A note on the Lévy distance

1975 ◽  
Vol 12 (02) ◽  
pp. 412-414
Author(s):  
J. W. Thompson
Keyword(s):  
Distribution Functions ◽  
Convergence In Distribution ◽  
Change Of Scale ◽  
Simple Bounds ◽  
Lévy Distance ◽  
Transformed Distribution ◽  
Levy Distance

The Lévy distance, L(F,G), between two distribution functions F and G has the important property that convergence of L(Fn,F) is equivalent to convergence in distribution. The fact that L(F,G) is not invariant under a change of scale has been thought to be a disadvantage. However, simple bounds on the Lévy distance between the transformed distribution functions can be found.

A note on the Lévy distance

1975 ◽  
Vol 12 (2) ◽  
pp. 412-414 ◽  
Author(s):  
J. W. Thompson
Keyword(s):  
Distribution Functions ◽  
Convergence In Distribution ◽  
Change Of Scale ◽  
Simple Bounds ◽  
Lévy Distance ◽  
Transformed Distribution ◽  
Levy Distance

The Lévy distance, L(F,G), between two distribution functions F and G has the important property that convergence of L(Fn,F) is equivalent to convergence in distribution. The fact that L(F,G) is not invariant under a change of scale has been thought to be a disadvantage. However, simple bounds on the Lévy distance between the transformed distribution functions can be found.

scholarly journals Collisionless distribution functions for force-free current sheets: using a pressure transformation to lower the plasma beta

2018 ◽  
Vol 84 (3) ◽  
Author(s):  
F. Wilson ◽  
T. Neukirch ◽  
O. Allanson
Keyword(s):  
Distribution Function ◽  
Current Sheet ◽  
Plasma Phys ◽  
Distribution Functions ◽  
Hermite Functions ◽  
Slow Convergence ◽  
Nonlinear Force ◽  
Free Current ◽  
Transformed Distribution ◽  
General Method

So far, only one distribution function giving rise to a collisionless nonlinear force-free current sheet equilibrium allowing for a plasma beta less than one is known (Allansonet al.,Phys. Plasmas, vol. 22 (10), 2015, 102116; Allansonet al.,J. Plasma Phys., vol. 82 (3), 2016a, 905820306). This distribution function can only be expressed as an infinite series of Hermite functions with very slow convergence and this makes its practical use cumbersome. It is the purpose of this paper to present a general method that allows us to find distribution functions consisting of a finite number of terms (therefore easier to use in practice), but which still allow for current sheet equilibria that can, in principle, have an arbitrarily low plasma beta. The method involves using known solutions and transforming them into new solutions using transformations based on taking integer powers ($N$) of one component of the pressure tensor. The plasma beta of the current sheet corresponding to the transformed distribution functions can then, in principle, have values as low as$1/N$. We present the general form of the distribution functions for arbitrary$N$and then, as a specific example, discuss the case for$N=2$in detail.

Application of a Weierstrass Theorem to the Convergence of Moments

1969 ◽  
Vol 12 (1) ◽  
pp. 86-90
Author(s):  
L.K. Chan ◽  
E.R. Mead
Keyword(s):  
Undergraduate Student ◽  
Complex Variables ◽  
Distribution Functions ◽  
Convergence In Distribution ◽  
Weierstrass Theorem ◽  
Introductory Course ◽  
Convergence Of Moments

In this note we apply a well-known theorem due to Weierstrass to show that under certain conditions convergence in distribution of a sequence of distribution functions implies the convergence of moments.This note may be understood by an undergraduate student who has an introductory course of complex variables and a second course of statistics.

Computer processing of high-resolution images of periodic specimens

1986 ◽  
Vol 44 ◽  
pp. 2-5
Author(s):  
W. Chiu ◽  
M.F. Schmid ◽  
T.-W. Jeng
Keyword(s):  
High Resolution ◽  
Fourier Transforms ◽  
Complex Structure ◽  
Distribution Functions ◽  
Multiple Image ◽  
Cryo Electron Microscopy ◽  
Structure Factors ◽  
Unit Cells ◽  
High Resolution Images ◽  
Phase Probability Distribution

Cryo-electron microscopy has been developed to the point where one can image thin protein crystals to 3.5 Å resolution. In our study of the crotoxin complex crystal, we can confirm this structural resolution from optical diffractograms of the low dose images. To retrieve high resolution phases from images, we have to include as many unit cells as possible in order to detect the weak signals in the Fourier transforms of the image. Hayward and Stroud proposed to superimpose multiple image areas by combining phase probability distribution functions for each reflection. The reliability of their phase determination was evaluated in terms of a crystallographic “figure of merit”. Grant and co-workers used a different procedure to enhance the signals from multiple image areas by vector summation of the complex structure factors in reciprocal space.

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