binary_global.C

/*
 * Methods of class Binary to compute global quantities
 *
 * (see file binary.h for documentation)
 */
 
/*
 *   Copyright (c) 2004 Francois Limousin
 *
 *   This file is part of LORENE.
 *
 *   LORENE is free software; you can redistribute it and/or modify
 *   it under the terms of the GNU General Public License as published by
 *   the Free Software Foundation; either version 2 of the License, or
 *   (at your option) any later version.
 *
 *   LORENE is distributed in the hope that it will be useful,
 *   but WITHOUT ANY WARRANTY; without even the implied warranty of
 *   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *   GNU General Public License for more details.
 *
 *   You should have received a copy of the GNU General Public License
 *   along with LORENE; if not, write to the Free Software
 *   Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
 *
 */
 
 
char binary_global_C[] = "$Header: /cvsroot/Lorene/C++/Source/Binary/binary_global.C,v 1.16 2014/10/13 08:52:45 j_novak Exp $" ;
 
/*
 * $Id: binary_global.C,v 1.16 2014/10/13 08:52:45 j_novak Exp $
 * $Log: binary_global.C,v $
 * Revision 1.16  2014/10/13 08:52:45  j_novak
 * Lorene classes and functions now belong to the namespace Lorene.
 *
 * Revision 1.15  2006/08/01 14:26:50  f_limousin
 * Small changes
 *
 * Revision 1.14  2006/04/11 14:25:15  f_limousin
 * New version of the code : improvement of the computation of some
 * critical sources, estimation of the dirac gauge, helical symmetry...
 *
 * Revision 1.13  2005/09/18 13:13:41  f_limousin
 * Extension of vphi in the compactified domain for the computation
 * of J_ADM by a volume integral.
 *
 * Revision 1.12  2005/09/15 14:41:04  e_gourgoulhon
 * The total angular momentum is now computed via a volume integral.
 *
 * Revision 1.11  2005/09/13 19:38:31  f_limousin
 * Reintroduction of the resolution of the equations in cartesian coordinates.
 *
 * Revision 1.10  2005/04/08 12:36:45  f_limousin
 * Just to avoid warnings...
 *
 * Revision 1.9  2005/02/17 17:35:00  f_limousin
 * Change the name of some quantities to be consistent with other classes
 * (for instance nnn is changed to nn, shift to beta, beta to lnq...)
 *
 * Revision 1.8  2004/07/21 11:46:24  f_limousin
 * Add function mass_adm_vol() to compute the ADM mass of the system
 * with a volume integral instead of a surface one.
 *
 * Revision 1.7  2004/05/25 14:25:53  f_limousin
 * Add the virial theorem for conformally flat configurations.
 *
 * Revision 1.6  2004/03/31 12:44:54  f_limousin
 * Minor modifs.
 *
 * Revision 1.5  2004/03/25 10:29:01  j_novak
 * All LORENE's units are now defined in the namespace Unites (in file unites.h).
 *
 * Revision 1.4  2004/02/27 10:25:30  f_limousin
 * Modif. to avoid an error in compilation.
 *
 * Revision 1.3  2004/02/27 10:03:04  f_limousin
 * The computation of mass_adm() and mass_komar() is now OK !
 *
 * Revision 1.2  2004/01/20 15:21:36  f_limousin
 * First version
 *
 *
 * $Header: /cvsroot/Lorene/C++/Source/Binary/binary_global.C,v 1.16 2014/10/13 08:52:45 j_novak Exp $
 *
 */
 
 
// Headers C
#include "math.h"
 
// Headers Lorene
#include "nbr_spx.h"
#include "binary.h"
#include "unites.h"
#include "metric.h"
 
            //---------------------------------//
            //      ADM mass           //
            //---------------------------------//
 
namespace Lorene {
double Binary::mass_adm() const {
    
  using namespace Unites ;
  if (p_mass_adm == 0x0) {      // a new computation is requireed
    
    p_mass_adm = new double ; 
        
    *p_mass_adm = 0 ; 
    
    const Map_af map0 (et[0]->get_mp()) ;
    const Metric& flat = (et[0]->get_flat()) ;
 
    Vector dpsi(0.5*(et[0]->get_lnq() - 
                 et[0]->get_logn()).derive_cov(flat)) ;
  
    Vector ww (0.125*(contract(et[0]->get_hij().derive_cov(flat), 1, 2) 
                  - (et[0]->get_hij().trace(flat)).derive_con(flat))) ;
    
    dpsi.change_triad(map0.get_bvect_spher()) ;
    ww.change_triad(map0.get_bvect_spher()) ;
 
    // ww = 0 in Dirac gauge (Eq 174 of BGGN)
    Scalar integrand (dpsi(1) + 0*ww(1)) ;
 
    *p_mass_adm = map0.integrale_surface_infini (integrand) / (-qpig/2.) ;
    
    }   // End of the case where a new computation was necessary
    
    return *p_mass_adm ; 
    
}
 
double Binary::mass_adm_vol() const {
 
  using namespace Unites ;
 
  double massadm ;
  massadm = 0. ;
 
  for (int i=0; i<=1; i++) {        // loop on the stars
 
    // Declaration of all fields
      const Scalar& psi4 = et[i]->get_psi4() ;
      Scalar psi (pow(psi4, 0.25)) ;
      psi.std_spectral_base() ;
      const Scalar& ener_euler = et[i]->get_ener_euler() ;
      const Scalar& kcar_auto = et[i]->get_kcar_auto() ;
      const Scalar& kcar_comp = et[i]->get_kcar_comp() ;
      const Metric& gtilde = et[i]->get_gtilde() ;
      const Metric& flat = et[i]->get_flat() ;
      const Sym_tensor& hij = et[i]->get_hij() ;
      const Sym_tensor& hij_auto = et[i]->get_hij_auto() ;
      const Vector& dcov_logn = et[i]->get_dcov_logn() ;
      const Vector& dcov_phi = et[i]->get_dcov_phi() ;
      const Vector& dcov_lnq = 2*dcov_phi + dcov_logn ;
      const Scalar& lnq_auto = et[i]->get_lnq_auto() ;
      const Scalar& logn_auto = et[i]->get_logn_auto() ;
      const Scalar& phi_auto = 0.5 * (lnq_auto - logn_auto) ;
 
      const Tensor& dcov_hij_auto = hij_auto.derive_cov(flat) ;
      const Tensor& dcov_gtilde = gtilde.cov().derive_cov(flat) ;
      const Tensor& dcov_phi_auto = phi_auto.derive_cov(flat) ;
      const Tensor& dcov_logn_auto = logn_auto.derive_cov(flat) ;
      const Tensor& dcov_lnq_auto = lnq_auto.derive_cov(flat) ;
      Tensor dcovdcov_lnq_auto = lnq_auto.derive_cov(flat).derive_cov(flat) ;
      dcovdcov_lnq_auto.inc_dzpuis() ;
      Tensor dcovdcov_logn_auto = logn_auto.derive_cov(flat).derive_cov(flat) ;
      dcovdcov_logn_auto.inc_dzpuis() ;
 
      // Source in IWM approximation 
      Scalar source = - psi4 % (qpig*ener_euler + (kcar_auto + kcar_comp)/4.) 
    - 0*2*contract(contract(gtilde.con(), 0, dcov_phi, 0), 
               0, dcov_phi_auto, 0, true) ;
      
      // Source = 0 in IWM 
      source += 4*contract(hij, 0, 1, dcov_logn * dcov_phi_auto, 0, 1) +
    2*contract(hij, 0, 1, dcov_phi * dcov_phi_auto, 0, 1) +
    0.0625 * contract(gtilde.con(), 0, 1, contract(
               dcov_hij_auto, 0, 1, dcov_gtilde, 0, 1), 0, 1) - 
           0.125 * contract(gtilde.con(), 0, 1, contract(dcov_hij_auto, 
                  0, 1, dcov_gtilde, 0, 2), 0, 1) -
     contract(hij,0,1,dcovdcov_lnq_auto + dcov_lnq_auto*dcov_lnq,0,1) +
     contract(hij,0,1,dcovdcov_logn_auto + dcov_logn_auto*dcov_logn,0,1) ;
 
      source = source * psi ;
 
      source.std_spectral_base() ;
 
      massadm += - source.integrale()/qpig ;
  }
 
  return massadm ;
}
 
            //---------------------------------//
            //      Komar mass         //
            //---------------------------------//
 
double Binary::mass_kom() const {
    
  using namespace Unites ;
 
  if (p_mass_kom == 0x0) {      // a new computation is requireed
    
    p_mass_kom = new double ; 
      
    *p_mass_kom = 0 ; 
    
    const Tensor& logn = et[0]->get_logn() ;
    const Metric& flat = (et[0]->get_flat()) ;
    const Sym_tensor&  hij = (et[0]->get_hij()) ;
    Map_af map0 (et[0]->get_mp()) ; 
    
    Vector vect = logn.derive_con(flat) + 
                       contract(hij, 1, logn.derive_cov(flat), 0) ;
    vect.change_triad(map0.get_bvect_spher()) ;
    Scalar integrant (vect(1)) ;
    
    *p_mass_kom = map0.integrale_surface_infini (integrant) / qpig ;
    
  } // End of the case where a new computation was necessary
    
  return *p_mass_kom ; 
    
}
 
double Binary::mass_kom_vol() const {
    
  using namespace Unites ;
 
  double masskom ;
  masskom = 0. ;
 
  for (int i=0; i<=1; i++) {        // loop on the stars
 
     // Declaration of all fields
      const Scalar& psi4 = et[i]->get_psi4() ;
      const Scalar& ener_euler = et[i]->get_ener_euler() ;
      const Scalar& s_euler = et[i]->get_s_euler() ;
      const Scalar& kcar_auto = et[i]->get_kcar_auto() ;
      const Scalar& kcar_comp = et[i]->get_kcar_comp() ;
      const Metric& gtilde = et[i]->get_gtilde() ;
      const Metric& flat = et[i]->get_flat() ;
      const Sym_tensor& hij = et[i]->get_hij() ;
      const Scalar& logn = et[i]->get_logn_auto() + et[i]->get_logn_comp() ;
      const Scalar& logn_auto = et[i]->get_logn_auto() ;
      Scalar nn = exp(logn) ;
      nn.std_spectral_base() ;
      
      const Tensor& dcov_logn_auto = logn_auto.derive_cov(flat) ;
      const Vector& dcov_logn = et[i]->get_dcov_logn() ;
      const Vector& dcon_logn = et[i]->get_dcon_logn() ;
      const Vector& dcov_phi = et[i]->get_dcov_phi() ;
      Tensor dcovdcov_logn_auto = (logn_auto.derive_cov(flat))
    .derive_cov(flat) ;
      dcovdcov_logn_auto.inc_dzpuis() ;
 
      Scalar source = qpig * psi4 % (ener_euler + s_euler) ;
      source += psi4 % (kcar_auto + kcar_comp) ;
      source += - 0*contract(dcov_logn_auto, 0, dcon_logn, 0, true) 
      - 2. * contract(contract(gtilde.con(), 0, dcov_phi, 0), 0, 
              dcov_logn_auto, 0, true) ;
      source += - contract(hij, 0, 1, dcovdcov_logn_auto + 
               dcov_logn_auto*dcov_logn, 0, 1) ;
 
      source = source / qpig * nn  ;
  
      source.std_spectral_base() ;
 
      masskom += source.integrale() ;
      
  }
 
  return masskom ;
 
}
 
 
            //---------------------------------//
            //   Total angular momentum        //
            //---------------------------------//
 
const Tbl& Binary::angu_mom() const {
 
  using namespace Unites ;
 
  /*
    if (p_angu_mom == 0x0) {        // a new computation is requireed
    
      p_angu_mom = new Tbl(3) ; 
      
      p_angu_mom->annule_hard() ;   // fills the double array with zeros
  
      const Sym_tensor& kij_auto = et[0]->get_tkij_auto() ;
      const Sym_tensor& kij_comp = et[0]->get_tkij_comp() ;
      const Tensor& psi4 = et[0]->get_psi4() ;
      const Map_af map0 (kij_auto.get_mp()) ;
 
      Sym_tensor kij = (kij_auto + kij_comp) / psi4 ;
      kij.change_triad(map0.get_bvect_cart()) ;
 
      // X component
      // -----------
 
      Vector vect_x(et[0]->get_mp(), CON, map0.get_bvect_cart()) ;      
       
      for (int i=1; i<=3; i++) {
 
      Scalar kij_1 = kij(3, i) ;
      Scalar kij_2 = kij(2, i) ;
          
      kij_1.mult_rsint() ;
      Valeur vtmp = kij_1.get_spectral_va().mult_sp() ;
      kij_1.set_spectral_va() = vtmp ; 
      
      kij_2.mult_r() ;
      vtmp = kij_2.get_spectral_va().mult_ct() ;
      kij_2.set_spectral_va() = vtmp ; 
 
      vect_x.set(i) = kij_1 - kij_2 ;
      }
 
      vect_x.change_triad(map0.get_bvect_spher()) ;
      Scalar integrant_x (vect_x(1)) ;
      
      p_angu_mom->set(0) = map0.integrale_surface_infini (integrant_x) 
                      / (8*M_PI) ;
      
      // Y component
      // -----------
      
      Vector vect_y(et[0]->get_mp(), CON, map0.get_bvect_cart()) ;      
   
      for (int i=1; i<=3; i++) {
 
      Scalar kij_1 = kij(1, i) ;
      Scalar kij_2 = kij(3, i) ;      
      
      kij_1.mult_r() ;
      Valeur vtmp = kij_1.get_spectral_va().mult_ct() ;
      kij_1.set_spectral_va() = vtmp ; 
      
      kij_2.mult_rsint() ;
      vtmp = kij_2.get_spectral_va().mult_cp() ;
      kij_2.set_spectral_va() = vtmp ; 
     
      vect_y.set(i) = kij_1 - kij_2 ;
      }
 
      vect_y.change_triad(map0.get_bvect_spher()) ;
      Scalar integrant_y (vect_y(1)) ;
      
      p_angu_mom->set(1) = map0.integrale_surface_infini (integrant_y) 
                      / (8*M_PI) ;
      
      // Z component
      // -----------
 
      Vector vect_z(et[0]->get_mp(), CON, map0.get_bvect_cart()) ;      
 
      for (int i=1; i<=3; i++) {
 
      Scalar kij_1 = kij(2, i) ;
      Scalar kij_2 = kij(1, i) ;          
 
      kij_1.mult_rsint() ;
      Valeur vtmp = kij_1.get_spectral_va().mult_cp() ;
      kij_1.set_spectral_va() = vtmp ; 
 
      kij_2.mult_rsint() ;
      vtmp =  kij_2.get_spectral_va().mult_sp() ;
      kij_2.set_spectral_va() = vtmp ;
 
      vect_z.set(i) = kij_1 - kij_2 ;
      }
       
      vect_z.change_triad(map0.get_bvect_spher()) ;
      Scalar integrant_z (vect_z(1)) ;
      
      p_angu_mom->set(2) = map0.integrale_surface_infini (integrant_z) 
                     ;// (8*M_PI) ;
      
      
    }   // End of the case where a new computation was necessary
  */
  
 
    /*  
  if (p_angu_mom == 0x0) {      // a new computation is requireed
    p_angu_mom = new Tbl(3) ; 
    p_angu_mom->annule_hard() ; // fills the double array with zeros
    p_angu_mom->set(0) = 0. ;
    p_angu_mom->set(1) = 0. ;
 
    // Alignement 
    double orientation_un = et[0]->get_mp().get_rot_phi() ;
    assert ((orientation_un==0) || (orientation_un==M_PI)) ;
    double orientation_deux = et[1]->get_mp().get_rot_phi() ;
    assert ((orientation_deux==0) || (orientation_deux==M_PI)) ;
    int same_orient = (orientation_un == orientation_deux) ? 1 : -1 ;
    
    // Construction of an auxiliar grid and mapping
    int nzones = et[0]->get_mp().get_mg()->get_nzone() ;
    double* bornes = new double [nzones+1] ;
    double courant = (et[0]->get_mp().get_ori_x()-et[0]->get_mp().get_ori_x())+1 ;
    for (int i=nzones-1 ; i>0 ; i--) {
      bornes[i] = courant ;
      courant /= 2. ;
    }
    bornes[0] = 0 ;
    bornes[nzones] = __infinity ;
    
    Map_af mapping (*(et[0]->get_mp().get_mg()), bornes) ;
    
    delete [] bornes ; 
    
    // Construction of k_total
    Sym_tensor k_total (mapping, CON, mapping.get_bvect_cart()) ;
    
    Vector shift_un (mapping, CON, mapping.get_bvect_cart()) ;
    Vector shift_deux (mapping, CON, mapping.get_bvect_cart()) ;
    
    Vector beta_un (et[0]->get_beta_auto()) ;
    Vector beta_deux (et[1]->get_beta_auto()) ;
    beta_un.change_triad(et[0]->get_mp().get_bvect_cart()) ;
    beta_deux.change_triad(et[1]->get_mp().get_bvect_cart()) ;
    beta_un.std_spectral_base() ;
    beta_deux.std_spectral_base() ;
    
    shift_un.set(1).import(beta_un(1)) ;
    shift_un.set(2).import(beta_un(2)) ;
    shift_un.set(3).import(beta_un(3)) ;
 
    shift_deux.set(1).import(same_orient*beta_deux(1)) ;
    shift_deux.set(2).import(same_orient*beta_deux(2)) ;
    shift_deux.set(3).import(beta_deux(3)) ;
    
    Vector shift_tot (shift_un+shift_deux) ;
    shift_tot.std_spectral_base() ;
    shift_tot.annule(0, nzones-2) ;
    
    
    // Substract the residuals
    shift_tot.inc_dzpuis(2) ;
    shift_tot.dec_dzpuis(2) ;
    
    
    Sym_tensor temp_gamt (et[0]->get_gtilde().cov()) ;
    temp_gamt.change_triad(mapping.get_bvect_cart()) ;
    Metric gamt_cart (temp_gamt) ;
    
    k_total = shift_tot.ope_killing_conf(gamt_cart) / 2. ;
    
    for (int lig=1 ; lig<=3 ; lig++)
    for (int col=lig ; col<=3 ; col++)
      k_total.set(lig, col).mult_r_ced() ;
    
    Vector vecteur_un (mapping, CON, mapping.get_bvect_cart()) ;
    for (int i=1 ; i<=3 ; i++)
      vecteur_un.set(i) = k_total(1, i) ;
    vecteur_un.change_triad (mapping.get_bvect_spher()) ;
    Scalar integrant_un (vecteur_un(1)) ;
    
    Vector vecteur_deux (mapping, CON, mapping.get_bvect_cart()) ;
    for (int i=1 ; i<=3 ; i++)
      vecteur_deux.set(i) = k_total(2, i) ;
    vecteur_deux.change_triad (mapping.get_bvect_spher()) ;
    Scalar integrant_deux (vecteur_deux(1)) ;
    
    // Multiplication by y and x :
    integrant_un.set_spectral_va() = integrant_un.get_spectral_va()
      .mult_st() ;
    integrant_un.set_spectral_va() = integrant_un.get_spectral_va()
      .mult_sp() ;
    
    integrant_deux.set_spectral_va() = integrant_deux.get_spectral_va()
      .mult_st() ;
    integrant_deux.set_spectral_va() = integrant_deux.get_spectral_va()
      .mult_cp() ;
    
    p_angu_mom->set(2) = mapping.integrale_surface_infini (-integrant_un
                     +integrant_deux) / (2*qpig) ;
 
  }
 
    */
    
    if (p_angu_mom == 0x0) {        // a new computation is requireed
    
    p_angu_mom = new Tbl(3) ; 
    p_angu_mom->annule_hard() ; // fills the double array with zeros
 
    // Reference Cartesian vector basis of the Absolute frame
    Base_vect_cart bvect_ref(0.) ;  // 0. = parallel to the Absolute frame
    
    for (int i=0; i<=1; i++) {      // loop on the stars
 
        const Map& mp = et[i]->get_mp() ; 
        int nzm1 = mp.get_mg()->get_nzone() - 1 ; 
        
        // Function exp(-(r-r_0)^2/sigma^2)
        // --------------------------------
        
        double r0 = mp.val_r(nzm1-1, 1, 0, 0) ;
        double sigma = 1.*r0 ;
        
        Scalar rr (mp) ;
        rr = mp.r ;
        
        Scalar ff (mp) ;
        ff = exp( -(rr - r0)*(rr - r0)/sigma/sigma ) ;
        for (int ii=0; ii<nzm1; ii++)
          ff.set_domain(ii) = 1. ;
        ff.set_outer_boundary(nzm1, 0) ;
        ff.std_spectral_base() ;
 
        // Azimuthal vector d/dphi 
        Vector vphi(mp, CON, bvect_ref) ;       
        Scalar yya (mp) ;
        yya = mp.ya ;
        Scalar xxa (mp) ;
        xxa = mp.xa ;
        vphi.set(1) = - yya * ff ;  // phi^X
        vphi.set(2) = xxa * ff ; 
        vphi.set(3) = 0 ;  
 
        vphi.set(1).set_outer_boundary(nzm1, 0) ;
        vphi.set(2).set_outer_boundary(nzm1, 0) ;
    
        vphi.std_spectral_base() ; 
        vphi.change_triad(mp.get_bvect_cart()) ; 
        
        // Matter part
        // -----------
        const Scalar& ee = et[i]->get_ener_euler() ;  // E
        const Scalar& pp = et[i]->get_press() ; // p
        const Scalar& psi4 = et[i]->get_psi4() ; // Psi^4
        Scalar rho = pow(psi4, double(2.5)) * (ee+pp) ; 
        rho.std_spectral_base() ;
 
        Vector jmom = rho * (et[i]->get_u_euler()) ; 
                
        const Metric& gtilde = et[i]->get_gtilde() ; 
        const Metric_flat flat (mp.flat_met_cart()) ; 
        
        Vector vphi_cov = vphi.up_down(gtilde) ;
        
        Scalar integrand = contract(jmom, 0, vphi_cov, 0) ; 
              
        p_angu_mom->set(2) += integrand.integrale() ;
 
        // Extrinsic curvature part (0 if IWM)
        // -----------------------------------
        
        const Sym_tensor& aij = et[i]->get_tkij_auto() ;
        rho = pow(psi4, double(1.5)) ;  
        rho.std_spectral_base() ;
        
        // Construction of D_k \Phi^i
        Itbl type (2) ;
        type.set(0) = CON ;
        type.set(1) = COV ;
        
        Tensor dcov_vphi (mp, 2, type, mp.get_bvect_cart()) ;
        dcov_vphi.set(1,1) = 0. ;
        dcov_vphi.set(2,1) = ff ;
        dcov_vphi.set(3,1) = 0. ;
        dcov_vphi.set(2,2) = 0. ;
        dcov_vphi.set(3,2) = 0. ;
        dcov_vphi.set(3,3) = 0. ;
        dcov_vphi.set(1,2) = -ff ;
        dcov_vphi.set(1,3) = 0. ;
        dcov_vphi.set(2,3) = 0. ;
        dcov_vphi.inc_dzpuis(2) ;
        
        Connection gamijk (gtilde, flat) ;
        const Tensor& deltaijk = gamijk.get_delta() ;
        
        // Computation of \tilde D_i \tilde \Phi_j
        Sym_tensor kill_phi (mp, COV, mp.get_bvect_cart()) ;
        kill_phi = contract(gtilde.cov(), 1, dcov_vphi +
                    contract(deltaijk, 2, vphi, 0), 0) +
          contract(dcov_vphi + contract(deltaijk, 2, vphi, 0), 0,
               gtilde.cov(), 1) ; 
 
        integrand = rho * contract(aij, 0, 1, kill_phi, 0, 1) ; 
        
        p_angu_mom->set(2) += integrand.integrale() / (4*qpig) ;
        
        
    }  // End of the loop on the stars
 
    }   // End of the case where a new computation was necessary
  
    return *p_angu_mom ; 
  
}
 
 
 
            //---------------------------------//
            //      Total energy           //
            //---------------------------------//
 
double Binary::total_ener() const {
    /*
    if (p_total_ener == 0x0) {      // a new computation is requireed
    
    p_total_ener = new double ; 
        
        *p_total_ener = mass_adm() - star1.mass_b() - star2.mass_b() ; 
        
    }   // End of the case where a new computation was necessary
    
    */
    return *p_total_ener ; 
    
}
 
 
            //---------------------------------//
            //   Error on the virial theorem   //
            //---------------------------------//
 
double Binary::virial() const {
    
    if (p_virial == 0x0) {      // a new computation is requireed
    
    p_virial = new double ; 
        
        *p_virial = 1. - mass_kom() / mass_adm() ; 
        
    }
    
    return *p_virial ; 
    
}
}