14,227 research outputs found

    Well-balanced and asymptotic preserving schemes for kinetic models

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    In this paper, we propose a general framework for designing numerical schemes that have both well-balanced (WB) and asymptotic preserving (AP) properties, for various kinds of kinetic models. We are interested in two different parameter regimes, 1) When the ratio between the mean free path and the characteristic macroscopic length ϵ\epsilon tends to zero, the density can be described by (advection) diffusion type (linear or nonlinear) macroscopic models; 2) When ϵ\epsilon = O(1), the models behave like hyperbolic equations with source terms and we are interested in their steady states. We apply the framework to three different kinetic models: neutron transport equation and its diffusion limit, the transport equation for chemotaxis and its Keller-Segel limit, and grey radiative transfer equation and its nonlinear diffusion limit. Numerical examples are given to demonstrate the properties of the schemes

    Novel biodegradable polyesters derived from carbohydrates

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    This thesis concerns the synthesis of new biodegradable polyesters from carbohydrates. All the new products have been characterised using nuclear magnetic resonance (NMR) spectroscopy and mass spectrometry; some have been studied using X-ray crystallography. In Chapter 1, which is the introduction, some of the leading biodegradable polymers are introduced, in particular attention is paid to polyesters which are synthesised from carbohydrates. In addition, the standard polymerisation methods for the polyester synthesis - ring opening polymerisation and step polymerisation - are described. Chapter 2 details the synthesis and polymerisation of acetic acid 5-acetoxy-6-oxo-tetrahydro-pyran-2-yl methyl ester. Three different initiating systems have been used for ring opening polymerisations; the kinetics and the polymer product of the polymerisation are discussed. L-Lactide was used in copolymerisations with acetic acid 5-acetoxy-6-oxo-tetrahydro-pyran-2-yl methyl ester; the copolymers show distinct thermal properties and accelerated degradation rates compared with poly(L-Lactide) (PLLA). Chapter 3 involves applying functionalised carbohydrates, with one free hydroxyl group, as the co-initiator for the ring opening polymerisation (ROP) of L-lactide. The polymerisations were well controlled with linear relationships between the percentage lactide conversion and the polymer’s molecular weight. The carbohydrates form the functionalised end group of the PLLA, which, in turn, improve the hydrophilicity of the resulting PLLA. These end-group functionalised polylactides have been used as the matrix for human-osteoblast-derived osteosarcoma cells (SaOS-2 cells) culturing, and these studies proved that the carbohydrate end groups were non toxic. Chapter 4 describes the synthesis of a series of other carbohydrate lactones derived from D-glucono-1,5-lactone, D-xylose and 2-deoxy-Dribose; the lactones include 2,3,4-tri-O-benzyl-D-xylonolactone, 6-methyl-2-oxotetrahydro- 2H-pyran-3-yl acetate, (S)-6-(benzyloxymethyl)-tetrahydropyran-2-one. The ROP of these lactones resulted in formation of low molecular weight oligomers, except for 2,3,4-tri-O-benzyl-D-xylonolactone, which can not be polymerised. Chapter 5 describes the overall conclusions resulting from the experiments described in the thesis. Chapter 6 is the experimental section and thus provides a detailed description of the synthesis of all the compounds prepared in the thesis and their analytical data. The Appendices include two papers that have already been published, describing some of the work in the thesis; important additional NMR spectra and MALDI-ToF spectra; and the complete X-ray crystallography data

    Full-counting statistics of charge and spin transport in the transient regime: A nonequilibrium Green's function approach

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    We report the investigation of full-counting statistics (FCS) of transferred charge and spin in the transient regime where the connection between central scattering region (quantum dot) and leads are turned on at t=0t=0. A general theoretical formulation for the generating function (GF) is presented using a nonequilibrium Green's function approach for the quantum dot system. In particular, we give a detailed derivation on how to use the method of path integral together with nonequilibrium Green's function technique to obtain the GF of FCS in electron transport systems based on the two-time quantum measurement scheme. The correct long-time limit of the formalism, the Levitov-Lesovik's formula, is obtained. This formalism can be generalized to account for spin transport for the system with noncollinear spin as well as spin-orbit interaction. As an example, we have calculated the GF of spin-polarized transferred charge, transferred spin, as well as the spin transferred torque for a magnetic tunneling junction in the transient regime. The GF is compactly expressed by a functional determinant represented by Green's function and self-energy in the time domain. With this formalism, FCS in spintronics in the transient regime can be studied. We also extend this formalism to the quantum point contact system. For numerical results, we calculate the GF and various cumulants of a double quantum dot system connected by two leads in transient regime. The signature of universal oscillation of FCS is identified. On top of the global oscillation, local oscillations are found in various cumulants as a result of the Rabi oscillation. Finally, the influence of the temperature is also examined

    Derivation of the bacterial run-and-tumble kinetic equation from a model with biochemical pathway

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    Kinetic-transport equations are, by now, standard models to describe the dynamics of populations of bacteria moving by run-and-tumble. Experimental observations show that bacteria increase their run duration when encountering an increasing gradient of chemotactic molecules. This led to a first class of models which heuristically include tumbling frequencies depending on the path-wise gradient of chemotactic signal. More recently, the biochemical pathways regulating the flagellar motors were uncovered. This knowledge gave rise to a second class of kinetic-transport equations, that takes into account an intra-cellular molecular content and which relates the tumbling frequency to this information. It turns out that the tumbling frequency depends on the chemotactic signal, and not on its gradient. For these two classes of models, macroscopic equations of Keller-Segel type, have been derived using diffusion or hyperbolic rescaling. We complete this program by showing how the first class of equations can be derived from the second class with molecular content after appropriate rescaling. The main difficulty is to explain why the path-wise gradient of chemotactic signal can arise in this asymptotic process. Randomness of receptor methylation events can be included, and our approach can be used to compute the tumbling frequency in presence of such a noise
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