An Extension of Attouch's Theorem and Its Application to Second-Order Epi-Differentiation of Convexly Composite Functions

1992 ◽  
Vol 332 (2) ◽  
pp. 861 ◽  
Author(s):  
Rene A. Poliquin
Keyword(s):  
Second Order ◽  
Composite Functions ◽  
Attouch’S Theorem

Convergence of Subdifferentials of Convexly Composite Functions

1999 ◽  
Vol 51 (2) ◽  
pp. 250-265 ◽  
Author(s):  
C. Combari ◽  
R. Poliquin ◽  
L. Thibault
Keyword(s):  
Banach Space ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Second Order ◽  
Constrained Optimization Problems ◽  
Composite Functions ◽  
Kuratowski Convergence ◽  
General Banach Space

AbstractIn this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlevé-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

Disappearance Voltage Determinations Using a Convergent Beam

1971 ◽  
Vol 29 ◽  
pp. 184-185
Author(s):  
W. L. Bell
Keyword(s):  
Second Order ◽  
Objective Method ◽  
Uniform Thickness ◽  
Rocking Curve ◽  
Crystal Perfection ◽  
Kikuchi Line ◽  
Central Maximum ◽  
Diffraction Patterns ◽  
Convergent Beam ◽  
Beam Technique

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.

Second order interference in two photon absorption

1996 ◽  
Vol 43 (9) ◽  
pp. 1765-1771 ◽  
Author(s):  
M. W. HAMILTON and D. S. ELLIOTT
Keyword(s):  
Second Order ◽  
Photon Absorption ◽  
Two Photon Absorption ◽  
Two Photon
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