Uniform approximation of systems: A Banach space approach

1973 ◽  
Vol 12 (2) ◽  
pp. 203-217 ◽  
Author(s):  
M. Milanese ◽  
A. Negro
Keyword(s):  
Banach Space ◽  
Uniform Approximation ◽  
Space Approach

Stochastic differential equations

1955 ◽  
Vol 51 (4) ◽  
pp. 663-677 ◽  
Author(s):  
D. A. Edwards ◽  
J. E. Moyal
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Differential Equations ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Random Variables ◽  
Finite Variance ◽  
Mean Square ◽  
General Banach Space ◽  
Space Approach

The work of which this paper is an account began as a study of differential equations for functions whose values are random variables of finite variance. It was intended that all questions of convergence should be treated from the standpoint of strong convergence in Hilbert space—familiar to probabilists from the writings of Karhunen(11) and Loève(13) as mean-square convergence. The more general Banach-space approach now adopted was made possible by the discovery of a theorem (Theorem 1 of this paper) which Mr D. G. Kendall, its apparent author, kindly communicated to us.

Computer Assisted Image Reconstruction of Oligomer Structure

1976 ◽  
Vol 34 ◽  
pp. 472-473
Author(s):  
A.M. Jones ◽  
A. Max Fiskin
Keyword(s):  
Three Dimensional ◽  
Depth Of Field ◽  
Computer Assisted ◽  
Projection Data ◽  
Two Dimensional ◽  
Single Particles ◽  
Dimensional Reconstruction ◽  
Computer Assisted Image ◽  
The Fourier Transform ◽  
Space Approach

If the tilt of a specimen can be varied either by the strategy of observing identical particles orientated randomly or by use of a eucentric goniometer stage, three dimensional reconstruction procedures are available (l). If the specimens, such as small protein aggregates, lack periodicity, direct space methods compete favorably in ease of implementation with reconstruction by the Fourier (transform) space approach (2). Regardless of method, reconstruction is possible because useful specimen thicknesses are always much less than the depth of field in an electron microscope. Thus electron images record the amount of stain in columns of the object normal to the recording plates. For single particles, practical considerations dictate that the specimen be tilted precisely about a single axis. In so doing a reconstructed image is achieved serially from two-dimensional sections which in turn are generated by a series of back-to-front lines of projection data.

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