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Computational Proximity pp 141–173Cite as

Homotopic Maps, Shapes and Borsuk–Ulam Theorem

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Part of the book series: Intelligent Systems Reference Library ((ISRL,volume 102))

Abstract

This chapter introduces object spaces, where objects are located in a visual field.

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Notes

  1. 1.

    Many thanks to Andrzej Skowron, who contributed the picture of Karol Bursuk in Fig. 5.1.

  2. 2.

    E.W. Weisstein, antipodal points, http://mathworld.wolfram.com/AntipodalPoints.html.

  3. 3.

    Many thanks to A. Tozzi for contributing this section.

References

  1. Borsuk, K.: Drei sätze über die n-dimensionale euklidische sphäre. Fundamenta Mathematicae XX, 177–190 (1933)

    MATH  Google Scholar 

  2. Weisstein, E.: Antipode. WolframMathWorld (2015). http://mathworld.wolfram.com/Antipode.html

  3. Aichholzer, O., Caraballo, L., Díaz-Bánez, J., Fabila-Monroy, R., Ochoq, C., Nigsch, P.: Characterization of extremal antipodal polygons. Graphs Comb. 31, 321–333 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Weisstein, E.: Hypersphere. Wolfram MathWorld (2015). http://mathworld.wolfram.com/Hypersphere.html

  5. Munkres, J.: Topology, 2nd edn. Prentice-Hall, Englewood Cliffs (2000). Xvi + 537 pp. 1st edn. in 1975, MR0464128

    Google Scholar 

  6. Borsuk, K.: Concerning the classification of topological spaces from the stand-point of the theory of retracts. Fundamenta Mathematicae XLVI, 177–190 (1958–1959)

    Google Scholar 

  7. Borsuk, K.: Fundamental retracts and extensions of fundamental sequences. Fundamenta Mathematicae 64(1), 55–85 (1969)

    MathSciNet  MATH  Google Scholar 

  8. Borsuk, K., Gmurczyk, A.: On homotopy types of 2-dimensional polyhedra. Fundamenta Mathematicae 109(2), 123–142 (1980)

    MathSciNet  MATH  Google Scholar 

  9. Dodson, C., Parker, P.: A User’s Guide to Algebraic Topology. Kluwer, Dordrecht (1997). (Xii+405 pp. ISBN: 0-7923-4292-5, MR1430097)

    Book  MATH  Google Scholar 

  10. Tozzi, A., Peters, J.: Our thoughts follow a donut-like trajectory in the brain. ResesarchGate Preprint, pp. 1–13 (2015). doi:10.13140/RG.2.1.3305.8008

  11. Peters, J., Tozzi, A.: The Borsuk-Ulam theorem explains quantum entanglement. ResesarchGate Preprint, pp. 1–7 (2015). doi:10.13140/RG.2.1.3860.1685

  12. Crabb, M., Jaworowski, J.: Aspects of the borsuk-ulam theorem. J. Fixed Point Theory Appl. 13, 459–488 (2013). doi:10.1007/s11784-013-0130-7

    Google Scholar 

  13. Lusternik, L., Schnirelmann, L.: Topological methods in calculus of variations [Russian]. Gosudarstv. Izdat. Tehn.-Teor, Lit (1930)

    Google Scholar 

  14. Su, F.: Borsuk-Ulam implies Brouwer: A direct construction. Am. Math. Mon. 104(9), 855–859 (1997). MR1479992

    Article  MathSciNet  MATH  Google Scholar 

  15. Burak, G., Karaca, I.: Digital borsuk-ulam theorem. Bull. Iran. Math. Soc. (2015) (to appear)

    Google Scholar 

  16. Ege, O., Karaca, I.: Banach fixed point theorem for digital images. J. Nonlinear Sci. Appl. 8(3), 237–245 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Willard, S.: General Topology. Dover Pub. Inc, Mineola (1970). Xii + 369pp, ISBN: 0-486-43479-6 54-02, MR0264581

    MATH  Google Scholar 

  18. Krantz, S.: A Guide to Topology. The Mathematical Association of America, Washington, D.C. (2009). Ix + 107 pp

    Google Scholar 

  19. Manetti, M.: Topology. Springer, Heidelberg (2015). doi:10.1007/978-3-319-16958-3. Xii+309 pp

    Google Scholar 

  20. Cohen, M.: A Course in Simple Homotopy Theory. Springer, New York (1973). X+144 pp., MR0362320

    Book  MATH  Google Scholar 

  21. Collins, G.: The shapes of space. Sci. Am. 291, 94–103 (2004)

    Article  MathSciNet  Google Scholar 

  22. Malgouyres, R.: Homotopy in 2-dimensional digital images. Theor. Comput. Sci. 230(1–2), 221–233 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Murrell, H., Hashimoto, K., Takatori, D.: Fisher discrimination with kernels. Mathematica J. 11, 1–17 (2011)

    Google Scholar 

  24. Naimpally, S., Peters, J.: Topology with Applications. Topological Spaces via Near and Far. World Scientific, Singapore (2013). Xv + 277 pp, Am. Math. Soc. MR3075111

    Book  MATH  Google Scholar 

  25. Renze, J., Uznanski, D., Weisstein, E.: Half-plane. Wolfram MathWorld (2015). http://mathworld.wolfram.com/Half-Plane.html

  26. Peters, J., Guadagni, C.: Strongly proximal continuity & strong connectedness. arXiv 1504(02740), 1–11 (2015)

    Google Scholar 

  27. Di Concilio, A., Gerla, G.: Quasi-metric spaces and point-free geometry. Math. Struct. Comput. Sci. 16(1), 115–137 (2006). MR2220893

    Article  MathSciNet  MATH  Google Scholar 

  28. Di Concilio, A.: Point-free geometries: Proximities and quasi-metrics. Math. Comp. Sci. 7(1), 31–42 (2013). MR3043916

    Google Scholar 

  29. Whitehead, J.: Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. 45, 243–327 (1939)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 31, 2nd edn. Springer, Heidelberg (2006). Xviii+644 pp. ISBN: 978-3-642-05157-9, MR2840298

    Google Scholar 

  31. Hairer, E.: Important aspects of geometric numerical integration. J. Sci. Comput. 25(1–2), 67–81 (2005). MR2231943

    Article  MathSciNet  MATH  Google Scholar 

  32. Ehresmann, C.: Les connexions infinitésimales dans un espace fibré différentiable. (french) [infinitesimal connections in a differentiable fiber space]. Séminaire Bourbaki 1(24), 153–168 (1995). MR1605161

    MathSciNet  Google Scholar 

  33. Boothby, W.: An introduction to differentiable manifolds and Riemannian geometry. Pure and Applied Mathematics, vol. 120, 2nd edn. Academic Press Inc, Orlando (1986). Xvi+430 pp. ISBN: 0-12-116052-1, MR0861409

    Google Scholar 

  34. Jost, J.: Riemannian geometry and geometric analysis. Pure and Applied Mathematics, vol. 120, 6th edn. Springer, Heidelberg (2002). Xiv+611 pp. ISBN: 978-3-642-21297-0, MR2829653

    Google Scholar 

  35. Snyman, J.: Practical mathematical optimization. An introduction to basic optimization theory and classical and new gradient-based algorithms. Applied Optimization, vol. 97. Springer, New York (2005). Xx+257 pp. ISBN: 0-387-24348-8, MR2120543

    Google Scholar 

  36. Sengupta, B., Friston, K., Penny, W.: Efficient gradient computation for dynamical models. Neuroimage 98, 521–527 (2014). doi:10.1016/j.neuroimage.2014.04.040

    Article  Google Scholar 

  37. Beyer, W., Zardecki, A.: The early history of the ham sandwich theorem. Am. Math. Mon. 111(1), 58–61 (2004). MR2212093

    Article  MathSciNet  MATH  Google Scholar 

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  1. University of Manitoba, Winnipeg, MB, Canada

    James F. Peters

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Peters, J.F. (2016). Homotopic Maps, Shapes and Borsuk–Ulam Theorem. In: Computational Proximity. Intelligent Systems Reference Library, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-30262-1_5

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  • DOI: https://doi.org/10.1007/978-3-319-30262-1_5

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