Entropy in dissimilarity and chirality measures

Journal of Mathematical Chemistry

Volumn 19, Issue 2, 1996, Pages 147-174

  Chauvin, Remi ^a  

^a UNIVERSITÉ PAUL SABATIER   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030532235     PISSN: 02599791     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF01165181     Document Type: Article

References (53)

1. H. Bock, K. Ruppert, C. Näther, Z. Havlas, H.F. Herrmann, C. Arad, I. Göbel, A. John, J. Meuret, S. Nick, A. Rauschenbach, W. Seitz, T. Vaupel and B. Solouki, Angew. Chem. Int. Ed. Engl. 31 (1992) 550.
2. For a general introduction, see for example: H.B. Kagan, Pour la Science 172 (1992) 42; K. Mislow, Bull. Soc. Chim. Fr. 131 (1994) 534.
3. For a general introduction, see for example: H.B. Kagan, Pour la Science 172 (1992) 42; K. Mislow, Bull. Soc. Chim. Fr. 131 (1994) 534.
4. R. Chauvin, J. Phys. Chem. 96 (1992) 4701.
5. See for example: H.A. Feldman, Anal. Biochem. 48 (1972) 317.
6. E. Ruch and L. Ugi, Topics in Stereochemistry, Vol. 4 (Wiley-Interscience, New York, 1969) p. 99.
7. 2.
8. R. Chauvin, J. Math. Chem. 16 (1994) 257.
9. R. Chauvin, J. Phys. Chem. 96 (1992) 4706.
10. Generally speaking G must be a compact group and "dg" is its Haar measure: J.P. Serre, Représentations Linéaires des Groupes Finis, 3d Ed. (Hermann, Paris, 1967).
11. See: Concepts and Applications of Molecular Similarity, eds. M. A. Johnson and G.M. Maggiora (Wiley, New York, 1990); Topics in Current Chemistry I and II, ed. K.D. Sen, Vols. 173 and 174 (Springer, Berlin, 1995), and references therein.
12. See: Concepts and Applications of Molecular Similarity, eds. M. A. Johnson and G.M. Maggiora (Wiley, New York, 1990); Topics in Current Chemistry I and II, ed. K.D. Sen, Vols. 173 and 174 (Springer, Berlin, 1995), and references therein.
13. R. Chauvin, J. Math. Chem. 16 (1994) 269.
14. p (u,v)≥0.
15. p (u, v) = 0 ⇒ ∃h ∈ G, v = gu.
16. R. Chauvin, J. Math. Chem. 16 (1994) 285.
17. See for example: P.G. Mezey, J. Math. Chem. 7 (1991) 39; A.Y. Meyer and W.G. Richards, J. Computer-Aided Mol. Design 5 (1991) 427; G.D. Purvis III, J. Computer-Aided Mol. Design 5 (1991) 55.
18. See for example: P.G. Mezey, J. Math. Chem. 7 (1991) 39; A.Y. Meyer and W.G. Richards, J. Computer-Aided Mol. Design 5 (1991) 427; G.D. Purvis III, J. Computer-Aided Mol. Design 5 (1991) 55.
19. See for example: P.G. Mezey, J. Math. Chem. 7 (1991) 39; A.Y. Meyer and W.G. Richards, J. Computer-Aided Mol. Design 5 (1991) 427; G.D. Purvis III, J. Computer-Aided Mol. Design 5 (1991) 55.
20. F) has been calculated for achiral skeletons such as a ball, a sphere, a regular tetrahedron, a trigonal bipyramid, a regular octahedron, a cube, etc.: see text further and ref. [7].
21. (a) R. Chauvin, J. Math. Chem. 16 (1994) 245.
22. (b) P.G. Mezey and J. Maruani, Mol. Phys. 69 (1990) 97.
23. (a) For a general survey, see ref. [18] and: New Developments in Molecular Chirality, ed. P.G. Mezey (Kluwer, Dordrecht, 1991). See also ref. [8], and for more recent examples: V.E. Kuz'min, I.B. Stel'makh, M.B. Bekker and D.V. Pozigun, J. Phys. Org. Chem. 5 (1992) 295; V.E. Kuz'min, I.B. Stel'makh, I.V. Yudanova, D.V. Pozigun and M.B. Bekker, J. Phys. Org. Chem. 5 (1992) 299; A. Seri-Levy and W.G. Richards, Tetrahedron: Asymm. 4 (1993) 1917.
24. (a) For a general survey, see ref. [18] and: New Developments in Molecular Chirality, ed. P.G. Mezey (Kluwer, Dordrecht, 1991). See also ref. [8], and for more recent examples: V.E. Kuz'min, I.B. Stel'makh, M.B. Bekker and D.V. Pozigun, J. Phys. Org. Chem. 5 (1992) 295; V.E. Kuz'min, I.B. Stel'makh, I.V. Yudanova, D.V. Pozigun and M.B. Bekker, J. Phys. Org. Chem. 5 (1992) 299; A. Seri-Levy and W.G. Richards, Tetrahedron: Asymm. 4 (1993) 1917.
25. (a) For a general survey, see ref. [18] and: New Developments in Molecular Chirality, ed. P.G. Mezey (Kluwer, Dordrecht, 1991). See also ref. [8], and for more recent examples: V.E. Kuz'min, I.B. Stel'makh, M.B. Bekker and D.V. Pozigun, J. Phys. Org. Chem. 5 (1992) 295; V.E. Kuz'min, I.B. Stel'makh, I.V. Yudanova, D.V. Pozigun and M.B. Bekker, J. Phys. Org. Chem. 5 (1992) 299; A. Seri-Levy and W.G. Richards, Tetrahedron: Asymm. 4 (1993) 1917.
26. (a) For a general survey, see ref. [18] and: New Developments in Molecular Chirality, ed. P.G. Mezey (Kluwer, Dordrecht, 1991). See also ref. [8], and for more recent examples: V.E. Kuz'min, I.B. Stel'makh, M.B. Bekker and D.V. Pozigun, J. Phys. Org. Chem. 5 (1992) 295; V.E. Kuz'min, I.B. Stel'makh, I.V. Yudanova, D.V. Pozigun and M.B. Bekker, J. Phys. Org. Chem. 5 (1992) 299; A. Seri-Levy and W.G. Richards, Tetrahedron: Asymm. 4 (1993) 1917.
27. (b) H. Zabrodsky and D. Avnir, J. Am. Chem. Soc. 117 (1995) 462.
28. A.B. Buda, T. Auf der Heyde and K. Mislow, Angew. Chem. Int. Ed. Engl. 31 (1992) 989.
29. (a) A.B. Buda and K. Mislow, J. Am. Chem. Soc. 114 (1992) 6006.
30. (b) N. Weinberg and K. Mislow, J. Math. Chem. 14 (1993) 427.
31. Distance is here understood with its strong mathematical meaning. This term was used in inverted commas by Mislow to qualify the underlying concept of "chirality measure of the second kind": see ref. [18].
32. Note 13 in ref. [18].
33. V. Prelog, Science 193 (1976) 17.
34. In a somewhat different spirit, the use of the Hausdorff distance was first propounded by Rassat in: A. Rassat, C.R. Acad. Sc. Paris II 299 (1984) 53.
35. D. Andelman and P.G. de Gennes, C.R. Acad. Sci. Paris 307 (II) (1988) 233; D. Andelman, J. Am. Chem. Soc. 111 (1989) 6536.
36. D. Andelman and P.G. de Gennes, C.R. Acad. Sci. Paris 307 (II) (1988) 233; D. Andelman, J. Am. Chem. Soc. 111 (1989) 6536.
37. C.E. Wintner, J. Chem. Educ. 60 (1983) 550.
38. E.L. Eliel, Topics in Current Chemistry, Vol. 105 (Springer, Berlin, 1982) p. 70.
39. For chiral simplexes and multipexes, see: A.S. Dreiding and K. Wirth, MATCH 8 (1980) 341.
40. G. Gilat, J. Math. Chem. 15 (1994) 197. G. Gilat and Y. Gordon, J. Math. Chem. 16 (1994) 37.
41. G. Gilat, J. Math. Chem. 15 (1994) 197. G. Gilat and Y. Gordon, J. Math. Chem. 16 (1994) 37.
42. p(T, T′).
43. p(T) would become sheerer and tend to the "0-1" (discrete) chirality measure. The problem is open.
44. p+1.
45. 2(g)dg.
46. 2(λ(T)T, σλ(T)T) = 1.
47. qkT), where a/Δ(T) is the interaction constant normalized to the size of T.
48. H. Zabrodsky, S. Peleg and D. Avnir, J. Am. Chem. Soc. 31 (1992) 989.
49. E.L. Eliel, Isr. J. Chem. 15 (1977) 7.
50. For a discussion of this statement see: K. Mislow and J. Siegel, J. Am. Chem. Soc. 106 (1984) 3319.
51. m(u, v) = 1.
52. 2(y,y′)/2d(x′,y)d(x,y′)≤1 is true as soon as x = x′ (or y = y′). If this inequality stands for any points x, x′, y, y′, C would formally define a kind of "cosine" in a non-Euclidean metric space.
53. Discrete Geometry and Convexity, Annals of the New York Academy of Sciences, Vol. 440 (The New York Academy of Sciences, New York, 1985).