Mapping point

To calculate the profiles and the differential capacitance of the interface numerically we have to choose a differential equation solver. However, the usual packages require that the problem is posed on a finite interval rather than on a semi-infinite interval as in our problem. In principle, we can transform the semi-infinite interval into a finite one, but the price to pay is a loss of translational invariance of the equations and the point mapped from that at infinity is singular, which may pose a problem on the solver. Most of the solvers are designed for initial-value problems while in our case we deal with a boundary-value problem. To circumvent these inconveniences we follow a procedure strongly influenced by the Lie group description. 

The imaging detectors, whether for point mapping, line scanning, or array detection, can be coupled with different types of spectrometers. Instrument types are classified by wavelength selection modality into imaging Fourier transform (FT) and tunable filter (TF) spectrometers, both of which are presented below, and dispersive spectrometers. FT imaging systems are classical laboratory instruments while TF spectrometers are compact and robust systems for chemical imaging. 

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

The speed of data acquisition in point mapping protocols generally precludes the examination of large sample regions at high spatial and spectral resolution. In addition to the usual trading rules of FTIR spectroscopy, there are tradeoffs in data... 

The SNR analysis for a single-element detector in a point mapping configuration was adapted to analyse the SNR of a pixel in an imaging FPA detector as42... 

FTIR point-to-point mapping of cervical tissue... 

Suppose for illustration that A and B are rings of functions on closed sets in k", with k = k. The maximal ideals P in A then correspond to points x in the set. If PB 4 B, some maximal ideal of B contains P, and the corresponding point maps to x. Thus when A - B is flat, the extra condition involved in faithful flatness is precisely surjectivity on the closed sets. Condition (2) is the generalization of that to arbitrary rings. 

Figu re n.6 Schematic diagrams for (a) point mapping and (b) DuoScan mapping the same area at the same step size. The colored circles represent materials of interest, (a) Small circles represent laser focus (b) Small squares represent pixels. 

Fig. 4 Left Cartoon of the image of a dog as the stimulus stream is abstracted and modeled in the brain. Right Naive learning requires high point-to-point mapping to define a single object (dog). A folded knowledge surface can categorize and differentiate a class of objects...

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