scholarly journals Saddle point solutions for non-local elliptic operators

2016 ◽  
Vol 44 (2) ◽  
pp. 527 ◽  
Author(s):  
Alessio Fiscella
Keyword(s):  
Saddle Point ◽  
Elliptic Operators ◽  
Non Local

scholarly journals A Resonance Problem for Non-Local Elliptic Operators

2013 ◽  
Vol 32 (4) ◽  
pp. 411-431 ◽  
Author(s):  
Alessio Fiscella ◽  
Raffaella Servadei ◽  
Enrico Valdinoci
Keyword(s):  
Elliptic Operators ◽  
Resonance Problem ◽  
Non Local

scholarly journals Worldline theories with towers of internal states

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Steven Abel ◽  
Daniel Lewis
Keyword(s):  
String Theory ◽  
Saddle Point ◽  
Invariant Theory ◽  
Degrees Of Freedom ◽  
Geometric Interpretation ◽  
Modular Invariance ◽  
Modular Invariant ◽  
Internal States ◽  
Non Local ◽  
Internal Degrees Of Freedom

Abstract We study particle theories that have a tower of worldline internal degrees of freedom. Such a theory can arise when the worldsheet of closed strings is dimensionally reduced to a worldline, in which case the tower is infinite with regularly spaced masses. But our discussion is significantly more general than this, and there is scope to consider all kinds of internal degrees of freedom carried by the propagating particle. For example it is possible to consider towers corresponding to other geometries, or towers with no obvious geometric interpretation that still yield a modular invariant theory. Truncated towers generate non-local particle theories that share with string theory the property of having a Gross-Mende-like saddle point in their amplitudes. This provides a novel framework for constructing exotic theories which may have desirable properties such as finiteness and modular invariance.

Toward High-Resolution Low-Voltage SEM

1988 ◽  
Vol 46 ◽  
pp. 218-219
Author(s):  
Zhifeng Shao
Keyword(s):  
Low Voltage ◽  
Spherical Aberration ◽  
Chromatic Aberration ◽  
Limiting Factor ◽  
Operating Voltage ◽  
Probe Radius ◽  
Non Local ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

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