Determinants algebra

Thermoeconomic accounting using algebraically determined prices permitting comparisons of subsystems and their costing items as though they were relatively independent. 

The paralinear or log-det transformation corrects for nonstationarity (see Swofford et al 1996). In this method, which is applicable only to distance tree building, the numbers of raw substitutions of each type and in each direction are tallied for each sequence pair in a fom-by-fom matrix as shown in Figure 14.7. Each matrix has an algebraic determinant, the log of which becomes a factor in estimating sequence divergence, hence the name log-det. Pairwise comparisons of sequences having various and assorted patterns of base frequencies will yield a variety of matrix patterns, giving a variety of determinant values. Thus, each estimated pairwise distance will be affected by the determinant particular to each pair, which effectively... 

In the course of study, students should master material that is both simple and complex. Much of this involves familiarity with the set of mathematical tools repeatedly used throughout this book. The appendices provide ample reference to such a toolbox. These include matrix algebra, determinants, vector spaces, vector orthogonalization. secular equations, matrix diagonaUzation. 

Because this process requires a httle knowledge of linear algebra (determinants and matrices), a limited review is given in Appendix A (matrix algebra). Here, we introduce the concept through a simple example. 

As pointed out in Section XVII-8, agreement of a theoretical isotherm equation with data at one temperature is a necessary but quite insufficient test of the validity of the premises on which it was derived. Quite differently based models may yield equations that are experimentally indistinguishable and even algebraically identical. In the multilayer region, it turns out that in a number of cases the isotherm shape is relatively independent of the nature of the solid and that any equation fitting it can be used to obtain essentially the same relative surface areas for different solids, so that consistency of surface area determination does not provide a sensitive criterion either. 

Since exchanging two columns in a determinant changes its sign, simple algebra... 

Ac Che limic of Knudsen screaming Che flux relacions (5.25) determine Che fluxes explicitly in terms of partial pressure gradients, but the general flux relacions (5.4) are implicic in Che fluxes and cheir solution does not have an algebraically simple explicit form for an arbitrary number of components. It is therefore important to identify the few cases in which reasonably compact explicit solutions can be obtained. For a binary mixture, simultaneous solution of the two flux equations (5.4) is straightforward, and the result is important because most experimental work on flow and diffusion in porous media has been confined to pure substances or binary mixtures. The flux vectors are found to be given by... 

The algebra of determining these relations is straightforward and brief, since the consequently the coefficients just given... 

Using the expanded determinants from Problem 6, write explicit algebraic expressions for the three minimization parameters a, b, and c for a parabolic curve fit. 

Some systems can give quantitative results from known pieces of data complete with proper units. For example, these systems can take all the starting information and then determine a set of equations from the available list that can yield the desired result. The program could subsequently convert units or algebraically solve the equations if necessary. 

The example demonstrates that not all the B-numbers of equation 5 are linearly independent. A set of linearly independent B-numbers is said to be complete if every B-number of Dis a product of powers of the B-numbers of the set. To determine the number of elements in a complete set of B-numbers, it is only necessary to determine the number of linearly independent solutions of equation 13. The solution to the latter is well known and can be found in any text on matrix algebra (see, for example, (39) and (40)). Thus the following theorems can be stated. 

Internal Return Rate. Another rate criterion, the internal return rate (IRR) or discounted cash flow rate of return (DCERR), is a popular ranking criterion for profitabiUty. The IRR is the annual discounting rate that makes the algebraic sum of the discounted annual cash flows equal to zero or, more simply, it is the total return rate at the poiat of vanishing profitabiUty. This is determined iteratively. 

Here m < 5, n = 8, p > 3. Choose D, V, i, k, and as the primary variables. By examining the 5x5 matrix associated with those variables, we can see that its determinant is not zero, so the rank of the matrix is m = 5 thus, p = 3. These variables are thus a possible basis set. The dimensions of the other three variables h, p, and Cp must be defined in terms of the primary variables. This can be done by inspection, although linear algebra can be used, too. 

Values of thermal-expansion coefficients to be used in determining total displacement strains for computing the stress range are determined from Table 10-52 as the algebraic difference between the value at design maximum temperature and that at the design minimum temperature for the thermal cycle under analysis. 

Multiple-Effect Evaporators A number of approximate methods have been published for estimating performance and heating-surface requirements of a multiple-effect evaporator [Coates and Pressburg, Chem. Eng., 67(6), 157 (1960) Coates, Chem. Eng. Prog., 45, 25 (1949) and Ray and Carnahan, Trans. Am. Inst. Chem. Eng., 41, 253 (1945)]. However, because of the wide variety of methods of feeding and the added complication of feed heaters and condensate flash systems, the only certain way of determining performance is by detailed heat and material balances. Algebraic soluflons may be used, but if more than a few effects are involved, trial-and-error methods are usually quicker. These frequently involve trial-and-error within trial-and-error solutions. Usually, if condensate flash systems or feed heaters are involved, it is best to start at the first effect. The basic steps in the calculation are then as follows ... 

Aeeurate measurement of harmonic quantities and their cumulative effect is a complex subject, and is not possible to determine theoretically with the help of algebraic equations. But it can be easily measured with the help of a harmonic analyser. Cigre (1989) has made a comprehensive study of the likely harmonics and their amplitudes that may be present in a power system. These are briefly described in Table 23.1. 

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