Spatially resolved average

Compared with that of common analyses which determine an average composition of the sample, the information amount of spatially resolved analyses increases by the factor of geometrical resolving power RA that may be in the concrete case... 

Unfortunately, the A-representability constraints from the orbital representation are not readily generalized to the spatial representation. A first clue that the A-representability problem is more complicated for the spatial basis is that while every A-representable Q-density can be written as a weighted average of Slater determinantal Q-densities in the orbital resolution (cf. Eq. (54)), this is clearly not true in the spatially resolved formulation. For example, the pair density (Q = 2) of any real electronic system will have a cusp where electrons of opposite spin coincide but a weighted average of Slater determinantal pair densities,... 

To elucidate the contribution of the different histological structures to the average Raman spectral signatures from intact pollen grains, we conducted spatially resolved measurements on sections of rye pollen grains. Such mapping experiments can help separation of predominant spectral contributions from different substructures. 

Third, there may be a concentration gradient of reactants and products along the length of the catalyst bed. If the structure of the catalyst depends upon the composition of the gas phase, then an average of the various structures will be measured. There is little discussion of this topic in the literature of XAFS spectroscopy of working catalysts. An extreme example of structural variations within a sample is discussed in Section 6, where there is a discussion of XAFS spatially resolved spectra recorded to allow direct observation of the axial distribution of phases present. If the XAFS data are not measured with spatial resolution, then it is recommended that XAFS data be measured under differential conversion conditions. However, if the aim of the experiment is to relate the catalyst structure directly to that in some industrial catalytic processes, then differential conversion conditions will only reflect the structure of the catalyst at the inlet of the bed. To learn about the structure of the catalyst near the outlet of the bed, the reaction has to be conducted at high conversions. If it is anticipated that this operation will lead to variations in the catalyst structure along the bed, then the feed to the micro-reactor should be one that mimics the concentration of reactants toward the downstream end of the bed (i.e., products should be added to the reactants). 

The techniques discussed so far give sufficiently high time resolutions but are spatially averaging. Since the discovery that a catalyst does not necessarily oscillate in synchrony over its entire surface, spatially resolving techniques have become increasingly important. The simplest spatially resolving... 

We call the choice Fa = Fioc a local control scheme in contrast to the global control scheme = Fgio which requires only the global average and does not depend explicitly on the spatial variable. The second option has considerable experimental advantages since the spatial average is related to the total charge in the quantum well and does not require a spatially resolved measurement. 

Goerke U and Kimmich R 1998 NMR-imaging techniques for quantitative characterization of periodic motions incoherent averaging and spectral side band analysis Spatially Resolved Magnetic Resonance, Proc. 4th Int. Conf. on Magnetic Resonance Microscopy and Macroscopy ed P Blumler, B Blumich, R E Botto and E Fukushima (Weinheim Wiley-VCH) pp 499-506... 

How then does this influence the flow of catalyst particles in the vessel and the liquid -solid mass transfer Figure 2c shows the particle velocities, for a particle density of 1,500 kg/m under identical conditions to Figure 2b. By subtraction of the temporally averaged liquid and particle velocities, the spatially resolved slip velocities can be obtained. Figure 3. This shows that the relative velocity of the liquid and particle (and hence the mass transfer) also varies considerably over the vessel, with the highest values confined to the impeller discharge region. 

In this work we employed near-field scanning optical microscopy (NSOM) to study spatially resolved optical properties of a monolayer of silver nanoparticles and their effect on the fluorescence signal of Rh6G dye and fluorescently labeled polyelectrolyte PAH-F1TC. Results of this work indicate that up to 30 times fluorescence enhancement can be achieved in small clusters with average lateral dimension between 100 and 150 nm depending on the excitation wavelength. 

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