- 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. - 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) - 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.* - 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.* - 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. - 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 - 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. - 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. - 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.* - 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 - 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 - 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.* - 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.* - 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.* - 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.* - 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.* - 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.* - 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).* - 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 - Kogan, I. A.

**Two algorithms for a moving frame construction.**

*Canadian Journal of Math., Vol. 55, No. 2, (2003), 266 - 291.*

pdf - 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 - Kogan, I. A. and Olver, P.

**The invariant variational bicomplex.***Contemporary Mathematics, AMS, Vol. 285, (2001), 131 - 144.*

Corrected version Corrections to published version - Kogan, I. A.

**Inductive construction of moving frames.***Contemporary Mathematics, AMS, Vol. 285, (2001), 157 - 170.*

pdf - 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

Kogan, I. A.

Inductive Approach to Cartan's Moving Frame Method with Applications to Classical Invariant Theory

*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 x^{n}+y^{n}+z^{n}.