• The Kobayashi pseudodistance on almost complex manifolds 

      Overholt, Marius; Kruglikov, Boris S. (Working paper; Arbeidsnotat, 1997-03-06)
      We extend the definition of the Kobayashi pseudodistance to almost complex manifolds and show that its familiar properties are for the most part preserved. We also study the automorphism group of an almost complex manifold and finish with some examples.
    • Theory of linear G-difference equations 

      Lychagin, Valentin V.; Jakobsen, Per K. (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 ...
    • Nijenhuis tensors in pseudoholomorphic curves neighborhoods 

      Kruglikov, Boris (Working paper; Arbeidsnotat, 2000)
      In this paper we consider the normal forms of almost complex structures in a neighborhood of pseudoholomorphic curve. We define normal bundles of such curves and study the properties of linear bundle almost complex structures. We describe 1-jet of the almost complex structure along a curve in terms of its Nijenhuis tensor. For pseudoholomorphic tori we investigate the problem of pseudoholomorphic ...
    • Existence of close pseudoholomorphic disks for almost complex manifolds and an applications to Kobayashi-Royden pseudonorm 

      Kruglikov, Boris (Journal article; Tidsskriftartikkel; Peer reviewed, 2000-02-23)
      It is proved in the paper that near every pseudoholomorphic disk on an almost complex manifold a disk of almost the same size in any close direction passes. As an application the Kobayashi-Royden pseudonorm for almost complex manifolds is defined and studied.
    • Examples of integrable sub-Riemannian geodesic flows 

      Kruglikov, Boris (Journal article; Tidsskriftartikkel; Peer reviewed, 2001-10-22)
      We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy.
    • The categorical theory of relations and quantization 

      Lychagin, Valentin V.; Jakobsen, Per K. (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 ...
    • The Capabilities of the EISCAT Svalbard Radar for Inter-hemispheric Coordinated Studies 

      van Eyken, Tony; Grydeland, Tom; Strømme, Anja; La Hoz, Cesar (Journal article; Tidsskriftartikkel; Peer reviewed, 2002)
      In this article we want to present the EISCAT Svalbard Radar (ESR) in some detail, as well as some of the instruments of interest for ionospheric and magnetospheric research that are located in the vicinity of it. We particularly describe how this instrument cluster, close to the geomagnetic conjugate point of the Chinese Antarctic Zhongshan station, can contribute to inter-hemispheric coordinated ...
    • Matematisk kulturhistorie : artikkelsamling 

      Thorvaldsen, Steinar (Book; Bok, 2002-08)
    • Deformation of big pseudoholomorphic disks and application to the Hanh pseudonorm 

      Kruglikov, Boris (Journal article; Tidsskriftartikkel; Peer reviewed, 2003-04-14)
      We simplify proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh pseudodistances. It is shown they coincide in dimensions higher than four. The result is new even in the complex case.
    • Linearizability of d-webs, d ≥ 4, on two-dimensional manifolds 

      Goldberg, Vladislav V.; Lychagin, Valentin V.; Akivis, Maks A. (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 ...
    • Factorisation patterns of division polynomials 

      Verdure, Hugues (Journal article; Tidsskriftartikkel; Peer reviewed, 2004-05)
      The choice of an elliptic curve for the implementa- tion of an elliptic curve cryptosystem requires count- ing the number of points on such a curve over a fi- nite field. An improvement of Schoof’s algorithm for counting the number of rational points on an ellip- tic curve defined over a finite field takes advantage of some factor of the division polynomials. In this paper, we study the ...
    • Strictly non-proportional geodesically equivalent metrics have htop(g) = 0 

      Kruglikov, Boris; Matveev, Vladimir S. (Journal article; Tidsskriftartikkel; Peer reviewed, 2004-10-24)
      If a closed manifold M possesses two Riemannian metrics which have the same unparameterized geodesics and are not strictly proportional at each point, then the topological entropy of both geodesic flows is zero. This is the main result of the paper and it has many dynamical and topological corollaries. In particular, such a manifoldM should be finitely covered by the product of a rationally ...
    • On the Blaschke Conjecture for 3-Webs 

      Goldberg, Vladislav V.; Lychagin, Valentin V. (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 

      Kruglikov, Boris; Lychagin, Valentin V. (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 ...
    • A piece-wise affine contracting map with positive entropy 

      Kruglikov, Boris; Rypdal, Martin (Working paper; Arbeidsnotat, 2005-04-10)
      We construct the simplest chaotic system with a two-point attractor.
    • Tangent and normal bundles in almost complex geometry 

      Kruglikov, Boris (Journal article; Tidsskriftartikkel; Peer reviewed, 2005-06-10)
      We define and study pseudoholomorphic vector bundles structures, particular cases of which are tangent and normal bundle almost complex structures. As an application we deduce normal forms of 1-jets of almost complex structures along a pseudoholomorphic submanifold. In dimension four we relate these normal forms to the problem of pseudoholomorphic foliation of a neighborhood of a curve and the ...
    • Dynamics and entropy in the Zhang model of Self-Organized Criticality 

      Kruglikov, Boris; Rypdal, Martin (Working paper; Arbeidsnotat, 2005-09-12)
      We give a detailed study of dynamical properties of the Zhang model, including evaluation of topological entropy and estimates for the Lyapunov exponents and the dimension of the attractor. In the thermodynamic limit the entropy goes to zero and the Lyapunov spectrum collapses.1
    • Entropy via multiplicity 

      Kruglikov, Boris; Rypdal, Martin (Working paper; Arbeidsnotat, 2005-09-30)
      The topological entropy of piecewise affine maps is studied. It is shown that singularities may contribute to the entropy only if there is angular expansion and we bound the entropy via the expansion rates of the map. As a corollary we deduce that non-expanding conformal piecewise affine maps have zero topological entropy. We estimate the entropy of piecewise affine skew-products. Examples of ...
    • Invariants of pseudogroup actions: Homological methods and Finiteness theorem 

      Kruglikov, Boris; Lychagin, Valentin V. (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 ...