Now showing items 1-17 of 17
Abstract: | 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 of rank two and one and prove that in general such webs are not linearizable. This solves the Blaschke problem “to find invariant conditions for a planar 4-web to be of rank 1 or 2 or 3”. Finally, we find invariant characterization of planar 5-webs of maximum rank and prove than in general such webs are not linearizable. |
Description: | Dette er forfatternes aksepterte versjon. |
URI: | http://hdl.handle.net/10037/2049 |
Abstract: | 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 evaluate the formal functional dimension of the solutions space. The results are applied to establish new integration methods and solve several differential-geometric problems. |
Description: | Dette er forfatternes aksepterte versjon |
URI: | http://hdl.handle.net/10037/2048 |
Abstract: | 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^{n} acting on the full (unconstraint) jet-space as well as on some invariant equations. |
URI: | http://hdl.handle.net/10037/2051 |
Abstract: | 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. |
URI: | http://hdl.handle.net/10037/2050 |
Abstract: | 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. |
URI: | http://hdl.handle.net/10037/1853 |
Abstract: | 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. |
URI: | http://hdl.handle.net/10037/1852 |
Abstract: | 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 equations, and for flat connections we give an explicit construction of their solutions. |
Description: | Dette er forfatternes aksepterte versjon |
URI: | http://hdl.handle.net/10037/2046 |
Abstract: | 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 web foliations are totally geodesic, and affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem. |
URI: | http://hdl.handle.net/10037/1855 |
Abstract: | 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 results are obtained for planar d-webs,d >4, provided that additional d −4 second-order invariants vanish. |
Description: | Dette er forfatternes aksepterte versjon |
URI: | http://hdl.handle.net/10037/2047 |
Abstract: | 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 and prove the finiteness theorem. This implies the well-known Lie-Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations. |
URI: | http://hdl.handle.net/10037/2053 |
Abstract: | 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 with respect to f and g4, and d − 4 PDEs of the second order with respect to f and g4, ..., gd. For d = 4, this result confirms Blaschke’s conjecture on the nature of conditions for the linearizabilty of a 4-web. We also give Mathematica codes for testing 4- and d-webs (d > 4) for linearizability and examples of their usage. |
Description: | Dette er forfatternes aksepterte versjon. |
URI: | http://hdl.handle.net/10037/2056 |
Abstract: | We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb. |
URI: | http://hdl.handle.net/10037/1854 |
Abstract: | We present some old and recent results on rank problems and linearizability of geodesic planar webs |
Description: | Rettighetshaver (Forfatter) har gitt oss tillatelse til å legge ut artikkelen (forfatternes aksepterte versjon) |
URI: | http://hdl.handle.net/10037/2044 |
Abstract: | 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. |
Description: | Dette er forfatternes aksepterte versjon. |
URI: | http://hdl.handle.net/10037/2055 |
Abstract: | 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 Spencer δ-cohomology of a symbolic system and its restriction to a non-characteristic subspace. |
URI: | http://hdl.handle.net/10037/2052 |
Abstract: | 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 equations, symmetries of an equation are module endomorphisms and conserved structures are invariants in the tensor algebra of the given equation. We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions incluse these as a special case. |
Description: | This is the author’s final accepted manuscript |
URI: | http://hdl.handle.net/10037/2058 |
Abstract: | 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 defined in terms of the symmetric monoidal structure. Equivalence relations are shown to be commutative monoids in the category of relations. Quantization in our view is a property of functors between monoidal categories. This notion of quantization induce a deformation of all algebraic structures in the category, in particular the ones defining properties of relations like transitivity and symmetry. |
URI: | http://hdl.handle.net/10037/2057 |
Now showing items 1-17 of 17
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