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Bijections of #x211D;n onto itself

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Abstract

We characterize affine and continuous maps within the class of bijections of #x211D;n onto itself by the preservation of various geometric or topological figures. A characterization of similarity maps of Hilbert space is given.

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References

  1. Acźel, J. and McKiernan, M. A., ‘On the Characterization of Plane Projective and Complex Möbius-Transformations’, Math. Nachr. 33 (1967), 315–337.

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexandrov, A. D., ‘A Contribution to Chronogeometry’, Canad. J. Math. 19 (1967), 1119–1128.

    Article  MathSciNet  MATH  Google Scholar 

  3. Aleksandrov, A. D., ‘Mappings of Families of Sets’, Soviet Math. Dokl. 11 (1970), 376–380.

    MATH  Google Scholar 

  4. Beckman, F. S. and Quarles, D. A., ‘On Isometries of Euclidean Spaces’, Proc. Amer. Math. Soc. 4 (1953), 810–815.

    Article  MathSciNet  MATH  Google Scholar 

  5. Benz, W., ‘A Characterization of Plane Lorentz Transformations’, J. Geom. 10 (1977), 45–56.

    Article  MathSciNet  MATH  Google Scholar 

  6. Borchers, H. J. and Hegerfeldt, G. C., ‘The Structure of Space-Time Transformations’, Comm. Math. Phys. 28 (1972), 259–266.

    Article  MathSciNet  MATH  Google Scholar 

  7. Cacciafesta, F., ‘An Observation about a Theorem of A. D. Alexandrov Concerning Lorentz Transformations’, J. Geom. 18 (1982), 5–8.

    Article  MathSciNet  MATH  Google Scholar 

  8. Carathéodory, C., ‘The Most General Transformations of Plane Regions which Transform Circles into Circles’, Bull Amer. Math. Soc. 43 (1937), 573–579.

    Article  MathSciNet  MATH  Google Scholar 

  9. Darboux, G., ‘Sur le théorème fondamental de la géométrie projective’, Math. Ann. 17 (1880), 55–61.

    Article  MathSciNet  MATH  Google Scholar 

  10. Gibbons, J. and Webb, C., ‘Circle-Preserving Functions of Spheres’, Trans. Amer. Math. Soc. 248 (1979), 67–83.

    Article  MathSciNet  MATH  Google Scholar 

  11. Guc, A. K., ‘On Mappings of Families of Sets’, Soviet Math. Dokl. 14 (1973), 506–508.

    MathSciNet  MATH  Google Scholar 

  12. Kuz'minykh, A. V., ‘On the Characterization of Isometric and Similarity Mappings’, Soviet Math. Dokl. 20 (1979), 82–84.

    MATH  Google Scholar 

  13. Kuz'minykh, A. V., ‘Affineness of Convex-Invariant Mappings’, Siberian Math. J. 16 (1975), 918–922.

    Article  MathSciNet  Google Scholar 

  14. Lester, J. A., ‘Euclidean Plane Point-Transformations Preserving Unit Area or Unit Perimeter’, Arch. Math. 45 (1985), 561–564.

    Article  MathSciNet  MATH  Google Scholar 

  15. Pfeffer, W. F., ‘Lorentz Transformations of a Hilbert Space’, Amer. J. Math. 103 (1981), 691–709.

    Article  MathSciNet  MATH  Google Scholar 

  16. Shaidenko, A. V., ‘Mappings of Families of Cones’, Siberian Math. J. 20 (1979), 118–125.

    Article  MathSciNet  Google Scholar 

  17. Wegner, B., ‘Über eine charakteristische Eigenschaft affiner Abbildungen’, Math.-Phys. Semesterber. 19 (1972), 68–72.

    MathSciNet  MATH  Google Scholar 

  18. Zeeman, E. C., ‘Causality Implies the Lorentz Group’, J. Math. Phys. 5 (1964), 490–493.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, King Fahd University of Petroleum and Minerals, 31261, Dhahran, Saudi Arabia

    R. J. Gardner

  2. Department of Mathematics, North Texas State University, TX76203-5116, Denton, USA

    R. Daniel Mauldin

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  1. R. J. Gardner
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  2. R. Daniel Mauldin
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Supported in part by a grant from the National Science Foundation.

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Gardner, R.J., Mauldin, R.D. Bijections of #x211D;n onto itself. Geom Dedicata 26, 323–332 (1988). https://doi.org/10.1007/BF00183024

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