Approximating the Radial Distribution Function of the Electron in a Hydrogen Atom by a Normal Distribution Suggests That Magnetic Confinement Fusion Would Be Less Energy Efficient than Inertial Confinement Fusion

Since the position of the electron in a hydrogen atom cannot be determined, the region in which it resides is said to be determined stochastically and forms an electron cloud. The probability density function of the single electron in 1s orbit is expressed as φ2, a function of distance from the nucleus. However, the probability of existence of the electron is expressed as a radial distribution function at an arbitrary distance from the nucleus, so it is estimated as the probability of the entire spherical shape of that radius. In this study, it has been found that the electron existence probability approximates the radial distribution function by assuming that the probability of existence of the electron being in the vicinity of the nucleus follows a normal distribution for arbitrary x-, y-, and z-axis directions. This implies that the probability of existence of the electron, which has been known only from the distance information, would follow a normal distribution independently in the three directions. When the electrons’ motion is extremely restricted in a certain direction by the magnetic field of both tokamak and helical fusion reactors, the probability of existence of the electron increases with proximity to the nucleus, and as a result, it is less likely to be liberated from the nucleus. Therefore, more and more energy is required to free the nucleus from the electron in order to generate plasma.

Simulation of Haloxylon Ammodendron Stand Basic Diameter Structure Based on Fuzzy Distribution Function

Based on the investigation data of 12 Haloxylon ammodendron plots in the south edge of Gurbantunggut Desert, Fuzzy distribution was introduced into the study of Haloxylon ammodendron base diameter structure fitting according to the consistency between the characteristics of Fuzzy distribution function and the distribution series of cumulative percentage of stand base diameter, and the fitting precision and effect of Fuzzy distribution function were discussed. The root mean square error RMSE and determination coefficient R2 values showed that Fuzzy-Γ1, Fuzzy-Γ2, Fuzzy-Γ3, Fuzzy-Γ4 had good fitting performance, among which Fuzzy-Γ1 had relatively high fitting precision, and its parameters were closely related to stand age and density, Fuzzy-Γ2 distribution function was the second, and Fuzzy-Γ4 distribution function had the worst fitting effect. By introducing a parameter c from the similarity of four distribution function formulas, a generalized Fuzzy distribution function Fuzzy-Γ5 is obtained. This function shows the highest fitting accuracy. Most of the values of parameter c are near 1 or 2, which shows that the diameter distribution is mainly approximate to Fuzzy-Γ1 and Fuzzy-Γ2.

Some Notes of Property of Distribution Function of Many Variables

In this paper we discuss a step further some convergence and continuity problems of distribution function on R^i. We give the following results: (1)distribution function F(x_1,…,x_k) on R^k is continuous if and only if all marginal distribution functions of F is continuous on R^1. (2)If limF_n(x_1,……,x_k)=F(x_1,…,x_k) and limF_n(x_1—0,…,x_k—0)=F(x_1—0,…,x_k—0) at all non-continuity points of F, then

A New Function for Modelling Diameter Frequency Distribution in the Tropical Rain Forest of Xishuangbanna,Southwest of China

Permanent plots in the montane tropical rain forests in Xishuangbanna, southwest China, were established, and different empirical models, based on observation data of these plots in 1992, were built to model diameter frequency distributions. The focus of this study is on predicting accuracy of stem number in the larger diameter classes, which is much more important than that of the smaller trees, from the view of forest management, and must be adequately considered in the modelling and estimate. There exist 3 traditional ways of modelling the diameter frequency distribution: the negative exponential function model, limiting line function model, and Weibull distribution model. In this study, a new model, named as the logarithmic J-shape function, together with the others, was experimented and was found as a more suitable model for modelling works in the tropical forests.

Analysis of Ocean Wave Characteristic Distributions Modeled by Two Different Transformed Functions

The probability distributions of wave characteristics from three groups of sampled ocean data with different significant wave heights have been analyzed using two transformation functions estimated by non-parametric and parametric methods. The marginal wave characteristic distribution and the joint density of wave properties have been calculated using the two transformations, with the results and accuracy of both transformations presented here. The two transformations deviate slightly between each other for the calculation of the crest and trough height marginal wave distributions, as well as the joint densities of wave amplitude with other wave properties. The transformation methods for the calculation of the wave crest and trough height distributions are shown to provide good agreement with real ocean data. Our work will help in the determination of the most appropriate transformation procedure for the prediction of extreme values.

Stability of the Dividing Distribution Function Method for ParticleSize Distribution Analysis in Small Angle X－Ray Scattering

The Dividing Distribution Function (DDF) method is one of the methods by which the particle size distribution of ultrafine powder can be evaluated from its small angle X-ray scattering data. In this paper, the stability of the solution obtained from DDF method has been investigated through optimizing the coefficient matrix, introducing a damping factor and a least square treatment. All calculations were accomplished with a microcomputer. It was shown that the average deviations of the size distribution obtained are not larger than the assigned random errors to the scattering intensities as long as the corresponding requirements are satisfied.

Temporal and spatial distribution of phytoplankton functional groups and role of environment factors in a deep subtropical reservoir

Phytoplankton and environment factors were investigated in 2015 and phytoplankton functional groups were used to understand their temporal and spatial distribution and their driving factors in Wanfeng Reservoir. Seventeen functional groups(B, D, E, F, G, J, Lo, MP, P, S1, T, W1, W2, X1, X2, Xph, Y) were identified based on 34 species. The dominant groups were: J/B/P/D in dry season, X1/J/Xph/G/T in normal season and J in flood season. Phytoplankton abundance ranged from 5.33×10~4 cells/L to 3.65×10~7 cells/L, with the highest value occurring in flood season and lowest in dry season. The vertical profi le of dominant groups showed little differentiation except for P, which dominated surface layers over 20 m as a result of mixing water masses and higher transparency during dry season. However, the surface waters presented higher values of phytoplankton abundance than other layers, possibly because of greater irradiance. The significant explaining variables and their ability to describe the spatial distribution of the phytoplankton community in RDA diff ered seasonally as follows: dry season, NH4-N, NO_3-N, NO_2-N, TN:TP ratio and transparency(SD); normal season, temperature(WT), water depth, TN, NH4-N and NO_3-N; flood season, WT, water depth, NO_3-N and NO_2-N. Furthermore, nitrogen, water temperature, SD and water depth were significant variables explaining the variance of phytoplankton communities when datasets included all samples. The results indicated that water physical conditions and hydrology were important in phytoplankton community dynamics, and nitrogen was more important than phosphorus in modifying phytoplankton communities. Seasonal differences in the relationship between the environment and phytoplankton community should be considered in water quality management.

Representation of three-dimensional mass distribution of the Earth's interior by biorthogonal series and its use for studying internal structure of the planet

This paper utilizes the mathematical concept of approximation within an ellipsoid from a single viewpoint to present the spatial mass distribution function of the Earth's interior and its internal potential.The primary focus lies in constructing the volume distribution of masses in the planet's interior, with the expansion coefficients being linear combinations of the Stokes constants. Several possible approaches are suggested for determining accurately these coefficients employing three-dimensional(biorthogonal)polynomials. By expressing the mass distribution function of the Earth's interior and its internal potential as a series, an algorithm is introduced for the calculation of gravitational energy. It allows us to estimate fluctuations in gravitational energy. The implementation of this algorithm offers the means of establishing the extent to which the Earth deviates from a state of hydrostatic equilibrium as a celestial body.Due to the aforementioned method, calculations have been conducted to validate its effectiveness and reliability. This example is given as an illustration of a given method for studying the internal structure of planets.

Adaptive Time Frequency Distribution Based on Linear Chirp Modulated Gaussian Functions

We propose a method that uses linear chirp modulated Gaussian functions as the elementary functions, by adaptively adjusting variances, time frequency centers and sweep rates, to decompose signals. By taking WVD, an improved adaptive time frequency distribution is developed, which is non negative, free of cross term interference, and of better time frequency resolution. The paper presents an effective numerical algorithm to estimate the optimal parameters of the basis. Simulations indicate that the proposed approach is effective in analyzing signal's time frequency behavior.

Influence of porosity distribution on nonlinear free vibration and transient responses of porous functionally graded skew plates

This article deals with the investigation of the effects of porosity distributions on nonlinear free vibration and transient analysis of porous functionally graded skew(PFGS)plates.The effective material properties of the PFGS plates are obtained from the modified power-law equations in which gradation varies through the thickness of the PFGS plate.A nonlinear finite element(FE)formulation for the overall PFGS plate is derived by adopting first-order shear deformation theory(FSDT)in conjunction with von Karman’s nonlinear strain displacement relations.The governing equations of the PFGS plate are derived using the principle of virtual work.The direct iterative method and Newmark’s integration technique are espoused to solve nonlinear mathematical relations.The influences of the porosity distributions and porosity parameter indices on the nonlinear frequency responses of the PFGS plate for different skew angles are studied in various parameters.The effects of volume fraction grading index and skew angle on the plate’s nonlinear dynamic responses for various porosity distributions are illustrated in detail.

Research on the behavior of fiber orientation probability distribution function in the planar flows

The equation of two-dimensional fiber direction vector was solved theoretically to give the fiber orientation distri- bution in simple shear flow, flow with two direction shears, extensional flow and arbitrary planar incompressible flow. The Fok- ker-Planck equation was solved numerically to validify the theoretical solutions. The stable orientation and orientation period of fiber were obtained. The results showed that the fiber orientation distribution is dependent on the relative not absolute magnitude of the matrix rate-of-strain of flow. The effect of fiber aspect ratio on the orientation distribution of fiber is insignificant in most conditions except the simple shear case. It was proved that the results for a planar flow could be generalized to the case of 3-D fiber direction vector.

A prediction model for the effective thermal conductivity of nanofluids considering agglomeration and the radial distribution function of nanoparticles

Conventional heat transfer fluids usually have low thermal conductivity, limiting their efficiency in many applications. Many experiments have shown that adding nanosize solid particles to conventional fluids can greatly enhance their thermal conductivity. To explain this anomalous phenomenon, many theoretical investigations have been conducted in recent years. Some of this research has indicated that the particle agglomeration effect that commonly occurs in nanofluids should play an important role in such enhancement of the thermal conductivity, while some have shown that the enhancement of the effective thermal conductivity might be accounted for by the structure of nanofluids, which can be described using the radial distribution function of particles. However, theoretical predictions from these studies are not in very good agreement with experimental results. This paper proposes a prediction model for the effective thermal conductivity of nanofluids, considering both the agglomeration effect and the radial distribution function of nanoparticles. The resulting theoretical predictions for several sets of nanofluids are highly consistent with experimental data.

PROBABILITY DISTRIBUTION FUNCTION OF NEAR-WALL TURBULENT VELOCITY FLUCTUATIONS

By large eddy simulation （LES）, turbulent databases of channel flows at different Reynolds numbers were established. Then, the probability distribution functions of the streamwise and wall-normal velocity fluctuations were obtained and compared with the corresponding normal distributions. By hypothesis test, the deviation from the normal distribution was analyzed quantitatively. The skewness and flatness factors were also calculated. And the variations of these two factors in the viscous sublayer, buffer layer and log-law layer were discussed. Still illustrated were the relations between the probability distribution functions and the burst events-sweep of high-speed fluids and ejection of low-speed fluidsIin the viscous sub-layer, buffer layer and loglaw layer. Finally the variations of the probability distribution functions with Reynolds number were examined.

Three-Component Model for Bidirectional Reflection Distribution Function of Thermal Coating Surfaces

We present a bidirectional reflection distribution function （BRDF） model for thermal coating surfaces based on a three-component reflection assumption, in which the specular reflection is given according to the microfacet theory and Snell＇s law, the multiple reflection is considered Nth cosine distributed, and the volume scattering is uniformly distributed in reflection angles according to the experimental results. This model describes the reflection characteristics of thermal coating surfaces more completely and reasonably. Simulation and measurement results of two thermal coating samples SR107 and S781 are given to validate that this three-component model significantly improves the modeling accuracy for thermal coating surfaces compared with the existing BRDF models.

Field-Induced Structure and Pair Distribution Function of Fractals

The structure transitions in cluster-cluster aggregation (CCA) and diffusion-limited aggregation (DLA) under external electric fields have been investigated by computer simulations. With the increase of external electric field, there exists a structure transition from disorder to order, i. e., the aggregates change from fractals of diffusion-limited CCA and DLA to the electrorheological chains parallel to electric field. Pair distribution shows, the system changes from local order to long-range order gradually with the external field rising.

Influence of Generalized （r, q） Distribution Function on Electrostatic Waves

Non-Maxwellian particle distribution functions possessing high energy tail and shoulder in the profile of distribution function considerably change the damping characteristics of the waves. In the present paper Landau damping of electron plasma （Langmuir） waves and ion-acoustic waves in a hot, isotropic, unmagnetized plasma is studied with the generalized （r, q） distribution function. The results show that for the Langmuir oscillations Landau damping becomes severe as the spectral index r or q reduces. However, for the ion-acoustic waves Landau damping is more sensitive to the ion temperature than the spectral indices.

A new early warning method for dam displacement behavior based on non-normal distribution function

Traditional methods for early warning of dam displacements usually assume that residual displacements follow a normal distribution.This assumption deviates from the reality,thereby affecting the reliability of early warning results and leading to misjudgments of dam displacement behavior.To solve this problem,this study proposed an early warning method using a non-normal distribution function.A new early warning index was developed using cumulative distribution function(CDF)values.The method of kernel density estimation was used to calculate the CDF values of residual displacements at a single point.The copula function was used to compute the CDF values of residual displacements at multiple points.Numerical results showed that,with residual displacements in a non-normal distribution,the early warning method proposed in this study accurately reflected the dam displacement behavior and effectively reduced the frequency of false alarms.This method is expected to aid in the safe operation of dams.

Vertical distribution characteristics of soil saturated hydraulic conductivity under different land use patterns in the Mu Us sandy land

Background,aim,and scope Soil saturated hydraulic conductivity(K_(s))is a key parameter in the hydrological cycle of soil;however,we have very limited understanding of K_(s) characteristics and the factors that inf luence this key parameter in the Mu Us sandy land(MUSL).Quantifying the impact of changes in land use in the Mu Us sandy land on K_(s) will provide a key foundation for understanding the regional water cycle,but will also provide a scientific basis for the governance of the MUSL.Materials and methods In this study,we determined K_(s) and the basic physical and chemical properties of soil(i.e.,organic matter,bulk density,and soil particle composition)within the first 100 cm layer of four different land use patterns(farmland,tree,shrub,and grassland)in the MUSL.The vertical variation of K_(s) and the factors that influence this key parameter were analyzed and a transfer function for estimating K_(s) was established based on a multiple stepwise regression model.Results The K_(s) of farmland,tree,and shrub increased gradually with soil depth while that of grassland remained unchanged.The K_(s) of the four patterns of land use were moderately variable;mean K_(s)values were ranked as follows:grassland(1.38 mm·min^(-1))<tree(1.76 mm·min^(-1))<farmland(1.82 mm·min^(-1))<shrub(3.30 mm·min^(-1)).The correlation between K_(s) and organic matter,bulk density,and soil particle composition,varied across different land use patterns.A multiple stepwise regression model showed that silt,coarse sand,bulk density,and organic matter,were key predictive factors for the K_(s) of farmland,tree,shrub,and grassland,in the MUSL.Discussion The vertical distribution trend for K_(s) in farmland is known to be predominantly influenced by cultivation,fertilization,and other factors.The general aim is to improve the water-holding capacity of shallow soil on farmland(0-30 cm in depth)to conserve water and nutrients;research has shown that the K_(s) of farmland increases with soil depth.The root growth of tree and shrub in sandy land exerts mechanical force on the soil due to biophysical processes involving rhizospheres,thus leading to a significant change in K_(s).We found that shallow high-density fine roots increased the volume of soil pores and eliminated large pores,thus resulting in a reduction in shallow K_(s).Therefore,the K_(s) of tree and shrub increased with soil depth.Analysis also showed that the K_(s) of grassland did not change significantly and exhibited the lowest mean value when compared to other land use patterns.This finding was predominantly due to the shallow root system of grasslands and because this land use pattern is not subject to human activities such as cultivation and fertilization;consequently,there was no significant change in K_(s) with depth;grassland also had the lowest mean K_(s).We also established a transfer function for K_(s) for different land use patterns in the MUSL.However,the predictive factors for K_(s) in different land use patterns are known to be affected by soil cultivation methods,vegetation restoration modes,the distribution of soil moisture,and other factors,thus resulting in key differences.Therefore,when using the transfer function to predict K_(s) in other areas,it will be necessary to perform parameter calibration and further verification.Conclusions In the MUSL,the K_(s) of farmland,tree,and shrub gradually increased with soil depth;however,the K_(s) of grassland showed no significant variation in terms of vertical distribution.The mean K_(s) values of different land use patterns were ranked as follows:shrub>farmland>tree>grassland;all land use patterns showed moderate levels of variability.The K_(s) for different land use patterns exhibited differing degrees of correlation with soil physical and chemical properties;of these,clay,silt,sand,bulk density,and organic matter,were identified as important variables for predicting K_(s) in farmland,tree,shrub,and grassland,respectively.Recommendations and perspectives In this study,we used a stepwise multiple regression model to establish a transfer function prediction model for K_(s) for different land use patterns;this model possessed high estimation accuracy.The ability to predict K_(s) in the MUSL is very important in terms of the conservation of water and nutrients.

Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.

Wigner Function of Density Operator for Negative Binomial Distribution

By using the technique of integration within an ordered product （IWOP） of operator we derive Wigner function of density operator for negative binomial distribution of radiation field in the mixed state case, then we derive the Wigner function of squeezed number state, which yields negative binomial distribution by virtue of the entangled state representation and the entangled Wigner operator.