Institutt for matematikk og statistikk: Nye registreringer
Viser treff 361-380 av 394
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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 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. -
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 ... -
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 ... -
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. -
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 ... -
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 ... -
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 ... -
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 ... -
Estimation of Ambiguity Functions With Limited Spread
(Research report; Forskningsrapport, 2008-04-07)This paper proposes a new estimation procedure for the ambiguity function of a non-stationary time series. The stochastic properties of the empirical ambiguity function calculated from a single sample in time are derived. Different thresholding procedures are introduced for the estimation of the ambiguity function. Such estimation methods are suitable if the ambiguity function is only non-negligible ... -
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 -
Statistical analysis of CGPS time series
(Master thesis; Mastergradsoppgave, 2007)All points on the surface of the Earth are moving. To define the velocity of a given point, we can place a GPS receiver there and measure the coordinates every day. After collecting enough data, we can generate a time series of three coordinates, North, East and Height directions. The most used technique to determine such displacements, is the linear model. The main objective of this thesis ... -
A piece-wise affine contracting map with positive entropy
(Working paper; Arbeidsnotat, 2005-04-10)We construct the simplest chaotic system with a two-point attractor. -
Modeling temporal fluctuations in avalanching system
(Working paper; Arbeidsnotat, 2008-07-22)We demonstrate how to model the toppling activity in avalanching systems by stochastic differential equations (SDEs). The theory is developed as a generalization of the classical mean field approach to sandpile dynamics by formulating it as a generalization of Itoh’s SDE. This equation contains a fractional Gaussian noise term representing the branching of an avalanche into small active clusters, ... -
A stochastic theory for temporal fluctuations in self-organized critical systems
(Working paper; Arbeidsnotat, 2008-07-22)A stochastic theory for the toppling activity in sandpile models is developed, based on a simple mean-field assumption about the toppling process. The theory describes the process as an anti-persistent Gaussian walk, where the diffusion coefficient is proportional to the activity. It is formulated as a generalization of the Itˆo stochastic differential equation with an anti-persistent fractional ... -
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 ... -
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. -
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.