Theory of Zonoids: I A Mathematical Summary

  • Patrick R. Valentin
Chapter
Part of the NATO ASI Series book series (ASIC, volume 383)

Abstract

Requirements to be fulfilled by a meaningful measure of the separation state of a physico-chemical system are set up and it is shown that the separation state can be represented by a geometric entity called a zonoid. A brief account is given of zonoid mathematical properties, necessary for chromatographic applications. Several new concepts are introduced such as regular selectivity and a classification of transformation processes in four classes: separation, mixing, null and sepmix processes, with the last two being new types of processes. Zonoid volume gives a useful (although degraded) measure of separation. For problems of evolution modelled by scalar partial differential equations in form of conservation laws, theorems on non increase of the separation state are given and the fundamental importance of the increase in the minimum volume enclosing a given quantity of species is stressed. Evolution equation for the zonoid itself as well as its volume show that the dimension of the Euclidean space where a process occurs has a deep impact on separation.

Keywords

Separation State Hausdorff Distance Isoperimetric Inequality Conservative Species Quantity Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Valentin, P., Is chromatography a separation process, The zonoid answer, J. Chromatog., 556, 25–80 (1991)CrossRefGoogle Scholar
  2. [2]
    Valentin, P., Design and optimisation of preparative chromatographic separations, in Percolation Process and Applications, Rodrigues and Tondeur (ed) NATO-ASI Series E33, Sijthoff and Noordhoff, Netherlands, pp 141–195, 1980Google Scholar
  3. [3]
    Rony, P. R., The extent of separation, a universal separation index, Separation Science, 3, 239, (1968)CrossRefGoogle Scholar
  4. [4]
    Rony, P. R., The extent of separation: application to elution chromatography, Separation Science, 3 357, 1968CrossRefGoogle Scholar
  5. [5]
    Karlin, S., Total Positivity, Stanford University Press 1968Google Scholar
  6. [6]
    Bandle, C., Isoperimetric Inequalities and applications, Pitman, New-York, 1980Google Scholar
  7. [7]
    Vol’pert A.I., Hudjaev S.I., Analysis in classes of discontinuous functions, and equations of mathematical physics, Martinus Nijhhoff Publishers, Dordrecht, 1985Google Scholar
  8. [8]
    Duff, G.F.D., Integral Inequalities for Equimeasurable Rearrangements, Can. J. Math., 22, 408–430, (1970)Google Scholar
  9. [9]
    Fleming, W.H., Rishel R., An Integral formula for total gradient variation, Arch. Math., 11, 218–222, (1960)Google Scholar
  10. [10]
    Federer, H., Geometric Measure Theory, Springer Verlag, Berlin (1969)Google Scholar
  11. [11]
    Dafermos, C.M., The entropy rate admissibility criterion for solutions of hyperbolic conservation laws, J. Diff. Equ., 14, 202–212 (1973)CrossRefGoogle Scholar
  12. [12]
    Krushkov, S.N., First order quasilinear equations in several independent variables, Math. USSR Sb., 10, 217–273 (1970)CrossRefGoogle Scholar
  13. [13]
    Oleinik, O. A., Uniqueness and Stability of the general solution of the Cauchy problem for a quasi-linear equation, Amer. Math. Soc. Transi. Ser., 2, 33, 285–290 1963 )Google Scholar
  14. [14]
    Alvino, A., Lions P.L., Trombetti, G., On optimization problems with prescribed rearrangements, Nonlinear Analysis, TMA., 13 185–220 (1989)Google Scholar
  15. [15]
    Ferone, V., Posterario, M. R., Maximization on classes of functions with fixed rearrangement, Diff. and Integral Equ., 4, 707–718 (1991)Google Scholar
  16. [16]
    Coxeter, H.S.M., The classification of zonohedra by means of projective diagrams, J. Math. Pures Appl., (9) 41, 137–156, 1962Google Scholar
  17. [17]
    Coxeter, H.S.M., Regular Polytopes, 2nd ed., Mac Millan, New-York, 1963Google Scholar
  18. [18]
    Bolker, E.D., A class of convex bodies, Trans. Amer. Math. Soc., 145, 323–345, 1969CrossRefGoogle Scholar
  19. [19]
    McMullen, P., On zonotopes, Trans. Amer. Math. Soc., 159 91–110 1971CrossRefGoogle Scholar
  20. [20]
    Shephard, G.C., Combinatorial properties of associated zonotopes, Canad. J. Math., 26 302–321, 1974Google Scholar
  21. [21]
    Schneider, R., Weil, W., Zonoids and related topics, in: Applications of Convexity, Birkhauser Verlag, Basel, pp 296–317 (1982)Google Scholar
  22. [22]
    Blackwell, D., Comparison of experiments, in Proceedings of the second Berkeley Symposium on Mathematical Statistics and Probability, [1950, Berkeley], pp. 93–102 Berkeley, University Press, Los Angeles, (1951)Google Scholar
  23. [23]
    Szafran, N., Zonoèdres: de la Géométrie Algorithmique à la Théorie de la Séparation, Thèse de Doctorat en Mathématiques Appliquées, University J. Fourier, Grenoble, 25 Oct. 91.Google Scholar
  24. [24]
    J. Mossino, J.M. Rakotoson, Isoperimetric Inequalities in Parabolic Equations, Ann. Scuola Norm. Sup. Pisa, ser. IV, XIII, pp 56–71 (1986).Google Scholar
  25. [25]
    J.M. Rakotoson, Réarrangement relatif: propriétés et applications aux équations aux dérivées partielles, Thèse Doctorat en Sciences Mathématiques, Univ. Paris Sud, Orsay (1987).Google Scholar
  26. [26]
    Glansdorff, P., Prigogine, I., Structure, Stabilité et Fluctuations, Masson, Paris, 1971.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  • Patrick R. Valentin
    • 1
  1. 1.Elf- FranceElf- Solaize Research CenterFrance

Personalised recommendations