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  • 1.
    Andersson, Fredrik
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Edgar, SB
    Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden.
    Spin coefficients as Lanczos scalars: Underlying spinor relations2000In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 41, no 5, p. 2990-3001Article in journal (Refereed)
    Abstract [en]

    It has been conjectured by Lopez-Bonilla and co-workers that there is some linear relationship between the NP spin coefficients and the Lanczos scalars, and examples have been given for a number of different classes of space-times. We show that in each of those examples a Lanczos potential can be defined in a very simple way directly from the spinor dyad. Although some of these examples seem to have no deeper geometric meaning, we emphasize that there are structural links between Lanczos potential and spin coefficients which we highlight in some other examples. In particular we show that the direct identification of Lanczos potentials with spin coefficients is possible for some important classes of space-times while the direct identification of Lanczos potentials with the properly weighted spin coefficients is also possible for several important classes of space-times. In both of these cases we obtain the necessary and sufficient conditions on the spin coefficients for such identifications to be possible, which enables us to test space-times directly. (C) 2000 American Institute of Physics. [S0022-2488(00)03104-2].

  • 2.
    Arnlind, Joakim
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Curvature and geometric modules of noncommutative spheres and tori2014In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 55, no 4, p. 041705-Article in journal (Refereed)
    Abstract [en]

    When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connection from Euclidean space, and we compute its scalar curvature.

  • 3.
    Arnlind, Joakim
    Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden .
    Representation theory of C  -algebras for a higher-order class of spheres and tori2008In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 49, p. 053502-1-053502-13, article id 053502Article in journal (Refereed)
    Abstract [en]

    We construct C  -algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of “loop” and “string” representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the “Hénon algebras,” for which the dynamical map is a generalized Hénon map, and give an example where irreducible representations of all dimensions exist.

  • 4.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Grosse, Harald
    Mathematical Physics, Austria .
    Deformed noncommutative tori2012In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 53, no 7, p. 073505-Article in journal (Refereed)
    Abstract [en]

    We recall a construction of non-commutative algebras related to a one-parameter family of (deformed) spheres and tori, and show that in the case of tori, the *-algebras can be completed into C*-algebras isomorphic to the standard non-commutative torus. As the former was constructed in the context of matrix (or fuzzy) geometries, it provides an important link to the framework of non-commutative geometry, and opens up for a concrete way to study deformations of non-commutative tori. Furthermore, we show how the well-known fuzzy sphere and fuzzy torus can be obtained as formal scaling limits of finite-dimensional representations of the deformed algebras, and their projective modules are described together with connections of constant curvature.

  • 5.
    Arnlind, Joakim
    et al.
    Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
    Makhlouf, Abdenacer
    Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications, 4, rue des Frères Lumière F-68093 Mulhouse, France .
    Silvestrov, Sergei
    Mälardalen University, Division of Applied Mathematics, The School of Education, Culture and Communication, Box 883, 721 23 Västerås, Sweden och Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
    Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras2011In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, article id 123502Article in journal (Refereed)
    Abstract [en]

    As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.

  • 6.
    Arnlind, Joakim
    et al.
    Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
    Makhlouf, Abdenacer
    Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France .
    Silvestrov, Sergei
    Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden .
    Ternary Hom–Nambu–Lie algebras induced by Hom–Lie algebras2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, p. 043515-1-043515-11, article id 43515Article in journal (Refereed)
    Abstract [en]

    The need to consider n  -ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n  -ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n  -Lie structures) constructed from the binary multiplication of a Hom–Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom–Nambu–Lie algebras obtained using this construction.

  • 7.
    Basarab-Horwath, Peter
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Gungor, F.
    Istanbul Technical University, Turkey.
    Linearizability for third order evolution equations2017In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 58, no 8, article id 081507Article in journal (Refereed)
    Abstract [en]

    The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence of an infinite-dimensional symmetry group. Linearizing transformations for this class are found using symmetry structure and local conservation laws. A number of special cases as examples are discussed. Their transformation to equations within the same class by differential substitutions and connection with KdV and mKdV equations is also reviewed in this framework. Published by AIP Publishing.

  • 8.
    Basarab-Horwath, Peter
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Zhdanov, R.Z.
    Institute of Mathematics, 3 Tereshchenkivska Street, 252004 Kyiv, Ukraine.
    Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries2001In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 42, no 1, p. 376-389Article in journal (Refereed)
    Abstract [en]

    We suggest a new approach to the problem of dimensional reduction of initial/ boundary value problems for evolution equations in one spatial variable. The approach is based on higher-order (generalized) conditional symmetries of the equations involved. It is shown that reducibility of an initial value problem for an evolution equation to a Cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question. We also give some examples of the solution of initial value problems for second- and third-order nonlinear differential equations by reduction by their conditional symmetries. We give a systematic classification of general second-order partial differential equations admitting second-order conditional symmetries, based on Lie's classification of invariant second-order ordinary differential equations. This yields five classes of principally new initial value problems for nonlinear evolution equations which admit no Lie symmetries and are reducible via second-order conditional symmetries. © 2001 American Institute of Physics.

  • 9.
    Bengtsson, Ingemar
    et al.
    Fysikum Stockholms Universitet.
    Bruzda, Wojciech
    Instytut Fizyki im. Smoluchowskiego Uniwersytet Jagiellonski, Kraków.
    Ericsson, Åsa
    Fysikum Stockholms Universitet.
    Larsson, Jan-Åke
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Tadej, Wojciech
    Wydzial Matematyczno-Przyrodniczy Universytet Kardynala Stefana Wyszynskiego, Warszawa.
    Zyczkowski, Karol
    Instytut Fizyki im. Smoluchowskiego Uniwersytet Jagiellonski, Kraków.
    Mutually unbiased bases and Hadamard matrices of order six2007In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, p. 052106-1-21Article in journal (Refereed)
    Abstract [en]

    We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

  • 10.
    Birnir, B.
    et al.
    Department of Mathematics, UCSB, Santa Barbara, CA 93106, United States.
    Hou, S.
    Department of Mathematics, Michigan State University, East Lansing, MI 48824, United States.
    Wellander, N.
    Swedish Defence Research Agency, FOI, P.O. Box 1165, SE-581 11 Linkoping, Sweden.
    Derivation of the viscous Moore-Greitzer equation for aeroengine flow2007In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, no 6Article in journal (Refereed)
    Abstract [en]

    The viscous Moore-Greitzer equation modeling the airflow through the compression system in turbomachines, such as a jet engine, is derived using a scaled Navier-Stokes equation. The method utilizes a separation of scale argument based on the different spatial scales in the engine and the different time scales in the flow. The pitch and size of the rotor-stator pair of blades provides a small parameter, which is the size of the local cell. The motion of the stator and rotor blades in the compressor produces a very turbulent flow on a fast time scale. The leading order equation, for the fast time and local scales, describes this turbulent flow. The next order equations produce an axisymmetric swirl and a flow pattern analogous to Rayleigh-B´nard convection rolls in Rayleigh-B´nard convection. On a much larger spatial scale and a slower time scale, there exist modulations of the flow including instabilities called surge and stall. A higher order equation, in the small parameter, describes these global flow modulations, when averaged over the small (local) spatial scales, the fast time scale, and the time scale of the vortex rotations. Thus a more general system of spatially global, slow time equations is obtained. This system can be solved numerically without any approximations. The viscous Moore-Greitzer equation is obtained when small inertial terms are dropped from these slow time, spatially global equations averaged once more in the axial direction. The new equations are simulated with two different simplifying assumptions and the results are compared with simulations of the viscous Moore-Greitzer equations. © 2007 American Institute of Physics.

  • 11.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Necessary and sufficient conditions for n-dimensional conformal Einstein spaces via dimensionally dependent identities2005In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 46, no 1Article in journal (Refereed)
    Abstract [en]

    Listing has recently extended results of Kozameh, Newman, and Tod for fourdimensional space-times and presented a set of necessary and sufficient conditions for a metric to be locally conformally equivalent to an Einstein metric in all semi-Riemannian spaces of dimension n≥4-subject to a nondegeneracy restriction on the Weyl tensor. By exploiting dimensionally dependent identities we demonstrate how to construct two alternative versions of these necessary and sufficient conditions which we believe will be useful in applications. The four-dimensional case is discussed in detail and examples are also given in five and six dimensions.

  • 12.
    Edgar, SB
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Hoglund, A
    Dimensionally dependent tensor identities by double antisymmetrization2002In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 43, no 1, p. 659-677Article in journal (Refereed)
    Abstract [en]

    Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in n-dimensional space of a pair of fundamental identities involving trace-free (p,p)-forms where 2p greater than or equal ton. We generalize Lovelock's results, and by using the fact that associated with any tensor in n-dimensional space there is associated a fundamental tensor identity obtained by antisymmetrizing over n+1 indices, we establish a very general "master" identity for all trace-free (k,l)-forms. We then show how various other special identities are direct and simple consequences of this master identity, in particular we give direct application to Maxwell, Lanczos, Ricci, Bel, and Bel-Robinson tensors, and also demonstrate how relationships between scalar invariants of the Riemann tensor can be investigated in a systematic manner. (C) 2002 American Institute of Physics.

  • 13.
    Enblom, Alexandra
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Hardy-Carleman type inequalities for Dirac operators2015In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 56, no 10, p. 103503-Article in journal (Refereed)
    Abstract [en]

    General Hardy-Carleman type inequalities for Dirac operators are proved. New inequalities are derived involving particular traditionally used weight functions. In particular, a version of the Agmon inequality and Treve type inequalities is established. The case of a Dirac particle in a (potential) magnetic field is also considered. The methods used are direct and based on quadratic form techniques. (C) 2015 AIP Publishing LLC.

  • 14.
    Lundmark, H.
    et al.
    Dept. of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Sask., Canada.
    Rauch, Stefan
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Driven Newton equations and separable time-dependent potentials2002In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 43, no 12, p. 6166-6194Article in journal (Refereed)
    Abstract [en]

    We present a class of time-dependent potentials in Rn that can be integrated by separation of variables: by embedding them into so-called cofactor pair systems of higher dimension, we are led to a time-dependent change of coordinates that allows the time variable to be separated off, leaving the remaining part in separable Stäckel form. © 2002 American Institute of Physics.

  • 15.
    Marciniak, Krzysztof
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Science and Technology.
    Blaszak, Maciej
    Institution för fysik Adam Mickiewicz universitet, Poznan, Polen.
    From Stäckel systems to integrable hierarchies of PDE's: Beneti class of separation relations.2006In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 47, p. 0329041-03290426Article in journal (Refereed)
    Abstract [en]

    We propose a general scheme of constructing of soliton hierarchies from finite dimensional Stäckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e., certain Stäckel systems with quadratic in momenta integrals of motion.

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