Integration and Separation

  • Octavian IordacheEmail author
Part of the Lecture Notes in Intelligent Transportation and Infrastructure book series (LNITI)


Trees are combinatorial objects that describe multi-scale systems, classifications, separation schemes, schedules, automata self-reconfiguration, logical schemes, and organizational structures. Hopf algebras and dual graded graphs associated to trees generate separation schemes and highlight the separation-integration methods. Schemes of operations as middle vessel column MVC distillation, pressure swing adsorption, PSA, or simulated moving bed, SMB, of increasing importance in chemical, pharmaceutical and food industry or in environment protection are studied here. Innovative cyclic operations of separation have been proposed as polytopic projects.


  1. Aguiar, M., Sottile, F.: Cocommutative Hopf algebras of permutations and trees. J. Algebraic Comb. 22(4), 451–470 (2005)MathSciNetCrossRefGoogle Scholar
  2. Ammon, J.: Hypercube Connectivity within ccNUMA Architectures. SGI Origin Team, Mountain View, CA (1998)Google Scholar
  3. Brouder, C., Frabetti, A.: QED Hopf algebras on planar binary trees. J. Algebra 267(1), 298–322 (2003)MathSciNetCrossRefGoogle Scholar
  4. Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199(1), 203–242 (1998)MathSciNetCrossRefGoogle Scholar
  5. Connes, A., Kreimer, D.: Renormalization in QFT and the Riemann-Hilbert Problem I. Comm. Math. Phys. 210, 249–273 (2000)MathSciNetCrossRefGoogle Scholar
  6. Dăscălescu, S., Năstăsescu, C., Raianu, Ș.: Hopf Algebras. An introduction, Pure and Applied Mathematics, vol. 235 Marcel Dekker (2001)Google Scholar
  7. Demicoli, D., Stichlmair, J.: Novel operational strategy for the separation of ternary mixtures via cyclic operation of a batch distillation column with side withdrawal. Chem. Eng. Trans. 3, 361–366 (2003)Google Scholar
  8. Flodman, H.R., Timm, D.C.: Batch distillation employing cyclic rectification and stripping operations. ISA Trans. 51(3), 454–460 (2012)CrossRefGoogle Scholar
  9. Fomin, S.: Duality of graded graphs. J. Algebraic Combin. 3, 357–404 (1994)MathSciNetCrossRefGoogle Scholar
  10. Fulman, J.: Mixing time for a random walk on rooted trees. Electron. J. Combin., 16(1), Research Paper 139, 13(2009)Google Scholar
  11. Grossman, R., Larson, R.G.: Hopf algebraic structures of families of trees. J. Algebra 126, 184–210 (1989)MathSciNetCrossRefGoogle Scholar
  12. Grossman, R., Larson, R.G.: Solving nonlinear equations from higher order derivations in linear stages. Adv. Math. 82(2), 180–202 (1990)MathSciNetCrossRefGoogle Scholar
  13. Hoffman, M.E.: Combinatorics of rooted trees and Hopf algebras. Trans. Amer. Math. Soc. 355, 3795–3811 (2003)MathSciNetCrossRefGoogle Scholar
  14. Hoffman, M.E.: Rooted Trees and Symmetric Functions: Zhao’s Homomorphism and the commutative Hexagon. arXiv:0812.2419 (2008)
  15. Iordache, O.: Polytope Projects. Taylor & Francis CRC Press, Boca Raton, FL (2013)CrossRefGoogle Scholar
  16. Iordache, O.: Implementing Polytope Projects for Smart Systems. Springer, Cham, Switzerland (2017)CrossRefGoogle Scholar
  17. Kotai, B., Lang, P., Balazs, T.: Separation of maximum azeotropes in a middle vessel column. In: Institution of Chemical Engineers Symposium Series, vol. 152, pp. 699–708. Institution of Chemical Engineers, 1999 (2006)Google Scholar
  18. Loday, J.-L., Ronco, M.: Hopf algebra of the planar binary trees. Adv. Math. 139, 299–309 (1998)MathSciNetCrossRefGoogle Scholar
  19. Luo, L. (ed.): Heat and mass transfer intensification and shape optimization: A multi-scale approach. Springer, Berlin (2013)Google Scholar
  20. Pang, C.Y.: Markov Chains from Descent Operators on Combinatorial Hopf Algebras. arXiv preprint arXiv:1609.04312. 2016 Sep 14 (2016)
  21. Qing, Y.: Differential posets and dual graded graphs. Diss. MIT, Cambridge (2008)Google Scholar
  22. Ruthven, D.M.: Principles of Adsorption and Adsorption Processes. Wiley, New York (1984)Google Scholar
  23. Seitz, C.L.: The cosmic cube. Commun. ACM 28(1), 22–33 (1985)CrossRefGoogle Scholar
  24. Sloss, C.A.: Enumeration of walks on generalized differential posets. M.S. Thesis, University of Waterloo, Canada (2005)Google Scholar
  25. Stanley, R.P.: Differential posets. J. Amer. Math. Soc. 1, 919–961 (1988)MathSciNetCrossRefGoogle Scholar
  26. Stanley, R.P.: Enumerative Combinatorics, vol. 1. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  27. Yang, R.T.: Gas Separation by Adsorption Processes. Butherworths, Boston, MA (1987)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.PolystochasticMontrealCanada

Personalised recommendations