About difference electron densities and their properties

2017 ◽  
Vol 73 (6) ◽  
pp. 460-473 ◽  
Author(s):  
Maria Cristina Burla ◽  
Benedetta Carrozzini ◽  
Giovanni Luca Cascarano ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori
Keyword(s):  
Crystal Structure ◽  
Electron Density ◽  
Probability Distribution Function ◽  
Strong Support ◽  
Joint Probability ◽  
The Other ◽  
Joint Probability Distribution ◽  
Electron Densities ◽  
Structure Solution ◽  
Joint Probability Distribution Function

Difference electron densities do not play a central role in modern phase refinement approaches, essentially because of the explosive success of the EDM (electron-density modification) techniques, mainly based on observed electron-density syntheses. Difference densities however have been recently rediscovered in connection with theVLD(Vive la Difference) approach, because they are a strong support for strengthening EDM approaches and forab initiocrystal structure solution. In this paper the properties of the most documented difference electron densities, here denoted asF−Fp,mF−FpandmF−DFpsyntheses, are studied. In addition, a fourth new difference synthesis, here denoted as {\overline F_q} synthesis, is proposed. It comes from the study of the same joint probability distribution function from which theVLDapproach arose. The properties of the {\overline F_q} syntheses are studied and compared with those of the other three syntheses. The results suggest that the {\overline F_q} difference may be a useful tool for making modern phase refinement procedures more efficient.

scholarly journals Multiple-wavelength anomalous dispersion techniques applied to powder data: a probabilistic method for finding the substructureviajoint probability distribution functions

2008 ◽  
Vol 42 (1) ◽  
pp. 30-35 ◽  
Author(s):  
Angela Altomare ◽  
Benny Danilo Belviso ◽  
Maria Cristina Burla ◽  
Gaetano Campi ◽  
Corrado Cuocci ◽  
...  
Keyword(s):  
Probability Distribution ◽  
Probability Distribution Function ◽  
Probabilistic Method ◽  
Joint Probability ◽  
Direct Methods ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Probability Distribution Functions ◽  
Multiple Wavelength ◽  
Joint Probability Distribution Function

A new joint probability distribution function method is described to find the anomalous scatterer substructure from powder data. The method requires two wavelengths; the conclusive formulas provide estimates of the substructure structure factor moduli, from which the anomalous scatterer positions can be found by Patterson or direct methods. The theory has been preliminarily applied to two compounds, the first having Pt and the second having Fe as anomalous scatterer. Both substructures were correctly identified.

The probability distribution function of structure factors with non-integral indices. III. The joint probability distribution in the P1¯ case

1999 ◽  
Vol 55 (2) ◽  
pp. 322-331 ◽  
Author(s):  
Carmelo Giacovazzo ◽  
Dritan Siliqi ◽  
Angela Altomare ◽  
Giovanni Luca Cascarano ◽  
Rosanna Rizzi ◽  
...  
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Function Method ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Conditional Distributions ◽  
Structure Factors ◽  
Joint Probability Distribution Function ◽  
Atomic Parameters

The joint probability distribution function method has been developed in P1¯ for reflections with rational indices. The positional atomic parameters are considered to be the primitive random variables, uniformly distributed in the interval (0, 1), while the reflection indices are kept fixed. Owing to the rationality of the indices, distributions like P(F p 1 , F p 2 ) are found to be useful for phasing purposes, where p 1 and p 2 are any pair of vectorial indices. A variety of conditional distributions like P(|F p 1 | | |F p 2 |), P(|F p 1 | |F p 2 ), P(\varphi_{{\bf p}_1}|\,|F_{{\bf p}_1}|, F_{{\bf p}_2}) are derived, which are able to estimate the modulus and phase of F p 1 given the modulus and/or phase of F p 2 . The method has been generalized to handle the joint probability distribution of any set of structure factors, i.e. the distributions P(F 1, F 2,…, F n+1), P(|F 1| |F 2,…, F n+1) and P(\varphi1| |F|1, F 2,…, F_{n+1}) have been obtained. Some practical tests prove the efficiency of the method.

The joint probability distribution function of structure factors with rational indices. V. The estimates

1999 ◽  
Vol 55 (3) ◽  
pp. 525-532
Author(s):  
Carmelo Giacovazzo ◽  
Dritan Siliqi ◽  
Cristina Fernández-Castaño ◽  
Giuliana Comunale
Keyword(s):  
Probability Distribution Function ◽  
Joint Probability ◽  
Experimental Tests ◽  
Joint Probability Distribution ◽  
Integral Index ◽  
Acta Cryst ◽  
Reliability Factor ◽  
Practical Applications ◽  
Structure Factors ◽  
Joint Probability Distribution Function

The probabilistic formulae [Giacovazzo, Siliqi & Fernández-Castaño (1999). Acta Cryst. A55, 512–524] relating standard and half-integral index reflections are modified for practical applications. The experimental tests prove the reliability of the probabilistic relationships. The approach is further developed to explore whether the moduli of the half-integral index reflections can be evaluated in the absence of phase information; i.e. by exploiting the moduli of the standard reflections only. The final formulae indicate that estimates can be obtained, even though the reliability factor is a constant.

scholarly journals EVALUATION OF ELECTROMYOGRAPHIC NOISE STATISTICAL CHARACTERISTICS IN MULTICHANNEL ECG RECORDINGS

2019 ◽  
pp. 118-125
Author(s):  
Evgene B. Grigoriev ◽  
Alexander S. Krasichkov ◽  
Evgeny M. Nifontov
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Joint Probability ◽  
Bivariate Normal Distribution ◽  
Statistical Characteristics ◽  
Joint Probability Distribution ◽  
Electrocardiogram Signal ◽  
Ecg Leads ◽  
Joint Probability Distribution Function

Electromyographic noise is one of the most common noises in electrocardiogram. In case of several electrocardiogram leads, electromyographic noise affects each lead to different extent. It can be taken into account when developing algorithms for multilead electrocardiogram record processing. However, in the existing literature, there is no information about the relationship of electromyographic noise in various ECG leads and their joint probability distribution. The purpose of this paper is to study statistical characteristics of electromyographic noise in ECG signal, from which the electromyographic noise is extracted. The paper proposes a method for extracting electromyographic noise from electrocardiogram signal, based on a polynomial approximation of electrocardiogram signal fragments in sliding window with overlapping fragment subsequent weight averaging. Using this method, fragments of electromyographic noise are extracted from multichannel electrocardiogram records. Based on the obtained data, a joint probability distribution function of electromyographic noise in two adjacent leads is selected, and the correlation relationships between the electromyographic noise in various ECG leads are investigated. The results show that the joint probability distribution function of electromyographic noise in two adjacent leads in the first approximation can be described using bivariate normal distribution. In addition, between the samples of electromyographic noise from two adjacent leads quite strong correlation relationships can be observed.

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