Mathematical concept algebraic operation

In this chapter, we discuss symbolic mathematical operations, including algebraic operations on real scalar variables, algebraic operations on real vector variables, and algebraic operations on complex scalar variables. We introduce the concept of a mathematical function and discuss trigonometric functions, logarithms and the exponential function. 

The algebra of sets, or set theory, includes such fundamental mathematical concepts as set cardinality and subsets, which are a part of the study of various levels of mathematics from arithmetic to calculus and beyond. The algebra of propositions (logic or propositional calculus) was developed to fecilitate the reasoning process by providing a way to symbolically represent statements and to perform calculations based on defined operations, properties, and truth tables. Logic is studied in philosophy, as well... 

Only a few basic mathematical skills are required for the study of general chemistry. But to concentrate your attention on the concepts of chemistry, you will find it necessary to have a firm grasp of these basic mathematical skills. In this appendix, we will review scientific (or exponential) notation, logarithms, simple algebraic operations, the solution of quadratic equations, and the plotting of straight-line graphs. 

The definition of a mathematical space begins with the set of objects X, Y, Z,. .. that occupy the space (an intrinsically empty space being a physically problematic concept). Among the simplest algebraic structures that can characterize such objects is that of a linear manifold, also called a linear vector space, affine space, etc. By definition, such a manifold has only two operations— addition (X + Y) and multiplication by a scalar (AX)— resulting in each case in another element of the manifold. These operations have the usual distributive,... 

In recent years the ready availability of computers to handle the algebraic calculations, and the widespread use of the methods of operations research, have made it much easier for the engineer to use more complicated mathematical correlations on which to base his predictions. This approach is, however, less fundamental than is a theory in chemistry or physics which links previously unrelated concepts. 

We noted that Equation 4.77 is very important in the nonisothermal operation of a CSTR. This algebraic equation has more than one solution and leads to the concept of multiple steady states (MSS). On the other hand, the differential equation that characterizes a PER has only one solution, that is, the PER operates at a single steady state. Multiple steady states are of particular concern to us because they can occur in the physically realizable range of variables, between zero and infinity, and not at absurd values such as negative concentration or temperature (which would then be no more than a mathematical artifact). 

The transfer unit and theoretical plate concepts are quite similar. When equilibrium relationships and process operating lines can be reduced to simple mathematical forms, in fact, the number of transfer units and the number of theoretical plates can be related algebraically. 

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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