Series representation summation

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

In the X-ray diffraction experiment the number of scattered X-ray beams (Bragg reflections) that must be recombined (summed) is large. These would have been focused by an X-ray lens if such a lens could have been devised. This summation is done mathematically, and there are several algebraic representations of waves that are convenient for this. Values for the amplitudes F(hkl) of the waves necessary are obtained from the intensities of the Bragg reflections. Values for the relative phases of these waves, however, are not obtained experimentally. Unfortunately, in the summation of a series of waves, the contributions of the relative phase angles are as important as, and generally more important than, the contributions of the amplitudes of the diffracted beams. 

The diagrammatic representation of the centroid density enhances one s ability to approximately evaluate the full perturbation series [3]. For example, one can focus on a class of diagrams with the same topological characteristics. The sum of such a class results in a compact analytical expression that includes infinite terms in the summation. A very useful technique in such cases is the renormalization of diagrams [57,58]. This procedure can be applied to the vertices to define the effective potential theory diagrammatically [3, 21-23] and, in doing so, an accurate approximation to the centroid density [3]. 

In the geometric representation of the composition of the membrane, the volume fraction of each component in each layer is described quantitatively by corresponding geometric parameters. In the electrochemical part of the model, each layer is treated as a set of two resistors and all sets, whose number equals the number of layers, are arranged in series forming an equivalent electrical circuit. Summation of the resistances of the layers expressed with appropriate equations leads to the final formula on specific conductivity of the membrane. Calculations based on the model require measurements of the conductivity of the membrane in contact with electrolyte solutions of different concentration. 

The data contained in a digitally recorded image is an ordered finite array of discrete values of intensity (grayscale). To manipulate this data, the continuous integrals defining the Fourier transform and convolution must be expressed as approximating summations. For a series of discrete samples jc( t) of a continuous function x(t), a representation in the frequency domain can be written as... 

In their work related to the Promolecular Atom Shell Approximation (PASA), Amat and Carbo-Dorca used atomic Gaussian ED functions that were fitted on 6-31IG atomic basis set results [35]. In the PASA approach that is considered in the present work, a promolecular ED distribution Pa is represented analytically as a weighted summation over the nat atomic ED distributions p, which are described in terms of series of three squared Is Gaussian functions fitted from atomic basis set representations [36] ... 

The summations above are over an infinite number of terms, but if we tnmcate the series to some finite order TV, we obtain an approximate Fonrier representation of the function ... 

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