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Visual Topology and Variational Problems on Two-Dimensional Surfaces

  • Anatoly T. Fomenko
  • Alexandr O. Ivanov
  • Alexey A. Tuzhilin

Keywords

Minimal Surface Soap Film Soap Bubble Steiner Minimal Tree Minimal Network 
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.

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References

  1. [1]
    D. Cieslik, Steiner Minimal Trees, Kluwer Academic Publishers, Dordrecht-Boston-London, 1998.Google Scholar
  2. [2]
    Dao Chong Thi, A. T. Fomenko, Minimal Surfaces and Plateau Problem, Nauka, Moscow, 1987 (in Russian). (Engl. translation Amer. Math. soc., Providence, RI, 1991.)Google Scholar
  3. [3]
    B. A. Dubrovin, A. T. Fomenko, S. P. Novikov, Modern Geometry, Nauka, Moscow, 1986. English translation: Part 1, GTM 93, 1984; Part 2, GTM 104, 1985, Springer-Verlag, New York.Google Scholar
  4. [4]
    M. Emmer, Bolle di sapone: un viaggio tra matematica, arte e fantasia, La Nuova Italia, Firenze, 1991.Google Scholar
  5. [5]
    M. Emmer, Soap Bubbles in art and science, M. Emmer, ed., The Visual Mind, MIT press, 1993.Google Scholar
  6. [6]
    M. Emmer, Soap Bubbles, video, english version, 27 minutes, Rome (1984).Google Scholar
  7. [7]
    A. T. Fomenko, and A. A. Tuzhilin, Elements of geometry and topology of minimal surfaces in three-dimensional space, Translations of Math. Monographs, AMS, v. 93, 1992.Google Scholar
  8. [8]
    R. L. Francis, A note on the optimum location of new machines in existing plant layouts, J. Indust. Engrg., v. 14, pp. 57–59, 1963.Google Scholar
  9. [9]
    M. R. Garey, and D. S. Johnson, The Rectilinear Steiner Problem is NP-Complete, SIAM J. Appl. Math., v. 32, pp. 826–834, 1977.MathSciNetGoogle Scholar
  10. [10]
    M. Hanan, On Steiner’s Problem with Rectilinear Distance, SIAM J. Appl. Math., v. 14, pp. 255–265, 1966.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Hildebrandt, and A. Tromba, The Parsimonious Universe, Springer-Verlag, New York, 1996.Google Scholar
  12. [12]
    F. K. Hwang, On Steiner minimal trees with rectilinear distance, SIAM J. of Appl. Math., v. 30, pp. 104–114, 1976.MATHGoogle Scholar
  13. [13]
    F. K. Hwang, D. Richards, and P. Winter, The Steiners Tree Problem, Elsevier Science Publishers, 1992.Google Scholar
  14. [14]
    A. O. Ivanov, and A. A. Tuzhilin, Minimal Networks. The Steiner Problem and Its Generalizations, CRC Press, N.W., Boca Raton, Florida, 1994.Google Scholar
  15. [15]
    A. O. Ivanov, and A. A. Tuzhilin, Branching Solutions to One-Dimensional Variational Problems, World Scientific Publ., 2001.Google Scholar
  16. [16]
    V. Jarnik, and M. Kössler, O minimalnich grafeth obeahujicich n danijch bodu, Cas. Pest. Mat. a Fys., v. 63, pp. 223–235, 1934.Google Scholar
  17. [17]
    L. S. Polak, Variational principals in mechanics, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959 (in Russian).Google Scholar
  18. [18]
    J. E. Taylor, The structure of singularities in soap-bubble-like and soapfilm-like minimal surfaces, Ann. Math., v. 103, pp. 489–539, 1976.MATHGoogle Scholar
  19. [19]
    J. A. Thorpe, Elementary topics in differential geometry, Springer-Verlag, New York, 1979.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Anatoly T. Fomenko
    • 1
  • Alexandr O. Ivanov
    • 1
  • Alexey A. Tuzhilin
    • 1
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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