Algebraic operations, hierarchy

The linear operators on a vector space forms themselves a vector space, called operator space. In this context, the original vector space is called the carrier space for the operators. The operator space is sometimes normed, but usually not. Since operator products are defined, we have here a vector space where a product of vectors to give a vector is defined. Such a vector space is also called a linear algebra. Operations and functions can be defined in the operator space thus we can define superoperators for which the operator space is the carrier space. The hierarchy is not usually driven any further. Functions are usually named in analogy to their analytical counterparts. To be specific, assume that A has a spectral resolution... 

The left hand side (x) may contain any number of clauses combined by Boolean algebra operators. In all but the simplest expert systems it is possible to include the expert s confidence in the conclusion. A frame describes hierarchical dependencies between objects. In a frame, the upper level (or parent) object passes attributes to the objects beneath it in the hierarchy (children). In other words, children inherit attributes from their parents. For example ... 

Parentheses can be used (and should be used liberally) to reflect the exact hierarchy of operations. The operational hierarchy is the same as for standard algebraic expressions as follows ... 

Apply these rules according to the algebraic hierarchy of calculations (consider symbols of enclosure first, then multiplication/division operations and lastly addition/subtraction operations). 

Parentheses are used in the usual algebraic fashion to prevent errors caused by the hierarchy of arithmetic operations (multiplication or division is performed before addition or subtraction, for example). 

A basic requirement for the correct calculation of arithmetic and algebraic expressions is the adherence to the correct hierarchy, or order, of operations. Table 2.2 shows that parentheses have the... 

The beauty of this simple concept lies in the fact that an R-matrix may be applied to any ensemble of molecules (B) to show the products (E) characteristic of the reaction (R). Therefore, the basic irreducible R-matrices (the reaction core ) constitute the highest level of a hierarchy. These R-matrices are also equivalent to mathematical formulations of Arens operators placed on the matrices of the cyclic atoms, in order to be applied to the computer using matrix algebra. 

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