On the possible transition of the Gaussian model of an imperfect gas

1977 ◽  
Vol 357 (1690) ◽  
pp. 345-353 ◽  
Keyword(s):  
Gaussian Function ◽  
Gaussian Model ◽  
Radius Of Convergence ◽  
Virial Series ◽  
Imperfect Gas ◽  
Cyclomatic Number ◽  
Analytic Behaviour ◽  
Irreducible Graphs ◽  
Limiting Case ◽  
Third Derivative

It is known that the model of an imperfect gas for which the Mayer function is the Gaussian function - A exp ( - r 2 / a 2 ) of the distance r between two molecules is reducible to a problem in graph theory. It is shown that if the irreducible graphs occurring in the virial series are grouped according to their cyclomatic number we can expand the virial series in powers of the parameter A . The radius of convergence of the first few sub-series is the same. For the two-dimensional gas in the limiting case of A small, we find that, at a certain density, well outside the radius of convergence of the virial series, the third derivative of pressure vanishes. It is suggested that this may be the indication of a phase-transition of the model, and that its analytic behaviour is similar to that of lattice-type antiferromagnetic models.

Morphological Character and Statistical Property of Random Surface of ZnO Thin Film Prepared by RF Magnetron Sputtering

2010 ◽  
Vol 663-665 ◽  
pp. 1159-1162
Author(s):  
Ning Yu Zhang ◽  
Qing Song Huo ◽  
Li Xin Han ◽  
Gang Fu ◽  
Jun Qing Zhao ◽  
...  
Keyword(s):  
Thin Film ◽  
Magnetron Sputtering ◽  
Root Mean Square ◽  
Gaussian Function ◽  
Gaussian Model ◽  
Film Surface ◽  
Root Mean Square Roughness ◽  
Zno Thin Film ◽  
Mean Square ◽  
Auto Correlation

A method for characterizing the morphology property of ZnO film surface with Gaussian correlation is investigated. The parameters of root-mean-square roughness w and lateral correlation lengthξare introduced in Gaussian model to describe the correlation properties of the random film surfaces. In the experimental performance, ZnO thin films are grown on quartz glass and silicon substrates by the reactive radio-frequency magnetron sputtering method under different deposition pressure. The surface morphologies of the film surface are scanned by an atomic force microscopy. The height auto-correlation functions and root-mean-square roughness are obtained by using the numerical calculus method. Carried on the fitting with the Gaussian function to the height auto-correlation function data, the lateral correlation lengths are extracted to describe the statistical properties of ZnO thin film in mathematics with other parameters.

scholarly journals Improved single-molecule localization precision in astigmatism-based 3D superresolution imaging using weighted likelihood estimation

2018 ◽  
Author(s):  
Christopher H. Bohrer ◽  
Xinxing Yang ◽  
Zhixin Lyu ◽  
Shih-Chin Wang ◽  
Jie Xiao
Keyword(s):  
Single Molecule ◽  
Optical System ◽  
Gaussian Function ◽  
Gaussian Model ◽  
Likelihood Estimation ◽  
Three Dimensions ◽  
Weighted Likelihood ◽  
Superresolution Microscopy ◽  
Engineering Technique ◽  
Localization Precision

AbstractAstigmatism-based superresolution microscopy is widely used to determine the position of individual fluorescent emitters in three-dimensions (3D) with subdiffraction-limited resolutions. This point spread function (PSF) engineering technique utilizes a cylindrical lens to modify the shape of the PSF and break its symmetry above and below the focal plane. The resulting asymmetric PSFs at different z-positions for single emitters are fit with an elliptical 2D-Gaussian function to extract the widths along two principle x- and y-axes, which are then compared with a pre-measured calibration function to determine its z-position. While conceptually simple and easy to implement, in practice, distorted PSFs due to an imperfect optical system often compromise the localization precision; and it is laborious to optimize a multi-purpose optical system. Here we present a methodology that is independent of obtaining a perfect PSF and enhances the localization precision along the z-axis. By utilizing multiple calibration images of fluorescent beads at varying z-planes and characterizing experimentally measured background distributions, we numerically approximated the probability of observing a certain signal in a given pixel from a single emitter at a particular z-plane. We then used a weighted maximum likelihood estimator (WLE) to determine the 3D-position of the emitter. We demonstrate that this approach enhances z-axis localization precision in all conditions we tested, in particular when the PSFs deviate from a standard 2D Gaussian model.

GATE MODELING OF LATERAL SCATTERING OF PROTON PENCIL BEAMS

2020 ◽  
Vol 189 (1) ◽  
pp. 76-88
Author(s):  
Shiva Zarifi ◽  
Hadi Taleshi Ahangari ◽  
Sayyed Bijan Jia ◽  
Mohammad Ali Tajik-Mansoury ◽  
Milad Najafzadeh
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Energy Range ◽  
Ionization Potential ◽  
Gaussian Function ◽  
Gaussian Model ◽  
Absorbed Dose ◽  
Body Tissue ◽  
Simulation Code ◽  
Optimal Value

Abstract To validate the GATE Monte Carlo simulation code and to investigate the lateral scattering of proton pencil beams in the major body tissue elements in the therapeutic energy range. In this study, GATE Monte Carlo simulation code was used to compute absorbed dose and fluence of protons in a water cubic phantom for the clinical energy range. To apply the suitable physics model for simulation, different physics lists were investigated. The present research also investigated the optimal value of the water ionization potential as a simulation parameter. Thereafter, the lateral beam profile of proton pencil beams were simulated at different energies and depths in body tissue elements. The range results obtained using the QGSP_BIC_EMY physics showed the best compatibility with the NIST database data. Moreover, it was found that the 76 eV is the optimal value for the water ionization potential. In the next step, it was shown that the beam halo can be described by adding a supplementary Gaussian function to the standard single-Gaussian model, which currently is used by treatment planning systems (TPS).

AB INITIO ANALYTIC CUBIC FORCE CONSTANT IN GENERAL COORDINATE SPACE USING SPHERICAL GAUSSIAN FUNCTION

2002 ◽  
Vol 13 (03) ◽  
pp. 375-381
Author(s):  
M. SOLIMANNEJAD ◽  
A. H. PAKIARI
Keyword(s):  
Gaussian Function ◽  
Closed Shell ◽  
Computational Time ◽  
Coordinate Space ◽  
The Third ◽  
Analytic Third ◽  
Third Derivative ◽  
Shell Energy ◽  
Spherical Gaussian Function ◽  
Gaussian Orbital

The Floating Spherical Gaussian Orbital (FSGO) closed-shell energy analytic third derivative with respect to nuclear displacement is presented here. In order to save computational time, the formulas for the third derivative of electron nuclear attraction are written in terms of original integrals. Although the formulas were presented in general for any type of application, the FSGO method is applied to H 2 O and CH 2 molecules in singlet ground electronic state. Besides the accuracy of analytical derivative, the saving in computational time is significant.

scholarly journals An alternative form of the super-Gaussian wind turbine wake model

2020 ◽  
Vol 5 (3) ◽  
pp. 1225-1236 ◽  
Author(s):  
Frédéric Blondel ◽  
Marie Cathelain
Keyword(s):  
Wind Turbine ◽  
Shape Function ◽  
Gaussian Function ◽  
Gaussian Model ◽  
Alternative Form ◽  
Near Wake ◽  
Gaussian Shape ◽  
Wake Model ◽  
Turbine Wake ◽  
Wind Turbine Wake

Abstract. A new analytical wind turbine wake model, based on a super-Gaussian shape function, is presented. The super-Gaussian function evolves from a nearly top-hat shape in the near wake to a Gaussian shape in the far wake, which is consistent with observations and measurements of wind turbine wakes. Using such a shape function allows the recovery of the mass and momentum conservation that is violated when applying a near-wake regularization function to the expression of the maximum velocity deficit of the Gaussian wake model. After a brief introduction of the theoretical aspects, an easy-to-implement model with a limited number of parameters is derived. The super-Gaussian model predictions are compared to wind tunnel measurements, full-scale measurements, and a large-eddy simulation (LES), showing a good agreement and an improvement compared with predictions based on the Gaussian model.

scholarly journals An alternative form of the super-Gaussian wind turbine wake model

2020 ◽  
Author(s):  
Frédéric Blondel ◽  
Marie Cathelain
Keyword(s):  
Wind Turbine ◽  
Shape Function ◽  
Gaussian Function ◽  
Gaussian Model ◽  
Alternative Form ◽  
Near Wake ◽  
Gaussian Shape ◽  
Wake Model ◽  
Turbine Wake ◽  
Wind Turbine Wake

Abstract. A new analytical wind turbine wake model, based on a super-Gaussian shape function, is presented. The super-Gaussian function evolves from a nearly top-hat shape in the near wake to a Gaussian shape in the far wake, which is consistent with observations and measurements made on wind turbine wakes. Using such a shape function allows to recover the mass and momentum conservation that is violated when applying a near-wake regularization function to the expression of the maximum velocity deficit of the Gaussian wake model. After a brief introduction of the theoretical aspects, an easy-to-implement model with a limited number of parameters is derived. The super-Gaussian model predictions are compared to wind tunnel measurements, full-scale measurements and a LES simulation, showing a good agreement and an improvement compared with predictions based on the Gaussian model.

scholarly journals A Universal Model for Prediction of COVID-19 Pandemic Based on Machine Learning

2020 ◽  
Author(s):  
Lu Wang ◽  
De-wei Han ◽  
Kang Li ◽  
Xu Yang ◽  
Xiao-lei Yin ◽  
...  
Keyword(s):  
Gradient Descent ◽  
Gaussian Function ◽  
Gaussian Model ◽  
Real Data ◽  
Chi Square ◽  
Global Trends ◽  
Rate Of Spread ◽  
Gradient Descent Algorithm ◽  
Specific Factors ◽  
Country Specific

Abstract Background: With the current worldwide spreading of the coronary virus (COVID-19) pandemic, accurately predicting the rate of spread of the virus has become an urgent need. Methods: In this article we propose a universal COVID-19 prediction model that is independent of country-specific factors in this paper. By analyzing the pandemic data in China, we combined the advantages of Gaussian function with that of chi-square distribution function, to render an innovative mathematical model named the H-Gaussian with five parameters to be learned, and solved the parameters by a gradient descent algorithm. Results: We trained the model with partial historical pandemic data to predict subsequent pandemic trends in several regions, and validated the predictions with real data. The H-Gaussian model was experimentally shown to correctly predict the pandemic trends, and the parameters had good interpretability. Conclusions: On this basis, the global trends of the pandemic are given based on the data currently available, as well as suggestions for subsequent prevention strategies.

Freeze fracture imaging and computer simulation of liposome structure: Membrane geometry and defects

1983 ◽  
Vol 41 ◽  
pp. 646-647
Author(s):  
Joseph A. Zasadzinski
Keyword(s):  
Liquid Crystalline ◽  
Freeze Fracture ◽  
Continuum Theory ◽  
Closed Surfaces ◽  
Straight Line ◽  
Low Weight ◽  
Concentric Spheres ◽  
Membrane Geometry ◽  
Dupin Cyclides ◽  
Limiting Case

At low weight fractions, many surfactant and biological amphiphiles form dispersions of lamellar liquid crystalline liposomes in water. Amphiphile molecules tend to align themselves in parallel bilayers which are free to bend. Bilayers must form closed surfaces to separate hydrophobic and hydrophilic domains completely. Continuum theory of liquid crystals requires that the constant spacing of bilayer surfaces be maintained except at singularities of no more than line extent. Maxwell demonstrated that only two types of closed surfaces can satisfy this constraint: concentric spheres and Dupin cyclides. Dupin cyclides (Figure 1) are parallel closed surfaces which have a conjugate ellipse (r1) and hyperbola (r2) as singularities in the bilayer spacing. Any straight line drawn from a point on the ellipse to a point on the hyperbola is normal to every surface it intersects (broken lines in Figure 1). A simple example, and limiting case, is a family of concentric tori (Figure 1b).To distinguish between the allowable arrangements, freeze fracture TEM micrographs of representative biological (L-α phosphotidylcholine: L-α PC) and surfactant (sodium heptylnonyl benzenesulfonate: SHBS)liposomes are compared to mathematically derived sections of Dupin cyclides and concentric spheres.

Suppression of γ’ Precipitation in a Laser-Melted Nickel-Base Alloy

1977 ◽  
Vol 35 ◽  
pp. 270-271
Author(s):  
J. M. Walsh ◽  
J. C. Whittles ◽  
B. H. Kear ◽  
E. M. Breinan
Keyword(s):  
Cooling Rate ◽  
Base Alloy ◽  
Material Surface ◽  
Intimate Contact ◽  
Absorbed Power ◽  
Perfect Contact ◽  
Nickel Base ◽  
Rapidly Solidified ◽  
Limiting Case ◽  
The Heat Transfer Coefficient

Conventionally cast γ’ precipitation hardened nickel-base superalloys possess well-defined dendritic structures and normally exhibit pronounced segregation. Splat quenched, or rapidly solidified alloys, on the other hand, show little or no evidence for phase decomposition and markedly reduced segregation. In what follows, it is shown that comparable results have been obtained in superalloys processed by the LASERGLAZE™ method.In laser glazing, a sharply focused laser beam is traversed across the material surface at a rate that induces surface localized melting, while avoiding significant surface vaporization. Under these conditions, computations of the average cooling rate can be made with confidence, since intimate contact between the melt and the self-substrate ensures that the heat transfer coefficient is reproducibly constant (h=∞ for perfect contact) in contrast to the variable h characteristic of splat quenching. Results of such computations for pure nickel are presented in Fig. 1, which shows that there is a maximum cooling rate for a given absorbed power density, corresponding to the limiting case in which melt depth approaches zero.

Multidimensional, Nonthreshold Gaussian Model of Source Memory

2004 ◽  
Author(s):  
William P. Banks ◽  
Clayton Stephenson ◽  
Nisha Gottfredson ◽  
Andrew A. Sparrow
Keyword(s):  
Source Memory ◽  
Gaussian Model
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