Box 8-1 Quadratic Equations

For constant parameters k, C,in, kj2, the solution of the nonlinear differential Equation 21-24 is given in Box 21.5. The steady-state, Cim results from the solution of a quadratic equation. It is controlled by the characteristic concentration... 

With this change in chemical composition, a series of statistically designed experiments was performed. Variables of interest were identified as reaction time and the concentrations of rSLPI, cystine and BME. Using the Box-Behnken statistical design, a series of 27 experiments was done. Activity assays and reversed phase HPLC analyses generated data on yield of active rSLPI, relative purity, and relative levels of specific contaminants of interest. The results were modeled with quadratic equations by the X-Stat program and projections of maximum yield and purity, and minimum production of the contaminants, were obtain. ... 

Mathematically, the quadratic equation also always has a negative answer for X (which we are not interested in since it makes no chemical sense). That value of X can be obtained by constraining the answer to be less than zero in the Subject to the Constraints box. Click on Add. For Cell Reference, click on Cl. Adjust the arrow to move to <=. In the Constraint dialogue box, type 0. Then click OK. Click on Solve, and you see the answer x = —0.40565 (and formula = —9E-07). 

Box 8.1). This is called the secular determinant. Expanding the determinant gives a quadratic equation for in terms of /f2i, /f 11 and H22- The quadratic equatirm has... 

The curve up to the point where the last data point was taken appears to be an inverted parabola therefore, a quadratic polynomial will be chosen to represent the curve. From equations in the box on linear least squares,... 

The technique used to measure absorptive properties of fluff pulp systems containing different super absorbents was to perform multivariate experiments (of the Box Wilson type [1]) and fit a quadratic regression equation in the parameters studied. This procedure reduces the danger of drawing conclusions that are valid for only a narrow set of conditions which can occur when univariate experimental methods are used. 

A quantum mechanical treatment of translation corresponds to a particle experiencing a zero potential as it translates but being constrained to be within some volume. It is the treatment of a particle in a box. We have found that for a one-dimensional box of length I, the quantum mechanical energy expression. Equation 8.22, is quadratic in the quantum number n. 

Figure 2.9 illustrates the approximate dependence of the energy on the wave vector. The picture is very similar to the parabolic form of a free electron (see Eq. (2.23)) however, there are deviations (see the thick lines) as a result of the obstacles we have inserted (a,2a,3a etc.) . We remember that the Schrodinger equation is a wave equation. We expect diffraction effects at the relevant positions in the reciprocal space (k space) marked in Fig. 2.9 . In the case of a small box, it is true that e quadratically depends on a, but there are only a few discrete points. For a large box the function becomes continuous. Since we imagine our periodic soUd as composed (cf. Fig. 2.2) of small boxes forming a large box , we expect a behaviour according to Fig. 2.9.

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