commonly used with Fast Marching. The performance of Fast Marching is compared that of Fast Sweeping, another eikonal solver. The second order method for solving moving boundary problems is applied to some simple examples. Finally, a globally defined free boundary problem inspired by fluid dynamics, the Bernoulli problem, is solved using the hybrid method.

Various models are developed to describe several different aspects of the dynamics of polymers. First, a model is developed which describes the dynamics of a single polymer chain, residing in a viscous fluid, in response to various stimuli. Next, constitutive models leading to reptation are developed, which determine the stress-strain relations for a system of physically constrained (PC) molecules residing within a chemically cross-linked (CC) system of molecules acting as a constraining tube for the PC system. These reptation models treat relaxation within the context of both intra-molecular and inter-molecular variability. Finally, a model is developed which combines the Rouse model of relaxing polymers and the stick-slip reptation model. This stick-slip/Rouse hybrid model incorporates parameters which were ignored in models developed in the earlier parts of the dissertation.

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The family of sets with the Baire property of a topological space X, i.e., sets which differ from open sets by meager sets, has different nice properties, like being closed under countable unions and differences. On the other hand, the family of sets without the Baire property of X is, in general, not closed under finite unions and intersections. This thesis focuses on the algebraic set-theoretic aspect of the families of sets without the Baire property which are not empty. It is composed of an introduction and five papers. In the first paper, we prove that the family of all subsets of ℝ of the form (C \ M) ∪ N, where C is a finite union of Vitali sets and M, N are meager, is closed under finite unions. It consists of sets without the Baire property and it is invariant under translations of ℝ. The results are extended to the space ℝn for n ≥ 2 and to products of ℝn with finite powers of the Sorgenfrey line. In the second paper, we suggest a way to build a countable decomposition <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Clbrace%5Ctext%20X_i%5Crbrace%20_%7Bi=1%7D%5E%5Cinfty" /> of a topological space X which has an open subset homeomorphic to (ℝn, τ), n ≥ 1, where τ is some admissible extension of the Euclidean topology, such that the union of each non-empty proper subfamily of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Clbrace%5Ctext%20X_i%5Crbrace%20_%7Bi=1%7D%5E%5Cinfty" /> does not have the Baire property in X. In the case when X is a separable metrizable manifold of finite dimension, each element of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Clbrace%5Ctext%20X_i%5Crbrace%20_%7Bi=1%7D%5E%5Cinfty" /> can be chosen dense and zero-dimensional. In the third paper, we develop a theory of semigroups of sets with respect to the union of sets. The theory is applied to Vitali selectors of ℝ to construct diverse abelian semigroups of sets without the Baire property. It is shown that in the family of such semigroups there is no element which contains all others. This leads to a supersemigroup of sets without the Baire property which contains all these semigroups and which is invariant under translations of ℝ. All the considered semigroups are enlarged by the use of meager sets, and the construction is extended to Euclidean spaces ℝn for n ≥ 2. In the fourth paper, we consider the family V1(Q) of all finite unions of Vitali selectors of a topological group G having a countable dense subgroup Q. It is shown that the collection <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Clbrace%5Ctext%20G%20%5Csetminus%20%5Ctext%20U%20:%20%5Ctext%20U%20%5Cin%20%5C%20%5Cmathcal%7BV%7D_1(%5Ctext%7BQ%7D)%5Crbrace" /> is a base for a topology τ(Q) on G. The space (G, τ (Q)) is T1, not Hausdorff and hyperconnected. It is proved that if Q1 and Q2 are countable dense subgroups of G such that Q1 ⊆ Q2 and the factor group Q2/Q1 is infinite (resp. finite) then τ(Q1) <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Ctiny%5Cnot%5Csubseteq" /> τ(Q2) (resp. τ (Q1) ⊆ τ (Q2)). Nevertheless, we prove that all spaces constructed in this manner are homeomorphic. In the fifth paper, we investigate the relationship (inclusion or equality) between the families of sets with the Baire property for different topologies on the same underlying set. We also present some applications of the local function defined by the Euclidean topology on R and the ideal of meager sets there.

This licentiate thesis is a part of an effort to run large electronic structure calculations in modern computational environments with distributed memory. The ultimate goal is to model materials consisting of millions of atoms at the level of quantum mechanics. In particular, the thesis focuses on different aspects of a computational problem of inverse factorization of Hermitian positive definite matrices. The considered aspects are numerical properties of the algorithms and parallelization. Not only is an efficient and scalable computation of inverse factors necessary in order to be able to run large scale electronic computations based on the Hartree–Fock or Kohn–Sham approaches with the self-consistent field procedure, but it can be applied more generally for preconditioner construction. Parallelization of algorithms with unknown load and data distributions requires a paradigm shift in programming. In this thesis we also discuss a few parallel programming models with focus on task-based models, and, more specifically, the Chunks and Tasks model. / eSSENCE

A Central Counterparty (CCP) handles clearing between its members and can mutualise and reduce the counterparty and credit risk in a network. In the case of a clearing member defaulting on its obligations, the defaulted portfolio will be taken over by the CCP, which will attempt to close out the positions as quickly as possible. It is vital that the CCP minimises the losses they may suffer during the period between default and close out, the so called holding period. This thesis investigates and tests several potential hedging strategies to minimise value fluctuations during the holding period. These include neutralising the exposure to different risk factors, as well as finding the ideal hedging position using principal component analysis. The defaulted portfolio can contain different instruments, such as options, interest rate swaps and bonds, which requires different approaches to neutralise exposure. To determine the performance of the different strategies, backtesting was performed on historical data from the years 2001 to 2013, and the results were analysed in order to determine the effectiveness and potential costs of the hedging. The results show that significant reduction in value fluctuations can be achieved by employing these strategies, while not exceeding an affordable level of cost. Based on the findings, a function was created in Java that can recommend optimal hedging positions given a defaulted portfolio of any composition.

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The inverse geothermal problem consist of estimating the temperature distribution below the earth’s surface using temperature and heat-flux measurements on the earth’s surface. The problem is important since temperature governs a variety of the geological processes including formation of magmas, minerals, fosil fuels and also deformation of rocks. Mathematical this problem is formulated as a Cauchy problem for an non-linear elliptic equation and since the thermal properties of the rocks depend strongly on the temperature, the problem is non-linear. This problem is ill-posed in the sense that it does not satisfy atleast one of Hadamard’s definition of well-posedness. We formulated the problem as an ill-posed non-linear operator equation which is defined in terms of solving a well-posed boundary problem. We demonstrate existence of a unique solution to this well-posed problem and give stability estimates in appropriate function spaces. We show that the operator equation is well-defined in appropriate function spaces. Since the problem is ill-posed, regularization is needed to stabilize computations. We demostrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a well- posed problem related to the operator equation. In this study we demostrate that the algorithm works efficiently for 2D calculations but can also be modified to work for 3D calculations.