Irina Kogan
Associate Professor
Department of MathematicsNorth Carolina State University
SAS Hall, room 3146
Campus Box 8205 , Raleigh, NC, 27695-8205
Phone: (919) 513-7437, Fax: (919) 513-7336
Brief Vita
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Irina KoganAssociate ProfessorDepartment of MathematicsNorth Carolina State University SAS Hall, room 3146 Campus Box 8205 , Raleigh, NC, 27695-8205 Phone: (919) 513-7437, Fax: (919) 513-7336 |
Brief Vita |
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Abstract: 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.
Abstract: 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.
Abstract: 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.
Abstract: 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.
Abstract: 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.
Abstract: 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 to express any rational invariant in terms of the generators.
Abstract: In this paper, we derive an explicit group invariant formula for the Euler-Lagrange equations associated with an invariant variational problem. The method relies on a group invariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independent interest.
Abstract: The method of moving frames, introduced by Elie Cartan, is a powerful tool for the solution of various equivalence problems. The practical implementation of Cartan's method, however, remains to be challenging, despite its later significant development and generalization. This paper presents two variations on the Fels and Olver algorithm, which under some conditions on the group action, simplify a moving frame construction. In addition, the first algorithm leads to a better understanding of invariant differential forms on the jet bundles, while the second expresses the differential invariants for the entire group in terms of the differential invariants of its subgroup.
Abstract: 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.
Abstract: 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.
Abstract: 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.
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.
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.