English
norskBlar i forfatter "Lychagin, Valentin V."
Viser treff 1-17 av 17
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Abelian equations and rank problems for planar webs
(Journal article; Tidsskriftartikkel; Peer reviewed, 2006-05-04)We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincar´e’s theorem: a planar 4- web of maximum rank is linearizable. We also find an invariant intrinsic characterization of planar 4-webs ... -
The categorical theory of relations and quantization
(Working paper; Arbeidsnotat, 2001-10-30)In this paper we develops a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to be symmetric monoidal categories in their own right and are found to be isomorphic to certain categories of A−A bicomodules. Properties of relations are ... -
Compatibility, multi-brackets and integrability of systems of PDEs
(Journal article; Tidsskriftartikkel; Peer reviewed, 2008-02-20)We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi- Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer δ-cohomology of generalized complete intersections and ... -
Differential invariants of the motion group actions
(Working paper; Arbeidsnotat, 2007-12-20)Differential invariants of a (pseudo)group action can vary when restricted to invariant submanifolds (differential equations). The algebra is still governed by the Lie-Tresse theorem, but may change a lot. We describe in details the case of the motion group O(n) ⋉ R<sup>n</sup> acting on the full (unconstraint) jet-space as well as on some invariant equations. -
Dimension of the solutions space of PDEs
(Conference object; Konferansebidrag, 2006-10-26)We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities. -
Feedback Differential Invariants
(Journal article; Tidsskriftartikkel, 2008-12-07)The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found. -
Feedback Equivalence of 1-dimensional Control Systems of the 1-st Order
(Working paper; Arbeidsnotat, 2008-12-09)The problem of local feedback equivalence for 1-dimensional control systems of the 1-st order is considered. The algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found. -
Geodesic Webs and PDE Systems of Euler Equations
(Journal article; Tidsskriftartikkel; Peer reviewed, 2008-10-30)We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x1, ..., xn) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d ≥ n + 1) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler ... -
Geodesic Webs of Hypersurfaces
(Working paper; Arbeidsnotat, 2008-12-11)In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n + 2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provide that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all ... -
Geodesic Webs on a Two-Dimensional Manifold and Euler Equations
(Journal article; Tidsskriftartikkel; Peer reviewed, 2008-10-30)We prove that any planar 4-web defines a unique projective structure in the plane in such a way that the leaves of the web foliations are geodesics of this projective structure. We also find conditions for the projective structure mentioned above to contain an affine symmetric connection, and conditions for a planar 4-web to be equivalent to a geodesic 4-web on an affine symmetric surface. Similar ... -
Invariants of pseudogroup actions: Homological methods and Finiteness theorem
(Working paper; Arbeidsnotat, 2005-12-07)We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of l-variants and l-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: We introduce a complex for covariants, define their cohomology ... -
Linearizability of d-webs, d ≥ 4, on two-dimensional manifolds
(Journal article; Tidsskriftartikkel; Peer reviewed, 2004-03-31)We find d − 2 relative differential invariants for a d-web, d ≥ 4, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f(x, y) and g4(x, y), ..., gd(x, y), then necessary and sufficient conditions for the linearizabilty of a d-web are two PDEs of the fourth order ... -
On a class of linearizable planar geodesic webs
(Working paper; Arbeidsnotat, 2008-12-13)We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb. -
On Rank Problems for Planar Webs and Projective Structures
(Chapter; Bokkapittel, 2008-12-03)We present some old and recent results on rank problems and linearizability of geodesic planar webs -
On the Blaschke Conjecture for 3-Webs
(Journal article; Tidsskriftartikkel; Peer reviewed, 2004-11-21)We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. As a side result, we show that the number of linearizations in the Gronwall conjecture does not exceed fifteen and give criteria for rigidity of 3-webs. -
Spencer δ-cohomology, restrictions, characteristics and involutive symbolic PDEs
(Working paper; Arbeidsnotat, 2005-03-07)We generalize the notion of involutivity to systems of differential equations of different orders and show that the classical results relating involutivity, restrictions, characteristics and characteristicity, known for first order systems, extend to the general context. This involves, in particular, a new definition of strong characteristicity. The proof exploits a spectral sequence relating ... -
Theory of linear G-difference equations
(Journal article; Tidsskriftartikkel; Peer reviewed, 1997-12-17)We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other ...
