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Bergqvist, Göran
Publications (10 of 17) Show all publications
Bergqvist, G. (2013). Exact probabilities for typical ranks of 2 × 2 × 2 and 3 × 3 × 2 tensors. Linear Algebra and its Applications, 438(2), 663-667
Open this publication in new window or tab >>Exact probabilities for typical ranks of 2 × 2 × 2 and 3 × 3 × 2 tensors
2013 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, no 2, p. 663-667Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2013
Keywords
Tensors; Multi-way arrays; Typical rank; Random matrices
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-85549 (URN)10.1016/j.laa.2011.02.041 (DOI)000313226900004 ()
Available from: 2012-11-23 Created: 2012-11-23 Last updated: 2017-12-07Bibliographically approved
Bergqvist, G. & Forrester, P. J. (2011). Rank probabilities for real random NxNx2 tensors. Electronic Communications in Probability, 16, 630-637
Open this publication in new window or tab >>Rank probabilities for real random NxNx2 tensors
2011 (English)In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 16, p. 630-637Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Institute of Mathematical Statistics / Bernoulli society., 2011
Keywords
tensors, multi-way arrays, typical rank, random matrices
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-72024 (URN)10.1214/ECP.v16-1655 (DOI)000296162600001 ()
Note

Funding Agencies|Australian Research Council||

Available from: 2011-11-11 Created: 2011-11-11 Last updated: 2017-12-08
Bergqvist, G. & Larsson, E. G. (2010). Overview of recent advances in numerical tensor algebra. In: Proceedings of Asilomar Conference on Signals, Systems and Computers: . Paper presented at 44th Annual Asilomar Conference on Signals, Systems and Computers, November 7-10, Pacific Grove, California, USA (pp. 3-7).
Open this publication in new window or tab >>Overview of recent advances in numerical tensor algebra
2010 (English)In: Proceedings of Asilomar Conference on Signals, Systems and Computers, 2010, p. 3-7Conference paper, Published 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).

National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-62825 (URN)10.1109/ACSSC.2010.5757454 (DOI)978-1-4244-9722-5 (ISBN)
Conference
44th Annual Asilomar Conference on Signals, Systems and Computers, November 7-10, Pacific Grove, California, USA
Available from: 2010-12-06 Created: 2010-12-06 Last updated: 2016-08-31Bibliographically approved
Bergqvist, G. & Larsson, E. G. (2010). The Higher-Order Singular Value Decomposition Theory and an Application. IEEE signal processing magazine (Print), 27(3), 151-154
Open this publication in new window or tab >>The Higher-Order Singular Value Decomposition Theory and an Application
2010 (English)In: IEEE signal processing magazine (Print), ISSN 1053-5888, E-ISSN 1558-0792, Vol. 27, no 3, p. 151-154Article in journal (Other academic) Published
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.

Place, publisher, year, edition, pages
Institute of Electrical and Electronic Engineers, 2010
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-55525 (URN)10.1109/MSP.2010.936030 (DOI)000276819100017 ()
Note
©2009 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.: Göran Bergqvist and Erik G. Larsson, Higher-Order Singular Value Decomposition: Theory and an Application, 2010, IEEE SIGNAL PROCESSING MAGAZINE, (27)3, 151-154. Available from: 2010-04-30 Created: 2010-04-30 Last updated: 2017-12-12
Bergqvist, G. & Eriksson, I. (2007). The Chevreton tensor and Einstein-Maxwell spacetimes conformal to Einstein spaces. Classical and Quantum Gravity, 24(13), 3437-3455
Open this publication in new window or tab >>The Chevreton tensor and Einstein-Maxwell spacetimes conformal to Einstein spaces
2007 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 24, no 13, p. 3437-3455Article in journal (Refereed) Published
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.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-12680 (URN)10.1088/0264-9381/24/13/018 (DOI)
Available from: 2007-10-24 Created: 2007-10-24 Last updated: 2009-04-23
Bergqvist, G. & Lankinen, P. (2005). Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank. Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, 461(2059), 2181-2195
Open this publication in new window or tab >>Algebraic and differential Rainich conditions for symmetric trace-free tensors of higher rank
2005 (English)In: 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) Published
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.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-22453 (URN)10.1098/rspa.2004.1411 (DOI)1676 (Local ID)1676 (Archive number)1676 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2017-12-13
Bergqvist, G. (2005). Spinors and conformal curvature. Publications of the Spanish mathematical society
Open this publication in new window or tab >>Spinors and conformal curvature
2005 (English)In: Publications of the Spanish mathematical societyArticle in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-30682 (URN)16287 (Local ID)16287 (Archive number)16287 (OAI)
Available from: 2009-10-09 Created: 2009-10-09 Last updated: 2015-09-30
Bergqvist, G. & Lankinen, P. (2004). Algebraic and differential Rainich conditions for the Bel-Robinson tensor and other higher rank tensors. In: 17th International Conference on General Relativity and Gravitation,2004.
Open this publication in new window or tab >>Algebraic and differential Rainich conditions for the Bel-Robinson tensor and other higher rank tensors
2004 (English)In: 17th International Conference on General Relativity and Gravitation,2004, 2004Conference paper, Published paper (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-22709 (URN)2006 (Local ID)2006 (Archive number)2006 (OAI)
Available from: 2009-10-07 Created: 2009-10-07
Bergqvist, G., Eriksson, I. & Senovilla, J. M. (2004). New conservation laws in Einstein-Maxwell spacetimes. In: 17th International Conference on General Relativity and Gravitation,2004.
Open this publication in new window or tab >>New conservation laws in Einstein-Maxwell spacetimes
2004 (English)In: 17th International Conference on General Relativity and Gravitation,2004, 2004Conference paper, Published paper (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-22705 (URN)2002 (Local ID)2002 (Archive number)2002 (OAI)
Available from: 2009-10-07 Created: 2009-10-07
Bergqvist, G. & Lankinen, P. (2004). Unique characterization of the Bel-Robinson tensor. Classical and quantum gravity, 21(14), 3499-3503
Open this publication in new window or tab >>Unique characterization of the Bel-Robinson tensor
2004 (English)In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 21, no 14, p. 3499-3503Article in journal (Refereed) Published
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.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-22452 (URN)10.1088/0264-9381/21/14/012 (DOI)1675 (Local ID)1675 (Archive number)1675 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2017-12-13
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