Application of Fractional Moments for Comparing Random Variables with Varying Probability Distributions

New methods are being presented for statistical treatment of different random variables with unknown probability distributions. These include analysis based on the probability circles, probability ellipses, generalized mean values, generalized Pearson correlation coefficient and the beta-function analysis. Unlike other conventional statistical procedures, the main distinctive feature of these new methods is that no assumptions are made about the nature of the probability distribution of the random series being evaluated. Furthermore, the suggested procedures do not introduce uncontrollable errors during their application. The effectiveness of these methods is demonstrated on simulated data with extended and reduced sample sizes having different probability distributions

Fourier transform spectroscopy and coupled-channel deperturbation treatment of the A1Sigma+ ~ b3Pi complex of KCs molecule

The laser induced fluorescence (LIF) spectra A1Sigma ~ b3Pi --> X1Sigma+ of KCs dimer were recorded in near infrared region by Fourier Transform Spectrometer with a resolution of 0.03 cm-1. Overall more than 200 LIF spectra were rotationally assigned to 39K133Cs and 41K133Cs isotopomers yielding with the uncertainty of 0.003-0.01 cm-1 more than 3400 rovibronic term values of the strongly mixed singlet A1Sigma+ and triplet b3Pi states. Experimental data massive starts from the lowest vibrational level v_A=0 of the singlet and nonuniformly cover the energy range from 10040 to 13250 cm-1 with rotational quantum numbers J from 7 to 225. Besides of the dominating regular A1Sigma+ ~ b3P Omega=0 interactions the weak and local heterogenous A1S+ ~ b3P Omega=1 perturbations have been discovered and analyzed. Coupled-channel deperturbation analysis of the experimental 39K133Cs e-parity termvalues of the A1S+ ~ b3P complex was accomplished in the framework of the phenomenological 4 x 4 Hamiltonian accounting implicitly for regular interactions with the remote states manifold. The resulting diabatic potential energy curves of the interacting states and relevant spin-orbit coupling matrix elements defined analytically by Expanded Morse Oscillators model reproduce 95% of experimental data field of the 39K133Cs isotopomer with a standard deviation of 0.004 cm-1 which is consistent with the uncertainty of the experiment. Reliability of the derived parameters was additionally confirmed by a good agreement between the predicted and experimental termvalues of 41K133Cs isotopomer. Calculated intensity distributions in the A ~ b --> X LIF progressions are also consistent with their experimental counterparts.Comment: 17 pages, 14 figure

BLIND SOURCE SEPARATION USING MAXIMUM ENTROPY PDF ESTIMATION BASED ON FRACTIONAL MOMENTS

Abstract. Recovering a set of independent sources which are linearly mixed is the main task of the blind source separation. Utilizing different methods such as infomax principle, mutual information and maximum likelihood leads to simple iterative procedures such as natural gradient algorithms. These algorithms depend on a nonlinear function (known as score or activation function) of source distributions. Since there is no prior knowledge of source distributions, the optimality of the algorithms is based on the choice of a suitable parametric density model. In this paper, we propose an adaptive optimal score function based on the fractional moments of the sources. In order to obtain a parametric model for the source distributions, we use a few sampled fractional moments to construct the maximum entropy probability density function (PDF) estimation . By applying an optimization method we can obtain the optimal fractional moments that best fit the source distributions. Using the fractional moments (FM) instead of the integer moments causes the maximum entropy estimated PDF to converge to the true PDF much faster . The simulation results show that unlike the most previous proposed models for the nonlinear score function, which are limited to some sorts of source families such as sub-gaussian and super-gaussian or some forms of source distribution models such as generalized gaussian distribution, our new model achieves better results for every source signal without any prior assumption for its randomness behavior

Properties of complex-valued power means of random variables and their applications

We consider power means of independent and identically distributed (i.i.d.) non-integrable random variables. The power mean is an example of a homogeneous quasi-arithmetic mean. Under certain conditions, several limit theorems hold for the power mean, similar to the case of the arithmetic mean of i.i.d. integrable random variables. Our feature is that the generators of the power means are allowed to be complex-valued, which enables us to consider the power mean of random variables supported on the whole set of real numbers. We establish integrabilities of the power mean of i.i.d. non-integrable random variables and a limit theorem for the variances of the power mean. We also consider the behavior of the power mean as the parameter of the power varies. The complex-valued power means are unbiased, strongly-consistent, robust estimators for the joint of the location and scale parameters of the Cauchy distribution.Comment: 43 pages; v3: Section 2 for backgrounds and Section 8 for the mixture Cauchy model added. Introduction shortened. To appear in Acta Math. Hun

Exploring the mechanisms behind nondiffusive transport in a simple turbulence model

Thesis (Ph.D.) University of Alaska Fairbanks, 2017Elements for nondiffusive transport have been identified in a plasma turbulence model based on the slab drift-wave model. Motivated by the self-organized criticality paradigm, a standard set of drift-wave equations in doubly-periodic spatial domain has been elevated to include a flux-driven background profile with critical gradients. The profile is maintained by the turbulence induced flux from the source to the sink. Tracers that follow the Lagrangian trajectories are the primary transport characterization technique. The competition between down-gradient relaxations and self-generated flows highlights the dual reactions to local steepening of profile gradients, which leads to different transport regimes. An additional external sheared flow further inhibits down-gradient transfer and acts as another critical threshold condition that can lead to flow-driven instabilities. Superdiffusive transport is observed primarily when radial relaxation events dominate while subdiffusive character become more prominent with self-generated and external poloidal flows. Diffusive transport exists when the superdiffusive and subdiffusive components are in balance. The interplay between turbulent relaxation and self-generated sheared poloidal flows, that form the basis for the transport explored in this model, is absent unless a flux-driven setup is used. Most of the rich dynamics were not present when running the simplified model without an equation for background profile evolution. Nondiffusive transport characteristics can also be recovered from a passive scalar field that is advected by the turbulent flow with an inherent diffusivity. The spread of a highly localized cloud of tracers and a passive scalar field reasserts the equivalence between the Lagrangian and quasi-Lagrangian frames. The coincidence between the passive scalar field with the tracers provide a regime of validity where existing experimental technique can be used to characterize transport from two-dimensional experimental data. The results from this work highlight the key features of flux-driven turbulent transport leading to nondiffusive transport. Specifcally, the dual reactions to the local steepening of profile gradients exposes the multiscale feature of turbulent transport that becomes more apparent under a flux-driven profile. The quantification of nondiffusive transport characteristics from the evolution of a passive scalar can have important implication towards the fundamental understanding of fluid turbulence and turbulent transport

2D stellar population and gas kinematics of the inner 1.5 kpc of the post-starburst quasar SDSS J0210-0903

Post-Starburst Quasars (PSQs) are hypothesized to represent a stage in the evolution of massive galaxies in which the star formation has been recently quenched due to the feedback of the nuclear activity. In this paper our goal is to test this scenario with a resolved stellar population study of the PSQ J0210-0903, as well as of its emitting gas kinematics and excitation. We have used optical Integral Field Spectroscopy obtained with the Gemini GMOS instrument at a velocity resolution of ~120 km/s and spatial resolution of ~0.5 kpc. We find that old stars dominate the luminosity (at 4700 \AA) in the inner 0.3 kpc (radius), while beyond this region (at ~0.8 kpc) the stellar population is dominated by both intermediate age and young ionizing stars. The gas emission-line ratios are typical of Seyfert nuclei in the inner 0.3 kpc, where an outflow is observed. Beyond this region the line ratios are typical of LINERs and may result from the combination of diluted radiation from the nucleus and ionization from young stars. The gas kinematics show a combination of rotation in the plane of the galaxy and outflows, observed with a maximum blueshift of -670 km/s. We have estimated a mass outflow rate in ionized gas in the range 0.3--1.1 M_sun/yr and a kinetic power for the outflow of dE/dt ~ 1.4--5.0 x 10^40 erg/s ~0.03% - 0.1% x L_bol. This outflow rate is two orders of magnitude higher than the nuclear accretion rate of ~8.7 x 10^-3 M_sun/yr, thus being the result of mass loading of the nuclear outflow by circumnuclear galactic gas. Our observations support an evolutionary scenario in which the feeding of gas to the nuclear region has triggered a circumnuclear starburst 100's Myr ago, followed by the triggering of the nuclear activity, producing the observed gas outflow which may have quenched further star formation in the inner 0.3 kpc.Comment: 17 pages, 9 Figures, 2 Table

Electromagnetic interactions in one-dimensional metamaterials

All data created during this research is available in ORE at https://doi.org/10.24378/exe.630Metamaterials offer the freedom to tune the rich electromagnetic coupling between the constituent meta-atoms to tailor their collective electromagnetic response. Therefore, a comprehensive understanding of the nature of electromagnetic interactions between meta-atoms is necessary for novel metamaterial design, which is provided in the first part of this thesis. The subsequent work in the thesis applies the understanding from the first part to design and demonstrate novel one-dimensional metamaterials that overcome the limitations of metamaterials proposed in literature or exhibit electromagnetic responses not previously observed. Split-ring Resonators (SRRs) are a fundamental building block of many electromagnetic metamaterials. In the first part of the work in this thesis, it is shown that bianisotropic SRRs (with magneto-electric cross-polarisation) when in close proximity to each other, exhibit a rich coupling that involves both electric and magnetic interactions. The strength and nature of the coupling between two identical SRRs are studied experimentally and computationally as a function of their separation and relative orientation. The electric and magnetic couplings are characterised and it is found that, when SRRs are close enough to be in each other's near-field, the electric and magnetic couplings may either reinforce each other or act in opposition. At larger separations retardation effects become important. The findings on the electromagnetic interactions between bianisotropic resonators are next applied to developing a one-dimensional ultra-wideband backward-wave metamaterial waveguide. The key concept on which the metamaterial waveguide is built is electro-inductive wave propagation, which has emerged as an attractive solution for designing backward-wave supporting metamaterials. Stacked metasurfaces etched with complementary SRRs (CSRRs) have also been shown to exhibit a broadband negative dispersion. It is demonstrated through experiment and numerical modeling, that the operational bandwidth of a CSRR metamaterial waveguide can be improved by restricting the cross-polarisation effects in the constituent meta-atoms. The metamaterial waveguide constructed using the modified non-bianisotropic CSRRs are found to have a fractional bandwidth of 56.3\% which, based on a thorough search of relevant literature, is the broadest reported value for an electro-inductive metamaterial. A traditional coupled-dipole toy-model is presented as a tool to understand the field interactions in CSRR based metamaterials, and to explain the origin of their negative dispersion response. This metamaterial waveguide should be of assistance in the design of broadband backward-wave metamaterial devices, with enhanced electro-inductive waveguiding effects. In the final part of the thesis, a one-dimensional metamaterial prototype that permits simultaneous forward- and backward-wave propagation is designed. Such a metamaterial waveguide could act as a microwave analogue of nanoparticle chains that support electromagnetic energy transfer with a positive or a negative dispersion due to the excitation of their longitudinal or transverse dipole modes. The symmetry of the designed hybrid meta-atom permits the co-existence of two non-interfering resonances closely separated in frequency. It is experimentally and computationally shown that the metamaterial waveguide supports simultaneous non-interacting forward- and backward-wave propagation in an overlapping frequency band. The proposed metamaterial design should be suitable for realising bidirectional wireless power transfer applications.EPSRC Centre for Doctoral Training in Electromagnetic Metamaterial

Estimation of aquifers hydraulic parameters by three different tecniques: geostatistics, correlation and modeling

Characterization of aquifers hydraulic parameters is a difficult task that requires field information. Most of the time the hydrogeologist relies on a group of values coming from different test to interpret the hydrogeological setting and possibly, generate a model. However, getting the best from this information can be challenging. In this thesis, three cases are explored. First, hydraulic conductivities associated with measurement scale of the order of 10−1 m and collected during an extensive field campaign near Tübingen, Germany, are analyzed. Estimates are provided at coinciding locations in the system using: the empirical Kozeny-Carman formulation, providing conductivity values, based on particle size distribution, and borehole impeller-type flowmeter tests, which infer conductivity from measurements of vertical flows within a borehole. Correlation between the two sets of estimates is virtually absent. However, statistics of the natural logarithm of both sets at the site are similar in terms of mean values and differ in terms of variogram ranges and sample variances. This is consistent with the fact that the two types of estimates can be associated with different (albeit comparable) measurement (support) scales. It also matches published results on interpretations of variability of geostatistical descriptors of hydraulic parameters on multiple observation scales. The analysis strengthens the idea that hydraulic conductivity values and associated key geostatistical descriptors inferred from different methodologies and at similar observation scales (of the order of tens of cm) are not readily comparable and should not be embedded blindly into a flow (and eventually transport) prediction model. Second, a data-adapted kernel regression method, originally developed for image processing and reconstruction is modified and used for the delineation of facies. This non-parametric methodology uses both the spatial and the sample value distribution, to produce for each data point a locally adaptive steering kernel function, self-adjusting the kernel to the direction of highest local spatial correlation. The method is shown to outperform the nearest-neighbor classification (NNC) in a number of synthetic aquifers whenever the available number of data is small and randomly distributed. Still, in the limiting case, when the domain is profusely sampled, both the steering kernel method and the NNC method converge to the true solution. Simulations are finally used to explore which parameters of the locally adaptive kernel function yield optimal reconstruction results in typical field settings. It is shown that, in practice, a rule of thumb can be used to get suboptimal results, which are best when key prior information such as facies proportions is used. Third, the effect of water temperature fluctuation on the hydraulic conductivity profile of coarse sediments beneath an artificial recharge facility is model and compared with field data. Due to the high permeability, water travels at a high rate, and therefore also water with different temperature is also present on the sediment under the pond at different moments, this translates into different hydraulic conductivity values within the same layer, even though all the other parameters are the same for this layer. Differences of almost 79% in hydraulic conductivity were observed for the model temperatures (2 °C – 25 °C). This variation of hydraulic conductivity in the sediment below the infiltration pond when water with varying temperature enters the sediment, causes the infiltration velocity to change with time and produces the observed fluctuation on the field measurements.La caracterización de los parámetros hidráulicos de los acuíferos es una tarea difícil que requiere información de campo. La mayoría de las veces el hidrogeólogo se basa en un grupo de valores procedentes de diferentes pruebas para interpretar la configuración hidrogeológica y posiblemente , generar un modelo . Sin embargo, obtener lo mejor de esta información puede ser un reto. En esta tesis se analizan tres casos. Primero, se analizan las conductividades hidráulicas asociadas a una escala de medición del orden de 10 m− 1 y obtenidas durante una extensa campaña de campo cerca de Tübingen, Alemania. Las estimaciones se obtuvieron en puntos coincidentes en el sitio, mediante: la formulación empírica de Kozeny - Carman, proporcionando valores de conductividad, con base en la distribución de tamaño de partículas y las pruebas del medidor de caudal de tipo impulsor en el pozo, el cual infiere las medidas de conductividad a partir de los flujos verticales dentro de un pozo. La correlación entre los dos conjuntos de estimaciones es prácticamente ausente. Sin embargo, las estadísticas del logaritmo natural de ambos conjuntos en el lugar son similares en términos de valores medios y difieren en términos de rangos del variograma y varianzas de muestra. Esto es consecuente con el hecho de que los dos tipos de estimaciones pueden estar asociados con escalas de apoyo de medición diferentes (aunque comparables). También coincide con los resultados publicados sobre la interpretación de la variabilidad de los descriptores geoestadísticos de parámetros hidráulicos en múltiples escalas de observación . El análisis refuerza la idea de que los valores de conductividad hidráulica y descriptores geoestadísticos clave asociados al inferirse de diferentes metodologías y en las escalas de observación similares (en el caso del orden de decenas de cm) no son fácilmente comparables y debe ser utilizados con cuidado en la modelación de flujo (y eventualmente, el transporte) del agua subterránea. En segundo lugar, un método de regresión kernel adaptado a datos, originalmente desarrollado para el procesamiento y la reconstrucción de imágenes se modificó y se utiliza para la delimitación de las facies. Esta metodología no paramétrica utiliza tanto la distribución espacial como el valor de la muestra, para producir en cada punto de datos una función kernel de dirección localmente adaptativo, con ajuste automático del kernel a la dirección de mayor correlación espacial local. Se demuestra que este método supera el NNC (por su acrónimo en inglés nearest-neighbor classification) en varios casos de acuíferos sintéticos donde el número de datos disponibles es pequeño y la distribución es aleatoria. Sin embargo, en el caso límite, cuando hay un gran número de muestras, tanto en el método kernel adaptado a la dirección local como el método de NNC convergen a la solución verdadera. Las simulaciones son finalmente utilizadas para explorar cuáles parámetros de la función kernel localmente adaptado dan resultados óptimos en la reconstrucción de resultados en escenarios típicos de campo. Se demuestra que, en la práctica, una regla general puede ser utilizada para obtener resultados casi óptimos, los cuales mejoran cuando se utiliza información clave como la proporción de facies. En tercer lugar, se modela el efecto de la fluctuación de la temperatura del agua sobre la conductividad hidráulica de sedimentos gruesos debajo de una instalación de recarga artificial y se compara con datos de campo. Debido a la alta permeabilidad, el agua se desplaza a alta velocidad alta, y por lo tanto, agua con temperatura diferente también está presente en el sedimento bajo el estanque en diferentes momentos, esto se traduce en diferentes valores de conductividad hidráulica dentro de la misma capa, a pesar de que todos los demás parámetros son los mismos para esta capa. Se observaron diferencias de casi 79 % en la conductividad hidráulica en el modelo, para las temperaturas utilizadas (2 º C - 25 º C ). Esta variación de la conductividad hidráulica en el sedimento por debajo de la balsa de infiltración cuando el agua de temperatura variable entra en el sedimento, causa un cambio en la velocidad de infiltración con el tiempo y produce las fluctuacciones observadas en las mediciones de campo

Dynamics in Cold Atomic Gases: Resonant Behaviour of the Quantum Delta-Kicked Accelerator and Bose-Einstein Condensates in Ring Traps

In this thesis, the dynamics of cold, trapped atomic gases are investigated, and the prospects for exploiting their nonlinear dynamics for inertial sensing are discussed. In the first part, the resonant and antiresonant dynamics of the atom-optical quantum delta-kicked accelerator with an initial symmetric momentum distribution are considered. The system is modelled as an ideal, non-interacting atomic gas, with a temperature-dependence governed by the width of the initial momentum distribution. The existence of resonant and antiresonant behaviour is established, and analytic expressions describing the dynamics of momentum moments of the time-evolved momentum distribution are derived. In particular, the momentum moment dynamics in both the resonant and antiresonant regimes depend strongly on the width of the initial momentum distribution. The resonant dynamics of all even-ordered momentum moments are shown to exhibit a power-law growth with an exponent given by the order of the moment in the zero-temperature regime, whereas for a broad, thermal initial momentum distribution the exponent is reduced by one. The cross-over in the intermediate regime is also examined, and a characteristic time is determined up to which the system exhibits dynamics associated with the zero-temperature regime. A similar analysis is made for the temperature-dependence of the antiresonant dynamics. This general behaviour is demonstrated explicitly by considering a Maxwell-Boltzmann and uniform momentum distribution, allowing exact expressions describing the dynamics of the second- and fourth-order momentum moments, and momentum cumulants, to be obtained. The relevance of these results to the potential of using this system in accurate determinations of the local gravitational acceleration is discussed. In the second part, the dynamics of one- and two-component Bose-Einstein Condensates prepared in a counter-rotating superposition of flows in a quasi-1D toroidal trap are studied. Particular attention is paid to the dynamical stability of the initial state in the presence of atom-atom interactions, included via a mean-field description within the Gross-Pitaevskii equation. A broad regime of dynamical stability using a two-component BEC is identified, in which a typical implementation using Rb-87 is predicted to lie. A proof-of-principle Sagnac atom-interferometer using a two-component Rb-87 BEC is then presented, and the accumulation of the Sagnac phase is shown to be possible via relative population measurement or, alternatively, through the continuous monitoring the precession of atomic density fringes. In contrast to conventional Sagnac interferometers, the accumulation of the Sagnac phase is independent of the enclosed area of the interferometer. The prospects of using this system for high-precision determinations of rotation is discussed