At the beginning of this chapter we have mentioned that the processes are a class of special distributions which are characterized by the variation of an attribute, as a distributed amount. The type of the support attribute is another criterion for the processes classification. From this point of view, the processes which have the temporal attribute as their support are considered to be models for the real processes, and the ones with other support type (spatial, frequential etc.) are models for some of the abstract processes.
Comment 4.6.1: The assertion that the temporal support processes are models for the real processes must be accepted with tolerance by the reader, because it contains only a half-truth. The real processes, in which the material systems are involved may be actually, as we are about to see in the following chapters, spatio-temporal or frequency-temporal distributions. Otherwise speaking, the real situation is much more complex but in the scope of this chapter, a short simplified expression is allowed.
As we shall see in the chapter 9 which is focused on the abstract objects (objects with whom the IPS class can operate), there is also a class of processes in which the variations of the properties of some information support systems (ISS) are involved - these are the abstract processes. This kind of process occurs within ISS medium (which shall be minutely approached in chapter 8), either inside of an IPS or on the external ISS. In the purpose of this paragraph, we must mention that specific elementary processes also exist for the abstract processes and these are similar with the above-mentioned ones, being representable by means of vectors, but they can have both temporal, spatial or frequencial intervals as support ranges.
An example of such an abstract SEP is the one in which the attribute variation has a spatial support, when SEP represents the even state variation of an attribute between two points of different and invariant spatial location. This spatial distribution of the variation of an attribute is also representable by means of SEP (namely, by means of vectors), but those particular vectors are variations between two simultaneous states, but located in two different spatial positions. The density of this type of SEP with spatial support in case of a scalar primary distribution is known as the directional derivative of that distribution, according to the theory of vectorial fields, and the direction on which this derivative has maximum values represents the gradient direction of that distribution (see annex X.8). The gradient is a local S1 state of a scalar distribution with spatial support.
Comment 4.6.2: If the reader shall analyze the classic relation which defines the gradient from annex X.8 by comparing it with the definition relation of the first rank density of a multiple support distribution, he might find out the resemblance between the two abstract objects. In this way, it may be noticed that the gradient of a scalar spatial 3D distribution is a vector (namely a SEP) which represents the total variation of the attribute assigned between two points, that is the amount of the specific variations according to the three support coordinates. Obviously, the distance between the two points is selected so that the density of the specific distributions to be considered as even.
Within the same category of SEP distributions with spatial support, but with a much more complex definition, the curl and divergence of a vector field are also included. We shall not minutely review these kind of processes because they are depicted in the textbooks on the vector fields theory, the sole intention of this section is to include some of these abstract objects into their appropriate classes, according to the classification system of the objectual philosophy, and to point out once again the fact that any SEP may be represented by means of a vector.
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