Two-dimensional input

The two-dimensional input of chemical structures into the computer may be achieved by means of specialized typing machines, of visual display units (VDU), and of connectivity tables. 

In the system DARC [228], a special input device (called topocodeur) is used. In computer-assisted organic synthesis, the most widely used input/output method is by means of VDUs [229—232]. This type of input/output is the most familiar to a chemist, since molecules are drawn on the VDU in their usual shape. It is especially interesting for ring compounds and allows us also to deal with the stereochemistry by means of the usual symbols. Although the input via a VDU requires sophisticated hardware, it will probably be used more and more. 

A connectivity table or bond-electron matrix is a matrix the elements of which indicate the nature of the bonds between the atoms and the number of free electrons on each atom. An off-diagonal entry atj in the /th row and y th column is the formal covalent bond order between the ith and y th atoms. The ith diagonal entry is the number of free valence electrons which belong to the ith atom. Reactions can also be characterized by matrices deduced from the connectivity tables of the reactants and products (see, for example, ref. 233). 

As a compound is represented by a square matrix having a number of rows and columns equal to its number of atoms, it is clear that such a representation cannot be used for input purposes, but is rather an internal representation of structures and reactions. Furthermore, as the numbering of the atoms in a chemical compound is a priori arbitrary, several connectivity tables can be deduced from a given compound. Thus, special algorithms are required to obtain a canonical connectivity table. For example, the Chemical Abstracts Services have recourse to an algorithm devised by Morgan [234]. 

A reaction mechanism is a set of elementary processes. In most programs, an elementary process is described by a list of the following type Reaction label/Reactant 1/Reactant 2/Product 1/Product 2/Pre-expo-nential factor/Activation energy. For example, C6me and co-workers [182] describe the initiation process of the neopentane pyrolysis as 

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The performance of a neural network is explored here for classification of two-dimensional input data by means of a feedforward neural network. The simulated data are known from Example 5.12 and describe 200 cases from four classes (Figure 8.16). 

Figure 8.15 Artificial neural network for classification of four classes based on two-dimensional input data.

The Kohonen network or self-organizing map (SOM) was developed by Teuvo Kohonen [11]. It can be used to classify a set of input vectors according to their similarity. The result of such a network is usually a two-dimensional map. Thus, the Kohonen network is a method for projecting objects from a multidimensional space into a two-dimensional space. This projection keeps the topology of the multidimensional space, i.e., points which are close to one another in the multidimensional space are neighbors in the two-dimensional space as well. An advantage of this method is that the results of such a mapping can easily be visualized. 

In this illustration, a Kohonen network has a cubic structure where the neurons are columns arranged in a two-dimensional system, e.g., in a square of nx I neurons. The number of weights of each neuron corresponds to the dimension of the input data. If the input for the network is a set of m-dimensional vectors, the architecture of the network is x 1 x m-dimensional. Figure 9-18 plots the architecture of a Kohonen network. 

The two seven set fuzzy input windows shown in Figure 10.8 gives a possible 7x7 set of control rules of the form given in equation (10.21). It is convenient to tabulate the two-dimensional rulebase as shown in Figure 10.9. 

The integrals are over the full two-dimensional volume F. For the classical contribution to the free energy /3/d([p]) the Ramakrishnan-Yussouff functional has been used in the form recently introduced by Ebner et al. [314] which is known to reproduce accurately the phase diagram of the Lennard-Jones system in three dimensions. In the classical part of the free energy functional, as an input the Ornstein-Zernike direct correlation function for the hard disc fluid is required. For the DFT calculations reported, the accurate and convenient analytic form due to Rosenfeld [315] has been used for this quantity. 

One important direetion of study has been to use empirieal adsorption data, together with the preassumed model for loeal adsorption, and attempt to extraet information about the form of x(e) [13,14]. The ehoiee of the model for loeal adsorption, whieh is an important input here, has been eustomarily treated quite easually, assuming that it has rather limited influenee on the form and properties of the evaluated EADFs. Usually, one of so many existing equations developed for adsorption on uniform surfaees is used as the loeal adsorption isotherm. The most often used forms of 0 p, T,e) are the Langmuir [6] and the Fowler-Guggenheim [15] equations for loealized adsorption. Ross and Olivier [4] extensively used the equation for mobile adsorption, whieh results from the two-dimensional version of the van der Waals theory of fluids. The most radieal solution has been... 

Exact calculations have already been carried out for simple one and two dimensional Euclidean geometries by exploiting properties of polynomials (chapter 5.2.1) and circulant matrices (chapter 5.2.2) over the finite field J-[q, q p wherep is prime. We will here rely instead on the theory of input-free modular systems, which is more suitable for dealing with the dynamics of completely arbitrary lattices. 

Consider a simple perceptron with N continuous-valued inputs and one binary (— 1) output value. In section 10.5.2 we saw how, in general, an A -dimensional input space is separated by an (N — l)-dimensional hyperplane into two distinct regions. All of the points lying on one side of the hyperplane yield the output -)-l all the points on the other side of the hyperplane yield -1. 

The actual computation is broken down into two steps (1) A Collision Step, during which the value of a site is sent to a computation look-up table, with its input written to the display screen memory as in CAM-6. RAPl s screen memory consists of 16 256 x 512 planes, (2) A Propagation Step, during which the system is decomposed into a set of one-bit two-dimensional planes (one plane for each bit of each site), and the bits of each site are displaced to one of the site s neighbors by a translation of the entire plane. 

The HYDRUS computer model was developed by the Agricultural Research Service of the USDA to estimate water flow in unsaturated soils that support plant growth81 It was developed as a onedimensional model, and then modified to allow solution of two-dimensional problems.82 HYDRUS employs the Richards equation to solve water flow in unsaturated soil however, it uses different solution methods from those used in UNSAT-H. It also requires extensive data input. The available windows version simplifies data entry and model operation. 

The SOM displays intriguing behavior if the input data are drawn from a two-dimensional distribution and the SOM weights are interpreted as Cartesian coordinates so that the position of each node can be plotted in two dimensions. In Example 5, the sample pattern consisted of data points taken at random from within the range [x = 0 to 1, y = 0 to 1], In Figure 3.21, we show the development of that pattern in more detail from a different random starting point. 

A one-dimensional SOM is less effective at filling the space defined by input data that cover a two-dimensional space (Figure 3.22) and is rather vulnerable to entanglement, where the ribbon of nodes crosses itself. It does, however, make a reasonable attempt to cover the sample dataset. 

As we saw in the previous chapter, self-organizing maps (SOMs) are a powerful way to reveal the clustering of multidimensional samples. The two-dimensional SOM is often able to provide an informative separation of samples into classes and the learning in which it engages requires no input from the user, beyond the initial selection of parameters that define the scale of the mapping and the way that the algorithm operates. 

Figure 2 Readout pattern of a two-dimensional CCD. The bottom row corresponds to the shift register. Its output is connected to the preamplifier input. (Reproduced by permission, from the Book of Photon Tools, Oriel, Stafford, CT.)...

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