Theoretical Distribution Function Research Articles

EMP stars with high mass IMF and hierarchical galaxy formation

AbstractExtremely metal-poor (EMP) stars in the Galactic halo are stars formed in the very early stage of the chemical evolution of the Galaxy. In previous study, we proposed that typical mass of EMP stars are massive, based on observations of carbon-enhanced EMP stars. In this study, we build a merger tree of the Galaxy semi-analytically and follow the chemical evolution along the merger tree. We also consider the effect of binary and high-mass initial mass function(IMF). Resultant theoretical metallicity distribution function (MDF) and abundance distribution are compared with observed metal-poor halo stars.

Time enlargement law for gas pulse breakdown

This paper investigates the possibility of applying the time enlargement law for predicting how gas-insulated systems would behave when exposed to pulse voltage loads of different shapes. For this purpose, the validity of the time enlargement law in this case has first been tested and the most suitable theoretical distribution function of the breakdown time random variable established. Pulse characteristics of the investigated insulating system have subsequently been determined, by applying the time enlargement law to experimental values of the breakdown time random variable, obtained in measurements with predefined shapes of the voltage load. Pulse characteristics thus obtained were compared with the corresponding pulse characteristics derived from the area law. The results demonstrate the advantages of the time enlargement law method, especially in the case of a non-homogeneous electric field. The experiments were conducted with SF6 gas, at different values of the pd product (pressure × inter-electrode gap), in a wide frequency range of applied pulse voltages, for a homogeneous, radial and point–plane electrode configuration.

Distribution functions of a heteronuclear hard dumbbell at a hard wall. Pure fluid and mixtures with hard spheres

Distribution functions (angle-averaged site/wall correlation functions) in the inhomogeneous systems of a heteronuclear hard dumbbell (of the site diameter ratio and ) were studied theoretically by applying a simple method to determine the background (cavity) correlation function from the residual chemical potentials of the hard dumbbell, hard wall and the corresponding combined body. Pure fluid and approximately equimolar mixtures of dumbbells with hard spheres were considered at several densities. Theoretical distribution functions for the most important site–wall distances were calculated and compared with simulation data; fair agreement was found. Advantage of the proposed method is its simplicity (the residual chemical potential depends, besides the packing fraction, only on the geometric characteristics of the hard body–its volume, surface area and mean radius); the method provides quick information on the density profiles in different inhomogeneous systems.

Advanced method for estimating direct-to-scattered ratio of Rician channel in reverberation chamber

A simple and efficient method for estimating the direct-to-scattered ratio of a Rician channel in a reverberation chamber is presented. Both parameters of the Rician distribution function are evaluated indirectly using rigorous statistical tools. Good agreement is reported between the experimental distribution function and the theoretical distribution function.

A matching algorithm for generation of statistically dependent random variables with arbitrary marginals

Simulation has gained acceptance in the operations research community as a viable method for analyzing complex problems. While random generation of variables with various marginal distributions has been studied at length, developing ability to preserve a given degree of statistical dependence among them has been lagging behind. This paper includes a short summary of the previous work and a description of the proposed algorithm for efficient re-arranging of generated random variables such that a desired product moment correlation matrix is induced. The proposed approach is different from similar algorithms that induce a desired rank-order correlation among random variables. The algorithm is demonstrated using three numerical examples, one of which also includes a comparison with @RISK commercial package. Its main features are simplicity, ease of implementation and the ability to handle either theoretical or empirical distribution functions.

Small-angle X-ray scattering from phase-separating amorphous metallic alloys undergoing nanocrystallization

We performed a combined small- and wide-angle X-ray scattering (SAXS/WAXS) study of nanocrystallization during annealing of binary Al 92Sm 8 and Al 91Gd 9 melt-spun glassy alloys. In course of the transformation an interference maximum develops in the SAXS region while WAXS spectra show formation of fcc-Al nanocrystals in the glassy phase. In order to determine the origin of the SAXS maximum the theoretical SAXS intensity and distance distribution function were calculated considering two structural models. The first model represented a single nanocrystal with a solute layer on its surface and the second one a phase-separated alloy with spatially correlated compositional fluctuations. The results of the calculations are compared with the experimental data. It is demonstrated that only the model representing correlated fluctuations reasonably reproduces the experimental results. We conclude that nano-scale glassy phase separation occurs in the investigated alloys and the nanocrystals form inside the Al-rich amorphous regions.

Modelling of ‘thermal shocks’ in composite materials using a meshfree FEM

Abstract This work discusses an application of a meshfree, semidiscrete finite element method to solving the ‘thermal shock’ problem for a thin, cylindrical plate made of functionally graded ceramics. The unsteady heat conduction equation is discretized in space with the partition of unity FEM. The temporal discretization is realized by using an explicit finite difference method. The computations are performed on two model samples: the homogenous one, made of the pure Al 2 O 3 material, and the composite plate, made of the alumina/zirconium layers with variable weight content of ZrO 2 . The heat transfer coefficient on the surface subjected to the ‘thermal shock’, as well as thermal conductivity and thermal diffusivity of the material, are modelled with the theoretical distribution function based on the experimental findings. The discrete solution is first checked for accuracy against the analytical solution to an exemplary test problem, yielding very advantageous results. Finally, it is verified in comparison with the jet-impingement-cooling experimental data, showing good accordance.

Significance of periodogram peaks and a pulsation mode analysis of the Beta Cephei star V403 Car

We discuss some commonly used methods for determining the significance of peaks in the periodograms of time series. We review methods for constructing the classical significance tests, their corresponding false alarm probability functions and the role played in these by independent random variables and by empirical and theoretical cumulative distribution functions. We discuss the concepts of independent frequencies and oversampling in periodogram analysis. We then compare the results of new Monte Carlo simulations for evenly spaced time series with results obtained previously by other authors, and present the results of Monte Carlo simulations for a specific unevenly spaced time series obtained for V403 Car.

Computationally efficient algorithms for the two-dimensional Kolmogorov–Smirnov test

Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2d-1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of Ω(n2lgn), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test.

A general definition of the Leimkuhler curve

In this paper, we provide a general definition of the Leimkuhler curve in terms of the theoretical cumulative distribution function. The definition applies to discrete, continuous and mixed random variables. Several examples are given to illustrate the use of the formula.

Wind speed spatial estimation for energy planning in Sicily: A neural kriging application

One of the first steps for the exploitation of any energy source is necessarily represented by its estimation and mapping at the aim of identifying the most suitable areas in terms of energy potential. In the field of renewable energies this is often a very difficult task, because the energy source is in this case characterized by relevant variations over space and time. This implies that any temporal, but also spatial, estimation model has to be able to incorporate this spatial and temporal variability. The paper deals with the spatial estimation of the wind fields in Sicily (Italy) by following a data-driven approach. Starting from the results of a preliminary study, a novel technique resulting from the integration of neural and geostatistical techniques was developed in order to obtain the wind speed maps for the region at 10 and 50 meters above the ground level. The mean values of the theoretical Weibull distribution function describing the wind regime at each of the available measurement sites were used to train a multi-layer perceptron (MLP) whose goal is to compute the most of the wind spatial trends. Other pieces of information about the territory (altitude, land coverage) were also used as inputs of the network and organized into a geographic information system (GIS) environment. The remaining de-trended linear means have been computed by using a universal kriging (UK) estimator. The results of these steps were then summed up and it was thus possible to obtain a map of the estimated wind fields.

Carboranes: chemical concepts derived from the AIM study of the experimental and theoretical electron density distribution functions

On the basis of high-resolution X-ray diffraction studies as well as quantum-chemical calculations of five carborane derivatives the peculiarities of electron density distribution functions have been analyzed. The data obtained permitted a deeper insight into the nature of unusual properties of the C-C bond in o-carborane and investigating intermolecular H ... H interactions in crystal. It was shown that such an approach allows estimating the values of lattice energy for the crystals of carboranes.

Sub-Poissonian statistics of quantum jumps in single molecule or atomic ion

A theory for statistics of quantum jumps in single molecule or ion driven by continues wave laser field is developed. These quantum jumps can relate to nonradiative singlet–triplet transitions in a molecule or to on→off jumps in a single ion with shelving processes. Distribution function w N ( T) of quantum jumps in time interval T is found. Computer simulation of quantum jumps is realized. Statistical treatment of simulated jumps reveals sub-Poissonian statistics of quantum jumps. The theoretical distribution function w N ( T) fits well the distribution of jumps found from simulated data. Experimental data on quantum jumps found in experiments with single Hg + ion are described by the function w N ( T) well.

Evolution of interacting binaries with a B type primary at birth

We revisited the analytical expression for the mass ratio distribution for non-evolved binaries with a B type primary. Selection effects governing the observations were taken into account in order to compare theory with observations. Theory was optimized so as to fit best with the observed q -distribution of SB1s and SB2s. The accuracy of this theoretical mass ratio distribution function is severely hindered by the uncertainties on the observations. We present a library of evolutionary computations for binaries with a B type primary at birth. Some liberal computations including loss of mass and angular momentum during binary evolution are added to an extensive grid of conservative calculations. Our computations are compared statistically to the observed distributions of orbital periods and mass ratios of Algols. Conservative Roche Lobe Over Flow (RLOF) reproduces the observed distribution of orbital periods but fails to explain the observed mass ratios in the range q ∈ $\lbrack$0.4–1$\rbrack$. In order to obtain a better fit the binaries have to lose a significant amount of matter, without losing much angular momentum.

Electrooptics of β-FeOOH particle in aqueous media : 2. Field-strength dependence of decay and steady-state birefringence, and the hydrodynamic and electrooptic properties

Steady-state and decay birefringence, expressed in terms of the optical phase retardation per cell length δ / d , was measured on β-FeOOH in aqueous ionic media at 633 nm and at 25 °C by an electric square-pulse technique over a wide range of field strength E to ca. 6 kV/cm. The field-strength dependence of both δ / d and field-free rotational relaxation time τ was determined at the sample concentrations between 0.0011 and 0.055 g/L and in the 0.02–2.0 mM NaCl concentration range. Extrapolation of both δ / d and τ values to infinitely high fields ( E 2 → ∞ ) could yield birefringence- and weight-average quantities, respectively. Observed τ values were decreased at weak fields but leveled off to ca. 0.3 ms at very high fields due to a slight polydispersity regarding the length and volume of particles. The weight-average relaxation time 〈 τ 〉 w was calculated with Perrin's expressions theoretically from the length, width, and volume of β-FeOOH particles estimated in the dried state from electron micrograph. These quantities were variously averaged. The size distribution was discussed in terms of observed discrete histogram and theoretical (Weibull and Lansing–Kraemer) distribution functions. The sign of observed δ / d value was always positive. The infinitely high-field ( δ / d ) ∞ values and the reduced optical anisotropy factor Δ g / n were evaluated by fitting to theoretical orientation functions. The intrinsic birefringence ( n 3 − n 1 ) could be estimated with the mean refractive index n p reported in the literature. For the spindle-shaped particle with an axial ratio of ca. 4, the sign of Δ g / n is always positive, whereas the quantity ( n 3 − n 1 ) was either negative ( n p > 2.35 ) or positive ( n p < 2.05 ) in sign or nearly zero (ca. n p = 2.26 ), depending critically on the n p values.

Statistical distributions of optimal global alignment scores of random protein sequences

BackgroundThe inference of homology from statistically significant sequence similarity is a central issue in sequence alignments. So far the statistical distribution function underlying the optimal global alignments has not been completely determined.ResultsIn this study, random and real but unrelated sequences prepared in six different ways were selected as reference datasets to obtain their respective statistical distributions of global alignment scores. All alignments were carried out with the Needleman-Wunsch algorithm and optimal scores were fitted to the Gumbel, normal and gamma distributions respectively. The three-parameter gamma distribution performs the best as the theoretical distribution function of global alignment scores, as it agrees perfectly well with the distribution of alignment scores. The normal distribution also agrees well with the score distribution frequencies when the shape parameter of the gamma distribution is sufficiently large, for this is the scenario when the normal distribution can be viewed as an approximation of the gamma distribution.ConclusionWe have shown that the optimal global alignment scores of random protein sequences fit the three-parameter gamma distribution function. This would be useful for the inference of homology between sequences whose relationship is unknown, through the evaluation of gamma distribution significance between sequences.

Analysis of molecular weight distributions of styrene‐methyl methacrylate copolymers using size exclusion chromatography data

AbstractSize exclusion chromatography (SEC) is a common method for the determination of molecular weight distributions (MWD) of polymers. Using appropriate normalization procedures, MWD data may be put in a form suitable for the description by different theoretical distribution functions. In this paper such a procedure is described and applied to a model system. Statistical copolymers of styrene and methyl methacrylate (SMMA) of different compositions were synthesized by radical copolymerization in a batch isothermal reactor with toluene as solvent and azobisisobutyronitrile as initiator. Copolymer MWDs were obtained by SEC and average copolymer composition by elemental analysis. MWDs of SMMA copolymers were correlated with the lognormal, Flory, Schulz, and Tung theoretical function. The best match between experimental and calculated data was obtained by the Schulz distribution. The α‐parameter of Schulz's function was related to the average molecular weight and showed its maximum at the midpoint of the copolymer composition range, in connection with the observed minimal copolymerization rate. The variation of the b‐parameter was found to be less pronounced; the parameter was related to the polydispersity index. This feature indicated that there was no significant change in the copolymerization reaction mechanism throughout the entire copolymer composition range.

Tip-jump statistics of stick-slip friction

Friction maps with atomic unit cell periodicity on highly oriented graphite in ultra-high vacuum are measured with the friction force microscope. The well-known stick-slip phenomenon is observed where the tip jumps between individual equilibrium positions. We perform a statistical analysis of the frictional forces that are necessary to induce the tip jumps. The corresponding histograms give a direct fingerprint of the distribution function of the tip-jump events. The histograms depend strongly on the atomic structure of the surface, which is related to the two-dimensionality of the surface potential. We compare the experimental to the theoretical distribution function, which is based on the thermally activated Tomlinson model for atomic friction, and quantitative values for the effective energy barriers are extracted.

About randomness of aerosol size distributions

All aerosol formation and evolution processes, such as nucleation, condensation, fragmentation, etc., are understood and rationalized via fundamental probabilistic concepts such as probabilities of collision, coagulation, dispersion, etc. Therefore all theoretical size distribution functions (lognormal, modified gamma distribution, self-preserving particle size distribution for Brownian coagulation, etc.) are in fact size probability density functions pdf ( r ) . Any (e.g., measured) size distribution f ( r ) of an aerosol system is some random realization of its pertinent size probability density function pdf ( r ) . When pdf ( r ) is viewed as a continuous function, the corresponding size distribution vanishes almost everywhere excluding some randomly set of sizes where f ( r ) = 1 . We investigate proximity between f ( r ) and pdf ( r ) in finite size intervals and derive expressions for estimation of the standard deviations of several aerosol size-dependent properties arising from randomness of f ( r ) .

Determination of applicable local porosity distributions in a powder bed by the maximum entropy method

Abstract Wick debinding can reduce the debinding cycle time and is a crucial technique in metal powder injection molding (MIM). The porosity of the compact varies locally, but there are few literature reports on determining the porosity distribution of the compact in MIM. The need for measuring porosity distribution motivates this study. A local porosity distribution (LPD) is quoted and measured from the digital image of the pore structure. Applicable LPDs and typical length scales are then calculated by the maximum entropy method. Proper theoretical porosity distribution functions are adopted to fit the applicable LPDs. According to the theoretical distribution function, a random number generator is used to generate data of porosity with quantitative randomness for engineering research. Results indicate that the typical length scale is essential and important for the numerical simulation in determining the size of the control volume (or grid).