The problem is effect of electromagnetic wave. When electromagnetic wave obliquely transmitted through two different medias ,electromagnetic wave undergoes reflection and refraction at the interface and inside the media. Computation of electromagnetic wave is well known by Maxwell`s equation. There are two cases solving questions. One is constant of £g¡B£`¡B£m.Another is variable of £g¡B£`¡B£m. In case one, use D`Alembertian equation and Helmholtz equation transforming Maxwell`s equation. And solve by ILHI`s(incomplete Lips-chitz-Hankel integrals) and FFT(fast Fourier transform). In case two,if £g¡B£`¡B£m are variables ,we can simplify Maxwell`s equation. It is similar to wave equation with source. We use Finite Element Method(FEM) getting Numerical solution by setting absorbing boundary and mesh. Using results by ILHI`s would get exact solution obliquely incident on two medias. Proof numerical solution by exact solution.

Utilizando conceitos de macânica quântica e teoria cinética apresentamos uma rederivação da equação de Boltzmann para a polarização. Mostramos a equivalência entre a equação que derivamos e a equação de Boltzmann encontrada na literatura ( [1], [2], [3] ) além de mostrar que essas derivações correspondem a considerar-se o efeito, sobre a polarização dos fótons da radiação cósmica de fundo, de dois espalhamentos Thompson com elétrons durante recombinação. Conduzimo-nos, ainda, a descrever a polarização completamente no espaço real, como iniciado em [4] em um caso especial. Mostramos a possibilidade dessa conversão, recobramos a geometria que está associada ao estudo do problema no espaço real e verificamos satisfeitas as condições de causalidade. / Applying concepts of quantum mechanics and kinetic theory we show a re-derivation of Boltzmann equation for the Cosmic Microwave Background (CMB) polarization. We show the equivalence between our derivation and those already known ( [1], [2], [3] ) and also that these derivations correspond to take into account the effect, on the photon polarization, of two Thompson scattering on electrons while decoupling from matter. We adress ourselves, then, to give a complete formalism for the CMB polarization problem in real space, as started in [4] in a special case. Besides the possibility of complete treatment of the problem in real space, we recover the geometry that describes it and that tha causal relations are satisfied.

This study uses the PIC (Particle-in-cell) method to simulate unsteady three-dimensional dynamics of particles in argon plasma under low pressure, high density, and weak ionization between two planar electrodes subject to a sudden biased voltage. Plasma has been widely used in materials processing, film manufacturing, nuclear fusion, lamps, etc. Properties of plasmas are also becoming important area for research. This work includes elastic collisions between electrons and neutrals, ions and neutrals, and inelastic collisions resulting in ionization from impacting neutrals by electrons, and charge exchange between ions and neutrals, and Coulomb collisions between electrons and ions. The model ignores secondary electron emission, recombination between ions and electrons, and assumes uniform distribution of the neutrals having velocity of Maxwellian distribution. The computed results show the effects of elastic and inelastic collisions on the characteristics of plasma and sheath (space charge region) in front of the workpiece surface. Unsteady mass, momentum and energy transport from the bulk plasma through sheath to the workpiece is confirmatively and exploratorily studied after successful comparison between PIC prediction and experimental data has been made.

Cerebral cortex, classified as gray matter, is the superficial layer of the cerebrum. In recent years, many studies have shown the abnormality of cortical thickness is possibly correlated to the disease or disorder in central nervous system, such as Alzheimer¡¦s disease and lissencephaly. Therefore, this purpose of this work is to implement the measurement of the cortical thickness. In general, two approaches, surface-based and voxel-based methods, have been proposed to measure the cortical thickness. In this thesis, a procedure of the voxel-based method using Laplace¡¦s equation was developed on the basis of a 2008 publication reported by Chloe Hutton et al to obtain voxel-based cortical thickness (VBCT) map. The result of our home-made program was further compared with those calculated by Hutton¡¦s program, whic h was generously provided by the author. The difference between two implementations was consisted of four main parts. First of all, different strategies of the tissue classification were used to define boundary condition of Laplace¡¦s equation. When grey matter, white matter, and cerebrospinal fluid were classified by maximizing the tissue probability, Hutton¡¦s program tends to search more voxels of cerebrospinal fluid in sulci by skeletonizing the non-parenchyma area. Second, the algorithm of layer growing also differs. The single layer obtained by the 26-neighborhood algorithm in our program would be obviously thicker than that provided by Hutton¡¦s program using 6-neighborhood. Third, compared with a fixed step size (usually 0.5 mm) porposed in the main reference to track cortical streamline, we designed a variable step size, reducing the underestimation of cortical thickness. The last but not the least, the connecting points of the cortical streamline usually are not grid points, thus requiring interpolation to estimate the stepping gradient. We adapted the linear interpolation for better accuracy when Hutton et al searched for the closest grid point for replacement to achieve faster computation.

Adini¡¦s elements are applied to Poisson¡¦s eigenvalue problems in the unit square with periodical boundary conditions and the leading eigenvalues are obtained from the Rayleigh quotient. The penalty techniques are developed to copy with periodical boundary conditions, and superconvergence is also explored for leading eigenvalues. The optimal convergence O(h^6) are obtained for quasiuniform elements (see [2, 21]). When the uniform rectangular elements are used, the superconvergence O(h^6+p) with p = 1 or p = 2 of leading eigenvalues is proved, where h is the maximal boundary length of Adini¡¦s elements. Numerical experiments are carried to verify the analysis made. Keywords. Adini¡¦s elements, Poisson¡¦s equation, periodical boundary conditions, eigenvalue problems.

Non-Darcian flow in a finite fractured aquifer is studied in this thesis. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden or a gradual rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer&rsquo / s equation which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady state fracture-block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow both in fractures and blocks. Numerical and analytical solutions give same results when Reynold&rsquo / s number is less than 0.1. The effect of non-linearity on the flow appears when Reynold&rsquo / s number is greater than 0.1. The larger the piezometric head gradient, the higher the flow rate and, thus, higher the non-linearity is. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution is a unique application of non-linear flow models to the fractured aquifers. It can be used in predicting water levels in fractured aquifers and evaluating time dependent flow rates in the analysis of recession hydrographs.

In this thesis, we investigate Sturmian comparison theory and oscillation for second order impulsive differential equations with fixed moments of impulse actions. It is shown that impulse actions may greatly alter the oscillation behavior of solutions. In chapter two, besides Sturmian type comparison results, we give Leightonian type comparison theorems and obtain Wirtinger type inequalities for linear, half-linear and non-selfadjoint equations. We present analogous results for forced super linear and super half-linear equations with damping. In chapter three, we derive sufficient conditions for oscillation of nonlinear equations. Integral averaging, function averaging techniques as well as interval criteria for oscillation are discussed. Oscillation criteria for solutions of impulsive Hill&amp / #8217 / s equation with damping and forced linear equations with damping are established.

Utilizando conceitos de macânica quântica e teoria cinética apresentamos uma rederivação da equação de Boltzmann para a polarização. Mostramos a equivalência entre a equação que derivamos e a equação de Boltzmann encontrada na literatura ( [1], [2], [3] ) além de mostrar que essas derivações correspondem a considerar-se o efeito, sobre a polarização dos fótons da radiação cósmica de fundo, de dois espalhamentos Thompson com elétrons durante recombinação. Conduzimo-nos, ainda, a descrever a polarização completamente no espaço real, como iniciado em [4] em um caso especial. Mostramos a possibilidade dessa conversão, recobramos a geometria que está associada ao estudo do problema no espaço real e verificamos satisfeitas as condições de causalidade. / Applying concepts of quantum mechanics and kinetic theory we show a re-derivation of Boltzmann equation for the Cosmic Microwave Background (CMB) polarization. We show the equivalence between our derivation and those already known ( [1], [2], [3] ) and also that these derivations correspond to take into account the effect, on the photon polarization, of two Thompson scattering on electrons while decoupling from matter. We adress ourselves, then, to give a complete formalism for the CMB polarization problem in real space, as started in [4] in a special case. Besides the possibility of complete treatment of the problem in real space, we recover the geometry that describes it and that tha causal relations are satisfied.

The Henry's law constant is important in the gas-liquid mass transfer process. This study investigates the apparent dimensionless Henry's law constant, also known as the gas-liquid partition coefficient (K'H), of both hydrophilic (methanol, isopropanol and acetone) and hydrophobic (toluene and p-xylene) organic compounds. The K'H in deionized (DI) water, wastewater with a maximum total dissolved organic carbon (DOC) content of 700 mg/L, and DI water mixed with a maximum activated sludge suspended solid (SS) concentration of 40,000 mg/L, are measured by the single equilibrium technique (SET) at 288 K, 293 K, 298 K and 303 K. Experimental results indicate that the K'H of the three tested volatile organic compounds (VOCs) varies according to three situations. First, the K'H of the hydrophilic compounds in mixed liquor with the maximum suspended solid concentration is higher than that in DI water. The maximum rates of increase of K'H from KH in the SS range between 288 K and 303 K are 27.2% for methanol, 23.5% for isopropanol and 16.1% for acetone. Second, the K'H values for toluene and p-xylene are lower than those in DI water. Between 288 K and 303 K, the maximum reduce rate of K'H from KH in the SS range are 87.3% for toluene and 93.0% for p-xylene. Third, the K'H values of all of the test compounds in the wastewater are lower than those in DI water. The maximum rates of decrease of K'H from KH in the DOC concentration range between 288 K and 303 K are 1.2% for methanol, 1.1% for isopropanol, 1.7% for acetone, 14.3% for toluene and 20.1% for p-xylene. A model is presented to related K'H to wastewater DOC and concentration of SS in the activated sludge, using an organic carbon-water partition coefficient (KOC) and activated sludge-water partition coefficient (KSS) as model parameters. The model is verified, and model parameters for test compounds are estimated. Temperature variations of K'H can be regressed using van't Hoff's equation. The gas-water phase change enthalpy £GH'gw rises approximately linearly with increasing SS for hydrophobic toluene and p-xylene. K'H variation can be controlled by SS concentration and temperature in activated sludge system. The temperature effect of KOC and KSS are also observed. KOC and KSS decrease with as the temperature rises from 288 K to 303 K. The VOCs phase change enthalpy and entropy can be calculated using thermodynamic analysis. The partition of VOCs into the organic carbon phase releases energy, and thus prefers low temperatures. Lower temperatures and higher SS concentrations lead to lower K'H for hydrophobic VOCs, while higher SS concentrations result in higher K'H for hydrophilic VOCs. Organic-rich wastewater or low-biomass mixed liquor can successfully remove hydrophilic VOCs from contaminated air streams by bioscrubber. However, bioreactors scrubbed with high biomass-containing mixed liquor perform better than regular activated sludge concentration in removing hydrophobic VOCs. The experimental results thus support the design and operation of bioscrubber and suspended operable bioreactors.

<p>Fluid simulating is one of the most difficult problems in computer graphics. On the other hand, water appears in our life very frequently. This thesis focuses on water simulating. We have two main methods to do this in the thesis: the first is wave based water simulating; Sine wave summing based and Fast Fourier Transform based methods are all belong to this part. The other one is physics based water simulating. We make it based on Navier-Stokes Equation and it is the most realistic animation of water. It can deal with the boundary and spray which other method cannot express. Then we put our emphasis on implement by the physics method using Navier-Stokes Equation.</p>