@article{agudelo_dippold_klein_kokot_geiger_kogan_2024, title={Euclidean and affine curve reconstruction}, volume={17}, ISSN={["1944-4184"]}, DOI={10.2140/involve.2024.17.29}, number={1}, journal={INVOLVE, A JOURNAL OF MATHEMATICS}, author={Agudelo, Jose and Dippold, Brooke and Klein, Ian and Kokot, Alex and Geiger, Eric and Kogan, Irina}, year={2024} }
 @article{kogan_2023, title={Invariants: Computation and Applications}, url={https://doi.org/10.1145/3597066.3597149}, DOI={10.1145/3597066.3597149}, abstractNote={Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of invariants and designing methods and algorithms to compute them remains an active area of ongoing research with an abundance of applications. In this incredibly vast topic, we focus on two particular themes displaying a fruitful interplay between the differential and algebraic invariant theories. First, we show how an algebraic adaptation of the moving frame method from differential geometry leads to a practical algorithm for computing a generating set of rational invariants. Then we discuss the notion of differential invariant signature, its role in solving equivalence problems in geometry and algebra, and some successes and challenges in designing algorithms based on this notion.}, journal={PROCEEDINGS OF THE INTERNATIONAL SYMPOSIUM ON SYMBOLIC & ALGEBRAIC COMPUTATION, ISSAC 2023}, author={Kogan, Irina A.}, year={2023}, pages={40–49} }
 @article{geiger_kogan_2021, title={Non-congruent non-degenerate curves with identical signatures}, volume={63}, ISSN={["1573-7683"]}, url={https://doi.org/10.1007/s10851-020-01015-x}, DOI={10.1007/s10851-020-01015-x}, abstractNote={While the equality of differential signatures (Calabi et al, Int. J. Comput. Vis. 26: 107-135, 1998) is known to be a necessary condition for congruence, it is not sufficient (Musso and Nicolodi, J. Math Imaging Vis. 35: 68-85, 2009). Hickman (J. Math Imaging Vis. 43: 206-213, 2012, Theorem 2) claimed that for non-degenerate planar curves, equality of Euclidean signatures implies congruence. We prove that while Hickman's claim holds for simple, closed curves with simple signatures, it fails for curves with non-simple signatures. In the later case, we associate a directed graph with the signature and show how various paths along the graph give rise to a family of non-congruent, non-degenerate curves with identical signatures. Using this additional structure, we formulate congruence criteria for non-degenerate, closed, simple curves and show how the paths reflect the global and local symmetries of the corresponding curve.}, number={5}, journal={JOURNAL OF MATHEMATICAL IMAGING AND VISION}, publisher={Springer Science and Business Media LLC}, author={Geiger, Eric and Kogan, Irina A.}, year={2021}, month={Jun}, pages={601–625} }
 @article{geiger_kogan_2021, title={Non-congruent non-degenerate curves with identical signatures (Feb, 1007/s10851-020-01015-x, 2021)}, volume={63}, ISBN={1573-7683}, url={https://doi.org/10.1007/s10851-021-01028-0}, DOI={10.1007/s10851-021-01028-0}, abstractNote={A correction to this paper has been published: https://doi.org/10.1007/s10851-021-01028-0}, number={6}, journal={JOURNAL OF MATHEMATICAL IMAGING AND VISION}, publisher={Springer Science and Business Media LLC}, author={Geiger, Eric and Kogan, Irina A.}, year={2021}, pages={776–776} }
 @article{jenssen_kogan_2020, title={A mixed boundary value problem for u(xy) = f (x , y, u, u(x), u(y))}, volume={268}, ISSN={["1090-2732"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85076254291&partnerID=MN8TOARS}, DOI={10.1016/j.jde.2019.11.063}, abstractNote={Consider a single hyperbolic PDE uxy=f(x,y,u,ux,uy), with locally prescribed data: u along a non-characteristic curve M and ux along a non-characteristic curve N. We assume that M and N are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the (x,y)-plane. It is known that if M is located above N, then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when M lies below N, the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function f). The construction, via Picard iteration, makes use of a careful choice of additional u-data which are updated in each iteration step.}, number={12}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Jenssen, Helge Kristian and Kogan, Irina A.}, year={2020}, month={Jun}, pages={7535–7560} }
 @article{kogan_ruddy_vinzant_2020, title={Differential Signatures of Algebraic Curves}, url={https://doi.org/10.1137/19M1242859}, DOI={10.1137/19M1242859}, abstractNote={In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) in such a way that two generic curves have the same signatures if and only if they are $G$-equivalent. We prove that for any $G$-action, there exists a pair of rational differential invariants, called classifying invariants, that can be used to construct signatures. We derive a formula for the degree of a signature curve in terms of the degree of the original curve, the size of its symmetry group and some quantities depending on a choice of classifying invariants. For the full projective group, as well as for its affine, special affine and special Euclidean subgroups, we give explicit sets of rational classifying invariants and derive a formula for the degree of the signature curve of a generic curve as a quadratic function of the degree of the original curve. We show that this generic degree is the sharp upper bound.}, journal={SIAM Journal on Applied Algebra and Geometry}, author={Kogan, Irina A. and Ruddy, Michael and Vinzant, Cynthia}, year={2020}, month={Jan} }
 @article{benfield_jenssen_kogan_2019, title={Jacobians with prescribed eigenvectors}, volume={65}, ISBN={1872-6984}, ISSN={0926-2245}, url={http://dx.doi.org/10.1016/j.difgeo.2019.03.008}, DOI={10.1016/j.difgeo.2019.03.008}, abstractNote={Let Ω⊂Rn be open and let R be a partial frame on Ω; that is, a set of m linearly independent vector fields prescribed on Ω (m≤n). We consider the issue of describing the set of all maps F:Ω→Rn with the property that each of the given vector fields is an eigenvector of the Jacobian matrix of F. By introducing a coordinate independent definition of the Jacobian, we obtain an intrinsic formulation of the problem, which leads to an overdetermined PDE system, whose compatibility conditions can be expressed in an intrinsic, coordinate independent manner. To analyze this system we formulate and prove a generalization of the classical Frobenius integrability theorems. The size and structure of the solution set of this system depends on the properties of the partial frame; in particular, whether or not it is in involution. A particularly nice subclass of involutive partial frames, called rich frames, can be completely analyzed. The involutive, non-rich case is somewhat harder to handle. We provide a complete answer in the case of m=3 and arbitrary n, as well as some general results for arbitrary m. The non-involutive case is far more challenging, and we only obtain a comprehensive analysis in the case n=3, m=2. Finally, we provide explicit examples illustrating the various possibilities.}, journal={Differential Geometry and its Applications}, publisher={Elsevier BV}, author={Benfield, Michael and Jenssen, Helge Kristian and Kogan, Irina A.}, year={2019}, month={Aug}, pages={108–146} }
 @article{benfield_jenssen_kogan_2018, title={A Generalization of an Integrability Theorem of Darboux}, volume={29}, ISSN={1050-6926 1559-002X}, url={http://dx.doi.org/10.1007/s12220-018-00119-6}, DOI={10.1007/s12220-018-00119-6}, abstractNote={In his monograph “Systèmes Orthogonaux” (Darboux, in Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, Gauthier-Villars, Paris, 1910) Darboux stated three theorems providing local existence and uniqueness of solutions to first-order systems of the type $$\begin{aligned} \partial _{x_i} u_\alpha (x)=f^\alpha _i(x,u(x)), \quad i\in I_\alpha \subseteq \{1,\dots ,n\}. \end{aligned}$$For a given point $${\bar{x}}\in \mathbb {R}^n$$ it is assumed that the values of the unknown $$u_\alpha $$ are given locally near $${\bar{x}}$$ along $$\{x\,|\, x_i={\bar{x}}_i \, \text {for each}\, i\in I_\alpha \}$$. The more general of the theorems, Théorème III, was proved by Darboux only for the cases $$n=2$$ and 3. In this work we formulate and prove a generalization of Darboux’s Théorème III which applies to systems of the form $$\begin{aligned} {{\mathbf {r}}}_i(u_\alpha )\big |_x = f_i^\alpha (x, u(x)), \quad i\in I_\alpha \subseteq \{1,\dots ,n\} \end{aligned}$$where $${\mathcal {R}}=\{{{\mathbf {r}}}_i\}_{i=1}^n$$ is a fixed local frame of vector fields near $${\bar{x}}$$. The data for $$u_\alpha $$ are prescribed along a manifold $$\Xi _\alpha $$ containing $${\bar{x}}$$ and transverse to the vector fields $$\{{{\mathbf {r}}}_i\,|\, i\in I_\alpha \}$$. We identify a certain Stable Configuration Condition (SCC). This is a geometric condition that depends on both the frame $${\mathcal {R}}$$ and on the manifolds $$\Xi _\alpha $$; it is automatically met in the case considered by Darboux. Assuming the SCC and the relevant integrability conditions are satisfied, we establish local existence and uniqueness of a $$C^1$$-solution via Picard iteration for any number of independent variables n.}, number={4}, journal={The Journal of Geometric Analysis}, publisher={Springer Nature}, author={Benfield, Michael and Jenssen, Helge Kristian and Kogan, Irina A.}, year={2018}, month={Nov}, pages={3470–3493} }
 @article{hong_hough_kogan_2017, title={Algorithm for computing mu-bases of univariate polynomials}, volume={80}, ISSN={["0747-7171"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84995550983&partnerID=MN8TOARS}, DOI={10.1016/j.jsc.2016.08.013}, abstractNote={We present a new algorithm for computing a μ-basis of the syzygy module of n polynomials in one variable over an arbitrary field K. The algorithm is conceptually different from the previously-developed algorithms by Cox, Sederberg, Chen, Zheng, and Wang for n=3, and by Song and Goldman for an arbitrary n. The algorithm involves computing a "partial" reduced row-echelon form of a (2d+1)×n(d+1) matrix over K, where d is the maximum degree of the input polynomials. The proof of the algorithm is based on standard linear algebra and is completely self-contained. The proof includes a proof of the existence of the μ-basis and as a consequence provides an alternative proof of the freeness of the syzygy module. The theoretical (worst case asymptotic) computational complexity of the algorithm is O(d2n+d3+n2). We have implemented this algorithm (HHK) and the one developed by Song and Goldman (SG). Experiments on random inputs indicate that SG is faster than HHK when d is sufficiently large for a fixed n, and that HHK is faster than SG when n is sufficiently large for a fixed d.}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, publisher={Elsevier BV}, author={Hong, Hoon and Hough, Zachary and Kogan, Irina A.}, year={2017}, pages={844–874} }
 @article{degree-optimal moving frames for rational curves_2017, year={2017}, month={Mar} }
 @article{on two theorems of darboux_2017, year={2017}, month={Sep} }
 @article{kogan_olver_2015, title={Invariants of objects and their images under surjective maps}, volume={36}, ISSN={1995-0802 1818-9962}, url={http://dx.doi.org/10.1134/s1995080215030063}, DOI={10.1134/s1995080215030063}, abstractNote={We examine the relationships between the differential invariants of objects and of their images under a surjective maps. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of the entire group acts projectably. In the former case, we establish a constructible isomorphism between the algebra of differential invariants of the images and the algebra of fiber-wise constant (gauge) differential invariants of the objects. In the latter case, we describe residual effects of the full transformation group on the image invariants. Our motivation comes from the problem of reconstruction of an object from multiple-view images, with central and parallel projections of curves from three-dimensional space to the two-dimensional plane serving as our main examples.}, number={3}, journal={Lobachevskii Journal of Mathematics}, publisher={Pleiades Publishing Ltd}, author={Kogan, I. A. and Olver, P. J.}, year={2015}, month={Jul}, pages={260–285} }
 @article{burdis_kogan_hong_2013, title={Object-Image Correspondence for Algebraic Curves under Projections}, volume={9}, ISSN={["1815-0659"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84875148955&partnerID=MN8TOARS}, DOI={10.3842/sigma.2013.023}, abstractNote={We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters.The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters.A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution.The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve.Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations.To solve the latter problem we make an algebraic adaptation of signature construction that has been used to solve the equivalence problems for smooth curves.We introduce a notion of a classifying set of rational differential invariants and produce explicit formulas for such invariants for the actions of the projective and the affine groups on the plane.}, journal={SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS}, publisher={SIGMA (Symmetry, Integrability and Geometry: Methods and Application)}, author={Burdis, Joseph M. and Kogan, Irina A. and Hong, Hoon}, year={2013} }
 @article{jenssen_kogan_2012, title={Extensions for Systems of Conservation Laws}, volume={37}, ISSN={["1532-4133"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84861547404&partnerID=MN8TOARS}, DOI={10.1080/03605302.2011.626827}, abstractNote={Extensions are central in the theory of conservation laws by providing intrinsic selection criteria for weak solutions. Given a system u t + f(u) x = 0 the extensions solve certain second order PDEs which are typically overdetermined. Determining the size of the set of extensions can be a challenging task. Instead of analyzing this second order system directly, we consider the equations satisfied by the lengths β i of the eigenvectors r i of the Jacobian matrix Df, measured with the inner product defined by an extension. For a given eigen-frame {r i } the extensions are determined uniquely, up to trivial affine parts, by these lengths. The β i solve a first order algebraic-differential system (the β-system) to which standard integrability theorems can be applied. The number of extensions is determined by determining the number of free constants and functions in the general solution to the β-system. We provide a complete breakdown for 3 × 3-systems, and for rich frames whose β-system has trivial algebraic part. Our framework is motivated by the work [16] where the authors consider conservative systems with prescribed eigen-frames. It is natural to ask how many extensions the resulting systems have, and the answer depends in an essential manner on the prescribed frame.}, number={6}, journal={COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS}, author={Jenssen, Helge Kristian and Kogan, Irina A.}, year={2012}, pages={1096–1140} }
 @inproceedings{burdis_kogan_2012, title={Object-image correspondence for curves under central and parallel projections}, ISBN={9781450312998}, url={http://dx.doi.org/10.1145/2261250.2261306}, DOI={10.1145/2261250.2261306}, abstractNote={We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. The latter problem is then solved using a separating set of rational differential invariants. A similar approach can be used to solve the projection problem for finite lists of points. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters.}, booktitle={Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12}, publisher={ACM Press}, author={Burdis, Joseph M. and Kogan, Irina A.}, year={2012}, pages={373–382} }
 @article{feng_kogan_krim_2010, title={Classification of Curves in 2D and 3D via Affine Integral Signatures}, volume={109}, ISSN={["1572-9036"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77949264986&partnerID=MN8TOARS}, DOI={10.1007/s10440-008-9353-9}, abstractNote={We propose new robust classification algorithms for planar and spatial curves subjected to affine transformations. Our motivation comes from the problems in computer image recognition. To each planar or spatial curve, we assign a planar signature curve. Curves, equivalent under an affine transformation, have the same signature. The signatures are based on integral invariants, which are significantly less sensitive to small perturbations of curves and noise than classically known differential invariants. Affine invariants are derived in terms of Euclidean invariants. We present two types of signatures: the global and the local signature. Both signatures are independent of curve parameterization. The global signature depends on a choice of the initial point and, therefore, cannot be used for local comparison. The local signature, albeit being slightly more sensitive to noise, is independent of the choice of the initial point and can be used to solve local equivalence problem. An experiment that illustrates robustness of the proposed signatures is presented.}, number={3}, journal={ACTA APPLICANDAE MATHEMATICAE}, author={Feng, Shuo and Kogan, Irina and Krim, Hamid}, year={2010}, month={Mar}, pages={903–937} }
 @article{jenssen_kogan_2010, title={SYSTEMS OF HYPERBOLIC CONSERVATION LAWS WITH PRESCRIBED EIGENCURVES}, volume={7}, ISSN={["1793-6993"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-77954208923&partnerID=MN8TOARS}, DOI={10.1142/s021989161000213x}, abstractNote={We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically overdetermined system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed using appropriate integrability theorems (Frobenius, Darboux and Cartan–Kähler). We give a complete analysis of the possible scenarios, including examples, for systems of three equations. As an application we characterize conservative systems with the same eigencurves as the Euler system for 1-dimensional compressible gas dynamics. The case of general rich systems of any size (i.e. when the given eigenvector fields are pairwise in involution; this includes all systems of two equations) is completely resolved and we consider various examples in this class.}, number={2}, journal={JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS}, author={Jenssen, Helge Kristian and Kogan, Irina A.}, year={2010}, month={Jun}, pages={211–254} }
 @misc{jenssen_kogan_2009, title={Construction of conservative systems}, ISBN={9780821847305 9780821892831}, ISSN={0160-7634 2324-7088}, url={http://dx.doi.org/10.1090/psapm/067.2/2605263}, DOI={10.1090/psapm/067.2/2605263}, abstractNote={We consider the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves. This yields an overdetermined system of equations for the eigenvalues-to-be. These equations are analyzed with techniques from exterior differential systems.}, journal={Hyperbolic Problems: Theory, Numerics and Applications}, publisher={American Mathematical Society}, author={Jenssen, Helge Kristian and Kogan, Irina A.}, year={2009}, pages={673–682} }
 @inproceedings{feng_krim_kogan_2007, title={3D Face Recognition using Euclidean Integral Invariants Signature}, ISBN={9781424411979 9781424411986}, url={http://dx.doi.org/10.1109/ssp.2007.4301238}, DOI={10.1109/ssp.2007.4301238}, abstractNote={A novel 3D face representation and recognition approach is presented in this paper. We represent a 3D face by a set of level curves of geodesic function starting from the nose tip, which is invariant under isometric transformation of the surfaces. A pose change induces a special Euclidean transformation (a composition of a rotation and a translation) of the surface that represents a face and leads to the Euclidean transformation of the iso-geodesic curves. A change of facial expression induces isometric transformation of the iso-geodesic curves. Although the set of isometric transformations of a surface is larger than the set of Euclidean transformations in 3D, we assume that iso-geodesic curves undergo piecewise Euclidean transformations, i.e. the transformation of relatively small segments of the level curves is Euclidean. A Euclidean invariant integral signature for curves in 3D is presented in this paper. Euclidean invariant integral signature provides a classification of spatial curves which is independent of their position in 3D space and parameterization, and is not sensitive to noise. A recognition procedure based on comparing face feature in the invariant signature space is proposed. Substantiating examples are provided with an achieved classification accuracy of 95% for faces with various poses and facial expressions.}, booktitle={2007 IEEE/SP 14th Workshop on Statistical Signal Processing}, publisher={IEEE}, author={Feng, S. and Krim, H. and Kogan, I. A.}, year={2007}, month={Aug}, pages={156–160} }
 @inproceedings{feng_aouada_krim_kogan_2007, title={3D Mixed Invariant and its Application on Object Classification}, volume={1}, ISBN={1424407273}, url={http://dx.doi.org/10.1109/icassp.2007.366716}, DOI={10.1109/icassp.2007.366716}, abstractNote={A new integro-differential invariant for curves in 3D transformed by affine group action is presented in this paper. The derivatives involved are of the first order, and therefore this invariant is significantly less sensitive to noise than classical affine differential invariants, the simplest of which involves derivatives of order 5. A classification procedure based on characteristic curves of an object surface is considered using our proposed mixed invariants. Substantiating examples are provided to verify efficiency and discriminant power of the characteristic spatial curve based 3D object classification.}, booktitle={2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07}, publisher={IEEE}, author={Feng, S. and Aouada, D. and Krim, H. and Kogan, I.}, year={2007}, month={Apr} }
 @inproceedings{smith_hollebrands_iwancio_kogan_2007, title={College geometry students' uses of technology in the process of constructing arguments}, booktitle={Proceedings of the 29th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education.}, author={Smith, R. and Hollebrands, K. and Iwancio, K. and Kogan, I.A.}, editor={Lamberg, T.Editor}, year={2007}, pages={1153–1160} }
 @inproceedings{feng_kogan_krim_2007, title={Integral invariants for 3D curves: an inductive approach}, volume={6508}, url={http://dx.doi.org/10.1117/12.707278}, DOI={10.1117/12.707278}, abstractNote={In this paper we obtain, for the first time, explicit formulae for integral invariants for curves in 3D with respect to the special and the full affine groups. Using an inductive approach we first compute Euclidean integral invariants and use them to build the affine invariants. The motivation comes from problems in computer vision. Since integration diminishes the effects of noise, integral invariants have advantage in such applications. We use integral invariants to construct signatures that characterize curves up to the special affine transformations.}, number={PART 1}, booktitle={Visual Communications and Image Processing 2007}, publisher={SPIE}, author={Feng, Shuo and Kogan, Irina A. and Krim, Hamid}, editor={Chen, Chang Wen and Schonfeld, Dan and Luo, JieboEditors}, year={2007}, month={Jan} }
 @article{hubert_kogan_2007, title={Rational invariants of a group action. Construction and rewriting}, volume={42}, ISSN={["0747-7171"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-33751417574&partnerID=MN8TOARS}, DOI={10.1016/j.jsc.2006.03.005}, abstractNote={Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is added to the ideal of the graph. Zero-dimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gröbner basis allows us to express any rational invariant in terms of the generators.}, number={1-2}, journal={JOURNAL OF SYMBOLIC COMPUTATION}, author={Hubert, Evelyne and Kogan, Irina A.}, year={2007}, pages={203–217} }
 @article{hubert_kogan_2007, title={Smooth and algebraic invariants of a group action. Local and global constructions.}, volume={7}, number={4}, journal={Foundations of Computational Math.}, author={Hubert, E. and Kogan, I.A.}, year={2007}, pages={345–383} }
 @article{hubert_kogan_2007, title={Smooth and algebraic invariants of a group action: Local and global constructions}, volume={7}, ISSN={["1615-3375"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-36549030235&partnerID=MN8TOARS}, DOI={10.1007/s10208-006-0219-0}, abstractNote={We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational invariants and play a role analogous to Cartan's normalized invariants in the smooth theory. The algebraic algorithms can be used for computing fundamental sets of differential invariants.}, number={4}, journal={FOUNDATIONS OF COMPUTATIONAL MATHEMATICS}, author={Hubert, Evelyne and Kogan, Irina A.}, year={2007}, month={Nov}, pages={455–493} }
 @inproceedings{smith_hollebrands_iwancio_kogan_2007, title={The effects of a dynamic program for geometry on college students' understandings of properties of quadrilaterals in the Poincare Disk model}, booktitle={Proceedings of the 9th International Conference on Mathematics Education in a Global Community}, author={Smith, R. and Hollebrands, K. and Iwancio, K. and Kogan, I.A.}, year={2007}, pages={613–618} }
 @inproceedings{baloch_krim_kogan_zenkov_2005, title={3D object representation with topo-geometric shape models}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-84863687976&partnerID=MN8TOARS}, booktitle={Proceedings of European Signal Processing Conference (EUSIPCO)}, author={Baloch, S. and Krim, H. and Kogan, I.A. and Zenkov, D.V.}, year={2005}, pages={2386–2389} }
 @inproceedings{baloch_krim_kogan_zenkov_2005, title={Rotation invariant topology coding of 2D and 3D objects using Morse theory}, volume={3}, ISBN={0780391349}, url={http://dx.doi.org/10.1109/icip.2005.1530512}, DOI={10.1109/icip.2005.1530512}, abstractNote={In this paper, we propose a numerical algorithm for extracting the topology of a three-dimensional object (2 dimensional surface) embedded in a three-dimensional space /spl Ropf//sup 3/. The method is based on capturing the topology of a modified Reeb graph by tracking the critical points of a distance function. As such, the approach employs Morse theory in the study of translation, rotation, and scale invariant skeletal graphs. The latter are useful in the representation and classification of objects in /spl Ropf//sup 3/.}, booktitle={IEEE International Conference on Image Processing 2005}, publisher={IEEE}, author={Baloch, S. and Krim, H. and Kogan, I. and Zenkov, D.}, year={2005}, pages={796–799} }
 @article{kogan_olver_2003, title={Invariant Euler-Lagrange equations and the invariant variational bicomplex}, volume={76}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-0037393424&partnerID=MN8TOARS}, DOI={10.1023/A:1022993616247}, number={2}, journal={Acta Applicandae Mathematicae}, author={Kogan, I.A. and Olver, P.}, year={2003}, pages={137–193} }
 @article{kogan_2003, title={Two Algorithms for a Moving Frame Construction}, volume={55}, ISSN={0008-414X 1496-4279}, url={http://dx.doi.org/10.4153/cjm-2003-013-2}, DOI={10.4153/cjm-2003-013-2}, abstractNote={Abstract}, number={2}, journal={Canadian Journal of Mathematics}, publisher={Canadian Mathematical Society}, author={Kogan, Irina A.}, year={2003}, month={Apr}, pages={266–291} }
 @inproceedings{kogan_maza_2002, title={Computation of canonical forms for ternary cubics}, ISBN={1581134843}, url={http://dx.doi.org/10.1145/780506.780526}, DOI={10.1145/780506.780526}, abstractNote={In this paper we conduct a careful study of the equivalence classes of ternary cubics under general complex linear changes of variables. Our new results are based on the method of moving frames and involve triangular decompositions of algebraic varieties. We provide a computationally efficient algorithm that matches an arbitrary ternary cubic with its canonical form and explicitly computes a corresponding linear change of coordinates. We also describe a classification of the symmetry groups of ternary cubics.}, booktitle={Proceedings of the 2002 international symposium on Symbolic and algebraic computation  - ISSAC '02}, publisher={ACM Press}, author={Kogan, Irina A. and Maza, Marc Moreno}, year={2002}, pages={151–160} }
 @misc{kogan_2001, title={Inductive construction of moving frames}, ISBN={9780821829646 9780821878750}, ISSN={1098-3627 0271-4132}, url={http://dx.doi.org/10.1090/conm/285/04741}, DOI={10.1090/conm/285/04741}, abstractNote={This paper presents a useful variation on the moving frame construction, which allows us to use a moving frame for a subgroup A of a Lie group G to produce a moving frame for the entire group G. This algorithm is applicable when G factors as a product of two subgroups G = A · B and automatically produces functional relations among invariants of G and its factors.}, journal={The Geometrical Study of Differential Equations}, publisher={American Mathematical Society}, author={Kogan, Irina A.}, year={2001}, pages={157–170} }
 @misc{kogan_olver_2001, title={The invariant variational bicomplex}, ISBN={9780821829646 9780821878750}, ISSN={1098-3627 0271-4132}, url={http://dx.doi.org/10.1090/conm/285/04739}, DOI={10.1090/conm/285/04739}, abstractNote={We establish a group-invariant version of the variational bicomplex that is based on a general moving frame construction. The main application is an explicit group-invariant formula for the Euler-Lagrange equations of an invariant variational problem.}, journal={The Geometrical Study of Differential Equations}, publisher={American Mathematical Society}, author={Kogan, Irina A. and Olver, Peter J.}, year={2001}, pages={131–144} }
 @phdthesis{inductive approach to cartan's moving frame method with applications to classical invariant theory_2000, url={https://www-proquest-com.prox.lib.ncsu.edu/docview/304612470/fulltextPDF/23BAF25E0A874972PQ/8?accountid=12725}, year={2000} }
 @article{berchenko_olver_2000, title={Symmetries of Polynomials}, volume={29}, ISSN={0747-7171}, url={http://dx.doi.org/10.1006/jsco.1999.0307}, DOI={10.1006/jsco.1999.0307}, abstractNote={Abstract New algorithms for determining discrete and continuous symmetries of polynomials - also known as binary forms in classical invariant theory - are presented, and implemented in MAPLE. The results are based on a new, comprehensive theory of moving frames that completely characterizes the equivalence and symmetry properties of submanifolds under general Lie group actions.}, number={4-5}, journal={Journal of Symbolic Computation}, publisher={Elsevier BV}, author={Berchenko, Irina and Olver, Peter J.}, year={2000}, month={May}, pages={485–514} }