An Elementary Example

These four operations constitute a group in the mathematical sense, and provide a convenient example with which to illustrate some definitions and terminology of group theory. The operations are called the elements of the group. Because there are four elements, this group is said to be of order four. 

If we imagine a drill soldier carrying out the sequence of operations implied by LR, the result is a return of the soldier to his original position. That is, the product of 

TABLE 13-1 Products of Elements in the Group of Four Commands 

The reciprocal, or inverse of an operation is that subsequent operation which returns the soldier to his original position. Hence, L is the reciprocal of R, which is expressed as LR = E,ot L = R. Examination of Table 13-1 shows that every column has E appearing once, which means that every element of our group has an inverse in the group. 

The associative law is obeyed if, in general, the sequence of operations A(BC) = (AB)C, where the parentheses enclose the pair of elements the product of which is to be evaluated first. We can test whether our elements satisfy this law by trying out all the combinations. FoiexwapleL(RA) = LL = Amd(LR)A = EA = A. One can quickly show that this group satisfies the associative law for all combinations of elements. 

---

Polynomial Equations Here is an elementary example that... 

An elementary example illustrates the role of symmetry in determining when a matrix element must be equal to zero. Consider the one-dimensional integral ... 

Now let us assume that the true object is zero for all subscript values less than 1. When we do the convolution, then, im must have a value of zero for m < 1 — L. When we perform the numerical convolution starting with m 1 — L, we compute zeros as we slide the spread function toward the region of finite ok. Finally, when the end of the spread function encounters the first nonzero ok, we obtain the first nonzero value of ik. We illustrate this particular sum of products with an elementary example in which L = 1 and m — 0 ... 

Let us illustrate the application of Boltzmann s formula for an elementary example isothermal mixing of ideal gases (Sidebar 5.10). For this purpose, consider a system of Na = Nb = 4 particles. For a particular partition n 4 — n of the four A-type particles between the VA and VB containers, the number of possible ways H of choosing n A-type particles and 4 — n B-type particles for the first container is given by the product of binomial coefficients... 

The direct fluorination of almost all categories of inorganic hydrides leads to fluorine-containing analogues of the hydride precursors. The fluorination of simple inorganic hydrides (47, 48) is an elementary example of such syntheses (see Fig. 9). 

Without definite examples to focus our thinking, it is easy to get entangled in a quasi-philosophical discussion of just what a mathematical model might be. To avoid this, I present a dogmatic statement on modeling and proceed to consider an elementary example, returning later to the philosophical caveats and more general considerations. 

An elementary example of diis process is the reaction of an organometallic reactant widi a ketone (or aldehyde) followed by dehydration of the resulting alcohol to die olefin. This is truly a sequential process in that the product alcohol is dehydrated in a second, independent reaction step. It suffers as a useful synthetic method because regioisomers are often formed hi die elimination step. 

Let us illustrate previous claim with an elementary example for p = 0. In Cartesian coordinates, let... 

By now we have set up the basic machinery which permits the principles of irreversible thermodynamics to be applied to problems of interest. We next illustrate the method of procedure by an elementary example. The same approach will be used in later sections, with appropriate variations on the basic theme. 

As the matrix product is an elementary example of a noncommutative product, it is very natural to look for n x n matrices in order to satisy the anticommutation relations (5). Here we show that matrices a and 0 obeying the anti-commutation relations have to be at least four-dimensional. 

This is an elementary example where the results of linear regression and analysis of variance could have been obtained by apparently simpler methods. Nevertheless, we shall see that this approach will prove necessary for more complex models and further development of the analysis of variance is the basis of statistical tests for analysis and testing of models against the data. 

To illustrate the salient features of the method, we first consider the following boundary value problem in an abstract form, and then later attempt an elementary example of diffusion and reaction in a slab of catalyst material. We shall assume there exists an operator of the type discussed in Chapter 2 (Section 2.5) so that in compact form, we can write... 

Using an elementary example, we have introduced the methodology and the conceptual framework for the finite integral transform, most notably the concept of the integral transform pairs. The function y is called the object and the product pair (y,K ) is called the integral transform of y or the image of y. 

We refer to the AR(1) model as an elementary example, because in this case one can think of the process Xt as the actual demand realizations. But Xt may include more information than the demand history. The need for such extension arises in various cases, such as (i) when customers provide early information about their prospective orders (ii) when planned promotions are known to one or more of the decision-makers in the supply chain or (iii) when a detailed point-of-sale data (e.g., at a store-level) are provided directly to suppliers. 

As an illustration on the use of this method, an elementary example follows. 

As an elementary example of processes of this type, let us discuss first the electrocyclic transformation of butadiene to cyclobutene. 

This is an elementary example of how the expression for entropy production can be used to obtain linear relations between thermodynamic forces and flows, which often turn out to be empirically discovered laws such as Ohm s law. In section 10.3 we shall see that similar consideration of entropy production due to diffusion leads to another empirically discovered law called the Pick s law of diffusion. Modern thermodynamics enables us incorporate many such phenomenological laws into one unified formalism. 

Different definitions of the heat current Jq give somewhat different expressions for a. An elementary example is when Jq is defined to be J then the flow associated with V(l/r) will be the heat current. Another form of cr arises when the force associated with the matter flow J is written as — V ijt instead of —V( X fc/r). By separating the gradient of x from the gradient of (l/T), it is straightforward to show that (15.5.12) can be rewritten as... 

AN ELEMENTARY EXAMPLE OF BIFURCATION AND SYMMETRY BREAKING Consider the equation... 

An elementary example is alforded by the momentum operator p = —ihdjdq, which is Hermitian on an appropriately defined class of functions (q) for these functions it is self-adjoint on -oo < q < +oo but this property is lost if either of the oo limits is replaced by any finite value a - see, for example, Thirring (1981). 

Opinion Formation - an Elementary Example of Semi-Quantitative Sociology... 

---