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Masters Degrees (Applied Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7108

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Browsing Masters Degrees (Applied Mathematics) by Author "Banasiak, Jacek."

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    Analysis and numerical solutions of fragmentation equation with transport.
    (2012) Wetsi, Poka David.; Banasiak, Jacek.; Shindin, Sergey Konstantinovich.
    Fragmentation equations occur naturally in many real world problems, see [ZM85, ZM86, HEL91, CEH91, HGEL96, SLLM00, Ban02, BL03, Ban04, BA06] and references therein. Mathematical study of these equations is mostly concentrated on building existence and uniqueness theories and on qualitative analysis of solutions (shattering), some effort has be done in finding solutions analytically. In this project, we deal with numerical analysis of fragmentation equation with transport. First, we provide some existence results in Banach and Hilbert settings, then we turn to numerical analysis. For this approximation and interpolation theory for generalized Laguerre functions is derived. Using these results we formulate Laguerre pseudospectral method and provide its stability and convergence analysis. The project is concluded with several numerical experiments.
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    Categorical systems biology : an appreciation of categorical arguments in cellular modelling.
    (2012) Songa, Maurine Atieno.; Banasiak, Jacek.; Amery, Gareth.
    With big science projects like the human genome project, [2], and preliminary attempts to seriously study brain activity, e.g. [9], mathematical biology has come of age, employing formalisms and tools from most branches of mathematics. Recent results, [51] and [53], have extended the relational (or categorical) approach of Rosen [44], to demonstrate that (in a very general class of systems) cellular self-organization/self-replication is implicit in metabolism and repair/stability. This is a powerful philosophical statement and removes the need of teleological argument. However, the result carries a technical limitation to Cartesian closed categories, which excludes many mathematical languages. We review the relevant literature on metabolic-repair pathways, category theory and systems theory, before performing a critique of this work. We find that the restriction to Cartesian closed categories is purely for simplicity, and describe how equivalent arguments may be built for monoidal closed categories. Moreover, any symmetric monoidal category may be "embedded" in a closed one. We discuss how these constructions/techniques provide the formal structure to treat self-organization/self-replication in most contemporary mathematical (modelling) languages. These results signicantly soften the impact on current modelling paradigms while extending the philosophical implications.
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    Matrix models of population theory.
    (2013) Abdalla, Suliman Jamiel Mohamed.; Banasiak, Jacek.
    Non-negative matrices arise naturally in population models. In this thesis, we first study Perron- Frobenius theory of non-negative irreducible matrices. We use this theory to investigate the asymptotic behaviour of discrete time linear autonomous models. Then we discuss an application for this in age structured population. Furthermore, we study Liapunov stability of a general non-linear autonomous model. We consider a general nonlinear autonomous model that arises in structured population. We assume that the associated nonlinear matrix of this model is non-increasing at all density levels. Then, we show the existence of global extinction. In addition, we show the stability condition of the extinction equilibrium of the this model in the Liapunov sense.