White matter integrity measured by fractional anisotropy correlates poorly with actual individual fiber anisotropy

Fractional anisotropy (FA), a very widely used measure of fiber integrity based on diffusion tensor imaging (DTI), is a problematic concept as it is influenced by several quantities including the number of dominant fiber directions within each voxel, each fiber's anisotropy, and partial volume effects from neighboring gray matter. High-angular resolution diffusion imaging (HARDI) can resolve more complex diffusion geometries than standard DTI, including fibers crossing or mixing. The tensor distribution function (TDF) can be used to reconstruct multiple underlying fibers per voxel, representing the diffusion profile as a probabilistic mixture of tensors. Here we found that DTIderived mean diffusivity (MD) correlates well with actual individual fiber MD, but DTI-derived FA correlates poorly with actual individual fiber anisotropy, and may be suboptimal when used to detect disease processes that affect myelination. Analysis of the TDFs revealed that almost 40% of voxels in the white matter had more than one dominant fiber present. To more accurately assess fiber integrity in these cases, we here propose the differential diffusivity (DD), which measures the average anisotropy based on all dominant directions in each voxel.

Best individual template selection from deformation tensor minimization

We study the influence of the choice of template in tensor-based morphometry. Using 3D brain MR images from 10 monozygotic twin pairs, we defined a tensor-based distance in the log-Euclidean framework [1] between each image pair in the study. Relative to this metric, twin pairs were found to be closer to each other on average than random pairings, consistent with evidence that brain structure is under strong genetic control. We also computed the intraclass correlation and associated permutation p-value at each voxel for the determinant of the Jacobian matrix of the transformation. The cumulative distribution function (cdf) of the p-values was found at each voxel for each of the templates and compared to the null distribution. Surprisingly, there was very little difference between CDFs of statistics computed from analyses using different templates. As the brain with least log-Euclidean deformation cost, the mean template defined here avoids the blurring caused by creating a synthetic image from a population, and when selected from a large population, avoids bias by being geometrically centered, in a metric that is sensitive enough to anatomical similarity that it can even detect genetic affinity among anatomies.

Dynamics of a dilute sheared inelastic fluid. II. The effect of correlations

The effect of correlations on the viscosity of a dilute sheared inelastic fluid is analyzed using the ring-kinetic equation for the two-particle correlation function. The leading-order contribution to the stress in an expansion in epsilon=(1-e)(1/2) is calculated, and it is shown that the leading-order viscosity is identical to that obtained from the Green-Kubo formula, provided the stress autocorrelation function in a sheared steady state is used in the Green-Kubo formula. A systemmatic extension of this to higher orders is also formulated, and the higher-order contributions to the stress from the ring-kinetic equation are determined in terms of the terms in the Chapman-Enskog solution for the Boltzmann equation. The series is resummed analytically to obtain a renormalized stress equation. The most dominant contributions to the two-particle correlation function are products of the eigenvectors of the conserved hydrodynamic modes of the two correlated particles. In Part I, it was shown that the long-time tails of the velocity autocorrelation function are not present in a sheared fluid. Using those results, we show that correlations do not cause a divergence in the transport coefficients; the viscosity is not divergent in two dimensions, and the Burnett coefficients are not divergent in three dimensions. The equations for three-particle and higher correlations are analyzed diagrammatically. It is found that the contributions due to the three-particle and higher correlation functions to the renormalized viscosity are smaller than those due to the two-particle distribution function in the limit epsilon -> 0. This implies that the most dominant correlation effects are due to the two-particle correlations.

pi-conjugation and conformation in a semiconducting polymer: small angle x-ray scattering study

Small angle x-ray scattering (SAXS) in a poly[2-methoxy-5-(2'-ethyl-hexyloxy)-1,4-phenylene vinylene] (MEH-PPV) solution has shown the important role of pi-electron conjugation in controlling the chain conformation and assembly. By increasing the extent of conjugation from 30 to 100%, the persistence length (l(p)) increases from 20 to 66 angstrom. Moreover, a pronounced second peak in the pair distribution function has been observed in a fully conjugated chain, at larger length scales. This feature indicates that the chain segments tend to self-assemble as the conjugation along the chain increases. Xylene enhances the rigidity of the PPV backbone to yield extended structures, while tetrahydrofuran solvates the side groups to form compact coils in which the lp is much shorter.

Memory of past random wave conditions in submarine groundwater discharge

Submarine groundwater discharge (SGD) is an integral part of the hydrological cycle and represents an important aspect of land-ocean interactions. We used a numerical model to simulate flow and salt transport in a nearshore groundwater aquifer under varying wave conditions based on yearlong random wave data sets, including storm surge events. The results showed significant flow asymmetry with rapid response of influxes and retarded response of effluxes across the seabed to the irregular wave conditions. While a storm surge immediately intensified seawater influx to the aquifer, the subsequent return of intruded seawater to the sea, as part of an increased SGD, was gradual. Using functional data analysis, we revealed and quantified retarded, cumulative effects of past wave conditions on SGD including the fresh groundwater and recirculating seawater discharge components. The retardation was characterized well by a gamma distribution function regardless of wave conditions. The relationships between discharge rates and wave parameters were quantifiable by a regression model in a functional form independent of the actual irregular wave conditions. This statistical model provides a useful method for analyzing and predicting SGD from nearshore unconfined aquifers affected by random waves

Structure and dynamics of two-dimensional sheared granular flows

The structure and dynamics of the two-dimensional linear shear flow of inelastic disks at high area fractions are analyzed. The event-driven simulation technique is used in the hard-particle limit, where the particles interact through instantaneous collisions. The structure (relative arrangement of particles) is analyzed using the bond-orientational order parameter. It is found that the shear flow reduces the order in the system, and the order parameter in a shear flow is lower than that in a collection of elastic hard disks at equilibrium. The distribution of relative velocities between colliding particles is analyzed. The relative velocity distribution undergoes a transition from a Gaussian distribution for nearly elastic particles, to an exponential distribution at low coefficients of restitution. However, the single-particle distribution function is close to a Gaussian in the dense limit, indicating that correlations between colliding particles have a strong influence on the relative velocity distribution. This results in a much lower dissipation rate than that predicted using the molecular chaos assumption, where the velocities of colliding particles are considered to be uncorrelated.

Statistically steady turbulence in thin films: direct numerical simulations with Ekman friction

We present a detailed direct numerical simulation (DNS) of the two-dimensional Navier-Stokes equation with the incompressibility constraint and air-drag-induced Ekman friction; our DNS has been designed to investigate the combined effects of walls and such a friction on turbulence in forced thin films. We concentrate on the forward-cascade regime and show how to extract the isotropic parts of velocity and vorticity structure functions and hence the ratios of multiscaling exponents. We find that velocity structure functions display simple scaling, whereas their vorticity counterparts show multiscaling, and the probability distribution function of the Weiss parameter 3, which distinguishes between regions with centers and saddles, is in quantitative agreement with experiments.

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier-Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.

Dynamics of dense sheared granular flows. Part II. The relative velocity distributions

The distribution of relative velocities between colliding particles in shear flows of inelastic spheres is analysed in the Volume fraction range 0.4-0.64. Particle interactions are considered to be due to instantaneous binary collisions, and the collision model has a normal coefficient of restitution e(n) (negative of the ratio of the post- and pre-collisional relative velocities of the particles along the line joining the centres) and a tangential coefficient of restitution e(t) (negative of the ratio of post- and pre-collisional velocities perpendicular to line joining the centres). The distribution or pre-collisional normal relative velocities (along the line Joining the centres of the particles) is Found to be an exponential distribution for particles with low normal coefficient of restitution in the range 0.6-0.7. This is in contrast to the Gaussian distribution for the normal relative velocity in all elastic fluid in the absence of shear. A composite distribution function, which consists of an exponential and a Gaussian component, is proposed to span the range of inelasticities considered here. In the case of roughd particles, the relative velocity tangential to the surfaces at contact is also evaluated, and it is found to be close to a Gaussian distribution even for highly inelastic particles.Empirical relations are formulated for the relative velocity distribution. These are used to calculate the collisional contributions to the pressure, shear stress and the energy dissipation rate in a shear flow. The results of the calculation were round to be in quantitative agreement with simulation results, even for low coefficients of restitution for which the predictions obtained using the Enskog approximation are in error by an order of magnitude. The results are also applied to the flow down an inclined plane, to predict the angle of repose and the variation of the volume fraction with angle of inclination. These results are also found to be in quantitative agreement with previous simulations.

Random vibration of a second order non-linear elastic system

A method is presented for obtaining, approximately, the response covariance and probability distribution of a non-linear oscillator under a Gaussian excitation. The method has similarities with the hierarchy closure and the equivalent linearization approaches, but is different. A Gaussianization technique is used to arrive at the output autocorrelation and the input-output cross-correlation. This along with an energy equivalence criterion is used to estimate the response distribution function. The method is applicable in both the transient and steady state response analysis under either stationary or non-stationary excitations. Good comparison has been observed between the predicted and the exact steady state probability distribution of a Duffing oscillator under a white noise input.

A conditional random field-based downscaling method for assessment of climate change impact on multisite daily precipitation in the Mahanadi basin

Downscaling to station-scale hydrologic variables from large-scale atmospheric variables simulated by general circulation models (GCMs) is usually necessary to assess the hydrologic impact of climate change. This work presents CRF-downscaling, a new probabilistic downscaling method that represents the daily precipitation sequence as a conditional random field (CRF). The conditional distribution of the precipitation sequence at a site, given the daily atmospheric (large-scale) variable sequence, is modeled as a linear chain CRF. CRFs do not make assumptions on independence of observations, which gives them flexibility in using high-dimensional feature vectors. Maximum likelihood parameter estimation for the model is performed using limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization. Maximum a posteriori estimation is used to determine the most likely precipitation sequence for a given set of atmospheric input variables using the Viterbi algorithm. Direct classification of dry/wet days as well as precipitation amount is achieved within a single modeling framework. The model is used to project the future cumulative distribution function of precipitation. Uncertainty in precipitation prediction is addressed through a modified Viterbi algorithm that predicts the n most likely sequences. The model is applied for downscaling monsoon (June-September) daily precipitation at eight sites in the Mahanadi basin in Orissa, India, using the MIROC3.2 medium-resolution GCM. The predicted distributions at all sites show an increase in the number of wet days, and also an increase in wet day precipitation amounts. A comparison of current and future predicted probability density functions for daily precipitation shows a change in shape of the density function with decreasing probability of lower precipitation and increasing probability of higher precipitation.

Orientational ordering in sheared inelastic dumbbells

Using even driven simulations, we show that homogeneously sheared inelastic dumbbells in two dimensions are randomly orientated in the limit of low density. As the packing fraction is increased, particles first tend to orient along the extensional axis, and then as the packing fraction is further increased, the alignment shifts closer to the flow axis. The orientational order parameter displays a continuous increase with packing fraction and does not appear to exhibit a universal scaling with elongation. Except at the highest packing fractions, the orientational distribution function can be reconstructed with only the first coefficient of the Fourier expansion.

The virial expansion of a classical interacting system

We consider N particles interacting pairwise by an inverse square potential in one dimension (Calogero-Sutherland-Moser model). For a system placed in a harmonic trap, its classical partition function for the repulsive regime is recognised in the literature. We start by presenting a concise re-derivation of this result. The equation of state is then calculated both for the trapped and the homogeneous gas. Finally, the classical limit of Wu's distribution function for fractional exclusion statistics is obtained and we re-derive the classical virial expansion of the homogeneous gas using this distribution function.

Diagnostic Tests Based on Quantile Residuals for Nonlinear Time Series Models

This thesis studies quantile residuals and uses different methodologies to develop test statistics that are applicable in evaluating linear and nonlinear time series models based on continuous distributions. Models based on mixtures of distributions are of special interest because it turns out that for those models traditional residuals, often referred to as Pearson's residuals, are not appropriate. As such models have become more and more popular in practice, especially with financial time series data there is a need for reliable diagnostic tools that can be used to evaluate them. The aim of the thesis is to show how such diagnostic tools can be obtained and used in model evaluation. The quantile residuals considered here are defined in such a way that, when the model is correctly specified and its parameters are consistently estimated, they are approximately independent with standard normal distribution. All the tests derived in the thesis are pure significance type tests and are theoretically sound in that they properly take the uncertainty caused by parameter estimation into account. -- In Chapter 2 a general framework based on the likelihood function and smooth functions of univariate quantile residuals is derived that can be used to obtain misspecification tests for various purposes. Three easy-to-use tests aimed at detecting non-normality, autocorrelation, and conditional heteroscedasticity in quantile residuals are formulated. It also turns out that these tests can be interpreted as Lagrange Multiplier or score tests so that they are asymptotically optimal against local alternatives. Chapter 3 extends the concept of quantile residuals to multivariate models. The framework of Chapter 2 is generalized and tests aimed at detecting non-normality, serial correlation, and conditional heteroscedasticity in multivariate quantile residuals are derived based on it. Score test interpretations are obtained for the serial correlation and conditional heteroscedasticity tests and in a rather restricted special case for the normality test. In Chapter 4 the tests are constructed using the empirical distribution function of quantile residuals. So-called Khmaladze s martingale transformation is applied in order to eliminate the uncertainty caused by parameter estimation. Various test statistics are considered so that critical bounds for histogram type plots as well as Quantile-Quantile and Probability-Probability type plots of quantile residuals are obtained. Chapters 2, 3, and 4 contain simulations and empirical examples which illustrate the finite sample size and power properties of the derived tests and also how the tests and related graphical tools based on residuals are applied in practice.

Heat transfer in rarefied couette flow on the basis of discrete ordinate method

The method of discrete ordinates, in conjunction with the modified "half-range" quadrature, is applied to the study of heat transfer in rarefied gas flows. Analytic expressions for the reduced distribution function, the macroscopic temperature profile and the heat flux are obtained in the general n-th approximation. The results for temperature profile and heat flux are in sufficiently good accord both with the results of the previous investigators and with the experimental data.