Algebra factoring

Eq. (1.34) is the simplified version of the Fowler-Nordheim equation. In the original Fowler-Nordheim equation, there is an algebraic prefactor to the exponential factor. Experimentally, the exponential factor always dominates the functional dependence, leaving the existence and the specific form of the algebraic factor hardly distinguishable (see Fowler and Nordheim, 1928 Good and Muller, 1956). In practical units (work function cj> in eV, and the field intensity F in V/A), Eq. (1.34) becomes... 

At r—all the terms with r and r - are negligible. To take derivatives with respect to r, it is obvious that the derivative from the exponential factor of the Slater function, Eq. (6.2), is much larger than from the algebraic factor. Therefore, the Schrodinger equation implies... 

The spirit of the Slater atomic wavefunction is to use the term with the highest power r" to represent the entire algebraic factor in the hydrogenlike wavefunctions. This approach is particularly suitable for treating STM-related problems. In the processes pertinent to STM, only the values of atomic wavefunctions a few Angstroms away from the nucleus of the atom are relevant, not the values near the core. Within the same spirit, we can derive all the Slater functions from a single function... 

For a tridiagonal linear system, Gaussian elimination simplifies to a simple algebraic factorization followed by back-substitution. The time taken is linearly proportional to the number of equations. 

Constrained optimization versus unconstrained optimization. In the adaptive wavelet algorithm, it was possible to avoid using constraints which ensured orthogonality. This is due to some clever algebraic factorizations of the wavelet matrix for which much credit is due to [6]. However, one constraint which we have not discussed in very much... 

With increasing floe diameter (or in the case of biofilm processes increasing film thickness), the system becomes truly heterogeneous. Transport within and between different phases is significant, and differential equations must be formulated for the different situations. These complex cases often can be reduced to pseudohomogeneous ones by introducing an algebraic factor, and this is the concept of reaction rate efficiency (cf. Sect. 4.5). 

Here the brackets [ ] are known as commutator brackets, and the expression in the brackets, itself an operator, is called the commutator of the operators. It is important to note that the function fix, y) has not been algebraically factored out of the left-hand side of Equation (10-1). Equation (10-1) is an operator equation and signifies that when the operator Dy — Dy Dj ] acts on the fiinction/(x, y), it produces the number zero. 

It should be cautioned that the correctness of the factors involving z has not really been verified experimentally and that the algebraic forms involved are modelistic. Additional analyses have been made by Ruckenstein and Dadybur-jor[lll]. 

The algebraic form of the expression (9.24) for the enhancement factor is specific to the particular reaction rate expression we have considered, and corresponding results can easily be obtained for other reactions in binary mixtures, for example the irreversible cracking A—2B. ... 

At the limit of Knudsen diffusion control it is not reasonable to expect that any of the proposed approximation methods will perform well since, as we know, percentage variations in pressure are quite large. Nevertheless it is interesting to examine their results, which are shown in Figure 11 4 At this limit it is easy to check algebraically that equations (11.54) and (11.55) become the same, while (11.60) differs from the other two. Correspondingly the values of the effectiveness factor calculated using the approximation of Kehoe and Aris coincide with the results of Apecetche et al., and with the exact solution, ile Hite and Jackson s effectiveness factors differ substantially. 

This technique (also known as the Grout reduction or Cholesky factorization) is based on the transfonnation of the matrix of coefficients in a system of algebraic equations into the product of lower and upper triangular matrices as... 

Mathematically, two factors are independent if they do not appear in the same term in the algebraic equation describing the response surface. For example, factors A and B are independent when the response, R, is given as... 

This model then leads us through a thicket of statistical and algebraic detail to the satisfying conclusion that going from small solute molecules to polymeric solutes only requires the replacement of mole fractions with volume fractions within the logarithms. Note that the mole fraction weighting factors are unaffected. 

Algebraic Method for Concentrated Gases When the feed gas is concentrated, the absorption factor, which is defined in general as A = where K = y°/x, can vary throughout the tower owing... 

The Power Factor is obtained by tbe ratio of the algebraic sum of wattmeter rettdings to voh-atnpere readings. For a three-phase system ... 

When using a conversion factor, we treat the units just like algebraic quantities they are multiplied or canceled in the normal way. 

When using a conversion factor, we treat the units just like algebraic quantities they are multiplied or canceled in the normal way. Thus, the units in the denominator of the conversion factor cancel the units in the original data, leaving the units in the numerator of the conversion factor. 

Solution Repeat the calculations in Example 2.3 but now reduce At by a factor of 2 for each successive calculation rather than by the factor of 4 used in the examples. Calculate the corresponding changes in a(tmax) and denote these changes by A. Then A should decrease by a factor of 2 for each calculation of a(t ax)- (The reader interested in rigor will note that the error is halved and will do some algebra to prove that the A are halved as well.) If A was the change that just occurred, then we would expect the next change to be A/2, the one after that to be A/4, and so on. The total... 

The measurement uncertainty is transformed into a corresponding uncertainty of the final result due to algebraic distortions and weighting factors, even if the calculator s accuracy is irrelevant. 

Weighted Regression) requires the user to dehne a signal-dependent model of the measurement error, e.g., sy = a + b x, which is then used to calculate the weighting factors 1/Vy at every abscissa x,-. For an example on how to enter the model, see Algebraic Function, ... 

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