| 1. |
- Musonda, John
(author)
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Orthogonal Polynomials, Operators and Commutation Relations
- 2017
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Licentiate thesis (other academic/artistic)abstract
- Orthogonal polynomials, operators and commutation relations appear in many areas of mathematics, physics and engineering where they play a vital role. For instance, orthogonal functions in general are central to the development of Fourier series and wavelets which are essential to signal processing. In particular, as demonstrated in this thesis, orthogonal polynomials can be used to establish the L2-boundedness of singular integral operators which is a fundamental problem in harmonic analysis and a subject of extensive investigations. The Lp-convergence of Fourier series is closely related to the Lp-boundedness of singular integral operators. Many important relations in physical sciences are represented by operators satisfying various commutation relations. Such commutation relations play key roles in such areas as quantum mechanics, wavelet analysis, representation theory, spectral theory, and many others.This thesis consists of three main parts. The first part presents a new system of orthogonal polynomials, and establishes its relation to the previously studied systems in the class of Meixner–Pollaczek polynomials. Boundedness properties of two singular integral operators of convolution type are investigated in the Hilbert spaces related to the relevant orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on L2-spaces, and estimates of the norms are obtained.The second part extends the investigation of the boundedness properties of the two singular integral operators to Lp-spaces on the real line, both in the weighted and unweighted spaces. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for L2 and weak boundedness for L1, and then using interpolation to obtain boundedness for the intermediate spaces. To obtain boundedness for the remaining spaces, duality is used in the translation invariant case, while the weighted case is partly based on the methods developed by M. Riesz in his paper of 1928 for the conjugate function operator.The third and final part derives simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Centralizers and centers are computed as an example of an application of the formulas.
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| 2. |
- Abola, Benard
(author)
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Perturbed Markov Chains with Damping Component and Information Networks
- 2020
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Doctoral thesis (other academic/artistic)abstract
- This thesis brings together three thematic topics, PageRank of evolving tree graphs, stopping criteria for ranks and perturbed Markov chains with damping component. The commonality in these topics is their focus on ranking problems in information networks. In the fields of science and engineering, information networks are interesting from both practical and theoretical perspectives. The fascinating property of networks is their applicability in analysing broad spectrum of problems and well established mathematical objects. One of the most common algorithms in networks' analysis is PageRank. It was developed for web pages’ ranking and now serves as a tool for identifying important vertices as well as studying characteristics of real-world systems in several areas of applications. Despite numerous successes of the algorithm in real life, the analysis of information networks is still challenging. Specifically, when the system experiences changes in vertices /edges or it is not strongly connected or when a damping stochastic matrix and a damping factor are added to an information matrix. For these reasons, extending existing or developing methods to understand such complex networks is necessary.Chapter 2 of this thesis focuses on information networks with no bidirectional interaction. They are commonly encountered in ecological systems, number theory and security systems. We consider certain specific changes in a network and describe how the corresponding information matrix can be updated as well as PageRank scores. Specifically, we consider the graph partitioned into levels of vertices and describe how PageRank is updated as the network evolves.In Chapter 3, we review different stopping criteria used in solving a linear system of equations and investigate each stopping criterion against some classical iterative methods. Also, we explore whether clustering algorithms may be used as stopping criteria.Chapter 4 focuses on perturbed Markov chains commonly used for the description of information networks. In such models, the transition matrix of an information Markov chain is usually regularised and approximated by a stochastic (Google type) matrix. Stationary distribution of the stochastic matrix is equivalent to PageRank, which is very important for ranking of vertices in information networks. Determining stationary probabilities and related characteristics of singularly perturbed Markov chains is complicated; leave alone the choice of regularisation parameter. We use the procedure of artificial regeneration for the perturbed Markov chain with the matrix of transition probabilities and coupling methods. We obtain ergodic theorems, in the form of asymptotic relations. We also derive explicit upper bounds for the rate of convergence in ergodic relations. Finally, we illustrate these results with numerical examples.
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| 3. |
- Albuhayri, Mohammed
(author)
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Asymptotics of implied volatility in the Gatheral double stochastic volatility model
- 2022
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Doctoral thesis (other academic/artistic)abstract
- We consider a market model of financial engineering with three factors represented by three correlated Brownian motions. The volatility of the risky asset in this model is the sum of two stochastic volatilities. The dynamic of each volatility is governed by a mean-reverting process. The first stochastic volatility of mean-reversion process reverts to the second volatility at a fast rate, while the second volatility moves slowly to a constant level over time with the state of the economy.The double mean-reverting model by Gatheral (2008) is motivated by empirical dynamics of the variance of the stock price. This model can be consistently calibrated to both the SPX options and the VIX options. However due to the lack of an explicit formula for both the European option price and the implied volatility, the calibration is usually done using time consuming methods like Monte Carlo simulation or the finite difference method.To solve the above issue, we use the method of asymptotic expansion developed by Pagliarani and Pascucci (2017). In paper A, we study the behaviour of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. We calculate explicitly the asymptotic expansions of implied volatility within a parabolic region up the second order. In paper B we improve the results obtain in paper A by calculating the asymptotic expansion of implied volatility under the Gatheral model up to order three. In paper C, we perform numerical studies on the asymptotic expansion up to the second order. The Monte-Carlo simulation is used as the benchmark value to check the accuracy of the expansions. We also proposed a partial calibration procedure using the expansions. The calibration procedure is implemented on real market data of daily implied volatility surfaces for an underlying market index and an underlying equity stock for periods both before and during the COVID-19 crisis. Finally, in paper D we check the performance of the third order expansion and compare it with the previous results.
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| 4. |
- Bäck, Per, 1987-
(author)
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On Hom-associative Ore Extensions
- 2022
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Doctoral thesis (other academic/artistic)abstract
- In this thesis, we introduce and study hom-associative Ore extensions. These are non-unital, non-associative, non-commutative polynomial rings in which the associativity condition is “twisted” by an additive group homomorphism. In particular, they are examples of hom-associative algebras, and they generalize the classical non-commutative polynomial rings introduced by Ore known as Ore extensions to the non-unital, hom-associative setting. At the same time, when the twisted associativity condition is null, they also generalize to the general non-unital, non-associative setting. We deduce necessary and sufficient conditions for hom-associative Ore extensions to exist, and construct concrete examples thereof. These include hom-associative generalizations of the quantum plane, the universal enveloping algebra of the two-dimensional non-abelian Lie algebra, and the first Weyl algebra, to name a few. The aforementioned algebras turn out to be formal hom-associative deformations of their associative counterparts, the latter two which cannot be formally deformed in the associative setting. Moreover, these are all weakly unital algebras, and we provide a way of embedding any multiplicative, non-unital hom-associative algebra into a multiplicative, weakly unital hom-associative algebra. This generalizes the classical unitalization of non-unital, associative algebras. We then study the hom-associative Weyl algebras in arbitrary characteristic, classify them up to isomorphism, and in the zero characteristic case, we prove that an analogue of the Dixmier conjecture is true. We also study hom-modules over hom-associative rings, and by doing so, we are able to prove a Hilbert's basis theorem for hom-associative Ore extensions. Our theorem includes as special cases both the classical Hilbert's basis theorem for Ore extensions and a Hilbert's basis theorem for unital, non-associative Ore extensions. Last, we construct examples of both hom-associative and non-associative Ore extensions which are all Noetherian by our theorem.
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| 5. |
- Seleka Biganda, Pitos, 1981-
(author)
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Analytical and Iterative Methods of Computing PageRank of Networks
- 2020
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Doctoral thesis (other academic/artistic)abstract
- This thesis is about variants of PageRank, methods of PageRank computation and perturbation analysis of a PageRank vector as a stationary distribution of a kind of perturbed Markov chain model. Chapter 2 of this thesis gives closed form formulae for ordinary and lazy PageRanks for some specific simple line graphs. Different cases of changes made to the simple line graph are considered and for each case, a corresponding formula for each of the two variants of PageRank is provided.Chapter 3 is dedicated to the exploration of relationships that exist between three known variants of PageRank: ordinary PageRank, lazy PageRank and random walk with backstep PageRank in terms of their convergence and consistency in rank scores for different graph structures with reference to PageRank parameters, the damping factor c and backstep parameter β. In Chapter 4, we discuss numerical methods used in solving the PageRank problem as a linear system and evaluate some stopping criteria that can be employed in such methods. Finally, in Chapter 5, we address the PageRank problem as a first order perturbed Markov chain problem and study the perturbation analysis for stationary distributions of Markov chains with damping component. We illustrate our results on asymptotic perturbation analysis by using different computational examples.
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| 6. |
- Thapa, Gyan, 1970-
(author)
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Optimization of Just-in-Time Sequencing Problems and Supply Chain Logistics
- 2015
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Doctoral thesis (other academic/artistic)abstract
- This dissertation presents a comprehensive and comparative progress in sequencing approaches of mixed-model just-in-time (JIT) sequencing problem together with discrete apportioment problem (DAP). The goal of JIT sequencing problem (JITSP) is to keep the rate of usage of parts as constant as possible along the assembly lines, and the goal of DAP is to divide a given integer number of delegates proportionally among the states or the parties according to their population or votes. Furthermore, the supply chain logistics problem is also reported in here with some real life applications.The single-level JITSP, known as the product rate variation problem (PRVP), is pseudo-polynomially solvable. The total PRVP minimizes sum deviation and the bottleneck PRVP minimizes the maximum deviation between the actual production and the ideal production. The assignment approach solves total PRVP whereas the perfect matching works for bottleneck PRVP solving the problem in pseudo-polynomial time. The multi-level JITSP, known as the output rate variation problem (ORVP), is NP-hard in most of the cases. However, some sequencing heuristics and dynamic programming are devised for near optimal solutions. And the pegging assumption reduces the ORVP into weighted case of PRVP. In this dissertation, the total PRVP with square and absolute deviations are considered and mean-based divisor methods are devised for the equitably efficient solution. The simultaneous dealing to the PRVP and DAP establishes the interlink between the production sequencing problem and integer seat allocating problem. The new upper bottlenecks are investigated and the problems are solved comparatively. The bottleneck PRVP instances for small deviations and cyclic sequences for total PRVP are shown to be optimal. The bicriterion sequencing is discussed with Pareto optimal solutions.The production sequencing problem is simultaneously dealt with supply chain logistics to balance overall supply chain system. The cross-docking supply chain logistics problem is formulated with a proposition to be solved. The real-world applications of JITSP and supply chain are listed and some open problems are pointed out as the closing of the dissertation.
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| 7. |
- Abola, Benard, et al.
(author)
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Chapter 2. Nonlinearly Perturbed Markov Chains and Information Networks
- 2021
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In: Applied Modeling Techniques and Data Analysis 1. - Hoboken, NJ : John Wiley & Sons. - 9781786306739 - 9781119821564 ; , s. 23-55
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Book chapter (peer-reviewed)abstract
- This chapter is devoted to studies of perturbed Markov chains, commonly used for the description of information networks. In such models, the matrix of transition probabilities for the corresponding Markov chain is usually regularized by adding aspecial damping matrix, multiplied by a small damping (perturbation) parameter ε. In this chapter, we present the results of detailed perturbation analysis of Markov chains with damping component and numerical experiments supporting and illustrating the results of this perturbation analysis.
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| 8. |
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| 9. |
- Aye, Tin Nwe, 1989-
(author)
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POPULATION DYNAMICS AND TREE GROWTH STRUCTURE IN MATHEMATICAL ECOLOGY
- 2021
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Doctoral thesis (other academic/artistic)abstract
- This thesis is based on four papers related to mathematical biology, where three papers focus on population dynamics and one paper concerns tree growth and stem structure. The first two papers are mainly devoted to studying the dynamics of physiologically structured population models by using Escalator Boxcar Train (EBT) method. The third paper concerns a class of stage-structured population systems, in both deterministic and stochastic settings. The fourth paper explores how a branch thinning model can be utilized to describe the cross-sectional area of the stem of a tree, thus generalizing the classical pipe model.In Paper I, we present a merging procedure to reduce the increasing system of ordinary differential equations generated by the EBT method. In particular, we modify the EBT method to include merging of cohorts. The accuracy of this model is explored on a colony of Daphnia Pulex.In Paper II, we study the convergence rate of the modified EBT model, allowing a general class of non-linear merging procedures. We show that this modified EBT method induces a bounded number of cohorts, independent of the number of time steps. This in turn, improves the speed of the numerical algorithm for solving the population dynamics from polynomial time to linear time, that is, the time consumption to find the solution is proportional to the number of time steps.In Paper III, a class of non-linear two-stage structured population models is studied with different growth rates for the unstructured food resource under different harvesting rates in both deterministic and stochastic settings. In the stochastic setting, we develop methods to evaluate emergent properties equivalent to the properties investigated in the deterministic case. In addition, new emergent properties, e.g. probability of extinction, are also investigated.In Paper IV, we explore the stem model which is developed by combining the pipe model and the branch thinning model. The stem model provides estimates of the heartwood, sapwood and stem cross-sectional area at any height. We corroborate the accuracy of our model with empirical data and the cross validation of our results shows a very high goodness of fit for the stem model.
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| 10. |
- Lundengård, Karl, 1987-
(author)
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Extreme points of the Vandermonde determinant and phenomenological modelling with power exponential functions
- 2019
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Doctoral thesis (other academic/artistic)abstract
- This thesis discusses two topics, finding the extreme points of the Vandermonde determinant on various surfaces and phenomenological modelling using power-exponential functions. The relation between these two problems is that they are both related to methods for curve-fitting. Two applications of the mathematical models and methods are also discussed, modelling of electrostatic discharge currents for use in electromagnetic compatibility and modelling of mortality rates for humans. Both the construction and evaluation of models is discussed.In the first chapter the basic theory for later chapters is introduced. First the Vandermonde matrix, a matrix whose rows (or columns) consists of monomials of sequential powers, its history and some of its properties are discussed. Next, some considerations and typical methods for a common class of curve fitting problems are presented, as well as how to analyse and evaluate the resulting fit. In preparation for the later parts of the thesis the topics of electromagnetic compatibility and mortality rate modelling are briefly introduced.The second chapter discusses some techniques for finding the extreme points for the determinant of the Vandermonde matrix on various surfaces including spheres, ellipsoids and cylinders. The discussion focuses on low dimensions, but some results are given for arbitrary (finite) dimensions.In the third chapter a particular model called the p-peaked Analytically Extended Function (AEF) is introduced and fitted to data taken either from a standard for electromagnetic compatibility or experimental measurements. The discussion here is entirely focused on currents originating from lightning or electrostatic discharges.The fourth chapter consists of a comparison of several different methods for modelling mortality rates, including a model constructed in a similar way to the AEF found in the third chapter. The models are compared with respect to how well they can be fitted to estimated mortality rate for several countries and several years and the results when using the fitted models for mortality rate forecasting is also compared.
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