In many hypothesis testing problems such as one-sample and two-sample test problems, the test statistics are degenerate U-statistics. One of the challenges in practice is the computation of U-statistics for a large sample size. Besides, for degenerate U-statistics, the limiting distribution is a mixture of weighted chi-squares, involving the eigenvalues of the kernel of the U-statistics. As a result, it’s not straightforward to construct the rejection region based on this asymptotic distribution. In this research, we aim to reduce the computation complexity of degenerate U-statistics and propose an easy-to-calibrate test statistic by using the divide-and-conquer method. Specifically, we randomly partition the full n data points into kn even disjoint groups, and compute U-statistics on each group and combine them by averaging to get a statistic Tn. We proved that the statistic Tn has the standard normal distribution as the limiting distribution. In this way, the running time is reduced from O(n^m) to O( n^m/km_n), where m is the order of the one sample U-statistics. Besides, for a given significance level , it’s easy to construct the rejection region. We apply our method to the goodness of fit test and two-sample test. The simulation and real data analysis show that the proposed test can achieve high power and fast running time for both one and two-sample tests.

We consider the system of equations arising from mantle dynamics introduced by McKenzie (J. Petrology, 1985). In this multi-phase model, the fluid melt velocity obeys Darcy's law while the deformable "solid" matrix is governed by a highly viscous Stokes equation. The system is then coupled through mass conservation and compaction relations. Together these equations form a coupled Darcy-Stokes system on a continuous single-domain mixture of fluid and matrix. The porosity φ, representing the relative volume of fluid melt to the bulk volume, is assumed to be much smaller than one. When coupled with solute transport and thermal evolution in a time-dependent problem, the model transitions dynamically from a non-porous single phase solid to a two-phase porous medium. Such mixture models have an advantage for numerical approximation since the free boundary between the one and two-phase regions need not be determined explicitly. The equations of mantle dynamics apply to a wide range of applications in deep earth physics such as mid-ocean ridges, subduction zones, and hot-spot volcanism, as well as to glacier dynamics and other two-phase flows in porous media. Mid-ocean ridges form when viscous corner flow of the solid mantle focuses fluid toward a central ridge. Melt is believed to migrate upward until it reaches the lithospheric "tent" where it then moves toward the ridge in a high porosity band. Simulation of this physical phenomenon required confidence in numerical methods to handle highly heterogeneous porosity as well as the single-phase to two-phase transition. In this work we present a standard mixed finite element method for the equations of mantle dynamics and investigate its limitations for vanishing porosity. While stable and optimally convergent for porosity bounded away from zero, the stability estimates we obtain suggest, and numerical results show, the method becomes unstable as porosity approaches zero. Moreover, the fluid pressure is no longer a physical variable when the fluid phase disappears and thus is not a good variable for numerical methods. Inspired by the stability estimates of the standard method, we develop a novel stable mixed method with uniqueness and existence of solutions by studying a linear degenerate elliptic sub-problem akin to the Darcy part of the full model: [mathematical equation], where a and b satisfy a(0)=b(0)=0 and are otherwise positive, and the porosity φ ≥ 0 may be zero on a set of positive measure. Using scaled variables and mild assumptions on the regularity of φ, we develop a practical mass-conservative method based on lowest order Raviart-Thomas finite elements. Finally, we adapt the numerical method for the sub-problem to the full system of equations. We show optimal convergence for sufficiently smooth solutions for a compacting column and mid-ocean ridge-like corner flow examples, and investigate accuracy and stability for less regular problems / text

The observation of solar and atmospheric neutrino oscillations place bounds on the mass differences. However, these probes are insensitive to the absolute mass. To date, cosmology has provided the best bounds on the total neutrino mass. These bounds are based on a degenerate mass model. With the increasing precision of cosmological data, we investigate the effect of the neutrino mass hierarchy. The precision of the parameter estimates stems from precise observations of the cosmic microwave background. However, the effect of neutrino mass hierarchy on this observation is smaller than the cosmic variance. Therefore, we rely on the measurement of the matter power spectrum for hierarchy effects. We propose the use of importance sampling rather than the commonly used Markov chain Monte Carlo. Importance sampling takes advantage of the microwave background's statistical insensitivity to hierarchy. We present forecasted bounds due to Planck and the proposed CMBPol. We also discuss the needed precision for future galaxy surveys in detecting the effect of neutrino mass hierarchy. / text

In this Thesis we discussed ultracold Fermi gas with an s-wave interaction and synthetic spin-orbit coupling under a variety of conditions. We considered the system in both three and two spatial dimensions, with equal-Rashba-Dresselhaus type or Rashba-only type of spin-orbit-coupling, and with or without an artificial Zeeman field. We found competing effects on Fermionic superfluidity from spin-orbit coupling and Zeeman fields, and topologically non-trivial states in the presence of both fields. We gave an outlook on the many-body physics in the last.

The main aim of this thesis is to generate 3D models of human heads from uncalibrated images. In order to extract geometric values of a human head, we find camera parameters using camera auto calibration. However, some image sequences generate non-unique (degenerate) solutions. An algorithm for removing degeneracy from the most common form of camera movement in face image acquisition is described. The geometric values of main facial features are computed initially. The model is then generated by gradual deformation of a generic polygonal model of a head. The accuracy of the models is evaluated using ground truth data from a range scanner. 3D models are covered with cylindrical texture values obtained from images. The models are appropriate for animation or identification applications.

Two experimental methods, optical limiting and degenerate four-wave mixing, are employed to study the nonlinear optical properties of various novel fullerenes structures. Optical limiting refers to decreased transmittance of a material with increased incident light intensity. Detailed measurements of the wavelength-dependence of fullerene optical limiters have illustrated several key features of reverse saturable absorption. Most important among these is the requirement of weak but non-negligible ground state absorption. We have shown that the optical limiting performance of C₆₀ can be extended into the near infrared range by appropriate modifications of the structure such as higher cage fullerenes or derivatization of the basic C₆₀ molecule. The higher cage fullerene C₇₆ shows improved optical limiting behavior compared to C₆₀, for wavelengths higher than 650 nm, but becomes a weak limiter in the 800 nm range. C₈₄, even at high concentrations in [alpha]-chloronaphthalene, does not reach the good performance of C₆₀, but instead shows weak optical limiting in the 800 nm range. We also demonstrate that by attaching various groups to the C₆₀ molecule, we can extend the optical limiting performance in the near infrared regime. The C₆₀ derivatives studied, (C₆₀ cyclic ketone, C₆₀ secondary amine, C₆₀CHC₆H₄CO₂H, and C₆₀C₄H₄(CH₃)CH₂O₂C(CH₂)CO₂H), have a similar characteristic: the attached groups cause a symmetry-breaking of the C₆₀ sphere and, therefore, there are new allowed transitions that appear as absorption features up to 750 nm. The optical limiting measurements show that these materials, even for low input energies, have an exceptionally strong optical limiting response in the 640 to 750 nm spectral region. For wavelengths higher than 800 nm, however, they become transparent and no optical limiting is observed. Excited state absorption cross-sections obtained from analysis of the optical limiting data reveal that the C₆₀ derivatives have a maximum triplet-triplet absorption cross-section at 700 nm, which is shifted from the 750 nm value for the C₆₀ molecule. For the first time, optical limiting measurements are performed on five separate C₈₄ isomers. These intriguing results show that the optical limiting behavior is strongly dependent on the cage symmetry. It is also found that the most abundant isomer does not have the strongest optical limiting performance, but is in fact one of the weaker optical limiters of the isomers isolated so far. The endohedral metallofullerenes are a unique class of fullerene materials and consist of one or more metal atoms encapsulated inside the buckyball cage. An important characteristic of these materials is the charge-transfer from the dopant atoms to the fullerene cage, which has a high electron affinity. The charge-transfer is similar to the optical excitation in a material, but although the electrons are placed in the lowest unoccupied molecular orbital (LUMO), there are no holes produced in the highest occupied molecular orbital (HOMO). This is an important analogy, since it has been previously shown that optical excitation enhances the nonlinear optical properties of a material. The nonresonant degenerate four-wave mixing experiments performed on the endohedral metallofullerene Er₂@C₈₂, at 1064 nm, show that the third order nonlinear susceptibility value is increased by orders of magnitude relative to the empty cage fullerenes, thus, confirming the charge-transfer process from the encapsulated atoms to the fullerene cage. We obtain a value [gamma]_xyyx⁽³⁾( &#173 [omega]; [omega], [omega], &#173 [omega])= &#173 8.65 &#215 10⁻³² esu for the molecular second order hyperpolarizability, which is almost three orders of magnitude larger than the values reported in literature for an empty cage fullerene. / Ph. D.

<p> Investigations on the stability and reactions of various isomers of the benzohomotropylium/bridged [11]-annulene system are presented in this thesis. As 1,6-methano-[11]-annulenium cations are isomeric with benzohomotropylium cations in principle it is possible for these systems to be interconverted by a circumambulatory migration of the 'bridging' methano group. If this reaction were to proceed to equilibrium it would allow the examination of the thermodynamic stability of the various isomers.</p> <p> Homotropylium cations with two methyl groups at C8 are known to undergo facile circumambulatory rearrangements. Thus in this work the 1-hydroxy 8,8-dimethyl-2,3-benzohomotropylium and 1-hydroxy 8,8-dimethyl-4,5-benzohomotropylium cations have been prepared by protonation of the corresponding 2,3- and 4,5- benzohomotropones in FSO3H. On the basis of their 1H NMR spectra it is concluded that they can properly be regarded as homoaromatic cations. The thermal isomerizations of these cations have been studied in detail. The thermal isomerization of these cations led to the formation of a series of products but does not yield the corresponding [11]-annulenium isomer. In addition the barriers to ring inversion in these cations were found to be much lower than those of other homotropylium cations. This suggests that the former cations have a very open structure. Besides this experimental approach, the relative stability of these isomers has been explored using MNDO calculations. The calculated heats of formation (ΔHf) showed that the 1-benzohomotropylium cation is more stable than the corresponding [11]-annulenium ion by 21 kcal/mol.</p> / Thesis / Master of Science (MSc)

No description available.

The purpose of this thesis is to investigate the nonlinear phenomena known as doubly-stimulated, non-degenerate two-photon emission (ND-2PE) in Gallium Arsenide (GaAs). 2PE refers to the simultaneous emission of two-photons as electrons move from the conduction band in a direct gap semiconductor to the valence band. Following the same path for describing one-photon emission (1PE) we describe 2PE as a product of the irradiance, and the negative of the loss which in this case is two-photon absorption, , the negative coming from the population inversion. We attempt to observe 2PE by using a frequency non-degenerate pump-probe experiment in which a third beam optically excites a 4 µm thick GaAs sample. We use nondegenerate beams in hopes of utilizing the 3-orders of magnitude enhancement seen in twophoton absorption (2PA) by going to extreme nondegeneracy (END) to enhance 2PE. GaAs is chosen due to the availability of the appropriate wavelengths, the maturity of the GaAs technology, its use in optoelectronic devices and its ability to be electrically pumped. During the experimental development we learn how to effectively etch and manipulate thin GaAs samples and model the transmission spectrum of these samples using thin film transmission matrices. We are able to match the measured transmission spectrum with the theoretical transmission spectrum. Here we etch the bulk GaAs left on the sample leaving only the 4 µm thickness of molecular beam epitaxial grown GaAs plus additional layers of aluminum gallium arsenide (AlGaAs). These samples were grown for us by Professor Gregory Salamo of the University of Arkansas. iv Using the pump-probe experiment on the 4 µm GaAs sample, we measure the change of the 2PA due to the presence of optically excited carriers. The goal is to reduce the 2PA signal to zero and then invert the 2PA signal indicating an increase in transmission indicative of 2PE when the population is inverted. Our results show that we achieve a 45% reduction in the 2PA signal in a 4 μm thick GaAs sample due to the excited carriers. Unfortunately, we currently cannot experimentally determine whether the reduction is strictly due to free-carrier absorption (FCA) of our pump or possibly due to a change in the two-photon absorption coefficient. We measure the transmission of various wavelengths around the bang gap of GaAs as a function of excitation wavelength and achieve a transmittance of ~80% which we attribute to possibly be one photon gain (1PG) at 880 nm. We also go to cryogenic temperatures to concentrate the carriers near the bottom of the conduction band and improve the theoretical gain coefficient for 2PE. Unfortunately, we do not observe a measurable change in 2PA with the addition of optically excited carriers. Along with FCA of our infrared pump we suspect that the difficulties in this first set of experiments are also a result or radiative recombination due to amplified spontaneous emission reducing our free carrier density along with the fact that 4 m is too thick for uniform excitation. We now have 1 m samples from Professor Gregory Salamo which we hope will give better and more definitive results

This work is concerned with sampling and computation of rare events in molecular systems. In particular, we present new methods for sampling the canonical ensemble corresponding to the Boltzmann-Gibbs probability measure. We combine an equation for controlling the kinetic energy of the system with a random noise to derive a highly degenerate diffusion (i.e. a diffusion equation where diffusion happens only along one or few degrees of freedom of the system). Next the concept of hypoellipticity is used to show that the corresponding Fokker-Planck equation of the highly degenerate diffusion is well-posed, hence we prove that the solution of the highly degenerate diffusion is ergodic with respect to the Boltzmann-Gibbs measure. We find that the new method is more efficient for computation of dynamical averages such as autocorrelation functions than the commonly used Langevin dynamics, especially in systems with many degrees of freedom. Finally we study the computation of free energy using an adaptive method which is based on the adaptive biasing force technique.