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The influences of quantum effects on the structure of the phase-space of generalized Gauss-Bonnet theory, introduced by the Lagrangian F(R,G), have been studied. G is the Gauss-Bonnet invariant, and the quantum effects are described via the account of conformal anomaly. It has been shown that the quantum effects change many aspects of the attractors of F(R,G) gravity models in the R-H plane, including the location of the attractors, the number of them and their stability properties. These variations are not, in general, from the type of small perturbations, but instead, it can induce the great, so not ignorable, variations which have root in the "singular perturbation" nature of this effect. In other words, one can not ignore the quantum corrections and must be always considered. The influences of the perfect barotropic fluids on this problem have been studied, and it has been shown that this kind of matters do not alter the quantum effects. It has been shown that the classical contribution of the coupled-quintessence model, which is responsible for inducing the quantum effects, is of this type, that is a perfect barotropic fluid, and therefore can not change our results., Comment: 20 pages, 10 figures, LaTeX

The generalized scalar-tensor models with Lagrangian $F(\phi,R)-U(\phi)(\nabla\phi)^2$ are considered. It is shown that the phantom-divide-line crossing and the deceleration to acceleration transition generally occurr in these models. Two specific examples, the coupled quintessence model and the Brans-Dicke model are considered. For the first example, it is shown that for the models with $\xi>3/16$, the $\omega=-1$ transition exists. This is verified numerically for some special cases. For the Brans-Dicke model, it is shown that the transition does not occur, a result which can be verified by using the exact solution of this model. Finally the contribution of quantum effects on these phenomena is investigated. It is shown that for some special cases where the $\omega=-1$ transition is classically forbidden, the quantum effects can induce transition. The $\xi=1/6$ of coupled quintessence model is an example of this. The quantum effects are described via the account of conformal anomaly., Comment: 20 pages, 5 figures, typos corrected, reference added

The generalized Gauss-Bonnet theory, introduced by Lagrangian F(R,G), has been considered as a general modified gravity for explanation of the dark energy. G is the Gauss-Bonnet invariant. For this model, we seek the situations under which the late-time behavior of the theory is the de-Sitter space-time. This is done by studying the two dimensional phase space of this theory, i.e. the R-H plane. By obtaining the conditions under which the de-Sitter space-time is the stable attractor of this theory, several aspects of this problem have been investigated. It has been shown that there exist at least two classes of stable attractors : the singularities of the F(R,G), and the cases in which the model has a critical curve, instead of critical points. This curve is R=12H^2 in R-H plane. Several examples, including their numerical calculations, have been discussed., Comment: 19 pages, 11 figures, typos corrected, a reference added

Considering the most general one-species reaction-diffusion processes on a Cayley tree, it has been shown that there exist two integrable models. In the first model, the reactions are the various creation processes, i.e. $\circ\circ\to\bullet\circ$, $\circ\circ\to\bullet\bullet$ and $\circ\bullet\to\bullet\bullet$, and in the second model, only the diffusion process $\bullet\circ\to\circ\bullet$ exists. For the first model, the probabilities $P_l(m;t)$, of finding $m$ particles on $l$-th shell of Cayley tree, have been found exactly, and for the second model, the functions $P_l(1;t)$ have been calculated. It has been shown that these are the only integrable models, if one restricts himself to $L+1$-shell probabilities $P(m_0,m_1,...,m_L;t)$s., Comment: 9 pages, to be appeared in Physica A

The modified gravity with F(R,G) Lagrangian, G is the Gauss-Bonnet invariant, is considered. It is shown that the phantom-divide-line crossing and the deceleration to acceleration transition generally occur in these models. Our results coincide with the known results of f(R)-gravity and f(G)-gravity models. The contribution of quantum effects to these transitions is calculated, and it is shown that in some special cases where there are no transitions in classical level, quantum contributions can induce transitions. The quantum effects are described via the account of conformal anomaly., Comment: 11 pages, LaTeX, a paragraph added, to be appeared in Phys. Rev. D

Considering the single scalar field models of dark energy, i.e. the quintessence and phantom models, it is shown that the quantum effects can cause the system crosses the w = -1 line. This phenomenon does not occur in classical level. The quantum effects are described via the account of conformal anomaly., Comment: LaTex, 7 pages, the version to be appeared in JCAP (2009)

The Klein-Gordon and Dirac equations are considered in a semi-infinite lab ($x > 0$) in the presence of background metrics $ds^2 =u^2(x) \eta_{\mu\nu} dx^\mu dx^\nu$ and $ds^2=-dt^2+u^2(x)\eta_{ij}dx^i dx^j$ with $u(x)=e^{\pm gx}$. These metrics have non-constant scalar-curvatures. Various aspects of the solutions are studied. For the first metric with $u(x)=e^{gx}$, it is shown that the spectrums are discrete, with the ground state energy $E^2_{min}=p^2c^2 + g^2c^2\hbar^2$ for spin-0 particles. For $u(x)=e^{-gx}$, the spectrums are found to be continuous. For the second metric with $u(x)=e^{-gx}$, each particle, depends on its transverse-momentum, can have continuous or discrete spectrum. For Klein-Gordon particles, this threshold transverse-momentum is $\sqrt{3}g/2$, while for Dirac particles it is $g/2$. There is no solution for $u(x)=e^{gx}$ case. Some geometrical properties of these metrics are also discussed., Comment: 14 pages, LaTeX, to be published in Int. Jour. Mod. Phys. A

In two-dimensional Yang-Mills and generalized Yang-Mills theories for large gauge groups, there is a dominant representation determining the thermodynamic limit of the system. This representation is characterized by a density the value of which should everywhere be between zero and one. This density itself is determined through a saddle-point analysis. For some values of the parameter space, this density exceeds one in some places. So one should modify it to obtain an acceptable density. This leads to the well-known Douglas-Kazakov phase transition. In generalized Yang-Mills theories, there are also regions in the parameter space where somewhere this density becomes negative. Here too, one should modify the density so that it remains nonnegative. This leads to another phase transition, different from the Douglas-Kazakov one. Here the general structure of this phase transition is studied, and it is shown that the order of this transition is typically three. Using carefully-chosen parameters, however, it is possible to construct models with phase-transition orders not equal to three. A class of these non-typical models are also studied., Comment: 11 pages, accepted for publication in Eur. Phys. J. C

An interacting dark energy model with interaction term $Q= \lambda_m H\rho_m+\lambda_dH\rho_d$ is considered. By studying the model near the transition time, in which the system crosses the w=-1 phantom-divide-line, the conditions needed to overcome the coincidence problem is investigated. The phantom model, as a candidate for dark energy, is considered and for two specific examples, the quadratic and exponential phantom potentials, it is shown that it is possible the system crosses the w=-1 line, meanwhile the coincidence problem is alleviated, the two facts that have root in observations., Comment: 15 pages, LaTeX. Some minor explanations are added. To be published in Phys. Rev. D

Considering the quintom model with arbitrary potential, it is shown that there always exists a solution which evolves from w > -1 region to w < -1 region. The problem is restricted to the slowly varying potentials, i.e. the slow-roll approximation. It is seen that the rate of this phase transition only depends on the energy density of matter at transition time, which itself is equal to the kinetic part of quintom energy density at that time. The perturbative solutions of the fields are also obtained., Comment: 11 pages, LaTeX, some explanations are added and some of the calculations are transfered to the appendix, to be appeared in Phys. Letts. B

Assuming the Hubble parameter is a continuous and differentiable function of comoving time, we investigate necessary conditions for quintessence to phantom phase transition in quintom model. For power-law and exponential potential examples, we study the behavior of dynamical dark energy fields and Hubble parameter near the transition time, and show that the phantom-divide-line w=-1 is crossed in these models., Comment: LaTeX, 19 pages, four figures, some minor changes in Introduction, two figures added and the references updated, accepted for publication in Phys. Rev. D

The large-N behavior of Yang-Mills and generalized Yang-Mills theories in the double-scaling limit is investigated. By the double-scaling limit, it is meant that the area of the manifold on which the theory is defined, is itself a function of N. It is shown that phase transitions of different orders occur, depending on the functional dependence of the area on N. The finite-size scalings of the system are also investigated. Specifically, the dependence of the dominant representation on A, for large but finite N is determined., Comment: 11 pages, to appear in Eur. Phys. J. C

As a candidate for the dark energy, the hessence model has been recently introduced. We discuss the critical points of this model in almost general case, that is for arbitrary hessence potential and almost arbitrary hessence-background matter interaction. It is shown that in all models, there always exist some stable late-time attractors. It is shown that our general results coincide with those solutions obtained earlier for special cases, but some of them are new. These new solutions have two unique characteristics. First the hessence field has finite value in these solutions and second, their stabilities depend on the second derivative of the hessence potential., Comment: 11 pages. Add some explanations about the autonomousity of the equations, and also a conclusion section was added. To appear in Phys. Rev. D (2006)

For reaction-diffusion processes without exclusion, in which the particles can exist in the same site of a one-dimensional lattice, we study all the integrable models which can be obtained by imposing a boundary condition on the master equation of the asymmetric diffusion process. The annihilation process is also added. The Bethe ansatz solution and the exact N-particle conditional probabilities are obtained., Comment: 13 pages, accepted for publication in Jour. Math. Phys

The large-N limit of the expectation values of the Wilson loops corresponding to two-dimensional U(N) Yang-Mills and generalized Yang-Mills theories on a sphere are studied. The behavior of the expectation values of the Wilson loops both near the critical area and for large areas are investigated. It is shown that the expectation values of the Wilson loops at large areas behave exponentially with respect to the area of the smaller region the boundary of which is the loop; and for the so called typical theories, the expectation values of the Wilson loops exhibit a discontinuity in their second derivative (with respect to the area) at the critical area., Comment: 11 pages,some typos corrected, to be appeared in Nucl. Phys. B

We obtain the electrostatic energy of two opposite charges near the horizon of stationary black-holes in the massive Schwinger model. Besides the confining aspects of the model, we discuss the Bekenstein entropy upper bound of a charged object using the generalized second law. We show that despite the massless case, in the massive Schwinger model the entropy of the black hole and consequently the Bekenstein bound are altered by the vacuum polarization., Comment: 14 pages, accepted for publication in "Gen. Rel. Grav. (2005)"

By considering the master equation of the partially asymmetric diffusion process on a one-dimensional lattice, the most general boundary condition (i.e. interactions) for the multi-species reaction-diffusion processes is considered. Resulting system has various interactions including diffusion to left and right, two-particle interactions $A_a A_b --> A_c A_d $ and the extended n-particle drop-push interactions to left and right. We obtain three distinct new models. The conditions on reaction rates to ensure the solvability of the resulting models are obtained. The two-particle conditional probabilities are calculated exactly., Comment: 17 pages, accepted for publication in J. Math. Phys. (2005)

The large-N limit of the two-dimensional U$(N)$ Yang-Mills theory on an arbitrary orientable compact surface with boundaries is studied. It is shown that if the holonomies of the gauge field on boundaries are near the identity, then the critical behavior of the system is the same as that of an orientable surface without boundaries with the same genus but with a modified area. The diffenece between this effective area and the real area of the surface is obtained and shown to be a function of the boundary conditions (holonomies) only. A similar result is shown to hold for the group SU$(N)$ and other simple groups., Comment: 11 pages, LaTeX2e

We study the Klein-Gordon and Dirac equations in the presence of a background metric ds^2 = -dt^2 + dx^2 + e^{-2gx}(dy^2 + dz^2) in a semi-infinite lab (x>0). This metric has a constant scalar curvature R=6g^2 and is produced by a perfect fluid with equation of state p=-\rho /3. The eigenfunctions of spin-0 and spin-1/2 particles are obtained exactly, and the quantized energy eigenvalues are compared. It is shown that both of these particles must have nonzero transverse momentum in this background. We show that there is a minimum energy E^2_{min}=m^2c^4 + g^2c^2\hbar^2$ for bosons E_{KG} > E_{min}, while the fermions have no specific ground state E_{Dirac}>mc^2., Comment: LateX, 10 pages

The Klein-Gordon and Dirac equations in a semi-infinite lab ($x > 0$), in the background metric $\ds^2 = u^2(x) (-\dt^2 + \dx^2) + \dy^2 + \dz^2$, are investigated. The resulting equations are studied for the special case $ u(x) = 1 + g x$. It is shown that in the case of zero transverse-momentum, the square of the energy eigenvalues of the spin-1/2 particles are less than the squares of the corresponding eigenvalues of spin-0 particles with same masses, by an amount of $mg\hbar c$. Finally, for nonzero transverse-momentum, the energy eigenvalues corresponding to large quantum numbers are obtained, and the results for spin-0 and spin-1/2 particles are compared to each other., Comment: 12 pages, LaTeX 2e

Single-species reaction-diffusion systems on a one-dimensional lattice are considered, in them more than two neighboring sites interact. Constraints on the interaction rates are obtained, that guarantee the closedness of the time evolution equation for $E_n(t)$'s, the probability that $n$ consecutive sites are empty at time $t$. The general method of solving the time evolution equation is discussed. As an example, a system with next-nearest-neighbor interaction is studied., Comment: 19 pages, LaTeX2e

The most general one dimensional reaction-diffusion model with nearest-neighbor interactions, which is exactly-solvable through the empty interval method, has been introduced. Assuming translationally-invariant initial conditions, the probability that $n$ consecutive sites are empty ($E_n$), has been exactly obtained. In the thermodynamic limit, the large-time behavior of the system has also been investigated. Releasing the translational invariance of the initial conditions, the evolution equation for the probability that $n$ consecutive sites, starting from the site $k$, are empty ($E_{k,n}$) is obtained. In the thermodynamic limit, the large time behavior of the system is also considered. Finally, the continuum limit of the model is considered, and the empty-interval probability function is obtained., Comment: 12 pages, LaTeX2e

We consider a process in which there are p-species of particles, i.e. A_1,A_2,...,A_p, on an infinite one-dimensional lattice. Each particle $A_i$ can diffuse to its right neighboring site with rate $D_i$, if this site is not already occupied. Also they have the exchange interaction A_j+A_i --> A_i+A_j with rate $r_{ij}.$ We study the range of parameters (interactions) for which the model is integrable. The wavefunctions of this multi--parameter family of integrable models are found. We also extend the 2--species model to the case in which the particles are able to diffuse to their right or left neighboring sites., Comment: 16 pages, LaTex

Using a covariant method to regularize the composite operators, we obtain the bosonized action of the massive Schwinger model on a classical curved background. Using the solution of the bosonic effective action, the energy of two static external charges with finite and large distance separation on a static curved space-time is obtained. The confining behavior of this model is also explicitly discussed., Comment: A disscussion about the infrared regularization and also two references are added. Accepted for publication in Phys. Rev. D (2001)

Berry phase for a spin--1/2 particle moving in a flat spacetime with torsion is investigated in the context of the Einstein-Cartan-Dirac model. It is shown that if the torsion is due to a dense polarized background, then there is a Berry phase only if the fermion is massless and its momentum is perpendicular to the direction of the background polarization. The order of magnitude of this Berry phase is discussed in other theoretical frameworks., Comment: 9 pages. Some typos corrected, a discussion on propagating torsion is added, accepted for publication in Eur. Phys. J. C (2001)

The phase structure of the generalized Yang--Mills theories is studied, and it is shown that {\it almost} always, it is of the third order. As a specific example, it is shown that all of the models with the interaction $\sum_j (n_j-j+N)^{2k}$ exhibit third order phase transition. ($n_j$ is the length of the $j$-th row of the Yang tableau corresponding to U($N$).) The special cases where the transition is not of the third order are also considered and, as a specific example, it is shown that the model $\sum_j (n_j-j+N)^2+g\sum_j (n_j-j+N)^{4}$ exhibits a third order phase transition, except for $g=27\pi^2/256$, where the order of the transition is 5/2., Comment: 14 pages, LaTex, one paragraph is added to introduction section, to describe the 5/2 phase transition, accepted for publication in Nucl. Phys. B (2001)

By generalizing the auxiliary field term in the Lagrangian of simplicial chiral models on a (d-1)-dimensional simplex, the generalized simplicial chiral models has been introduced in \c{Ali}. These models can be solved analytically only in d=0 and d=2 cases at large-N limit. In d=0 case, we calculate the eigenvalue density function in strong regime and show that the partition function computed from this density function is consistent with one calculated by path integration directly. In d=2 case, it is shown that all $V= {\rm Tr}(AA^{\d})^n$ models have a third order phase transition, same as the 2-dimensional Yang-Mills theory., Comment: 11 pages, 1 figure

We discuss the confining features of the Schwinger model on the Poincare half plane. We show that despite the fact that the expectation value of the large Wilson loop of massless Schwinger model displays the perimeter behavior, the system can be in confining phase due to the singularity of the metric at horizontal axis. It is also shown that in the quenched Schwinger model, the area dependence of the Wilson loop, in contrast to the flat case, is a not a sign of confinement and the model has a finite energy even for large external charges separation. The presence of dynamical fermions can not modify the screening or the confining behavior of the system. Finally we show that in the massive Schwinger model, the system is again in screening phase. The zero curvature limit of the solutions is also discussed., Comment: The results have been obtained for arbitrary scalar curvature, and the zero curvature limit of the solutions have been also discussed. Some references have been added. Has been accepted for publication in Int. J. Mod. Phys. A (2000)

We consider a process in which there are two types of particles, A and B, on an infinite one-dimensional lattice. The particles hop to their adjacent sites, like the totally asymmetric exclusion process (ASEP), and have also the following interactions: A+B -> B+B and B+A -> B+B, all occur with equal rate. We study this process by imposing four boundary conditions on ASEP master equation. It is shown that this model is integrable, in the sense that its N-particle S-matrix is factorized into a product of two-particle S-matrices and, more importantly, the two-particle S-matrix satisfy quantum Yang-Baxter equation. Using coordinate Bethe-ansatz, the N-particle wavefunctions and the two-particle conditional probabilities are found exactly. Further, by imposing four reasonable physical conditions on two-species diffusion-reaction processes (where the most important ones are the equality of the reaction rates and the conservation of the number of particles in each reaction), we show that among the 4096 types of the interactions which have these properties and can be modeled by a master equation and an appropriate set of boundary conditions, there are only 28 independent interactions which are integrable. We find all these interactions and also their corresponding wave functions. Some of these may be new solutions of quantum Yang-Baxter equation., Comment: LaTex,16 pages, some typos are corrected, will be appeared in Phys. Rev. E (2000)

In the context of the Einstein-Cartan-Dirac model, where the torsion of the space-time couples to the axial currents of the fermions, we study the effects of this quantum-gravitational interaction on a massless neutrino beam crossing through a medium with high number density of fermions at rest. We calculate the reflection amplitude and show that a specific fraction of the incident neutrinos reflects from this potential if the polarization of the medium is different from zero. We also discuss the order of magnitude of the fermionic number density in which this phenomenon is observable, in other theoretical contexts, for example the strong-gravity regime and the effective field theory approach., Comment: 8 pages, LaTex

Using the collective field theory approach of large-N generalized two-dimensional Yang-Mills theory on cylinder, it is shown that the classical equation of motion of collective field is a generalized Hopf equation. Then, using the Itzykson-Zuber integral at the large-N limit, it is found that the classical Young tableau density, which satisfies the saddle-point equation and determines the large-N limit of free energy, is the inverse of the solution of this generalized Hopf equation, at a certain point., Comment: 11 pages, add a paragraph after eq.(20) and add one reference, accepted for publication in: Nucl. Phys. B (2000)

Using the Einstein-Cartan-Dirac theory, we study the effect of torsion on neutrino oscillation. We see that torsion cannot induce neutrino oscillation, but affects it whenever oscillation exists for other reasons. We show that the torsion effect on neutrino oscillation is as important as the neutrino mass effect, whenever the ratio of neutrino number density to neutrino energy is $\sim 10^{69}$ cm$^{-3}$ /eV, or the number density of the matter is $\sim 10^{69}$ cm$^{-3}$., Comment: 7 pages, LaTex,Some typos corrected Journal: Int. J. Mod. Phys. A (1999) (will be appeared)

We find a general expression for the free energy of $G(\phi)=\phi^{2k}$ generalized 2D Yang-Mills theories in the strong ($A>A_c$) region at large $N$. We also show that in this region, the density function of Young tableau of these models is a three-cut problem. In the specific $\phi^6$ model, we show that the theory has a third order phase transition, like $\phi^2$ (YM_2) and $\phi^4$ models. We have some comments for $k \geq 4$ cases. At the end, we study the phase structure of $\phi^2 + g \phi^4$ model for $g \leq A/4$ region., Comment: 13 pages, LaTex,the introduction section was changed,will be appeared in: Eur. Phys. J. C (1999)

A two--parameter family of asymmetric exclusion processes for particles on a one-dimensional lattice is defined. The two parameters of the model control the driving force and an effect which we call pushing, due to the fact that particles can push each other in this model. We show that this model is exactly solvable via the coordinate Bethe Ansatz and show that its {\it N}-particle $S$-matrix is factorizable. We also study the interplay of the above effects in determining various steady state and dynamical characteristics of the system., Comment: The last version that will be appeared in " Jour. Stat. Phys. (1999) "

Using the correlation function of chiral vertex operators of the Coulomb gas model, we find the Laughlin wavefunctions of quantum Hall effect, with filling factor $\nu =1/m$, on Riemann sufaces with Poincare metric. The same is done for quasihole wavefunctions. We also discuss their plasma analogy., Comment: 10 pages, LaTex, the paper is completely rewritten, It will be appeared in : Jour. Phys. A 32 (1999)

We define a family of asymmetric processes for particles on a one-dimensional lattice, depending on a continuous parameter $\lambda \in [0,1] $, interpolating between the completely asymmetric processes [1] (for $\lambda =1$) and the n=1 drop-push models [2] (for $ \lambda =0$). For arbitrary $\la$, the model describes an exclusion process, in which a particle pushes its right neighbouring particles to the right, with rates depending on the number of these particles. Using the Bethe ansatz, we obtain the exact solution of the master equation ., Comment: 14 pages, LaTex

Using the standard saddle-point method, we find an explicit relation for the large-N limit of the free energy of an arbitrary generalized 2D Yang-Mills theory in the weak ($AA_c$) region, we investigate carefully the specific fourth Casimir theory, and show that the ordinary integral equation of the density function is not adequate to find the solution. There exist, however, another equation which restricts the parameters. So one can find the free energy in strong region and show that the theory has a third order phase transition., Comment: 10 pages, minor typos corrected, one reference updates

There has been some controversies at the large $N$ behaviour of the 2D Yang-Mills and chiral 2D Yang-Mills theories. To be more specific, is there a one parameter family of minima of the free energy in the strong region, or the minimum is unique. We show that there is a missed equation which, added to the known equations, makes the minimum unique., Comment: 8 pages,Latex

Using the path integral method, we calculate the partition function and the generating functional (of the field strengths) of the generalized 2D Yang-Mills theories in the Schwinger--Fock gauge. Our calculation is done for arbitrary 2D orientable, and also nonorientable surfaces., Comment: 6 pages, LaTex

Based on the technique of derivation of a theory, presented in our recent paper, we investigate the properties of the derived quantum system. We show that the derived quantum system possesses the (nonanomalous) symmetries of the original one, and prove that the exact Green functions of the derived theory are expressed in terms of the semiclassically approximated Green functions of the original theory., Comment: 8 pages,LaTex

By using the path integral method , we calculate the Green functions of field strength of Yang-Mills theories on arbitrary nonorientable surfaces in Schwinger-Fock gauge. We show that the non-gauge invariant correlators consist of a free part and an almost $x$-independent part. We also show that the gauge invariant $n$-point functions are those corresponding to the free part , as in the case of orientable surfaces., Comment: 6 pages,LaTex

Using the simple path integral method we calculate the $n$-point functions of field strength of Yang-Mills theories on arbitrary two-dimensional Riemann surfaces. In $U(1)$ case we show that the correlators consist of two parts , a free and an $x$-independent part. In the case of non-abelian semisimple compact gauge groups we find the non-gauge invariant correlators in Schwinger-Fock gauge and show that it is also divided to a free and an almost $x$-independent part. We also find the gauge-invariant Green functions and show that they correspond to a free field theory., Comment: 8 pages,latex

We find the Laughlin states of the electrons on the Poincare half-plane in different representations. In each case we show that there exist a quantum group $su_q(2)$ symmetry such that the Laughlin states are a representation of it. We calculate the corresponding filling factor by using the plasma analogy of the FQHE., Comment: 9 pages,Latex

The transformation of the $SL(2,R)/U(1)$ black hole under a boost of the subgroup U(1) is studied. It is found that the tachyon vertex operators of the black hole go into those of the $c=1$ conformal field theory coupled to gravity. The discrete states of the black hole also tend to the discrete states of the 2-d gravity theory. The fate of the extra discrete states of the black hole under boost are discussed., Comment: LaTeX file, 14 pages

After showing that the magnetic translation operators are not the symmetries of the QHE on non-flat surfaces , we show that there exist another set of operators which leads to the quantum group symmetries for some of these surfaces . As a first example we show that the $su(2)$ symmetry of the QHE on sphere leads to $su_q(2)$ algebra in the equator . We explain this result by a contraction of $su(2)$ . Secondly , with the help of the symmetry operators of QHE on the Pioncare upper half plane , we will show that the ground state wave functions form a representation of the $su_q(2)$ algebra ., Comment: 8 pages,latex,no figure