arXiv subjects

Irina A. Kogan

Publications and source records attributed to Irina A. Kogan.

18 recordsLinked to original sources

arXiv2025

Symmetry analysis and new partially invariant solutions for the gas dynamics system with a special equation of state

This paper is a contribution to the symmetry analysis of the gas dynamics system in the vein of the ''podmodeli'' (submodels) program outlined by Ovsyannikov (1994). We consider the case of the special state equation, prescribing pressure to be the sum of entropy and an arbitrary function of density. Such a system has a 12-dimensional symmetry Lie algebra. This work advances the study of its four-dimensional subalgebras, continuing the work started in Siraeva (2024). For a large subset of not previously considered, non-similar four-dimensional subalgebras from an optimal list in Siraeva (2014), we compute a complete set of generating invariants. For one of the subalgebras, we construct a partially symmetry-reduced system. We explicitly solve this reduced system (submodel). This leads to new families of explicit solutions of the original system. We analyze the trajectories of these solutions. Additionally, we match each of the subalgebras considered in this paper with its isomorphism class, planting a seed for future study of the hierarchy of the reduced systems.

math.AP↗

arXiv2024

Invariants: Computation and Applications

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.

cs.SC↗

arXiv2024

Equi-affine minimal-degree moving frames for polynomial curves

We develop a theory and an algorithm for constructing minimal-degree polynomial moving frames for polynomial curves in an affine space. The algorithm is equivariant under volume-preserving affine transformations of the ambient space and the parameter shifts. We show that any matrix-completion algorithm can be turned into an equivariant moving frame algorithm via an equivariantization procedure that we develop. We prove that if a matrix-completion algorithm is of minimal degree then so is the resulting equivariant moving frame algorithm. We propose a novel minimal-degree matrix-completion algorithm, complementing the existing body of literature on this topic.

math.AG↗

arXiv2021

Non-congruent non-degenerate curves with identical signatures

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.

math.DG↗

arXiv2019

A mixed boundary value problem for $u_{xy}=f(x,y,u,u_x,u_y)$

Consider a single hyperbolic PDE $u_{xy}=f(x,y,u,u_x,u_y)$, with locally prescribed data: $u$ along a non-characteristic curve $M$ and $u_x$ 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.

math.AP↗

arXiv2019

Differential signatures of algebraic curves

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.

math.AG↗

arXiv2019

Object-Image Correspondence for Algebraic Curves under Projections

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.

math.AG↗

arXiv2018

A generalization of an integrability theorem of Darboux

In his monograph "Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. Principes de géométrie analytique", 1910, Darboux stated three theorems providing local existence and uniqueness of solutions to first order systems of the type \[\partial_{x_i} u_α(x)=f^α_i(x,u(x)),\quad i\in I_α\subseteq\{1,\dots,n\}.\] For a given point $\bar x\in \mathbb{R}^n$ it is assumed that the values of the unknown $u_α$ are given locally near $\bar x$ along $\{x\,|\, x_i=\bar x_i \, \text{for each}\, i\in I_α\}$. 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 \[{\mathbf r}_i(u_α)\big|_x = f_i^α(x, u(x)), \quad i\in I_α\subseteq\{1,\dots,n\}\] where $\mathcal R=\{{\mathbf r}_i\}_{i=1}^n$ is a fixed local frame of vector fields near $\bar x$. The data for $u_α$ are prescribed along a manifold $Ξ_α$ containing $\bar x$ and transverse to the vector fields $\{{\mathbf r}_i\,|\, i\in I_α\}$. 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 $Ξ_α$; 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 $n$.

math.AP↗

arXiv2018

Degree-optimal moving frames for rational curves

A $\mathit{\text{moving frame}}$ at a rational curve is a basis of vectors moving along the curve. When the rational curve is given parametrically by a row vector $\mathbf{a}$ of univariate polynomials, a moving frame with important algebraic properties can be defined by the columns of an invertible polynomial matrix $P$, such that $\mathbf{a} P=[\gcd(\mathbf{a}),0\ldots,0]$. A $\mathit{\text{degree-optimal moving frame}}$ has column-wise minimal degree, where the degree of a column is defined to be the maximum of the degrees of its components. Algebraic moving frames are closely related to the univariate versions of the celebrated Quillen-Suslin problem, effective Nullstellensatz problem, and syzygy module problem. However, this paper appears to be the first devoted to finding an efficient algorithm for constructing a degree-optimal moving frame, a property desirable in various applications. We compare our algorithm with other possible approaches, based on already available algorithms, and show that it is more efficient. We also establish several new theoretical results concerning the degrees of an optimal moving frame and its components. In addition, we show that any deterministic algorithm for computing a degree-optimal algebraic moving frame can be augmented so that it assigns a degree-optimal moving frame in a $GL_n(\mathbb{K})$-equivariant manner. This crucial property of classical geometric moving frames, in combination with the algebraic properties, can be exploited in various problems.

math.AG↗

arXiv2017

On Two Theorems of Darboux

We provide precise formulations and proofs of two theorems from Darboux's lectures on orthogonal systems. These results provide local existence and uniqueness of solutions to certain types of first order PDE systems where each equation contains a single derivative for which it is solved: \[\frac{\partial u_i}{\partial x_j}(x)=f_{ij}(x,u(x)).\] The data prescribe values for the unknowns $u_i$ along certain hyperplanes through a given point $\bar x$. The first theorem applies to determined systems (the number of equations equals the number unknowns), and a unique, local solution is obtained via Picard iteration. While Darboux's statement of the theorem leaves unspecified "certaines conditions de continuité," it is clear from his proof that he assumes Lipschitz continuity of the maps $f_{ij}$. On the other hand, he did not address the regularity of the data. We provide a precise formulation and proof of his first theorem. The second theorem is more involved and applies to overdetermined systems of the same general form. Under the appropriate integrability conditions, Darboux used his first theorem to treat the cases with two and three independent variables. We provide a proof for any number of independent variables. While the systems are rather special, they do appear in applications; e.g., the second theorem contains the classical Frobenius theorem on overdetermined systems as a special case. The key aspect of the proofs is that they apply to non-analytic situations. In an analytic setup the results are covered by the general Cartan-Kähler theorem.

math.AP↗

arXiv2017

Jacobians with prescribed eigenvectors

Let $Ω\subset \mathbb{R}^n$ be open and let $\mathcal{R}$ be a partial frame on $Ω$, that is a set of $m$ linearly independent vector fields prescribed on $Ω$ ($m\leq n$). We consider the issue of describing the set of all maps $F:Ω\to\mathbb{R}^n$ 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, can be completely analyzed. Involutive, but 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. Our initial motivation for considering this problem comes from the geometric study of hyperbolic conservative systems in one spatial dimension.

math.DG↗

arXiv2017

Algorithm for computing $μ$-bases of univariate polynomials

We present a new algorithm for computing a $μ$-basis of the syzygy module of $n$ polynomials in one variable over an arbitrary field $\mathbb{K}$. 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 $n$. It involves computing a "partial" reduced row-echelon form of a $ (2d+1)\times n(d+1)$ matrix over $\mathbb{K}$, 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. It 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 $O(d^2n+d^3+n^2)$. We have implemented this algorithm (HHK) and the one developed by Song and Goldman (SG). Experiments on random inputs indicate that SG gets faster than HHK when $d$ gets sufficiently large for a fixed $n$, and that HHK gets faster than SG when $n$ gets sufficiently large for a fixed $d$.

math.AG↗

arXiv2015

Invariants of objects and their images under surjective maps

We examine the relationships between the differential invariants of objects and of their images under a surjective map. 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.

math.DG↗

arXiv2013

Object-image correspondence for curves under projections

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 main 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 by differential signature construction based on Cartan's moving frame method. A similar approach can be used to decide whether a given finite set of ordered points on a plane is an image of a given finite set of ordered points in R^3. 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.

math.AG↗

arXiv2011

Extensions for Systems of Conservation Laws

Extensions (entropies) play a central role in the theory of hyperbolic conservation laws by providing intrinsic selection criteria for weak solutions. For a given hyperbolic system u_t+f(u)_x=0, a standard approach is to analyze directly the second order PDE system for the extensions. Instead we find it advantageous to consider the equations satisfied by the lengths beta^i of the right eigenvectors r_i the Jacobian matrix Df, as measured with respect to the inner product defined by an extension. Our geometric formulation provides a natural and systematic approach to existence of extensions. By prescribing the eigen-fields r_i our results automatically apply to all systems with the same eigen-frame. The equations for the lengths beta^i form a first order algebraic-differential system (the beta-system) to which standard integrability theorems can be applied. The size of the set of extensions follows by determining the number of free constants and functions present in the general solution to the beta-system. We provide a complete breakdown of the various possibilities for systems of three equations, as well as for rich hyperbolic systems of any size.

math.AP↗

arXiv2011

Object-image correspondence for curves under finite and affine cameras

We provide criteria 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. These criteria reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. The latter problem can be addressed using Cartan's moving frame method. This leads to a novel algorithmic solution of the projection problem for curves. The computational advantage of the algorithms presented here, in comparison to algorithms based on a straightforward solution, lies in a significant reduction of a number of real parameters that has 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. The same approach can be used to decide whether a given finite set of ordered points on a plane is an image of a given finite set of ordered points in R^3. 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.

cs.CV↗

arXiv2009

Systems of hyperbolic conservation laws with prescribed eigencurves

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-Kahler). 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.

math.AP↗

arXiv2005

Rational, Replacement, and Local Invariants of a Group Action

The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame method in differential geometry. The generating set of rational invariants appears as the coefficients of a Groebner basis, reduction with respect to which allows to express a rational invariant in terms of the generators. The replacement invariants, introduced in the paper, are tuples of algebraic functions of the rational invariants. Any invariant, whether rational, algebraic or local, can be can be rewritten terms of replacement invariants by a simple substitution.

math.AC↗