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  • 1.
    Bergqvist, Göran
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Causal tensors and simple forms2002In: International Journal of Modern Physics A, ISSN 0217-751X, E-ISSN 1793-656X, Vol. 17, no 20, p. 2748-2748Article in journal (Refereed)
  • 2.
    Bergqvist, Göran
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Causal tensors in Lorentzian geometry2003In: Publications of the Spanish mathematical society, Vol. 5, p. 73-83Article in journal (Refereed)
  • 3.
    Bergqvist, Göran
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Exact probabilities for typical ranks of 2 × 2 × 2 and 3 × 3 × 2 tensors2013In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, no 2, p. 663-667Article in journal (Refereed)
    Abstract [en]

    We show that the probability to be of rank 2 for a 2×2×2 tensor with elements from a standard normal distribution is π/4, and that the probability to be of rank 3 for a 3×3×2 tensor is 1/2. In the proof results on the expected number of real generalized eigenvalues of random matrices are applied. For n×n×2 tensors with n≥4 we also present some new aspects of their rank.

  • 4.
    Bergqvist, Göran
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Monotone quantities for the Penrose inequality2003In: Penrose Inequalities,2003, 2003Conference paper (Other academic)
  • 5.
    Bergqvist, Göran
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Spinors and conformal curvature2005In: Publications of the Spanish mathematical societyArticle in journal (Refereed)
  • 6.
    Bergqvist, Göran
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    The Penrose inequality and null hypersurfaces2003In: Joint meeting of the American Mathematical Society and the Spanish Mathematical Society,2003, 2003Conference paper (Other academic)
  • 7.
    Bergqvist, Göran
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Eriksson, Ingemar
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    The Chevreton tensor and Einstein-Maxwell spacetimes conformal to Einstein spaces2007In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 24, no 13, p. 3437-3455Article in journal (Refereed)
    Abstract [en]

    In this paper, we characterize the source-free Einstein–Maxwell spacetimes which have a trace-free Chevreton tensor. We show that this is equivalent to the Chevreton tensor being of pure radiation type and that it restricts the spacetimes to Petrov type N or O. We prove that the trace of the Chevreton tensor is related to the Bach tensor and use this to find all Einstein–Maxwell spacetimes with a zero cosmological constant that have a vanishing Bach tensor. Among these spacetimes we then look for those which are conformal to Einstein spaces. We find that the electromagnetic field and the Weyl tensor must be aligned, and in the case that the electromagnetic field is null, the spacetime must be conformally Ricci-flat and all such solutions are known. In the non-null case, since the general solution is not known on a closed form, we settle by giving the integrability conditions in the general case, but we do give new explicit examples of Einstein–Maxwell spacetimes that are conformal to Einstein spaces, and we also find examples where the vanishing of the Bach tensor does not imply that the spacetime is conformal to a C-space. The non-aligned Einstein–Maxwell spacetimes with vanishing Bach tensor are conformally C-spaces, but none of them are conformal to Einstein spaces.

  • 8.
    Bergqvist, Göran
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Eriksson, Ingemar
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Senovilla, J. M. M.
    Departamento de Física Teórica, Universidad del País Vasco, Bilbao, Spain.
    New electromagnetic conservation laws2003In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 20, no 13, p. 2663-2668Article in journal (Refereed)
    Abstract [en]

    The Chevreton superenergy tensor was introduced in 1964 as a counterpart, for electromagnetic fields, of the well-known Bel–Robinson tensor of the gravitational field. We here prove the unnoticed facts that, in the absence of electromagnetic currents, Chevreton's tensor (i) is completely symmetric, and (ii) has a trace-free divergence if the Einstein–Maxwell equations hold. It follows that the trace of the Chevreton tensor is a rank-2, symmetric, trace-free, conserved tensor, which is different from the energy–momentum tensor, and nonetheless can be constructed for any test Maxwell field or any Einstein–Maxwell spacetime.

  • 9.
    Bergqvist, Göran
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Eriksson, Ingemar
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Senovilla, José M M
    New conservation laws in Einstein-Maxwell spacetimes2004In: 17th International Conference on General Relativity and Gravitation,2004, 2004Conference paper (Other academic)
  • 10.
    Bergqvist, Göran
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Forrester, Peter J.
    University of Melbourne, Victoria, Australia.
    Rank probabilities for real random NxNx2 tensors2011In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 16, p. 630-637Article in journal (Refereed)
    Abstract [en]

    We prove that the probability P_N for a real random Gaussian NxNx2 tensor to be of real rank N is P_N=(Gamma((N+1)/2))^N/G(N+1), where Gamma(x) and G(x) denote the gamma and the Barnes G-functions respectively. This is a rational number for N odd and a rational number multiplied by pi^{N/2} for N even. The probability to be of rank N+1 is 1-P_N. The proof makes use of recent results on the probability of having k real generalized eigenvalues for real random Gaussian N x N matrices. We also prove that log P_N= (N^2/4)log (e/4)+(log N-1)/12-zeta'(-1)+O(1/N) for large N, where zeta is the Riemann zeta function.

  • 11.
    Bergqvist, Göran
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Hoglund, A.
    Algebraic Rainich theory and antisymmetrization in higher dimensions2002In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 19, no 12, p. 3341-3355Article in journal (Refereed)
    Abstract [en]

    The classical Rainich(-Misner-Wheeler) theory gives necessary and sufficient conditions on an energy-momentum tensor T to be that of a Maxwell field (a 2-form) in four dimensions. Via Einstein's equations, these conditions can be expressed in terms of the Ricci tensor, thus providing conditions for a spacetime geometry to be an Einstein-Maxwell spacetime. One of the conditions is that T2 is proportional to the metric, and it has previously been shown in arbitrary dimension that any tensor satisfying this condition is a superenergy tensor of a simple p-form. Here we examine algebraic Rainich conditions for general p-forms in higher dimensions and their relations to identities by antisymmetrization. Using antisymmetrization techniques we find new identities for superenergy tensors of these general (non-simple) forms, and we also prove in some cases the converse: that the identities are sufficient to determine the form. As an example we obtain the complete generalization of the classical Rainich theory to five dimensions.

  • 12.
    Bergqvist, Göran
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lankinen, Paul
    Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank2005In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 461, no 2059, p. 2181-2195Article in journal (Refereed)
    Abstract [en]

    We present a study of Rainich-like conditions for symmetric and trace-free tensors T. For arbitrary even rank we find a necessary and sufficient differential condition for a tensor to satisfy the source-free field equation. For rank 4, in a generic case, we combine these conditions with previously obtained algebraic conditions to gain a complete set of algebraic and differential conditions on T for it to be a superenergy tensor of a Weyl candidate tensor, satisfying the Bianchi vacuum equations. By a result of Bell and Szekeres, this implies that in vacuum, generically, T must be the Bel-Robinson tensor of the spacetime. For the rank 3 case, we derive a complete set of necessary algebraic and differential conditions for T to be the superenergy tensor of a massless spin-3/2 field, satisfying the source-free field equation.

  • 13.
    Bergqvist, Göran
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lankinen, Paul
    Algebraic and differential Rainich conditions for the Bel-Robinson tensor and other higher rank tensors2004In: 17th International Conference on General Relativity and Gravitation,2004, 2004Conference paper (Other academic)
  • 14.
    Bergqvist, Göran
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lankinen, Paul
    Unique characterization of the Bel-Robinson tensor2004In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 21, no 14, p. 3499-3503Article in journal (Refereed)
    Abstract [en]

    We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson-type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.

  • 15.
    Bergqvist, Göran
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Larsson, Erik G
    Linköping University, Department of Electrical Engineering, Communication Systems. Linköping University, The Institute of Technology.
    Overview of recent advances in numerical tensor algebra2010In: Proceedings of Asilomar Conference on Signals, Systems and Computers, 2010, p. 3-7Conference paper (Other academic)
    Abstract [en]

    We present a survey of some recent developments for decompositions of multi-way arrays or tensors, with special emphasis on results relevant for applications and modeling in signal processing. A central problem is how to find lowrank approximations of tensors, and we describe some new results, including numerical methods, algorithms and theory, for the higher order singular value decomposition (HOSVD) and the parallel factors expansion or canonical decomposition (CP expansion).

  • 16.
    Bergqvist, Göran
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Larsson, Erik G.
    Linköping University, The Institute of Technology. Linköping University, Department of Electrical Engineering, Communication Systems.
    The Higher-Order Singular Value Decomposition Theory and an Application2010In: IEEE signal processing magazine (Print), ISSN 1053-5888, E-ISSN 1558-0792, Vol. 27, no 3, p. 151-154Article in journal (Other academic)
    Abstract [en]

    Tensor modeling and algorithms for computing various tensor decompositions (the Tucker/HOSVD and CP decompositions, as discussed here, most notably) constitute a very active research area in mathematics. Most of this research has been driven by applications. There is also much software available, including MATLAB toolboxes [4]. The objective of this lecture has been to provide an accessible introduction to state of the art in the field, written for a signal processing audience. We believe that there is good potential to find further applications of tensor modeling techniques in the signal processing field.

  • 17.
    Bergqvist, Göran
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Senovilla, José M. M.
    Structure of conformal Lorentz transformations2003In: Current Trends in Relativistic Astrophysics: theoretical, numerical, observational / [ed] L. Férnadez-Jambrina, L.M. González-Romero, Springer , 2003, p. 245-251Chapter in book (Other academic)
    Abstract [en]

    The present volume contains the expanded lectures of a meeting on relativistic astrophysics, the goal of which was to provide a modern introduction to specific aspects of the field for young researchers, as well as for nonspecialists from related areas. Particular emphasis is placed on the theory of black holes and evolution, relativistic stars and jet hydrodynamics, as well as the production and detection of gravitational waves. The book is complemented by further contributions and animation supplied on the accompanying CD-ROM.

1 - 17 of 17
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