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Muniain , Gauge Fields, Knots, and Gravity. Bardakci, E. Rabinovici, and B. Bauer and D. Belavin, A. Polyakov, and A. Zamolodchikov , Infinite conformal symmetry in two-dimensional quantum field theory , Nucl. Bellazzini, M. Mintchev, and P. Sorba , Bosonization and scale invariance on quantum wires , Journal of Physics A: Mathematical and Theoretical , vol. Bellissard, A. Elst, and H. Schulz-baldes , The noncommutative geometry of the quantum Hall effect , J. Phys , vol. Bernard and B. Doyon , A hydrodynamic approach to non-equilibrium conformal field theories , Journal of Statistical Mechanics: Theory and Experiment , vol.

Doyon , Energy flow in non-equilibrium conformal field theory , Journal of Physics A: Mathematical and Theoretical , vol. FS12 , pp. Doyon , Time-reversal symmetry and fluctuation relations in nonequilibrium quantum steady states , J. A , vol. Bernard, B. Doyon, and J.

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Viti , Non-equilibrium conformal field theories with impurities , Journal of Physics A: Mathematical and Theoretical , vol. Bernard , Conformal invariance in two-dimensional turbulence , Nature Physics , vol. Bernevig and T. Hughes , Topological insulators and topological superconductors , Bernevig, T. Hughes, and S. Bertlmann , Anomalies in quantum field theory , Bhaseen , Energy flow in quantum critical systems far from equilibrium , Nature Physics , vol. Blanter and M. Bott and J. Bourbaki , Groupes et Algebres de Lie, Chapitres 4, 5 et 6 , Cappelli, L.

### Volume 60, Issue 4

Georgiev, and I. Cappelli, C. Itzykson, and J. Zuber , Modular invariant partition functions in two dimensions , Nuclear Physics B , vol. Cardy , Conformal field theory comes of age , Phys. World 6N12 , pp. Cardy , Conformal invariance and surface critical behavior , Nuclear Physics B , vol. Cardy , Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories , Nuclear Physics B , vol.

Cardy , Operator content of two-dimensional conformally invariant theories , Nuclear Physics B , vol. Cardy ,??? Boltzmann law to quantum information , P Carpentier , Construction and properties of a topological index for periodically driven time-reversal invariant 2D crystals , Nuclear Physics B , vol. Charpentier and K. De-fromont, K. Francesco, P. Mathieu, and D. Vecchia, B. Durhuus, and J. Petersen , The Wess-Zumino action in two dimensions and non-abelian bosonization , Phys.

Doyon, M. Hoogeveen, and D. Bernard , Energy flow and fluctuations in nonequilibrium conformal field theory on star graphs , P Doyon , Lower bounds for ballistic current and noise in non-equilibrium quantum steady states , Nuclear Physics B , vol. Dynkin , The maximal subgroups of the classical groups , Trudy Moskovskogo Matematicheskogo Obshchestva , vol.

Dynkin , Semisimple subalgebras of semisimple Lie algebras , Matematicheskii Sbornik 72 , pp. Elitzur and G. Eugene and I.

Affleck , Tunneling in quantum wires: A boundary conformal field theory approach , Nucl. Feigin and D. Felder, K. Fendley, A. Ludwig, and H. Saleur , Exact nonequilibrium transport through point contacts in quantum wires and fractional quantum Hall devices , Physical Review B , vol. Figueroa-o-'farrill and S. Stanciu , Gauged Wess-Zumino terms and equivariant cohomology , Phys. Bowden, T. Okamoto, F. Kimura, H. Muta, K. Uchino et al. Appl and. Pap , vol. Warner and G. Jonkers, J. De-regt, J. Van-der-mullen, H.

Vos, F. De-groote et al.

Van-der-mullen, G. Boidin, and M. Van-de-sande , High-resolution electron density and temperature maps of a microwave plasma torch measured with a 2-D Thomson scattering system , Spectrochimica Acta Part B: Atomic Spectroscopy , vol. Van-der-mullen, M. Van-de-sande, N. De-vries, B. Broks, E. Iordanova et al. Marshall and G. Hieftje , Measurement of true gas kinetic temperatures in an inductively coupled plasma by laser-light scattering.

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Van-de-sande, and J. Van-der-mullen , Spatially resolved gas temperature measurements by Rayleigh scattering in a microwave discharge , Plasma Sources Science and Technology , vol. Gigosos and C. Gigosos, M. Gonzalez, and V. Iordanova, N.

De-vries, M. Guillemier, and J. Van-der-mullen , Absolute measurements of the continuum radiation to determine the electron density in a microwave-induced argon plasma , Journal of Physics D: Applied Physics , vol. To find an exact solution to the resulting system of nonlinear differential equation, we put the following constraints:. Using 10 in the resulting system of nonlinear differential equation, given in the appendix, we get. Therefore, we use the system of 11 - 14 plus the constraint given by 10 to find solution of those five unknown functions.

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One of the solutions to these equations is the following:. Substituting 15 into 5 we get a vanishing value of the scalar torsion. To understand the physical meaning of the integration constant that appears in solution 15 we discuss the physics related to this solution. First, the metric associated with the tetrad field given by 8 after using solution 15 in 2 has the form.

We rewrite the tetrad field 8 after using 15 in a way to understand its structure. To do this we use the fact that any tetrad field can be rewritten as. Using 18 to rewrite the tetrad field 8 after using 15 , we get. The matrices 21 and 22 give local Lorentz transformations; that is, they satisfy We have considered a solution in the vacuum case. Nine nonlinear differential equations have been obtained.

To solve these differential equations, we have imposed two constraints related to the derivative of the scalar torsion T. Then the number of the nonlinear differential equations reduced to four nonlinear differential equations. Wehavederived a solution which contains one constant of integration. To understand the physical meaning of the constant of integration, which exists in the solution, we have calculated its associated metric. To understand the properties of the derived solution, we have rewritten it as three matrices.

The first local Lorentz transformation local Lorentz transformation means that it satisfies A. By the use of 5 and 8 , we obtain the torsion scalar and its derivatives in terms of r and 0 as. The author declares that there is no conflict of interests regarding the publication of this paper.

## Book Physics Reports Vol340

From A. Equations A. If this term is vanishing, then A. Hehl, J. McCrea, E. Mielke, and Y. Ne'eman, "Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance," Physics Reports, vol. Wanas and H. Wanas, "The other side of gravity and geometry: anti-gravity and anticurvature," Advances in High Energy Physics, vol. Wanas, N. Youssef, and A. Youssef and A. Sid-Ahmed, "Linear connections and curvature tensors in the geometry of parallelizable manifolds," Reports on Mathematical Physics, vol.

Youssef and W. Elsayed, "A global approach to absolute parallelism geometry," Reports on Mathematical Physics, vol. Hayashi and T. Shirafuji, "New general relativity," Physical Review D, vol. Shirafuji, "Addendum to "new general relativity"," Physical Review D, vol. Bamba, S.