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  • 1.
    Bergqvist, Göran
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Causal tensors and simple forms2002Ingår i: International Journal of Modern Physics A, ISSN 0217-751X, E-ISSN 1793-656X, Vol. 17, nr 20, s. 2748-2748Artikel i tidskrift (Refereegranskat)
  • 2.
    Bergqvist, Göran
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Causal tensors in Lorentzian geometry2003Ingår i: Publications of the Spanish mathematical society, Vol. 5, s. 73-83Artikel i tidskrift (Refereegranskat)
  • 3.
    Bergqvist, Göran
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Curves and envelopes that bound the spectrum of a matrix2018Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 557Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A generalization of the method developed by Adam, Psarrakos and Tsatsomeros to find inequalities for the eigenvalues of a complex matrix A using knowledge of the largest eigenvalues of its Hermitian part H(A) is presented. The numerical range or field of values of A can be constructed as the intersection of half-planes determined by the largest eigenvalue of H(e(i theta) A.). Adam, Psarrakos and Tsatsomeros showed that using the two largest eigenvalues of H(A), the eigenvalues of A satisfy a cubic inequality and the envelope of such cubic curves defines a region in the complex plane smaller than the numerical range but still containing the spectrum of A. Here it is shown how using the three largest eigenvalues of H(A) or more, one obtains new inequalities for the eigenvalues of A and new envelope-type regions containing the spectrum of A. (C) 2018 Elsevier Inc. All rights reserved.

    Publikationen är tillgänglig i fulltext från 2020-07-24 14:59
  • 4.
    Bergqvist, Göran
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Exact probabilities for typical ranks of 2 × 2 × 2 and 3 × 3 × 2 tensors2013Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, nr 2, s. 663-667Artikel i tidskrift (Refereegranskat)
    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.

  • 5.
    Bergqvist, Göran
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Monotone quantities for the Penrose inequality2003Ingår i: Penrose Inequalities,2003, 2003Konferensbidrag (Övrigt vetenskapligt)
  • 6.
    Bergqvist, Göran
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Spinors and conformal curvature2005Ingår i: Publications of the Spanish mathematical societyArtikel i tidskrift (Refereegranskat)
  • 7.
    Bergqvist, Göran
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    The Penrose inequality and null hypersurfaces2003Ingår i: Joint meeting of the American Mathematical Society and the Spanish Mathematical Society,2003, 2003Konferensbidrag (Övrigt vetenskapligt)
  • 8.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Eriksson, Ingemar
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    The Chevreton tensor and Einstein-Maxwell spacetimes conformal to Einstein spaces2007Ingår i: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 24, nr 13, s. 3437-3455Artikel i tidskrift (Refereegranskat)
    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.

  • 9.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Eriksson, Ingemar
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Senovilla, J. M. M.
    Departamento de Física Teórica, Universidad del País Vasco, Bilbao, Spain.
    New electromagnetic conservation laws2003Ingår i: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 20, nr 13, s. 2663-2668Artikel i tidskrift (Refereegranskat)
    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.

  • 10.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Eriksson, Ingemar
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Senovilla, José M M
    New conservation laws in Einstein-Maxwell spacetimes2004Ingår i: 17th International Conference on General Relativity and Gravitation,2004, 2004Konferensbidrag (Övrigt vetenskapligt)
  • 11.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Forrester, Peter J.
    University of Melbourne, Victoria, Australia.
    Rank probabilities for real random NxNx2 tensors2011Ingår i: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 16, s. 630-637Artikel i tidskrift (Refereegranskat)
    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.

  • 12.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Hoglund, A.
    Algebraic Rainich theory and antisymmetrization in higher dimensions2002Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 19, nr 12, s. 3341-3355Artikel i tidskrift (Refereegranskat)
    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.

  • 13.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Lankinen, Paul
    Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank2005Ingår i: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 461, nr 2059, s. 2181-2195Artikel i tidskrift (Refereegranskat)
    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.

  • 14.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Lankinen, Paul
    Algebraic and differential Rainich conditions for the Bel-Robinson tensor and other higher rank tensors2004Ingår i: 17th International Conference on General Relativity and Gravitation,2004, 2004Konferensbidrag (Övrigt vetenskapligt)
  • 15.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Lankinen, Paul
    Unique characterization of the Bel-Robinson tensor2004Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 21, nr 14, s. 3499-3503Artikel i tidskrift (Refereegranskat)
    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.

  • 16.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Larsson, Erik G
    Linköpings universitet, Institutionen för systemteknik, Kommunikationssystem. Linköpings universitet, Tekniska högskolan.
    Overview of recent advances in numerical tensor algebra2010Ingår i: Proceedings of Asilomar Conference on Signals, Systems and Computers, 2010, s. 3-7Konferensbidrag (Övrigt vetenskapligt)
    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).

  • 17.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Larsson, Erik G.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för systemteknik, Kommunikationssystem.
    The Higher-Order Singular Value Decomposition Theory and an Application2010Ingår i: IEEE signal processing magazine (Print), ISSN 1053-5888, E-ISSN 1558-0792, Vol. 27, nr 3, s. 151-154Artikel i tidskrift (Övrigt vetenskapligt)
    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.

  • 18.
    Bergqvist, Göran
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Senovilla, José M. M.
    Structure of conformal Lorentz transformations2003Ingår i: Current Trends in Relativistic Astrophysics: theoretical, numerical, observational / [ed] L. Férnadez-Jambrina, L.M. González-Romero, Springer , 2003, s. 245-251Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    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.

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