References

APPLICATION OF ADJACENCY AND GENERALIZED INVERSE MATRICES TO ORDERING THE THREE OPTIMAL SEMI-LATIN SQUARES


[1] A. C. Atkinson and A. N. Donev, Optimum Experimental Designs, Oxford University Press, New York, 1992.

[2] R. A. Bailey, Semi-Latin squares, J. Statist. Plann. and Inf. 18 (1988), 299-312.

[3] R. A. Bailey, Efficient semi-Latin squares, Statistica Sinica 2 (1992), 413-437.

[4] R. A. Bailey and P. E. Chigbu, Enumeration of semi-Latin squares, Discrete Maths. 167/168 (1997), 73-84.

[5] R. A. Bailey and G. Royle, Optimal semi-Latin squares with side six and block size two, Pro. Roy. Soc. Ser. A 453 (1997), 1903-1914.

[6] P. J. Cameron, P. Dobcsányi, J. P. Morgan and L. H. Soicher, The external representation of block designs, Version 1.1 (2003).

http://designtheory.org /library/extrep/ext-rep.pdf

[7] P. E. Chigbu, Block Designs: Efficiency Factors and Optimality Criteria for Comparison, Linco Press Ltd., Enugu, Nigeria, 1998.

[8] P. E. Chigbu, Optimal semi-Latin squares for sixteen treatments in blocks of size four, J. Nig. Statist. Assoc. 13 (1999), 11-25.

[9] P. E. Chigbu, The best of the three optimal semi-Latin squares, The Indian J. Stats. 65(3) (2003), 641-648.

[10] P. E. Chigbu, A note on discriminating equally optimal semi-Latin squares for sixteen treatments in blocks of size four, J. Nig. Statist. Assoc. 17 (2004), 1-7.

[11] J. A. Fill and D. E. Fishkind, The Moore-Penrose generalized inverse for sums of matrices, SIAM J. Matrix Anal. and Appl. 21(2) (2000), 629-635.

[12] T. N. E. Greville, Some applications of the Pseudo inverse of a matrix, SIAM Review 2(1) (1960), 15-22.

[13] I. B. Onukogu, Foundations of Optimal Exploration of Response Surfaces, Ephrata Press, Nsukka, 1997.

[14] I. B. Onukogu and P. E. Chigbu (eds.), Super Convergent Line Series in Optimal Design of Experiment and Mathematical Programming, AP Express, Nsukka-Nigeria, 2002.

[15] L. Paterson, Circuits and efficiency in incomplete block designs, Biometrika 70 (1983), 215-225.

[16] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406-413.

[17] R. J. Plemmons and R. E. Cline, The generalized inverse of a nonnegative matrix, Proc. Amer. Mathcal. Soc. 31(1) (1972), 46-50.

[18] D. A. Preece and G. H. Freeman, Semi-Latin squares and related designs, J. Roy. Statist. Soc. Ser. B 45(2) (1983), 267-277.

[19] D. Raghavarao, Construction and Combinatorial Problems in Design of Experiments, Wiley, New York, 1971.

[20] M. A. Rakha, On the Moore-Penrose generalized inverse matrix, Applied Math. and Computation 158 (2004), 185-200.

[21] S. R. Searle, Matrix Algebra Useful for Statistics, John Wiley & Sons, Inc., Canada, 1982.

[22] N. P. Uto and P. E. Chigbu, On (M, S) and circuit properties of the three optimal semi-Latin squares, J. Math. Sci. 21(1) (2010a), 43-52.

[23] N. P. Uto and P. E. Chigbu, A near-regular graph design approach for ordering the three optimal semi-Latin squares, J. Nig. Stat. Assoc. 22 (2010b), 1-11.

[24] P. Wild, On circuits and optimality conjectures for block designs, J. Roy. Stat. Soc. Ser. B 49(1) (1987), 90-94.