Quadratic Closure

The following quadratic closure has been employed by Doi (1981), Lipscomb et al. (1988)  

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The linear closure approximations are exact for a completely random distribution of fiber orientation, while the quadratic closure approximations are exact for perfect uniaxial ahgnment of the fibers (Advani and Tucker 1987). 

Example 3.5. Startup of Simple Shear Flow Predictions for the Lipscomb-Denn Model, Eq. 3.102, and the Quadratic Closure Approximation... 

The Doi theory for rod-like molecules consists of two components. The first, calculating the rod orientation distribution and its evolution under external forces. The second, post calculating the stress tensor which is a function of the rod orientation. In both, the quadratic closure approximation is used. The rod orientation within the system is characterized by the deviatoric form of the orientation order parameter tensor (5), and is defined as ... 

The second-order rate constants for thiocyanate anation vs pH are shown in Fig. 1.13. The full line represents (1.216) with the values shown in scheme (1.217). This profile had been earlier recognized in the ring closure of the three analogous pH-related forms of Co(III)-edta to give Co(edta) in which the edta is completely coordinated.In the Co(lll) case the reactivities of the three forms are much closer. A plot of A [H+] -1-[H+] is a quadratic curve from which / ah2> ah be obtained. 

If J is a positive definite quadratic form with a closed extension, its closure J and hence the self-adjoint operator If belonging to J are also positive definite. 

The closure constraint has to be taken into account also in modelling results of mixture experiments. The closure means that the columns of the model matrix are linearly dependent making the matrix singular. One way to overcome this problem is to make the model using only N-1 variables, because we need to know only the values of N-1 variables, and the value of the N th variable is one minus the sum of the others. However, this may make the interpretation of the model coefficients quite difficult. Another alternative is to use the so-called Scheffe polynomials, i.e. polynomials without the intercept and the quadratic terms. It can be shown that Scheffe polynomials of N variables represent the same model as an ordinary polynomial of N-1 variables, naturally with different values for the polynomial coefficients. For example the quadratic polynomial of two... 

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