Structure Analysis and Application of Taylor Expansion Theorem

2021 ◽  
Vol 11 (04) ◽  
pp. 663-668
Author(s):  
国忠 崔
Keyword(s):  
Structure Analysis ◽  
Taylor Expansion ◽  
Expansion Theorem

q-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials

2019 ◽  
Vol 17 (06) ◽  
pp. 853-895
Author(s):  
Mourad E. H. Ismail ◽  
Zeinab S. I. Mansour
Keyword(s):  
Taylor Expansion ◽  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Bernoulli Polynomials ◽  
Classical Case ◽  
The Other ◽  
Euler Polynomials ◽  
Expansion Theorem ◽  
Bernoulli Polynomials And Numbers ◽  
Euler Polynomials And Numbers

In this paper, we introduce a generalization of the [Formula: see text]-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a [Formula: see text]-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in [Formula: see text]-analogs of Lidstone polynomials which are in fact [Formula: see text]-Bernoulli polynomials as in the classical case. The definitions of these [Formula: see text]-Bernoulli polynomials and numbers are introduced. We also introduce [Formula: see text]-analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points [Formula: see text] and [Formula: see text] either in terms of the polynomials [Formula: see text] or in terms of the polynomials [Formula: see text]. As an application, we introduce a new series expansion for the basic hypergeometric series [Formula: see text].

200kv superconducting cryo electron microscope

1987 ◽  
Vol 45 ◽  
pp. 642-643
Author(s):  
M. Iwatsuki ◽  
Y. Kokubo ◽  
Y. Harada ◽  
J. Lehman
Keyword(s):  
Electron Microscope ◽  
Structure Analysis ◽  
Organic Materials ◽  
Strong Interactions ◽  
Specimen Preparation ◽  
Great Effort ◽  
Electron Microscopic ◽  
Crystal Fields ◽  
Special Skill ◽  
Long Time

In recent years, the electron microscope has been significantly improved in resolution and we can obtain routinely atomic-level high resolution images without any special skill. With this improvement, the structure analysis of organic materials has become one of the interesting targets in the biological and polymer crystal fields.Up to now, X-ray structure analysis has been mainly used for such materials. With this method, however, great effort and a long time are required for specimen preparation because of the need for larger crystals. This method can analyze average crystal structure but is insufficient for interpreting it on the atomic or molecular level. The electron microscopic method for organic materials has not only the advantage of specimen preparation but also the capability of providing various information from extremely small specimen regions, using strong interactions between electrons and the substance. On the other hand, however, this strong interaction has a big disadvantage in high radiation damage.

RETRIEVE – a system for XRPD phase and structure analysis

2009 ◽  
Vol 2009 (30) ◽  
pp. 209-214 ◽  
Author(s):  
P. S. Dubinin ◽  
I. S. Yakimov ◽  
O. E. Piksina ◽  
Y. I. Yakimov ◽  
A. N. Zaloga
Keyword(s):  
Structure Analysis
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