Projection properties

Recalling the demands on the polymer architecture of a polymer brush and the projected properties in terms of swelling, wetting and friction, as described in the theoretical work, the brush has to consist of linear polymer chains of the same length at high grafting densities. The closest approximation to this can be obtained by the living anionic SIP (LASIP). The experimental difficulties outlined mean that only relatively few examples of LASIP are documented in the literature. 

Projection Properties of Factorial Designs for Factor Screening... 

When a regular design is used, any two factorial effects (main effects or interactions) are either orthogonal or are completely aliased. Partial aliasing of factorial effects makes the alias structures of nonregular designs much more complex (see Chapter 8 for information on the difficulties in the analysis when complex aliasing is present), but it also leads to some superior projection properties. 

Projection properties of Plackett-Burman designs were studied by Lin and Draper (1991, 1992). Their computer searches examined all of the projections of small Plackett-Burman designs onto a few factors. For example, each of the 165 projections of the 12-run design of Table 2 onto three factors consists of... 

Again, a 12-run Plackett-Burman design has the four-factor hidden projection property stated in Theorem 2 because it has no defining words of length three or four. On the other hand, it is not possible for a regular design to have the same property unless it is of resolution five. 

What happens if n is a multiple of 8 Cheng (1995) showed that, if the widely held conjecture is true that a Hadamard matrix exists for every order that is a multiple of 4, then for every n that is a multiple of 8, one can always construct an OA(n, 2, 2) that has defining words of length three or four. Such designs are not of projectivity 3, a fact also pointed out by Box and Tyssedal (1996), and do not have the desirable hidden projection properties mentioned above. However, this does not mean that, when n is a multiple of 8, there are no OA( ,. 2)s with good... 

Projection properties of designs with more than two levels have been relatively unexplored. Cheng and Wu (2001) found a nonregular OA(27, 38, 2) such that the quadratic model can be estimated in all four-factor projections. For/factors, each with three quantitative levels, the quadratic model specifies the mean response at factor level combination (xi,..., Xf) to be... 

Bulutoglu, D. A. and Cheng, C. S. (2003). Hidden projection properties of some nonregular fractional factorial designs and their applications. Annals of Statistics, 31, 1012— 1026. 

Chen, H. (1998). Some projective properties of fractional factorial designs. Statistics and Probability Letters, 40, 185-188. 

Cheng, C. S. (1998a). Hidden projection properties of orthogonal arrays with strength three. 

Lin, D. K. J. and Draper, N. R. (1991). Projection properties of Plackett and Burman designs. Technical Report 885, Department of Statistics, University of Wisconsin. 

Wang, J. C. and Wu, C. F. J. (1995). A hidden projection property of Plackett-Burman and related designs. Statistica Sinica, 5, 235-250. 

The critical need for chemical and physical property data is apparent to anyone who has worked on process-design projects. Property storage or predic-... 

Silvestri, A. J., Prater, C. D., and Wei, J., On the structure and analysis of complex systems of first order chemical reactions containing irreversible steps. H, Projection properties of the characteristic vectors. Chem. Eng. Set 23, 1191 (1968). 

---