Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-15T07:11:38.026Z Has data issue: false hasContentIssue false
  • English
  • Français

A Metrization for Power-Sets with Applications to Combinatorial Analysis

Published online by Cambridge University Press:  20 November 2018

Robert Silverman
Robert Silverman*
Affiliation:
Ohio State University and The National Bureau of Standards
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Combinatorial configurations may generally be phrased in terms of arrangements of objects into sets subject to certain conditions. In view of this, the question arises as to whether given a set S and its power-set Us (the class of all subsets of S), it might be possible to structure Us in a combinatorially significant manner. This paper proposes and investigates one such structuring achieved by defining a distance function over US.

Given A, B in Us, define their distance by

where N(E) denotes the number of elements in E, + ∞ being an admissible value.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 158 - 176
Copyright
Copyright © Canadian Mathematical Society 1960

References

References

1. Baer, Reinhold, Nets and groups, Trans. Amer. Math. Soc, 46 (1939), 110–141.Google Scholar
2. Rouse Ball, W. W., Mathematical Recreations and Essays (New York, 1947).Google Scholar
3. Bates, Grace E., Free loops and nets and their generalizations, Amer. J. Math., 69 (1947), 499–550.Google Scholar
4. Blumenthal, Leonard M., Distance geometries, University of Missouri Studies, vol. 13, no. 2 (1938).Google Scholar
5. Blumenthal, Leonard M. Theory and applications of distance geometry (New York, 1953).Google Scholar
6. Bose, R.C., On the application of the properties of Galois fields to the problem of construction of hyper-Graeco-Latin squares, Sankhya, 3 (1938), 323–338.Google Scholar
7. Bose, R.C. and Bush, K.A., Orthogonal arrays of strength two and three, Ann. Math. Stat., 23 (1952), 508–524.Google Scholar
8. Bush, K.A., Orthogonal arrays of index unity, Ann. Math. Stat., 23 (1952), 426–434.Google Scholar
9. Bruck, R.H., Finite nets, I. Numerical invariants, Can. J. Math., 3 (1951), 94–107.Google Scholar
10. Bruck, R.H. and Ryser, H.J., The nonexistence of certain finite projective planes, Can. J. Math., 1 (1949), 88–93.Google Scholar
11. Chowla, S. and Ryser, H.J., Combinatorial problems, Can. J. Math., 2 (1950), 93–99.Google Scholar
12. Euler, Leonard, Recherches sur une nouvelle espèce de quarrés magiques, Commentationes Algebraicae, Opera Omnia, series prima, 7 (1923).Google Scholar
12.1 Fisher, R.A. and Yates, F., Statistical tables for Biological, agricultural, and medical research (1st éd., Oliver and Boyd, 1938).Google Scholar
13. Hall, Marshall, Projective planes, Trans. Amer. Math. Soc, 54 (1943), 229–277.Google Scholar
14. Levi, F.W. Jr., Finite geometrical systems (University of Calcutta, 1942).Google Scholar
15. Lindenbaum, A., Contributions à Vétude de Vespèce métrique, I Fund. Math. 8 1926), 209–222.Google Scholar
16. MacNeish, H.F., Euler squares, Ann. Math., 23 (1921), 221–227.Google Scholar
17. Mann, H.B., On orthogonal Latin squares, Bull. Amer. Math. Soc, 50 (1944), 249–257.Google Scholar
18. Mann, H.B., Analysis and design of experiments (New York, 1949).Google Scholar
19. Norton, H.W., The 7X7 squares, Ann. Eugen., 9 (1939), 269–307.Google Scholar
20. Paley, R. E. A. C., On orthogonal matrices, J. Math, and Phys., 12 (1933), 311–320.Google Scholar
21. Rao, C.R., Hypercubes of strength d leading to confounded designs in factorial experiments, Bull. Calcutta Math. Soc, 38 (1946), 67–78.Google Scholar
22. Ryser, H.J., Geometries and incidence matrices, Amer. Math. Monthly, 62 (1955), 25–31.Google Scholar
23. Seiden, Esther, On the problem of construction of orthogonal arrays, Ann. Math. Stat., 25 (1954), 151–156.Google Scholar
24. Stevens, W.L., The completely orthogonalized Latin square, Ann. Eugen., 9 (1939), 82–93.Google Scholar
25. Tarry, G., Le problème de 36 officiers, Mathesis, 20 (1901).Google Scholar
26. Yates, F., Incomplete randomized blocks, Ann. Eugen., 7 (1936), 121–140.Google Scholar

Added in proof

27. Bose, R.C. and Shrikhande, S.S., On the falsity of Ruler's conjecture about the non-existence of two orthogonal Latin squares of order 4t + 2, Proc. N. A. S., 45 (1959), 734–737.Google Scholar
28. Bose, R.C. and Shrikhande, S.S., A note on a result in the theory of code construction, Inform, and Control, 2 (1959), 183–194.Google Scholar
29. Bose, R.C. and Shrikhande, S.S., On the falsity of Euler s conjecture for all orders exceeding 26, Amer. Math. Soc. Notices, 6 (1959), 379.Google Scholar
31. Parker, E.T., Construction of some sets of pairwise orthogonal Latin squares, Amer. Math. Soc. Notices, 5 (1958), 815.Google Scholar
30. Hamming, R.W., Error detecting and error correcting codes, Bell System Tech., 29 (1950), 147–160.Google Scholar
32. Parker, E.T., Orthogonal Latin squares, Proc. Nat. Acad. Sci., 45 (1959), 859–862.Google Scholar
33. Parker, E.T., Completion of disproof of Euler's conjecture, Am. Math. Soc. Notices, 6 (1959), 391.Google Scholar
34. Plotkin, M., Binary codes with specified minimum distance, Research Div. Rept. 51–20, University of Pennsylvania.Google Scholar
35. Varsamov, R.R., The evaluation of signals in codes with correction of errors, Math. Rev., 20 (1959), 262.Google Scholar