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CAPD::RedHom v2 - Homology Software Based on Reduction Algorithms

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Mathematical Software – ICMS 2014 (ICMS 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8592))

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Abstract

We present an efficient software package for computing homology of sets, maps and filtrations represented as cubical, simplicial and regular CW complexes. The core homology computation is based on classical Smith diagonalization, but the efficiency of our approach comes from applying several geometric and algebraic reduction techniques combined with smart implementation.

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References

  1. Dłotko, P., Kaczynski, T., Mrozek, M., Wanner, T.: Coreduction Homology Algorithm for Regular CW-Complexes. Discrete and Computational Geometry 46, 361–388 (2011), doi:10.1007/s00454-010-9303-y

    Article  MATH  MathSciNet  Google Scholar 

  2. Dłotko, P., Specogna, R., Trevisan, F.: Automatic generation of cuts on large-sized meshes for the T-Omega geometric eddy-current formulation. Computer Methods in Applied Mechanics and Engineering 198, 3765–3781 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological Persistence and Simplification. Discrete and Computational Geometry 28, 511–533 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Harker, S., Mischaikow, K., Mrozek, M., Nanda, V., Wagner, H., Juda, M., Dłotko, P.: The Efficiency of a Homology Algorithm based on Discrete Morse Theory and Coreductions. In: Diaz, R.G., Jurado, P.R. (eds.) Proceedings of the 3rd International Workshop on Computational Topology in Image Context, Chipiona, Spain, Image A, vol. 1, pp. 41–47 (November 2010) ISSN: 1885-4508

    Google Scholar 

  5. Juda, M., Mrozek, M.: Z 2-Homology of weak (p − 2)-faceless p-manifolds may be computed in O(n) time. Topological Methods in Nonlinear Analysis 40, 137–159 (2012)

    MATH  MathSciNet  Google Scholar 

  6. Kaczynski, T., Dłotko, P., Mrozek, M.: Computing the Cubical Cohomology Ring. In: Diaz, R.G., Jurado, P.R. (eds.) Proceedings of the 3rd International Workshop on Computational Topology in Image Context, Chipiona, Spain, Image A, vol. 3, pp. 137–142 (2010) ISSN: 1885-4508

    Google Scholar 

  7. Kaczynski, T., Mischaikow, M., Mrozek, M.: Computational Homology. Applied Mathematical Sciences, vol. 157. Springer-Verlag (2004)

    Google Scholar 

  8. Kaczynski, T., Mrozek, M., Ślusarek, M.: Homology computation by reduction of chain complexes. Computers and Math. Appl. 35, 59–70 (1998)

    Article  MATH  Google Scholar 

  9. Kapela, T., Mrozek, M., Pilarczyk, P., Wilczak, D., Zgliczyński, P.: CAPD - a Rigorous Toolbox for Computer Assisted Proofs in Dynamics, technical report, Jagiellonian University (2010)

    Google Scholar 

  10. Mischaikow, K., Mrozek, M.: Chaos in Lorenz equations: a computer assisted proof. Bull. AMS (N.S.) 33, 66–72 (1995)

    Article  MathSciNet  Google Scholar 

  11. Mischaikow, K., Mrozek, M.: Chaos in the Lorenz equations: a computer assisted proof. Part II: details. Mathematics of Computation 67, 1023–1046 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mischaikow, K., Mrozek, M., Pilarczyk, P.: Graph approach to the computation of the homology of continuous maps. Found. Comp. Mathematics 5, 199–229 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mrozek, M.: Index Pairs Algorithms. Found. Comp. Mathematics 6, 457–493 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mrozek, M.: Čech Type Approach to Computing Homology of Maps. Discrete and Computational Geometry 44(3), 546–576 (2010), doi:10.1007/s00454-010-9255-2.

    Article  MATH  MathSciNet  Google Scholar 

  15. Mrozek, M., Pilarczyk, P., Żelazna, N.: Homology algorithm based on acyclic subspace. Computers and Mathematics with Applications 55, 2395–2412 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mrozek, M., Batko, B.: Coreduction homology algorithm. Discrete and Computational Geometry 41, 96–118 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mrozek, M., Wanner, T.: Coreduction homology algorithm for inclusions and persistent homology. Computers and Mathematics with Applications 60(10), 2812–2833 (2010), doi:10.1016/j.camwa.2010.09.036

    Article  MATH  MathSciNet  Google Scholar 

  18. Mrozek, M., Żelawski, M., Gryglewski, A., Han, S., Krajniak, A.: Homological methods for extraction and analysis of linear features in multidimensional images. Pattern Recognition 45, 285–298 (2012)

    Article  Google Scholar 

  19. Pilarczyk, P.: Computer assisted method for proving existence of periodic orbits. Topological Methods in Nonlinear Analysis 13, 365–377 (1999)

    MATH  MathSciNet  Google Scholar 

  20. Koonin, J.: Topology of eigenspace posets for imprimitive reflection groups, http://arxiv.org/abs/1208.4435

  21. Computer Assisted Proofs in Dynamics, http://capd.ii.uj.edu.pl/

  22. Reduction Homology Algorithms, http://redhom.ii.uj.edu.pl/

  23. Computational Homology Project, http://chomp.rutgers.edu/

  24. GAP System, http://www.gap-system.org/

  25. Sage, http://www.sagemath.org/

  26. Linbox, http://www.eecis.udel.edu/~dumas/Homology/

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Author information

Authors and Affiliations

  1. Institute of Computer Science and Computational Mathematics, Jagiellonian University, Poland

    Mateusz Juda & Marian Mrozek

Authors
  1. Mateusz Juda
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  2. Marian Mrozek
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Editor information

Editors and Affiliations

  1. Department of Mathematics, North Carolina State University, Box 8205, 27695, Raleigh, NC, USA

    Hoon Hong

  2. Courant Institute, Dept. of Computer Science, New York Institute, 251 Mercer Street, 10012, New York, NY, USA

    Chee Yap

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Juda, M., Mrozek, M. (2014). CAPD::RedHom v2 - Homology Software Based on Reduction Algorithms. In: Hong, H., Yap, C. (eds) Mathematical Software – ICMS 2014. ICMS 2014. Lecture Notes in Computer Science, vol 8592. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44199-2_27

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  • DOI: https://doi.org/10.1007/978-3-662-44199-2_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44198-5

  • Online ISBN: 978-3-662-44199-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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