Complexity of object

To support understanding of an interactive complexity of object-oriented systems, the OF-FMEA method was developed (compare Section 4 of this presentation). 

It is sometimes helpful to distinguish between the complexity of objects and that of processes. If one seeks to compare different objects to one another in terms of their complexity that is object complexity theory. The objects might be molecules, people, social structures, poems, toxins, etc... An enormous literature exists with much of the chemical focus being on molecular complexity indices that attempt to order molecular graphs according to some measure of their complexity. While not yet directly concerned with complexity, much environmental focus has been on the relative toxicity of pollutants partially ordered by their effects on different things. 

For simulation the whole object can be presented as a complex of Dirichlet cells in three-dimensional cylindrical coordinates (R - tp - Z - geometry) [2] (Fig.2). 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

The reason why complexity and symmetry are linked together is quite straightforward. Indeed, a representation of highly symmetrical systems requires fewer characteristics than that of objects having low symmetry because, if we know the characteristics of one object, we can employ them to represent all those which are symmetrical with the given one. 

SONNIA can be employed for the classification and clustering of objects, the projection of data from high-dimensional spaces into two-dimensional planes, the perception of similarities, the modeling and prediction of complex relationships, and the subsequent visualization of the underlying data such as chemical structures or reactions which greatly facilitates the investigation of chemical data. 

The level of effort required for a frequency analysis is a function of the complexity of the system or process being analyzed and the level of detail required to meet the analysis objectives. Frequency analysis can typically require 25% to 50% of the total effort in a large-scale QRA study. If an uncertainty analysis is performed, the effort required for the frequency analysis can be much greater. 

On the basis of the values of AS° derived in this way it appears that the chelate effect is usually due to more favourable entropy changes associated with ring formation. However, the objection can be made that and /3l-l as just defined have different dimensions and so are not directly comparable. It has been suggested that to surmount this objection concentrations should be expressed in the dimensionless unit mole fraction instead of the more usual mol dm. Since the concentration of pure water at 25°C is approximately 55.5 moldm , the value of concentration expressed in mole fractions = cone in moldm /55.5 Thus, while is thereby increased by the factor (55.5), /3l-l is increased by the factor (55.5) so that the derived values of AG° and AS° will be quite different. The effect of this change in units is shown in Table 19.1 for the Cd complexes of L = methylamine and L-L = ethylenediamine. It appears that the entropy advantage of the chelate, and with it the chelate effect itself, virtually disappears when mole fractions replace moldm . ... 

Pt electrodeposits may also be produced from molten salt electrolytes. Such a high-temperature process has the advantage that the deposits are diffusion bonded to the titanium substrate and thus have good adhesion, and, if necessary, thick deposits can be produced. However, they have the disadvantage that because of the complexity of the process there is a limitation on the size and shape of the object to be plated, and the resultant deposits are softer and less wear resistant than those from aqueous solutions... 

Whereas pattern (b) is intuitively the most complex of the three patterns, it has neither the highest entropy (which belongs to pattern (c)) or the lowest (which belongs to pattern (a)). Indeed, were we to plot our intuitive sense of complexity as a function of the amount of order or disorder in a system, it would probably look something like that shown in figure 12.2. The problem is to find an objective measure of the complexity of a system that matches this intuition. 

Graph Complexity Static Captures intuitive feel of the complexity of a graph Not a-priori clear how to extend to more general objects... 

Given an object com[)osed of N interconnected and interacting [)arts, one might at first be te,m[)ted to equate the complexity of an object with its conventional information content, as defined l)y Shannon [.shann49] ... 

Another drawback to using Shannon information as a measure of complexity is the fact that it is based on an ensemble of all possible states of a system and therefore cannot describe the information content of a single state. Shannon information thus resembles traditional statistical mechanics - which describes the average or aggregate behavior of, say, a gas, rather than the motion of its constituent molecules - more so than it docs a complexity theory that must address the complexity of individual objects. 

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