Truncated Normal Distribution Research Articles

Distribution of fusion barriers

Heavy ion fusion cross sections and compound nucleus average spin values obtained from distribution of fusion barriers are discussed. Various shapes of distribution functions are studied using a truncated Gaussian distribution function (TGD). It is shown that fusion cross section and average spin values are less sensitive to different parametrization of TGD function, whereas the second derivative of the product of energy and fusion cross sections (w.r.t. energy), obtained from the corresponding TGD functions are significantly different depending on the shape of the barrier distribution function. It is also shown byχ 2 analysis of fusion cross section data that some systems favour a narrow Gaussian distribution function whereas others, for which the vibrational and rotational collective states are less important, favour a flat barrier distribution. A physical interpretation of the dynamical process that gives rise to different barrier distribution is given in the framework of microscopic coupled channel calculations.

Pressure Propagation in Pulsatile Flow Through Random Microvascular Networks

A microvascular network with random dimensions of vessels is built on the basis of statistical analysis of conjunctival beds reported in the literature. Our objective is to develop a direct method of evaluating the statistics of the pulsatile hydrodynamic field starting from a priori statistics which mimic the large-scale heterogeneity of the network. The model consists of a symmetric diverging-converging dentritic network of ten levels of vessels, each level described by a truncated Gaussian distribution of vessel diameters and lengths. In each vascular segment, the pressure distribution is given by a diffusion equation with random parameters, while the blood flow rate depends linearly on the pressure gradient. The results are presented in terms of the mean value and standard deviation of the pressure and flow rate waveforms at two positions along the network. It is shown that the assumed statistical variation of vessel lengths results in flow rate deviations as high as 50 percent of the mean, while the corresponding effect of vessel diameter variation is much smaller. For a given pressure drop, the statistical variation of lengths increases the mean flow while the effect on the mean pressure distribution is negligible.

National Plastics Exposition '91

A systematic numerical method has been presented to investigate the constitutive relationships between two-phase flow properties of horizontal fractures and aperture distributions. Based on fractal geometry, single rough-walled fractures are generated numerically by modified successive random addition (SRA) method and then aperture distributions with truncated Gaussian distribution are formed by shear displacement between lower and upper surfaces. (The truncated Gaussian distribution is used to describe aperture evolution under different normal stresses.) According to the assumption of two-dimensional porous media and local parallel plate model, invasion percolation approach is employed to model the two-phase flow displacement (imbibition) in generated horizontal fractures, in which capillary forces are dominant over viscous and gravity forces. For truncated Gaussian distributions, constitutive relationships from numerical simulation are compared to closed-form relationships and a good agreement is obtained. The simulation results indicate strong phase interference with the sum of two phase relative permeability values being less than one in the intermediate saturations. It is found that fracture properties related to residual saturations depend on spatial correlation of aperture distributions. Based on the simulation results, we proposed an empirical relationship between the fracture residual-saturation-rated parameters and the corresponding aperture distributions.

Combined equal and unequal element spacings for low sidelobe pattern of a symmetrical array with equal-amplitude elements

Numerical computations were made for a low-sidelobe-level 20-element symmetrical linear array whose equal-amplitude elements were spaced mostly equally (but some unequally) to yield aperture illumination densities along the array that approximate a truncated Gaussian distribution. It is argued that this design approach of using both equal and unequal interelement spacings and variations from the array analyzed should be investigated for large arrays and optimized for low sidelobe performance. Another design alternative for a large array is to have different size clusters of equally space elements with the intercluster spacings varied to attain desired tapered average illumination densities. >

Calculated Characteristics of Droplet Size and Velocity Distributions in liquid sprays

AbstractPredictions of the droplet size and velocity distributions in sprays under isothermal conditions are reported. The calculations are based on the maximum entropy formalism, complying with the conservation laws of liquid mass, momentum and energy. This theoretical approach considers only the macroscopic quantities about the atomization processes, without resorting to the details of the liquid breakup processes such as the onset and growth of instabilities. The derived joint droplet size and velocity distribution function depends on the Weber number as well as the liquid mass, momentum and energy source terms.These parameters represent the conditions under which the atomization occurs. The droplet velocity distributions are truncated Gaussian distributions for any specific sizes. The nondimensional Sauter mean diameter decreases slightly with the Weber number and then approaches an asymptotic constant. The calculated values of D21/D30 are very close to unity which agrees with the experimental observations. The computations also show that the atomization efficiency is very low; less than 2.6 percent.

SHORT COMMUNICATION

Abstract Abstract— This paper presents the prediction of the droplet size and velocity distributions in sprays based on the conservation laws of mass, momentum and energy through maximum entropy formalism, without taking into aecount the details of liquid atomization processes such as the characteristics of disturbances and behavior of liquid sheet. The exponential nature of the widely used empirical correlations for droplet size distributions gives credibility to the predicted PDF. The droplet velocity distributions are truncated Gaussian distributions, with the mean and variance dependent on the droplet size. The mean velocity for large droplets approaches a constant value, and the velocity variance becomes smaller as droplet size increases. Hence the velocity is more uniform for larger droplets in sprays.

Teletraffic performance of highway microcells with overlay macrocell

The teletraffic performance of a highway microcellular digital mobile radio system having an oversailing macrocell that spans many microcells is presented. The microcellular cluster is composed of concatenated segments of the highway where each segment is a microcell, typically 500-2000 m in length, with the base stations located at lamp-post elevations. A narrowband time-division-multiple-access arrangement supporting ten channels per carrier and one carrier per base station is used. The teletraffic analysis assumes there are n-up and n-down lanes, and that the vehicular speeds conform to a truncated Gaussian distribution whose mean speed is 100 or 50 km/h when the vehicular traffic is free-flowing or in traffic-congested conditions, respectively. >

Truncation Effects on Estimated Parameters of Tracer Distributions Sampled on Finite Domains

Abstract Methods are presented and demonstrated for compensating for the apparent effect of truncation of the sampled crosswind distribution of a tracer, at either or both boundaries of a surface sampling grid downwind from the point of release. Errors and correction on estimates of the apparent vertical mixing depth, h, the location of the centroid of the crosswind distribution ȳ, and the crosswind dispersion parameter, σy are those which would apply to a truncated Gaussian distribution having the observed moments (zeroth, first and second). For truncation on only one side of the tracer cloud, a linear relationship between certain measurable quantities based on moments of partial distributions is used to derive the parameters of a best-fit truncated Gaussian distribution by linear regression. The methods are illustrated by an example from the Cross Appalachian Tracer Experiment and by a Gaussian simulation.

Variations in the geomagnetic dipole 2: Statistical analysis of VDMs for the past 5 million years.

Virtual Dipole Moments (VDMs) are analysed to determine the distribution of True Dipole Moments (TDMs) from which they were derived. The data set used is all the available VDMs for the past 5 million years which have associated Virtual Geomagnetic Poles (VGPs) with latitudes greater than 45°. These data do not support a model which predicts a cyclic variation of the dipole moment during stable polarity periods. Neither do they support a model in which the field strength of the non-dipole components is a constant ratio of the mean dipole field strength. Instead they support a model in which the TDMs have a truncated Gaussian distribution and the field strength of the non-dipole components is linearly proportional to the TDM. These components introduce a Gaussian distributed scatter to the observed VDMs with standard deviation about 18.6% of the dipole moment, and the palaeointensity determinations themselves introduce a further Gaussian distributed scatter with standard deviation about 10%. The data give no reason to reject the hypothesis of a common mean and variance for the (untruncated) Gaussian distribution of the normal and reverse polarity TDMs but the hypothesis of a common truncation point can be rejected. This indicates a difference in the overall properties of the two polarity states.The term Palaeomagnetic Dipole Moment (PDM) is introduced for the value at which the truncated Gaussian distribution of TDMs peaks. As determined from the present data the PDM for the past 5 million years is 8.67±0.65×1022Am2 with 95% confidence. The estimated standard deviation of the untruncated distribution is 3.63±0.75×1022Am2 with 95% confidence, this variation (41.9%) relating solely to variation in the dipole moment itself. This reinforces the view that fluctuations in the dipole moment are much greater than are observed in data for the past 104 years, suggesting that they must occur over time scales of 105 or 106 years.

Theory of systems with random Ising bond interactions in the paramagnetic phase

A theory of the thermodynamic properties (spin correlation functions, susceptibility and specific heat) in the paramagnetic phase of Ising systems with various forms of random bond arrangement is developed. It represents an extension of the random phase approximation (RPA). The difference equation for the spin correlation functions which has random coefficients is solved using the coherent potential approximation (CPA), and the various parameters calculated self-consistently. Results for the quenched dilute bond case are in good agreement with other theories. For a truncated Gaussian distribution of bond strengths it is found that the transition temperature and other properties are fairly universal functions of the mean square of the distribution. The transition to a ferromagnetic phase disappears when the concentration of negative exchange bonds becomes large enough, apparently because of spin glass formation. Results for various binary distributions are also given.

Some computer experiments in picture processing for data compaction

This paper presents two digital methods for data compression of gray tone pictures of human faces by dividing the picture into many subpictures and transmitting the statistics of the subpictures, assuming a uniform or truncated gaussian distribution for the video values. First, a method of generating the video values (samples) of each subpicture at the receiving end is given. Then a scheme is presented to arrange these samples in the subpicture by making use of the differences between the means of video values of the subpicture and the neighboring subpictures. Also incorporated in the method is a feature to copy some subpictures from the input picture into the output picture, which have a wide variation of intensity levels, while keeping the data compression at 1 bit/pixel. Computer processed pictures are given as examples.

DATA TRANSFORMATION IN TWO‐WAY ANALYSIS OF VARIANCE

ABSTRACTOne purpose of data transformation is to better satisfy the fundamental assumptions of statistical analysis by linear models: (a) additivity, (b) homogeneity of variance, and (c) normality. If the original data do not satisfy (a)‐(c), a nonlinear transformation may improve approximation to these ideal conditions.This paper considers estimation of the parameter λ of the family of power transformationsfor data conforming to a replicated two‐way crossed classification. The transformation family y(λ) defined bythough having the advantage of being continuous through λ = 0, is shown to yield likelihood procedures which are scale invariant only if the design matrix allows for removal of an additive constant. Several alternative definitions for (ii) are given which yield both scale invariant likelihood procedures and continuity through λ = 0. By using (ii) in the case of a linear regression through the origin, numerical examples show how point estimates of λ can be shifted and how confidence intervals on λ can be shrunk and stretched almost arbitrarily, simply by rescaling the original observations.Two procedures for choosing a transformation for data from a replicated two‐way crossed classification are compared by empirical sampling. The statistic Lmax(λ), proposed by Box and Cox (1964), is based upon the likelihood of the transformed observations. The statistic Z(λ), proposed in this paper, is a linear combination of test statistics for removable nonadditivity and variance trending with mean. The null distribution of Z(λ) is approximately N(0,l).Two cases of empirical sampling from a 3 × 4 crossed classification with four replications were studied. In samples from the truncated Gaussian distribution, the coefficient of variation was roughly 0.23. In samples from the truncated Cauchy distribution, the coefficient of variation was roughly 0.91. The points of truncation depended upon the cell population means. The scale for the Cauchy distribution with location 0 was chosen so that the 95% points matched those of N(0,l). In Gaussian Samples: The 5% test based on Lmax(λ) was extremely conservative, the actual size being more like 0.3%. The estimated size of the nominal 5% test based on Z(λ) was 4.4% In Cauchy Samples: The 5% test based on Lmax(λ) was very non‐conservative. The estimated size was 15%. The estimated size of the nominal 5% test based on Z(λ) was 5.6%. A procedure for choosing a weighting for the components of Z(λ) based upon the sample data is also derived and investigated. Sampling shows that it works well for the two cases studied. This procedure should also work well in practice.