scholarly journals Approximations for the Electron Density in Meteor Trails

1958 ◽  
Vol 11 (4) ◽  
pp. 591 ◽  
Author(s):  
AA Weiss
Keyword(s):  
Exact Solution ◽  
Electron Density ◽  
Unit Length ◽  
Upper Atmosphere ◽  
Meteor Trails ◽  
Isothermal Atmosphere ◽  
Special Case ◽  
Meteor Particle

Herlofsen (1947) has shown that an exact solution of the equations governing the evaporation of a meteor particle during its flight in the upper atmosphere can be obtained in the special case of a spherical meteor in an isothermal atmosphere. In terms of the number n of meteor atoms evaporated in unit length of trail, the electron density a is . (1)

THE DISTRIBUTION OF IONIZATION ALONG UNDERDENSE METEOR TRAILS

1964 ◽  
Vol 42 (11) ◽  
pp. 2035-2047 ◽  
Author(s):  
D. W. Rice ◽  
P. A. Forsyth
Keyword(s):  
Upper Atmosphere ◽  
Sources Of Error ◽  
Signal Characteristics ◽  
Radio Signals ◽  
Meteor Trails

Attempts to use the decay of radio signals reflected from individual meteor trails to study the upper atmosphere have revealed a puzzling inconsistency in the signal behavior. An earlier paper pointed out that this inconsistency remained even when the previously postulated sources of error were eliminated. As a result, an irregularly ionized trail model was proposed and shown, by calculation of signal characteristics, to be capable of accounting for the observations. This paper presents results of a new experiment which permitted the determination of the ionization profiles as the meteor trails were formed. The predicted irregularities were found, even for trails which exhibited apparently "ideal" underdense signal characteristics.

Electron Density in Meteor Trails

Nature ◽  
1947 ◽  
Vol 160 (4072) ◽  
pp. 670-671 ◽  
Author(s):  
A. C. B. LOVELL
Keyword(s):  
Electron Density ◽  
Meteor Trails

On the asymptotic solution of the Orr–Sommerfeld equation by the method of multiple-scales

1968 ◽  
Vol 34 (1) ◽  
pp. 145-158 ◽  
Author(s):  
K. Kuen Tam
Keyword(s):  
Exact Solution ◽  
Velocity Profile ◽  
Asymptotic Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Order Equation ◽  
Zeroth Order ◽  
Independent Variables ◽  
Sommerfeld Equation ◽  
Special Case

The method of multiple-scales is used to obtain the asymptotic solution of the Orr–Sommerfeld equation. For the special case of a linear velocity profile, the solution so obtained agrees well with an approximation of the exact solution which is known. For the general case, transformations on both the dependent and independent variables are introduced to obtain a zeroth-order equation which differs from the inner equation studied so far. On the ground of the favourable comparison for the special case, the asymptotic solution constructed is expected to be uniformly valid.

scholarly journals Satisfying positivity requirement in the Beyond Complex Langevin approach

2018 ◽  
Vol 175 ◽  
pp. 11026 ◽  
Author(s):  
Adam Wyrzykowski ◽  
Błażej Ruba Ruba
Keyword(s):  
Exact Solution ◽  
Approximate Solutions ◽  
Groebner Basis ◽  
Matching Conditions ◽  
Quadratic Equations ◽  
Positive Distribution ◽  
Langevin Approach ◽  
Basis Method ◽  
Special Case ◽  
Complex Density

The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case.

scholarly journals An Approximation Theorem for Vector Equilibrium Problems under Bounded Rationality

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 45
Author(s):  
Wensheng Jia ◽  
Xiaoling Qiu ◽  
Dingtao Peng
Keyword(s):  
Exact Solution ◽  
Bounded Rationality ◽  
Equilibrium Problems ◽  
Approximation Theorem ◽  
Optimization Problems ◽  
Vector Equilibrium Problems ◽  
Special Cases ◽  
Full Rationality ◽  
Vector Equilibrium ◽  
Special Case

In this paper, our purpose is to investigate the vector equilibrium problem of whether the approximate solution representing bounded rationality can converge to the exact solution representing complete rationality. An approximation theorem is proved for vector equilibrium problems under some general assumptions. It is also shown that the bounded rationality is an approximate way to achieve the full rationality. As a special case, we obtain some corollaries for scalar equilibrium problems. Moreover, we obtain a generic convergence theorem of the solutions of strictly-quasi-monotone vector equilibrium problems according to Baire’s theorems. As applications, we investigate vector variational inequality problems, vector optimization problems and Nash equilibrium problems of multi-objective games as special cases.

A Theory of Lubrication With Turbulent Flow and Its Application to Slider Bearings

1960 ◽  
Vol 27 (1) ◽  
pp. 1-4 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Exact Solution ◽  
Reynolds Number ◽  
Turbulent Flow ◽  
Closed Form ◽  
Governing Equation ◽  
Reynolds Equation ◽  
Three Dimensions ◽  
Slider Bearing ◽  
Numerical Example ◽  
Special Case

The governing equation of turbulent lubrication in three dimensions, equivalent to the Reynolds equation of laminar lubrication, is derived. The problem of a slider bearing with no side leakage is then analyzed. An exact solution is found in closed form. Bearing characteristics are also established. It is found that the Reynolds number is an important parameter in the problem of turbulent lubrication. Furthermore, it is shown that the laminar lubrication may be considered as the special case of the present study. A numerical example is also included.

scholarly journals Free Axial Vibrations of Non-Uniform Rods

2011 ◽  
Vol 2-3 ◽  
pp. 882-889
Author(s):  
Shu Qi Guo ◽  
Shao Pu Yang
Keyword(s):  
Exact Solution ◽  
Material Properties ◽  
Taylor Expansion ◽  
Series Solution ◽  
Wkb Method ◽  
Kummer Functions ◽  
Axial Vibrations ◽  
Special Case

Free axial vibrations of non-uniform rods are investigated by a proposed method, which results in a series solution. In a special case, with the proposed method an exact solution with a concise form can be obtained, which imply four types of profiles with variation in geometry or material properties. However, the WKB (Wentzel-Kramers-Brillouin) method leads to a series solution, which is a Taylor expansion of the results of the pro-posed method. For the arbitrary non-uniform rods, the comparison indicates that the WKB method is simpler, but the convergent speed of the series solution resulting from the proposed method is faster than that of the WKB method, which is also validated numerically using an exact solution of a kind of non-uniform rods with Kummer functions.

scholarly journals Tracking Meteor Trails to Study the Mesosphere

Eos ◽  
2017 ◽  
Author(s):  
Emily Underwood
Keyword(s):  
Radar Data ◽  
Upper Atmosphere ◽  
Meteor Trails ◽  
New Phenomena

Twelve years of radar data reveal new phenomena in Earth’s upper atmosphere.

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