Jon-Magne Leinaas

Selected publications

• Leinaas, Jon Magne (2017). Luttinger liquids, Fermi liquids, and fractional statistics. Physical review B (PRB). ISSN 2469-9950. 95(15). doi: 10.1103/PhysRevB.95.155429. Full text in Research Archive
• Qin, M.; Leinaas, Jon Magne; Ryu, S; Ardonne, E; Xiang, Tiantian & Lee, D.-H. (2012). Quantum torus chain. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 86(13). doi: 10.1103/PhysRevB.86.134430. Full text in Research Archive Show summary

  We introduce a set of one-dimensional quantum lattice models which we refer to as the quantum torus chain. These models have discrete global symmetry and projective on-site representations. They possess an integer-valued parameter which controls the presence or absence of frustration. Depending on whether this parameter is even or odd, these models exhibit either gapped symmetry-breaking phases with isolated critical points or gapped symmetry-breaking phases separated by gapless phases. We discuss the property of these phases and phase transitions for two special values of the parameter and point out many open problems. © 2012 American Physical Society

• Horsdal, M; Hansson, Thors Hans; Leinaas, Jon Magne & Rypestøl, Marianne (2011). Charge fractionalization on quantum Hall edges. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 84(11). doi: 10.1103/PhysRevB.84.115313. Full text in Research Archive
• Sollid, Per Øyvind; Leinaas, Jon Magne & Myrheim, Jan (2011). Unextendible product bases and extremal density matrices with positive partial transpose. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 84(4). doi: 10.1103/PhysRevA.84.042325. Full text in Research Archive
• Leinaas, JM; Myrheim, Jan & Sollid, PO (2010). Numerical studies of entangled positive-partial-transpose states in composite quantum systems. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 81(6). doi: 10.1103/PhysRevA.81.062329. Full text in Research Archive Show summary

  We report here on the results of numerical searches for PPT states in a series of bipartite quantum systems of low dimensions. PPT states are represented by density matrices that remain positive semidefinite under partial transposition with respect to one of the subsystems, and our searches are for such states with specified ranks for the density matrix and its partial transpose. For a series of different ranks extremal PPT states and nonextremal entangled PPT states have been found. The results are listed in tables and charted in diagrams. Comparison of the results for systems of different dimensions reveals several regularities. We discuss lower and upper bounds on the ranks of extremal PPT states.

• Leinaas, Jon Magne; Myrheim, Jan & Sollid, Per Øyvind (2010). Low-rank extremal positive-partial-transpose states and unextendible product bases. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 81(6). doi: 10.1103/PhysRevA.81.062330. Full text in Research Archive
• Leinaas, Jon Magne; Horsdal, Mats & Hansson, Thors Hans (2009). Sharp fractional charges in Luttinger liquids. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 80(11). doi: 10.1103/PhysRevB.80.115327. Full text in Research Archive
• Leinaas, Jon Magne (2009). Anyons. In Greenberger, Daniel; Hentschel, Klaus & Weinert, Friedel (Ed.), Compendium in Quantum Physics. Springer Publishing Company. ISSN 9783540706229. p. 10–14. Full text in Research Archive
• Halvorsen, Tore Gunnar & Leinaas, Jon Magne (2008). Superluminal group velocity in a birefringent crystal. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 77. doi: 10.1103/PhysRevA.77.023808. Full text in Research Archive
• Dahl, Geir; Leinaas, Jon Magne; Myrheim, Jan & Ovrum, Eirik (2007). A tensor product matrix approximation problem in quantum physics. Linear Algebra and its Applications. ISSN 0024-3795. 420(2-3), p. 711–725. doi: 10.1016/j.laa.2006.08.026. Full text in Research Archive
• Leinaas, Jon Magne; Myrheim, Jan & Ovrum, Eirik (2007). Extreme points of the set of density matrices with positive partial transpose. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 76. doi: 10.1103/PhysRevA.76.034304. Full text in Research Archive
• Horsdal, Mats & Leinaas, Jon Magne (2007). Explicit mapping between a 2D quantum Hal l system and a 1D Luttinger liquid. II. Correlation functions. Physical Review B Condensed Matter. ISSN 0163-1829. 76. Full text in Research Archive
• Horsdal, Mats & Leinaas, Jon Magne (2007). Explicit mapping between a 2D quantum Hal l system and a 1D Luttinger liquid. I. Luttinger parameters. Physical Review B Condensed Matter. ISSN 0163-1829. 76. Full text in Research Archive
• Leinaas, JM; Myrheim, Jan & Ovrum, E (2006). Geometrical aspects of entanglement. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 74. Full text in Research Archive
• Leinaas, Jon Magne (2006). Nobelprisen i fysikk 2005. In Reisegg, Øyvind (Eds.), Aschehoug og Gyldedals Store Norske Årbok 2006. Kunnskapsforlaget. ISSN 9788257317331. p. 409–411. Full text in Research Archive
• Seidel, A.; Fu, H.; Lee, Dung-Hai; Leinaas, Jon Magne & More, J. (2005). Incompressible quantum liquids and new conservation laws. Physical Review Letters. ISSN 0031-9007. 95. Full text in Research Archive
• Lee, Dung-Hai & Leinaas, Jon Magne (2004). Role of interference in millimeter-wave-driven dc transport in a two-dimensional electron gas. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 69, p. art. no.–115336. Full text in Research Archive
• Korsbakken, JI & Leinaas, Jon Magne (2004). Fulling-Unruh effect in general stationary accelerated frames. Physical Review D. Particles and fields. ISSN 0556-2821. 70. Full text in Research Archive
• Korsbakken, Jan Ivar & Leinaas, Jon Magne (2004). The Fulling-Unruh effect in general stationary frames. Physical Review D. Particles and fields. ISSN 0556-2821. 70. Full text in Research Archive
• Leinaas, Jon Magne (2004). Kvantemekanikk 100 år etter Planck, Del 2. Fra Fysikkens Verden. ISSN 0015-9247. Full text in Research Archive
• Lee, Dung-Hai & Leinaas, Jon Magne (2004). Mott insulators without symmetry breaking. Physical Review Letters. ISSN 0031-9007. 92, p. art. no.–096401. Full text in Research Archive
• Chiao, Raymond Y.; Hansson, Thors Hans; Leinaas, Jon Magne & Viefers, Susanne F (2004). Effective photon-photon interaction in a two-dimensional "photon fluid". Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 69. Full text in Research Archive
• Chiao, Raymond Y.; Hansson, Thors Hans; Leinaas, Jon Magne & Viefers, Susanne F (2004). Two-dimensional 'photon fluid': effective photon-photon interaction and physical realizations. Journal of Physics B: Atomic, Molecular and Optical Physics. ISSN 0953-4075. 37, p. S81–S89. Full text in Research Archive
• Leinaas, Jon Magne (2002). Unruh effect in storage rings. In Chen, Pisin (Eds.), Quantum Aspects of Beam Physics. World Scientific, Singapore. Full text in Research Archive Show summary

  A uniformly accelerated system will get thermally excited due to interactions with the vacuum fluctuations of the quantum fields. This is the Unruh effect. Also a system accelerated in a circular orbit will be heated, but in this case complications arise relative to the linear case. An interesting question is in what sense the real quantum effects for orbital and spin motion of a circulating electron can be viewed as a demonstration of the Unruh effect. This question has been studied and debated. I review some of the basic points concerning the relation to the Unruh effect, and in particular look at how the electron can be viewed as a thermometer or detector that probes thermal and other properties of the vacuum state in the accelerated frame.

• Leinaas, Jon Magne (2002). Thermal excitations of accelerated electrons. In Bertlmann, Reinhold & Zeilinger, Anton (Ed.), Quantum (Un)speakables -- From Bell to Quantum Information. Springer. Full text in Research Archive
• Hansson, T.H.; Leinaas, Jon Magne & Viefers, S. (2001). Exclusion Statistics in a trapped two-dimensional Bose gas. Physical Review Letters. ISSN 0031-9007. 86, p. 2930–2933. Full text in Research Archive Show summary

  We study the statistical mechanics of a two-dimensional Bose gas with a repulsive delta-function interaction, using a mean-field approximation. By a direct counting of states we establish that this model obeys exclusion statistics and is equivalent to an ideal exclusion-statistics gas. We also show that this result is consistent with a full quantum-mechanical treatment of a quasi-two-dimensional system.

• Hansson, T.H.; Leinaas, Jon Magne & Viefers, S. (2001). Exclusion statistics in a two-dimensional trapped Bose gas. Journal of Physics B: Atomic, Molecular and Optical Physics. ISSN 0953-4075. 34, p. 1–8. Full text in Research Archive Show summary

  We briefly explain the notion of exclusion statistics and, in particular, discuss the concept of an ideal exclusion statistics gas. We then review recent work where it is demonstrated that a two-dimensional Bose gas with repulsive delta function interactions obeys ideal exclusion statistics, with a fractional parameter related to the interaction strength.

• Hansson, T.H.; Isakov, S.B.; Leinaas, Jon Magne & Lindström, Ulf (2001). Classical Phase Space and Statistical Mechanics of Identical Particles. Physical Review E. Statistical, Nonlinear, and Soft Matter Physics. ISSN 1539-3755. 63. Full text in Research Archive Show summary

  Starting from the quantum theory of identical particles, we show how to define a classical mechanics that retains information about the quantum statistics. We consider two examples of relevance for the quantum Hall effect: identical particles in the lowest Landau level, and vortices in the Chern-Simons Ginzburg-Landau model. In both cases the resulting classical statistical mechanics is shown to be a nontrivial classical limit of Haldane's exclusion statistics.

• Leinaas, Jon Magne (2000). Quantum Effects in Accelerator Physics. In Ellis, John & Amati, Daniele (Ed.), Quantum Reflections. Cambridge University Press. ISSN 9780521630085. p. 117–135. Full text in Research Archive Show summary

  Quantum effects for electrons in a storage ring are discussed, in particular the polarization effect due to spin flip synchrotron radiation. The electrons are treated as a simple quantum mechanical two-level system coupled to the orbital motion and the radiation field. The excitations of the spin system then are related to the Unruh effect, i.e. the effect that an accelerated radiation detector is thermally excited by vacuum fluctuations. This is a review of an earlier work that was done in collaboration with John Bell.

• Kjønsberg, Heidi & Leinaas, Jon Magne (1999). Charge and Statistics of Quantum Hall Quasi-Particles. A numerical study of mean values and fluctuations. Nuclear Physics B. ISSN 0550-3213. 559 [FS], p. 705–742. Full text in Research Archive Show summary

  We present Monte Carlo studies of charge expectation values and charge fluctuations for quasi-particles in the quantum Hall system. We have studied the Laughlin wave functions for quasi-hole and quasi-electron, and also Jain's definition of the quasi-electron wave function. The considered systems consist of from 50 to 200 electrons, and the filling fraction is 1/3. For all quasi-particles our calculations reproduce well the expected values of charge; -1/3 times the electron charge for the quasi-hole, and 1/3 for the quasi-electron. Regarding fluctuations in the charge, our results for the quasi-hole and Jain quasi-electron are consistent with the expected value zero in the bulk of the system, but for the Laughlin quasi-electron we find small, but significant, deviations from zero throughout the whole electron droplet. We also present Berry phase calculations of charge and statistics parameter for the Jain quasi-electron, calculations which supplement earlier studies for the Laughlin quasi-particles. We find that the statistics parameter is more well behaved for the Jain quasi-electron than it is for the Laughlin quasi-electron.

• Hansson, Hans; Karlhede, Anders; Leinaas, Jon Magne & Nilsson, Ulf (1999). Field Theory for Partially Polarized Quantum Hall States. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 60, p. 4866–4873. Full text in Research Archive

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• Leinaas, Jon Magne (2019). Solution Manual for Classical Mechanics and Electrodynamics. World Scientific. ISBN 9789811200700. 128 p. Full text in Research Archive
• Leinaas, Jon Magne (2019). Classical Mechanics and Electrodynamics. World Scientific. ISBN 9789813279988. 362 p. Full text in Research Archive

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• Leinaas, Jon-Magne & Myrheim, Jan (2023). Fractional statistics in low-dimensional systems. In Chakraborty, Tapash (Eds.), Encyclopedia of Condensed Matter Physics. Academic Press. ISSN 9780323908009. p. 394–401. doi: 10.1016/B978-0-323-90800-9.00187-6. Full text in Research Archive
• Leinaas, Jon-Magne (2022). Fermi Liquids and Fractional Statistics in One Dimension. In Niemi, Antti; Phua, K K & Shapere, Alfred (Ed.), Frank Wilczek: 50 Years of Theoretical Physics. World Scientific. ISSN 9789811255175. p. 141–161. doi: 10.1142/9789811251948_0012. Full text in Research Archive
• Leinaas, Jon Magne (2018). 100 år med Noethers teorem. Fra Fysikkens Verden. ISSN 0015-9247. Full text in Research Archive
• Leinaas, Jon Magne (2018). Relativitesteori, kvantefysikk og moderne teknologi. Full text in Research Archive
• Høgåsen, Hallstein; Brevik, Iver H.; Høye, Johan S. & Leinaas, Jon Magne (2018). Jan Myrheim 70 år. Fra Fysikkens Verden. ISSN 0015-9247. 80(1). Full text in Research Archive
• Leinaas, Jon Magne (2017). Fermi liquids, Luttinger liquids,and fractional statistics in one dimension. Full text in Research Archive
• Leinaas, Jon Magne (2016). Exchange statistics - Basic concepts. Full text in Research Archive
• Leinaas, Jon Magne (2015). Kvantemekanisk sammenfiltring. Full text in Research Archive
• Leinaas, Jon Magne (2015). Topology, gauge fields and topology. Full text in Research Archive
• Høgåsen, Hallstein & Leinaas, Jon Magne (2014). Jan Olav Eeg 70 år. Fra Fysikkens Verden. ISSN 0015-9247. 76(3), p. 74–74. Full text in Research Archive
• Leinaas, Jon Magne (2013). Anyoner, topologi og kvantedatamaskiner. Full text in Research Archive
• Leinaas, Jon Magne (2012). Quantum Computing, hva består det i og lar idéene seg realisere? Full text in Research Archive
• Leinaas, Jon Magne (2012). Hvordan få lys (tilsynelatende) til å gå fortere enn lyset? Full text in Research Archive
• Leinaas, Jon Magne (2012). Kvantefysikk. Full text in Research Archive
• Leinaas, Jon Magne (2012). Kvantefysikk og naturens tilfeldigheter. Full text in Research Archive
• Leinaas, Jon Magne (2012). Kvantefysikkens erkjennelsesmessige konsekvenser: Om uskarphet, EPR-paradokset og kvantemekanisk sammenfiltring. Full text in Research Archive
• Leinaas, Jon Magne (2011). Anyons, what are they, and where are they. Full text in Research Archive
• Leinaas, Jon Magne (2011). Kvanteteorien, hva forteller den oss? Full text in Research Archive
• Leinaas, Jon Magne (2011). Kvantemekanisk sammenfiltring. Full text in Research Archive
• Leinaas, Jon Magne (2011). Kvantemekanisk sammenfiltring. Full text in Research Archive
• Leinaas, Jon Magne (2010). Fractional charge in Luttinger liquids. Full text in Research Archive
• Leinaas, Jon Magne (2010). Fractional charge (and statistics) in Luttinger liquids. In Aldea, Alexandru & Barsan, Victor (Ed.), Trends in Nanophysics; Theory, Experiment and Technology. Springer Publishing Company. ISSN 9783642120695. p. 175–201. doi: 10.1007/978-3-642-12070-1_8. Full text in Research Archive
• Leinaas, Jon Magne (2009). Fractional Charges in Luttinger Liquids. Full text in Research Archive
• Leinaas, Jon Magne (2009). Schrödingers virus. [Radio]. NRK P2. Full text in Research Archive
• Leinaas, Jon Magne; Sollid, Per Øyvind & Myrheim, Jan (2009). Numerical and geometric studies of entanglement - Classification of lowest rank entangled PPT states in the 3x3-system. Full text in Research Archive
• Leinaas, Jon Magne (2008). Sammenfiltring av fotoner. Full text in Research Archive
• Leinaas, Jon Magne (2008). Sammenfiltrede fotoner. Full text in Research Archive
• Leinaas, Jon Magne (2008). Faster than light in a birefringent crystal. Full text in Research Archive
• Leinaas, Jon Magne (2008). Geometry of separable and entangled quantum states. Full text in Research Archive
• Leinaas, Jon Magne (2007). Sammenfiltrede fotoner. Full text in Research Archive
• Leinaas, Jon Magne (2007). Sammenfiltrede fotoner. Full text in Research Archive
• Leinaas, Jon Magne (2007). Black holes, Unruh effect and how accelerated electrons get hot. Full text in Research Archive
• Leinaas, Jon Magne (2007). Geometry of separable and entangled quantum states. Full text in Research Archive
• Dahl, Geir; Ovrum, Eirik; Leinaas, Jon Magne & Myrheim, Jan (2006). A tensor product matrix approximation problem in quantum physics. Full text in Research Archive
• Leinaas, Jon Magne (2006). Fractional statistics - basic principles and new developments. Full text in Research Archive
• Leinaas, Jon Magne (2006). Sammenfiltrede fotoner. Full text in Research Archive
• Leinaas, Jon Magne (2006). Fractional statistics, basic concepts and new developments. Full text in Research Archive
• Leinaas, Jon Magne (2006). Relativistic approach to entanglement. Full text in Research Archive
• Horsdal, Mats & Leinaas, Jon Magne (2006). Hartree-Fock calculation of density and correlation functions in the nu=1 quantum Hall regime. Full text in Research Archive
• Ovrum, Eirik; Leinaas, Jon Magne & Myrheim, Jan (2006). Numerical study of bound entanglement. Full text in Research Archive
• Leinaas, Jon Magne (2005). Fractional statistics - particles that are neither bosons nor fermions. Full text in Research Archive
• Leinaas, Jon Magne (2005). Fractional statistics. Full text in Research Archive
• Leinaas, Jon Magne (2005). Introduction to fractional statistics. Full text in Research Archive
• Leinaas, Jon Magne; Viefers, Susanne F & Vogt, Yngve (2005). Ny oppdagelse i Einstein-klasse: Ukjente partikler påvist etter 30 år. Full text in Research Archive
• Leinaas, Jon Magne (2005). Den mystiske kvantefysikken. Full text in Research Archive
• Leinaas, Jon Magne (2005). Kvantefysikk. In Einevold, Gaute & Newth, Eirik (Ed.), Naturens kode. Gyldendal Akademisk. ISSN 9788205345393. p. 109–121. Full text in Research Archive
• Ovrum, Eirik & Leinaas, Jon Magne (2005). The Geometry of Entanglement. Visualizations and calculation of distances II. Full text in Research Archive
• Ovrum, Eirik & Leinaas, Jon Magne (2005). The Geometry of Entanglement. Visualizations and calculation of distances. Full text in Research Archive
• Leinaas, Jon Magne (2004). Sorte Hull. Full text in Research Archive
• Ovrum, Eirik; Leinaas, Jon Magne & Hjorth-Jensen, Morten (2004). Quantum Computation of Energy Levels in a Spin Chain: A Detailed Simulation for a Small no of Spins. Full text in Research Archive
• Leinaas, Jon Magne (2004). Hva er kvantemekanisk sammenfiltring? Full text in Research Archive
• Leinaas, Jon Magne (2004). Black Holes, Unruh Effect and the Quantum Vacuum. Full text in Research Archive
• Leinaas, Jon Magne (2004). Kvantemekanikk 100 år etter Planck, Del 1. Fra Fysikkens Verden. ISSN 0015-9247. Full text in Research Archive
• Leinaas, Jon Magne (2003). Kvantemekanikk 100 år etter Planck. Full text in Research Archive
• Leinaas, Jon Magne (2002). Quantum vortices. Full text in Research Archive
• Leinaas, Jon Magne (2002). Generalized statistics in low-dimensional systems. Full text in Research Archive
• Leinaas, Jon Magne (2001). Exclusion statistics in an interacting 2D Bose gas. Full text in Research Archive
• Leinaas, Jon Magne (2001). Quantum information and quantum computation. Full text in Research Archive
• Leinaas, Jon Magne (2000). Lysets hastighet - tema i Verdt å vite. [Radio]. NRK, Oslo : NRK, programmet Verdt å vite (Erik Tunstad). Full text in Research Archive Show summary

  De fleste av oss har vel trodd at lyset beveger seg med en fast hastighet - sånn cirka 300.000 kilometer i sekundet. Det har ikke fysikerne trodd. De har visst at lyset har litt ulik hastighet, alt etter hvor det beveger seg. Men maksimalhastigheten har vært den vi finner i vakuum. Professor i fysikk ved Universitetet i Oslo, Jon Magne Leinaas, forteller litt om hva noen forskergrupper har klart, nemlig å få lys til å bevege seg raskere enn lyset. - Jeg er enig i at det høres merkelig ut. Men fysikere har vært klar over i lengre tid, og det har vært drevet eksperimenter med dette tidligere, blant annet har en gruppe ved Berkeley vist at man kan sende lyssignaler gjennom et medium, og få det raskere ut på andre siden. Riktignok med veldig små tidsdifferanser, sier Leinaas. Han forteller så litt videre om hva disse forskerne har kommet frem til. Han forteller dessuten om lys og hastighet mer inngående.

• Leinaas, Jon Magne (2000). Quantum and classical exclusion statistics. Full text in Research Archive
• Leinaas, Jon Magne (1999). Classical phase space and quantum statistics. Full text in Research Archive
• Leinaas, Jon Magne (1999). Accelerated electrons and the Unruh effect. Full text in Research Archive
• Leinaas, Jon Magne (1999). Spin and Statistics of Quantum Hall Quasi-Particles. Full text in Research Archive
• Leinaas, Jon Magne (1999). Phase space, statistics and vortices. Full text in Research Archive
• Horsdal, Mats; Leinaas, Jon-Magne & Viefers, Susanne Friederike (2008). The v=1 quantum Hall effect and its one-dimensional representation. Universitetet i Oslo. Full text in Research Archive Show summary

  The present work is a study of the relationship between a quantum Hall system and its one-dimensional representation. The standard theory, which is based on general arguments, predicts that the edge of the quantum Hall system can be described as a Luttinger liquid with universal parameters determined by the bulk state. In the asymptotic limit the correlation function on the edge should therefore approach a power law with a universal exponent. The Luttinger liquid behaviour has indeed been verified experimentally, in the sense of a power law behaviour. However, the exponent deviates from the value predicted by the theory.<br> The thesis is focused on the microscopical description of a quantum Hall system, in contrast to the general arguments usually employed. This approach makes it possible to study both the effects of the electron interaction and the interplay between the two edges of the system more directly. Both analytical and numerical techniques have been used to study special cases of the system. It is shown that the electron interaction can lead to renormalization of the exponent and to oscillations in the electron density.

• Ovrum, Eirik; Leinaas, Jon Magne & Hjorth-Jensen, Morten (2007). Entanglement and quantum simulators. Universitetet i Oslo. Full text in Research Archive
• Dahl, Geir; Leinaas, Jon Magne; Myrheim, Jan & Ovrum, Eirik (2006). A tensor product matrix approximation problem in quantum physics. Universitetet i Oslo. Full text in Research Archive Show summary

  We consider a matrix approximation problem arising in the study of entanglement in quantum physics. This notion represents a certain type of correlations between subsystems in a composite quantum system. The states of a system are described by a density matrix, which is a positive semidefinite matrix with trace one. The goal is to approximate such a given density matrix by a so-called separable density matrix, and the distance between these matrices gives information about the degree of entanglement in the system. Separability here is expressed in terms of tensor products. We discuss this approximation problem and show that it can be written as a convex optimization problem with special structure. We investigate related convex sets, and suggest an algorithm for this approximation problem which exploits the tensor product structure in certain subproblems. Finally some computational results and experiences are presented.

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