Preprints (2017):

  • Hong, H., Hough, Z., Kogan I. A. and Li, Z.
    Degree-optimal moving frames for rational curves,     Maple code and examples.

  • Benfield, M., Jenssen, H. K. and Kogan, I. A.
    Jacobians with prescribed eigenvectors.

  • Benfield, M., Jenssen, H. K. and Kogan, I. A.
    On two theorems of Darboux.

  • Benfield, M., Jenssen, H. K. and Kogan, I. A.
    A generalization of an integrability theorem of Darboux.

    Peer reviewed publications by Irina Kogan

    1. Hong, H., Hough, Z. and Kogan I. A.
      Algorithm for computing μ-bases of univariate polynomials,
      J. of Symbolic Comput., Vol. 8, No 3, (2017), 844 - 874. doi
      ArXiv preprint     Maple code and examples.

    2. Kogan I. A. and Olver P. J.
      Invariants of objects and their images under surjective maps

      Lobachevskii J. Math., Vol. 36, No 3, (2015), 260 - 285.
      Corrections and additions to the published version
      ArXiv (with correction and additions)

    3. Benfield, M., Jenssen, H. K. and Kogan, I. A.
      1-D Conservative Systems: A Geometric Approach
      Proceedings of the 14th International Conference on Hyperbolic Problems, held june 25-29, 2012, in Padova, Italy American Institute of Mathematics (AIMS) series on applied mathematics, 8 (2014), 749 - 757.

    4. Burdis, J. M., Kogan, I. A. and Hong, H.
      Object-image correspondence for algebraic curves under projections.
      SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, Vol. 9, (2013), 31 pp.

    5. Burdis, J. M. and Kogan, I. A.
      Object-image correspondence for curves under central and parallel projections.
      Proceedings of the Symposium on Computational Geometry (SoCG), ACM, New York (2012), 373 - 382.
      Maple code and supplementary materials.

    6. Jenssen, H. K. and Kogan, I. A.
      Extensions for systems of conservation laws.
      Communications in PDE's. No. 37 (2012), 1096 - 1140.
      Maple worksheet with examples     arXiv preprint

    7. Jenssen, H. K. and Kogan, I. A.
      Conservation laws with prescribed eigencurves.
      J. of Hyperbolic Differential Equations (JHDE) Vol. 7, No. 2., (2010), 211 - 254.
      Maple worksheet with examples     arXiv preprint.

    8. Feng, S., Kogan, I. A. and Krim, H.
      Classification of curves in 2D and 3D via affine integral signatures.
      Acta Applicandae Mathematicae, Vol. 109, No. 3, (2010), 903 - 937
      arXiv preprint.

    9. Jenssen, H. K. and Kogan, I. A.
      Construction of conservative systems.
      Proceedings of the 12th International Conference on Hyperbolic Problems, Proc. of Symposia in Applied Mathematics, 67 Part 2, AMS, (2009), 673 - 682.

    10. Hubert, E. and Kogan, I. A.
      Smooth and algebraic invariants of a group action. Local and global constructions.
      Foundations of Computational Math. J., Vol. 7, No. 4 (2007), 345 - 383.
      arXiv preprint

    11. Hubert, E. and Kogan, I. A.
      Rational invariants of a group action. Construction and rewriting.
      J. of Symbolic Comput., Vol 42, No 1-2, (2007) 203 - 217.
      arXiv preprint

    12. Feng, S., Kogan, I. A. and Krim, H.
      Integral invariants for curves in 3D: an inductive approach.
      Proceedings of IS&T/SPIE joint symposium, Visual Communication and Image Processing conference (VCIP), San Jose, CA, (2007), 65080I, 11 pp.

    13. Feng, S., Krim, H., and Kogan I. A.
      3D Face recognition using Euclidean integral invariant signature.
      Proceedings of the IEEE/SP 14th Workshop on Statistical Signal Processing (SSP), (2007), 156 -160.

    14. Aouada, D., Feng, S., Krim, H. and Kogan, I. A.
      3D mixed invariant and its application in object classification.
      Proceedings of International Conference on Acoustics, Speech, and Signal Processing (ICASSP), (2007), 461 - 464.

    15. Smith, R., Hollebrands, K., Iwancio, K. and Kogan, I. A.
      The effects of a dynamic program for geometry on college students' understandings of properties of quadrilaterals in the Poincare Disk model
      Proceedings of the 9th International Conference on Mathematics Education in a Global Community, (2007), 613 - 618.

    16. Smith, R., Hollebrands, K., Iwancio, K. and Kogan, I. A.
      College geometry students' uses of technology in the process of constructing arguments.
      Proceedings of the 29th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education. (T.~Lamberg, Ed.) (2007), 1153 - 1160.

    17. Baloch, S., Krim, H., Kogan, I. A. and Zenkov, D. V.
      Rotation invariant topology coding of 2D and 3D objects using Morse theory.
      Proceedings of IEEE International Conference on Image Processing (ICIP), (2005), 796 - 799.

    18. Baloch, S., Krim, H., Kogan, I. A. and Zenkov, D. V.
      3D object representation with topo-geometric shape models.
      Proceedings of European Signal Processing Conference (EUSIPCO), (2005) 4pp. (electronic).

    19. Kogan, I. A. and Olver, P.
      Invariant Euler-Lagrange equations and the invariant variational bicomplex.
      Acta Applicandae Mathematicae, Vol. 76, No. 2, (2003), 137 - 193.
      Corrected version     Corrections to published version.     Maple worksheet: iVB

    20. Kogan, I. A.
      Two algorithms for a moving frame construction.
      Canadian Journal of Math., Vol. 55, No. 2, (2003), 266 - 291.
      pdf

    21. Kogan, I. A. and Moreno Maza, M.
      Computation of canonical forms for ternary cubics.
      Proceedings of International Symposium on Symbolic and Algebraic Computation (ISSAC), ACM, (2002), 151 - 160.
      Corrected version     Corrections to published version     Maple worksheet

    22. Kogan, I. A. and Olver, P.
      The invariant variational bicomplex.
      Contemporary Mathematics, AMS, Vol. 285, (2001), 131 - 144.
      Corrected version     Corrections to published version

    23. Kogan, I. A.
      Inductive construction of moving frames.
      Contemporary Mathematics, AMS, Vol. 285, (2001), 157 - 170.
      pdf

    24. Berchenko (Kogan), I. A. and Olver, P.
      Symmetries of polynomials.
      J. of Symbolic Comput., Vol. 29, No 4-5, (2000), 485 - 514.
      Maple worksheet in html format     Maple worksheet in mws format
    Ph.D. Thesis:
    Kogan, I. A.
    Inductive Approach to Cartan's Moving Frame Method with Applications to Classical Invariant Theory
    University of Minnesota, 2000, advisor Peter Olver.

    Abstract: This thesis is devoted to algorithmic aspects of the implementation of Cartan's moving frame method to the problem of the equivalence of submanifolds under a Lie group action. We adopt a general definition of a moving frame as an equivariant map from the space of submanifolds to the group itself and introduce two algorithms, which simplify the construction of such maps. The first algorithm is applicable when the group factors as a product of two subgroups G=BA, allowing us to use moving frames and differential invariants for the groups A and B in order to construct a moving frame and differential invariants for G. This approach not only simplifies the computations, but also produces the relations among the invariants of G and its subgroups. We use the groups of the projective, the affine and the Euclidean transformations on the plane to illustrate the algorithm. We also introduce a recursive algorithm, allowing, provided the group action satisfies certain conditions, to construct differential invariants order by order, at each step normalizing more and more of the group parameters, at the end obtaining a moving frame for the entire group. The development of this algorithm has been motivated by the applications of the moving frame method to the problems of the equivalence and symmetry of polynomials under linear changes of variables. In the complex or real case these problems can be reduced and, in theory, completely solved as the problem of the equivalence of submanifolds. Its solution however involves algorithms based on the Gröbner basis computations, which due to their complexity, are not always feasible. Nevertheless, some interesting new results were obtained, such as a classification of ternary cubics and their groups of symmetries, and the necessary and sufficient conditions for a homogeneous polynomial in three variables to be equivalent to xn+yn+zn.