scholarly journals A Geometrical Explanation of Stein Shrinkage

2012 ◽  
Vol 27 (1) ◽  
pp. 24-30 ◽  
Author(s):  
Lawrence D. Brown ◽  
Linda H. Zhao
Keyword(s):  
Stein Shrinkage ◽  
Geometrical Explanation

scholarly journals Caterpillar solutions in coupled pendula

1988 ◽  
Vol 8 (8) ◽  
pp. 153-174 ◽  
Keyword(s):  
Dynamical Systems ◽  
Periodic Potential ◽  
Gordon Equation ◽  
Natural Extension ◽  
Sine Gordon Equation ◽  
Parameter Region ◽  
Intrinsic Interest ◽  
Finite Dimensional ◽  
Associated Solutions ◽  
Geometrical Explanation

AbstractThe single pendulum is one of the fundamental model problems in the theory of dynamical systems; coupled pendula, or equivalently, two elastically coupled particles in a periodic potential on a line, are a natural extension of intrinsic interest. The system arises in various physical applications and it inherits some rudiments of the behaviour exhibited by its finite-dimensional parent, the sine-Gordon equation. Among these phenomena are the so-called caterpillar solutions, whose behaviour is reminiscent of solitons. These solutions turn out to have a transparent geometrical explanation. There is an interesting bifurcation picture associated with the system: the parameter region is broken up into the set of ‘pyramids’ parametrized by pairs of integers; these integers characterize the behaviour of the associated solutions.

Geometrical explanation of parabolas and resonance in electron diffraction

1989 ◽  
Vol 47 ◽  
pp. 498-499
Author(s):  
M. Gajdardziska-Josifovska ◽  
J. M. Cowley
Keyword(s):  
Focal Point ◽  
Specular Reflection ◽  
Incident Beam ◽  
Simple Geometry ◽  
Line Parallel ◽  
Surface Resonance ◽  
Reflection Electron Microscopy ◽  
Diffraction Patterns ◽  
Geometrical Explanation ◽  
Transmission And Reflection

Reflection electron microscopy (REM) relies on the surface resonance (channeling) conditions for enhancement of the intensity of the specular reflection from a flat surface of a single crystal. The two most frequently cited geometries for attaining surface resonance conditions are: i) tilting the incident beam such that the specular beam in the RHEED pattern falls on an intersection of a K-line parallel to the surface with some oblique K-line; ii) positioning the specular beam on an intersection of a K-Iine parallel to the surface with some of the surface resonance regions bound by parabolas. Parabolas are also observed in the transmission diffraction patterns and have been explained as Kikuchi envelopes. Recent studies indicated a similarity between the CBED transmission and reflection patterns. We will describe a simple geometry which can be used to interpret the above observations.A parabola is by definition a curve of equal distance from a point (called focus) and a line (called directrix; see Fig.1 ).Simple previously unnoticed facs are that the zone axis is a focal point of all the parabolas belonging to a given zone, and that the directrix of each parabola corresponds to a K-line.

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