Summation layer

The network architecture of a PNN (Figure 8.1) is similar to that of a GRNN, except that its summation layer has a neuron for each data class and the neurons sum all the pattern neurons output corresponding to members of that summation neuron s data class to obtain the estimated probability density function for that data class. The single neuron in the output layer then... 

A 3D summation tomographic image is generated and layers in any direction may be viewed. Image quality is improved by filters. The method has been proved satisfactory on a high radiographic contrast cylindrical object and on more complex items. 

Tomographic summation images of vertical, horizontal and cylindrical layers were generated by positioning the film in suitable positions. The method has a deficiency when the films have to be positioned close one to the other. Hence for the generation of close or overlapping cuts there is a need for repeated exposures. 

Now in principle each layer will have its own values of a, q, and v, and consequently the summation of Equation (2.11) cannot be carried out unless simplifying assumptions are made. Brunauer, Emmett and Teller made three such assumptions (a) that in all layers except the first the heat of adsorption is equal to the molar heat of condensation q, (b) that in all layers except the first the evaporation-condensation conditions are identical, i.e. that... 

Because each layer is macroscopically homogeneous in its own region of space, the integrals in Equation (7.30) further simplify to summations ... 

For a laminate of N equal-thickness layers (N > 2) with orientation angles differing by ic/N as in Figures 7-53d and 7-53e, the summation for... 

The 8 functions limit the non-vanishing regions of / -space to discrete layer planes perpendicular to k. These layer planes are infinitely sharp, because the helix was assumed to be infinitely long. Limiting the summation to a finite length of the helix would lead to broadening of these layer planes. 

Here the summation is over molecules k in the same smectic layer which are neighbours of i and 0 is the angle between the intermolecular vector (q—r ) projected onto the plane normal to the director and a reference axis. The weighting function w(rjk) is introduced to aid in the selection of the nearest neighbours used in the calculation of PsCq). For example w(rjk) might be unity for separations less than say 1.4 times the molecular width and zero for separations greater than 1.8 times the width with some interpolation between these two. The phase structure is then characterised via the bond orientational correlation function... 

Therefore, after summation the total field due to all masses in the layer is... 

If adsorption occurs in pores limiting the number of layers then the summation in equation (4.27) is limited to n and the BET equation takes the form... 

Under some circumstances there may be a reason to restrict adsorption to a finite number of layers, that is, assign some specific value to n. In general, however, n - oo as p - p0 is usually taken as the upper limit for this summation. 

In the case of a simple system considered throughout Sect. 2, it is already clear that both the faradaic charge transfer and the non-faradaic double-layer charging will contribute to the impedance of the electrode-solution interface. In addition to this, we have to account for the ohmic resistances between the connections to indicator and counter electrodes. This was already illustrated in Sect. 1.1, Fig. 1. The first conclusion is therefore that the total impedance can be written as the summation... 

Thus, the bed surface covered by the non-erodible particles is calculated by successive summation on all the size ranges of the non-erodible particles in the increasing order of diameters. For a Ay depth of the bed given, the surface covered by the non-erodible particles can be determinate since the number of layers, included by Ay, is known for each size range. The evolution of the surface, covered by the non-erodible particles, according to the depth can be established. The depth from which the covered surface is equal to the bed surface, namely the value of the erosion depth erosion, is deduced from this evolution. We suppose that, if there are not non-erodible particles in the bed, the erosion depth is equal to infinity. The erosion depth is then divided by the diameter Dp of each size range. The number of layers likely to be affected by erosion is given by ... 

The total energy of interaction between the particles in a lyophobic sol is obtained by summation of the electric double layer and van der Waals energies, as illustrated in Figure 8.2. 

The idea about the summation of the times of consecutive steps of the examined solid-state process is of primary importance for understanding the peculiarities of multiphase growth of compound layers in binary heterogeneous systems. Moreover, even in the case of formation of a single compound layer, this idea makes it possible to reveal a few aspects of reaction... 

In the general case of varying thickness of the ApBq layer, the summation of dl and dt e m makes it possible to match the diffusional flux of the atoms across its bulk with the flux of the same atoms combined at the corresponding interface into the chemical compound. Actually, this is used instead of the continuity equation of any kind, which can hardly be employed in the case under consideration. 

Summation of the right-hand parts of equations (1.7) and (1.23) yields the general kinetic equation describing the rate of growth of the ApBq layer between initial substances A and B due to the simultaneous occurrence of partial chemical reactions (1.1) and (1.2) ... 

To establish differential equations relating At to the increases, AxBi, AxA2, AyB2 and Ay, 13, in thicknesses of the ApBq and ArBs layers, use is again made of the postulate about the summation of the time of diffusion of the A or B atoms and the time of subsequent chemical transformations for each of four partial chemical reactions taking place at phase interfaces 1, 2 and 3. This yields... 

Summation of the right-hand parts of equations (4.12) and (4.14) yields a general kinetic equation describing the growth rate of the ArBs layer between the ApBq and B phases in the case where both components are diffusing in its bulk... 

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