The reader is warned right from the start that the way how the distributions are approached in the present paper is different as compared to the style used in the works regarding the distributions theory, drawn-up by mathematicians3. This different approach was not chosen only for simply making all different, but due to other reasons:
The first from these reasons is the existence of a contradiction in the classic approach of the distributions, that is the rigor used within the differential and integral calculus for defining the notions such as derivative, differential, primitive, integral etc. which are valid only by mentioning the expression “for continuous functions only” along the continuous interval of their arguments, and the easy application of these notions also in case of the distributions with a clearly discontinuous character. Moreover, sometimes it is talking about a differentiation “in a classic, algebraic manner” and a differentiation “under the meaning of distributions”4
Another reason for the different approach of the distributions in the present paper comes from the specific organization of the information structure established by the objectual philosophy. According to this organization, the semantic information which is found in a message is mainly made-up from objects and processes (at which the objects are subjected to), the distributions being basic elements for defining these notions, as we are about to notice in the following chapters. Due to this reason, the definition of distributions had to be in compliance with this organization method.
Another reason which has determined the definition of distributions according to the selected mode is related to the purpose of their utilization. As I have already mentioned above, the distributions are in this paper a mathematical model used for representation of the objects (including the material ones), as entities which have a lot of properties (such as the shape, color, mass density, hardness, etc.), who are distributed on the object’s surface or in the volume of the object. These elements of the concrete distributions are the values of those attributes in a certain point (a certain space location) which belongs to the object; in other words, the value of the attribute depends through a certain relation, on the location, actual position of that point. We shall see next, if these dependence relations would be able to be independent from the actual location of the point, and, at least for certain zones of the object, we might use (only for those zones) the classic functions from the mathematical analysis. Unfortunately, for most of the real objects, the above mentioned dependence relations are not invariant, they can be even random, therefore, other mathematical instruments must be used for expressing these dependence relations, such as the distributions, which are more general than the functions.
The approach according to the objectual philosophy model of the distributions, clearly settles the conditions when the old differential and integral calculus can be used, and the conditions required for using the calculus with finite differences, which is more difficult and less elegant, but universally applicable. In the latter case, the notion of local derivative (such as it is defined within the differential calculus, as a derivative in a point), does not make sense anymore; it is replaced by a more general term which is called density, and the derived functions (any rank) are replaced by the derived distributions (any rank, too).
3 Such as, for example W. Kecs, P.P. Teodorescu - Introducere în teoria distribuțiilor cu aplicații în tehnică - Editura Tehnică, București 1975.
4 For example in the work of Emil Tocaci - Teoria câmpurilor, spațiul și energia - Editura Științifică și Enciclopedică, București 1984, with an entire chapter focused on distributions.
Copyright © 2006-2011 Aurel Rusu. All rights reserved.