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We present a new formulation for numerically obtaining axisymmetric equilibrium structures of rotating stars in two spatial dimensions. With a view to apply it to the secular evolution of rotating stars, we base it on the Lagrangian description, i.e. we solve the force-balance equations to find the spatial positions of fluid elements endowed individually with a mass, specific entropy and angular momentum. The system of non-linear equations obtained by finite-differencing the basic equations is solved with the W4 method, which is a new multidimensional root-finding scheme of our own devising. We augment it with a remapping scheme to avoid distortions of the Lagrangian coordinates. In this first one of a series of papers, we will give a detailed description of these methods initially. We then present the results of some test calculations, which include the construction of both rapidly rotating barotropic and baroclinic equilibrium states. We gauge their accuracies quantitatively with some diagnostic quantities as well as via comparisons with the counterparts obtained with an Eulerian code. For a demonstrative purpose, we apply the code to a toy-model cooling calculation of a rotating white dwarf.

We present a new formulation to construct numerically equilibrium configurations of rotating stars in general relativity. Having in mind the application to their quasi-static evolutions on a secular time-scale, we adopt a Lagrangian formulation of our own devising, in which we solve force-balance equations to seek for the positions of fluid elements corresponding to the grid points, instead of the ordinary Eulerian formulation. Unlike previous works in the literature, we do not employ the first integral of the Euler equation, which is not obtained analytically in general. We assign a mass, specific angular momentum and entropy to each fluid element in contrast to the previous Eulerian methods, in which the spatial distribution of the angular velocity or angular momentum is specified. These distributions are determined after the positions of all fluid elements (or grid points) are derived in our formulation. We solve the large system of algebraic non-linear equations that are obtained by discretizing the time-independent Euler and Einstein equations in the finite-element method by using our new multidimensional root-finding scheme, named the W4 method. To demonstrate the capability of our new formulation, we construct some rotational configurations, both barotropic and baroclinic. As toy models, we also solve three evolutionary sequences that mimic the cooling, mass-loss, and mass-accretion.

Realistic estimations on the elastic properties of neutron star matter are carried out with a large strain (ε≲0.5) in the framework of relativistic-mean-field model with Thomas-Fermi approximation, where various crystalline configurations are considered in a fully three-dimensional geometry with reflection symmetry. Our calculation confirms the validity of assuming Coulomb crystals for the droplet phase above neutron drip density, which nonetheless does not work at large densities since the elastic constants are found to be decreasing after reaching their peaks. Similarly, the analytic formulae derived in the incompressible liquid-drop model give excellent description for the rod phase at small densities, which overestimates the elastic constants at larger densities. For slabs, due to the negligence on the variations of their thicknesses, the analytic formulae from liquid-drop model agree qualitatively but not quantitatively with our numerical estimations. By fitting to the numerical results, these analytic formulae are improved by introducing dampening factors. The impacts of nuclear symmetry energy are examined adopting two parameter sets, corresponding to the slope of symmetry energy L=41.34 and 89.39 MeV. Even with the uncertainties caused by the anisotropy in polycrystallines, the elastic properties of neutron star matter obtained with L=41.34 and 89.39 MeV are distinctively different, results in detectable differences in various neutron star activities.

In the framework of the relativistic mean-field model with Thomas-Fermi approximation, we study the structures of low-density nuclear matter in a three-dimensional geometry with reflection symmetry. The numerical accuracy and efficiency are improved by expanding the mean fields according to fast cosine transformation and considering only one octant of the unit cell. The effect of finite cell size is treated carefully by searching for the optimum cell size. Typical pasta structures (droplet, rod, slab, tube, and bubble) arranged in various crystalline configurations are obtained for both fixed proton fractions and $\ensuremath{\beta}$ equilibration. It is found that the properties of droplets and bubbles are similar in body-centered cubic (bcc) and face-centered cubic (fcc) lattices, where the fcc lattice generally becomes more stable than bcc lattice as density increases. For the rod and tube phases, the honeycomb lattice is always more stable than the simple one. By introducing an $\ensuremath{\omega}\text{\ensuremath{-}}\ensuremath{\rho}$ cross coupling term, we further examine the pasta structures with a smaller slope of symmetry energy $L=41.34$ MeV, which predicts larger onset densities for core-crust transition and nonspherical nuclei. Such a variation due to the reduction of $L$ is expected to have impacts on various properties in neutron stars, supernova dynamics, and binary neutron star mergers.

We investigate systematically the quark-hadron mixed phase in dense stellar matter and its influence on compact star structures. The properties of quark matter and hadronic matter are fixed based on various model predictions. Beside adopting constant values, the surface tension Σ for the quark-hadron interface is estimated with the multiple reflection expansion method and equivparticle model. To fix the structures of quark-hadron pasta phases, a continuous dimensionality of the structure is adopted as proposed by Ravenhall et al. The corresponding properties of hybrid stars are then obtained and confronted with pulsar observations. It is found that the correlation between radius and tidal deformability in traditional neutron stars preserves in hybrid stars. For those permitted by pulsar observations, in almost all cases, the quark phase persists inside the most massive compact stars. The quark-hadron interface plays an important role in hybrid star structures once quark matter emerges. The surface tension Σ estimated with various methods increases with density, which predicts stiffer equation of states (EOSs) for the quark-hadron mixed phase and increases the maximum mass of hybrid stars. With or without the emergence of quark matter, the obtained EOSs of hybrid star matter are close to each other at densities n≲0.8 fm−3, while larger uncertainty is expected at higher densities.

Neutron stars are high-density objects formed by the gravitational collapse of massive stars, and the whole star can be likened to a giant nucleus. The interior of a neutron star is considered to contain exotic particles and states which do not appear in a normal nucleus. The internal states are constrained by observations of masses and radii via the equation of state of highly dense nuclear matter. Within these constraints, a variety of exotic states have been discussed. The internal state of neutron stars is closely related to its neutrino emission process, which cools the star from the inside. This effect can be compared with observations of the surface temperature of neutron stars. However, despite the wide range of observations of neutron stars, the nature of the neutron star matter remains uncertain. We consider quark matter as an exotic state and perform cooling calculations for neutron stars, incorporating the effects of nucleon superfluidity and quark colour superconductivity.We take into account the “quark-hadron continuity”, in which the neutron superfluidity is succeeded by thedquark pairing. Furthermore, we obtained the range of the neutron star cooling curve, taking into account the difference in surface temperature due to the composition of the surface layer. We found that the existence of quark matter causes strong neutrino emission from quarks, which is moderately suppressedbysuperfluidity and superconductivity, and canexplain the cold surface temperature of neutron stars.

We study the interface effects of a quark-hadron mixed phase in compact stars. The properties of nuclear matter are obtained based on the relativistic-mean-field model. For the quark phase, we adopt a perturbation model with running quark masses and coupling constant. At certain choices of parameter sets, it is found that varying the quark-hadron interface tension will have sizable effects on the radii (ΔR≈600 m) and tidal deformabilities (ΔΛ/Λ≈50%) of 1.36 solar mass hybrid stars. These provide possibilities for us to constrain the quark-hadron interface tension with future gravitational wave observations as well as the ongoing NICER mission.

The effect of pasta phases on the quark-hadron phase transition is investigated for a set of relativistic mean-field equations of state for both hadron and quark matter. The results of the full numerical solution with pasta phases are compared with those of an interpolating construction used in previous works, for which we demonstrate an adequate description of the numerical results. A one-to-one mapping of the free parameter of the construction to the physical surface tension of the quark-hadron interface is obtained for which a fit formula is given. For each pair of quark and hadron matter models the critical value of the surface tension is determined, above which the phase transition becomes close to the Maxwell construction. This result agrees well with earlier theoretical estimates. The study is extended to neutron star matter in beta equilibrium with electrons and muons and is applied to investigate the effect of pasta phases on the structure of hybrid compact stars and the robustness of a possible third family solution., Comment: 13 pages, 10 figures, 3 tables, corresponds to the version published in Phys. Rev. C

We have developed a new formulation to obtain self-gravitating, axisymmetric configurations in permanent rotation. The formulation is based on the Lagrangian variational principle, and treats not only barotropic but also baroclinic equations of state, for which angular momentum distributions are not necessarily cylindrical. We adopt a Monte Carlo technique, which is analogous to those employed in other fields, e.g. nuclear physics, in minimizing the energy functional, which is evaluated on a triangulated mesh. This letter is a proof of principle and detailed comparisons with existing results will be reported in the sequel, but some test calculations are presented, in which we have achieved an error of $O(10^{-4})$ in the Virial relation. We have in mind the application of this method to two-dimensional calculations of the evolutions of rotating stars, for which the Lagrangian formulation is best suited., Comment: 5 pages, 2 figures, accepted for publishing in MNRAS Letters

We discuss the possibility of a multidimensional structure of inhomogeneous chiral condensates, which might be realized in cold and dense quark matter or the core of compact stars. For one-dimensional structures, the system becomes unstable at finite temperature due to the Landau-Peierls instability attributed to the excitation of the Nambu-Goldstone modes, with the result that there is no long-range order, while quasi-long-range order is realized instead. On the other hand, chirally inhomogeneous phases with two- or three-dimensional modulations may be realized as a true long-range ordered phase at any temperature, as inferred from the Landau-Peierls theorem. We here consider possible strategies to explore a multidimensional structure of chiral crystals.

The existence of neutron stars with $2M_\odot$ requires the strong stiffness of the equation of state (EoS) of neutron-star matter. We introduce a multi-pomeron exchange potential (MPP) working universally among 3- and 4-baryons to stiffen the EoS. Its strength is restricted by analyzing the nucleus-nucleus scattering with the G-matrix folding model. The EoSs are derived using the Brueckner-Hartree-Fock (BHF) and the cluster variational method (CVM) with the nuclear interactions ESC and AV18. The mass-radius relations are derived by solving the Tolmann-Oppenheimer-Volkoff (TOV) equation, where the maximum masses over $2M_\odot$ are obtained on the basis of the terrestrial data. Neutron-star radii $R$ at a typical mass $1.5M_\odot$ are predicted to be $12.3\!\sim\!13.0$ km. The uncertainty of calculated radii is mainly from the ratio of 3- and 4-pomeron coupling constants, which cannot be fixed by any terrestrial experiment. Though values of $R(1.5M_\odot)$ are not influenced by hyperon-mixing effects, finely-observed values for them indicate degrees of EoS softening by hyperon mixing in the region of $M\!\sim\!2M_\odot$. If $R(1.5M_\odot)$ is less than about 12.4 km, the softening of EoS by hyperon mixing has to be weak. Useful information can be expected by the space mission NICER offering precise measurements for neutron-star radii within $\pm 5\%$., Comment: 8 pages, 7 figures

We have developed a new formulation to obtain self-gravitating, axisymmetric configurations in permanent rotation. The formulation is based on the Lagrangian variational principle with a triangulated mesh. It treats not only barotropic but also baroclinic equations of state. We compare the various stellar equilibria obtained by our new scheme with those by Hachisu's self-consistent field scheme for the barotropic case, and those by Fujisawa's self-consistent field scheme for the baroclinic case. Included in these rotational configurations are those with shellular-type rotations, which are commonly assumed in the evolution calculation of rotating stars. Although radiation processes, convections and meridional flows have not been taken into account in this study, we have in mind the application of this method to the two-dimensional evolution calculations of rotating stars, for which the Lagrangian formulation is best suited., Comment: 23 pages, 28 figures, accepted for publishing in MNRAS

We study the possibility of two types of inhomogeneous phases in core of neutron stars: one is the Coulomb crystal, which is known as quark-hadron pasta structures, and another one is chiral crystal. In the Coulomb crystal, the inhomogeneous phase appears as the result of the balance between the surface tension and the Coulomb interaction. In chiral crystal, we study the inhomogeneous chiral condensate, which has spatial modulation. In the simple model in 1+1 dimensions, this condensate has the same feature with the FFLO state, which is well known in the condensed matter physics.

This paper describes the fabrication process and device performance of complimentary metal oxide field effect transistor (CMOSFET) with direct silicon bonded (DSB) substrate. This works offers the first comprehensive evaluation of source/drain engineering for DSB devices. Scanning spreading resistance microscopy (SSRM) technique reveals specific dopant profile by lateral diffusion of boron along the bonding interface, in addition to the highly activated dopant at bonding interface in pMOSFET with DSB substrate. Key process condition, such as DSB thickness, hybrid formation process, source/drain engineering and optimization method are described.

Core-collapse supernovae have been supposed to be one of the most plausible sources of gravitational waves. Based on a series of our magnetohydrodynamic core-collapse simulations, we find that the gravitational amplitudes at core bounce can be within the detection limits for the currently running laser-interferometers for a galactic supernova if the central core rotates sufficiently rapidly. This is regardless of the difference of the realistic equations of state and the possible occurrence of the QCD phase transition near core bounce. Even if the core rotates slowly, we point out that the gravitational waves generated from anisotropic neutrino radiation in the postbounce phase due to the standing accretion shock instability (SASI) could be within the detection limits of the detectors in the next generation such as LCGT and the advanced LIGO for the galactic source. Since the waveforms significantly depend on the exploding scenarios, our results suggest that we can obtain the information about the long-veiled explosion mechanism from the gravitational wave signals. Furthermore we discuss the gravitational wave background (GWB) from the explosions of Pop III stars and show that the GWB from Pop III, depending on their formation rates, can be large enough to be within the detection limits of future planned interferometers such as DECIGO and BBO in the frequency interval of ~0.1-1 Hz. This means that the detections of GW background from Pop III stars can be an important tool to supply the information about the star formation history in the early universe.

Neutron stars with large masses $\sim 2M_{\odot}$ require the hard stiffness of equation of state (EoS) of neutron-star matter. On the other hand, hyperon mixing brings about remarkable softening of EoS. In order to solve this problem, a multi-pomeron exchange potential (MPP) is introduced as a model for the universal many-body repulsion in baryonic systems on the basis of the Extended Soft Core (ESC) baryon-baryon interaction. The strength of MPP is determined by analyzing the nucleus-nucleus scattering with the G-matrix folding model. The interactions in $\Lambda\!N$, $\Sigma\!N$ and $\Xi\!N$ channels are shown to be consistent with experimental indications. The EoS in neutron-star matter with hyperon mixing is obtained from ESC in addition of MPP, and mass-radius relations of neutron stars are derived. The maximum mass is shown to reach $2M_{\odot}$ even in the case of including hyperon mixing on the basis of model-parameters determined by terrestrial experiments., Comment: prepared for the 2015 EPJA Topical Issue on "Exotic Matter in Neutron Stars"

We propose a formation scenario of strange stars from rapidly rotating neutron stars after supernova explosions due to the spin-down of the compact stars. By considering a process where the total baryon mass is conserved but the angular momentum is lost due to the emission of gravitational waves and/or magnetic dipole radiation, we find that the transition from a rapidly rotating neutron star to a slowly rotating strange star is possible. The large amount of energy of 5 × 10 53 ergs can be released, which is large by some factors compared to the transition between spherical stars. The liberated energy might become a new energy source for a delayed explosion of supernova. Thus our scenario suggests that the supernova associated with gamma-ray bursts becomes promising targets of future observation of gravitational waves.

We calculate the neutrino production cross-section through the direct URCA process in proto-neutron star matter in the presence of a strong magnetic field. We assume isoentropic conditions and introduce a new equation of state parameter-set in the relativistic mean-field approach that can reproduce neutron stars with $M > 1.96$ M$_\odot$ as required by observations. We find that the production process increases the flux of emitted neutrinos along the direction parallel to the magnetic field and decreases the flux in the opposite direction. This means that the neutrino flux asymmetry due to the neutrino absorption and scattering processes in a magnetic field becomes larger by the inclusion of the neutrino production process., 9 pages, 3 figures

Tomoyuki Maruyama1, Jun Hidaka2, Toshitaka Kajino2, Nobutoshi Yasutake3, Takami Kuroda2, Myung-Ki Cheoun4, Chung-Yeol Ryu5 and Grant J. Mathews6 1College of Bioresource Sciences, Nihon University, Fujisawa 252-8510, Japan 2National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan 3Department of Physics, Chiba Institute of Technology, Narashino 275-0023, Japan 4Department of Physics, Soongsil University, Seoul, 156-743, Korea 5General Education Curriculum Center, Hanyang University, Seoul, 133-791, Korea 6Center of Astrophysics, University of Notre Dame, Notre Dame, IN 46556, USA

We calculate the spin deceleration of proto-neutron stars by asymmetric neutrino absorption in the context of a fully relativistic mean field theory. We calculate for the first time the spin deceleration of neutron stars due to asymmetric neutrino absorption in a toroidal magnetic field configurations. We find a surprising effect that the deceleration can much larger for asymmetric neutrino absorption in a toroidal magnetic field than the usually presumed braking due to magnetic dipole radiation. This may explain the observation that magnetars appear to have had a more rapid deceleration in the past., Comment: 13 pages, 1 figure, 1 table

We study the role of finite-size effects at the hadron-quark phase transition in a new hybrid equation of state constructed from an ab-initio Br\"uckner-Hartree-Fock equation of state with the realistic Bonn-B potential for the hadronic phase and a covariant non-local Nambu--Jona-Lasinio model for the quark phase. We construct static hybrid star sequences and find that our model can support stable hybrid stars with an onset of quark matter below $2 M_\odot$ and a maximum mass above $2.17 M_\odot$ in agreement with recent observations. If the finite-size effects are taken into account the core is composed of pure quark matter. Provided that the quark vector channel interaction is small, and the finite size effects are taken into account, quark matter appears at densities 2-3 times the nuclear saturation density. In that case the proton fraction in the hadronic phase remains below the value required by the onset of the direct URCA process, so that the early onset of quark matter shall affect on the rapid cooling of the star., Comment: version to match the one published in PRC

We calculate the neutrino production cross-section in the proto-neutron-star matter under a strong magnetic field in the relativistic mean-field approach. We introduce a new parameter-set which can reproduce the 1.96 solar mass neutron star. We find that the production process increases emitted neutrinos along the direction parallel to the magnetic field and decrease those along its opposite direction. It means that resultant asymmetry due to the neutrino absorption and scattering process in the magnetic field becomes larger by the addition of the neutrino production process.

A multi-pomeron exchange potential (MPP) is proposed as a model for the three-body repulsion indicated in neutron-star matter, which works universally among three- and four-baryons. Its strength is determined by analyzing the nucleus-nucleus scattering with the G-matrix folding model. The EoS in neutron matter is obtained including the MPP contribution. The neutron-star mass is calculated by solving the TOV equation. The maximum mass is obtained to be larger than the observed one $1.97 M_{solar}$ on the basis of the experimental data.

We study the hadron-quark mixed phase in protoneutron stars, where neutrinos are trapped and lepton number becomes a conserved quantity besides the baryon number and electric charge. Considering protoneutron-star matter as a ternary system, the Gibbs conditions are applied together with the Coulomb interaction. We find that there no crystalline (``pasta'') structure appears in the regime of high lepton-number fraction; the size of pasta becomes very large and the geometrical structure becomes mechanically unstable due to the charge screening effect. Consequently the whole system is separated into two bulk regions like an amorphous state, where the surface effect is safely neglected. There, the local charge neutrality is approximately attained, so that the equation of state is effectively reduced to the one for a binary system. Hence, we conclude that there is no possibility for the density discontinuity to appear in protoneutron-star matter, which is a specific feature in a pure system. These features are important when considering astrophysical phenomena such as supernova explosions or radiation of the gravitational wave from protoneutron stars.

We study the hadron-quark phase transition, taking into account the finite-size effects for neutron star matter. For the hadron phase, we adopt a realistic equation of state within the framework of the Brueckner-Hartree-Fock theory. For the quark phase, we apply the Dyson-Schwinger method. The properties of the mixed phase are clarified by considering the finite-size effects. We find that, if the surface tension is strong enough, the equation of state becomes to be close the one with the Maxwell condition, though we properly adopt the Gibbs conditions. This result is qualitatively the same with the one by the use of the simple bag model. We also find that the mass-radius relation by the EoS is consistent with the observations of massive neutron stars., Comment: 4 pages, 2 figures, the proceeding for "Nuclear Physics in Astrophysics VI"

We calculate neutrino scattering and absorption on the hot and dense neutron-star matter with hyperons under the strong magnetic field using a perturbative approach. We find that the absorption cross-sections show a remarkable angular dependence. Its strength is reduced in the direction parallel to the magnetic field and enhanced in the opposite direction. This asymmetric variation becomes maximally 2.2% of entire neutrino momentum when the magnetic field is assumed as about 2 × 1017G. Since the pulsar kick after the supernova explosion may have relationships to this asymmetry, detailed discussions about the pulsar kick and the asymmetry are presented with the comparison to the observed kick velocities in a fully relativistic approach.

We make a perturbative calculation of neutrino scattering and absorption in hot and dense hyperonic neutron-star matter in the presence of a strong magnetic Field. We calculate that the absorption cross-sections in a fully relativistic mean-field theory. We find that there is a remarkable angular dependence, i.e. the neutrino absorption strength is reduced in a direction parallel to the magnetic Field and enhanced in the opposite direction. This asymmetry in the neutrino absorption is estimated to be as much as 2.2 % of the entire neutrino momentum for an interior magnetic Field of 2 x 10^{17} G. The pulsar kick velocities associated with this asymmetry are shown to be comparable to observed velocities., Comment: arXiv admin note: substantial text overlap with arXiv:1009.0976

The quark-hadron mixed phase ("pasta" phase) with hyperons and neutrinos is studied at finite temperature. We take into account the finite-size effects to the mixed phase, imposing the Gibbs conditions. As a result, we find that the mixed phase becomes unstable at finite temperature. The presence of neutrinos makes the pasta structures more unstable. These results suggest that the quark-hadron mixed phase will appear during the evolution of protoneutron stars after supernova explosions.

We have developed an entirely new formulation to obtain self-gravitating, axisymmetric configurations in permanent rotation. It is based on the Lagrangian variational principle and, as a consequence, will allow us to apply it to stellar evolution calculations rather easily. We adopt a Monte Carlo technique, which is analogous to those employed in other fields, e.g. nuclear physics, in minimizing the energy functional. We also present the analogies between the study on rotating stellar configurations and the one on deformed nuclei. Possible applications are not limited to main sequence stars but will be extended to e.g. compact stars, proto-stars and planets. We believe that our formulation will be a major break-through then.

We investigate the properties of nuclear matter at the first-order phase transitions such as liquid-gas phase transition and hadron-quark phase transition. As a general feature of the first-order phase transitions of matter consisting of many species of charged particles, there appears a mixed phases with geometrical structures called ``pasta'' due to the balance of the Coulomb repulsion and the surface tension between two phases. The equation of state (EOS) of mixed phase is different from the one obtained by a bulk application of the Gibbs conditions or by the Maxwell construction due to the effects of the non-uniform structure. We show that the charge screening and strong surface tension make the EOS close to that of the Maxwell construction. The thermal effects are elucidated as well as the above finite-size effects., Presentation at NFQCD 2010 at Yukawa Inst. TO appear on Prog. Theor. Phys. Suppl

We calculate asymmetric neutrino absorption and scattering cross sections on hot and dense magnetized neutron-star matter including hyperons in fully relativistic mean field theory. The absorption/scattering cross sections are suppressed/enhanced incoherently in the direction of the magnetic fielld, B. The asymmetry is 2-4% at the matter density ��_0 < ��_B < 3��_0 and temperature T < 40 MeV for B=2 X 10^{17} G. This asymmetry is comparable to the effects owing to parity violation or asymmetric magnetic field topology proposed for the origin of pulsar kicks., 9 pages, 3 figure

Recently, the central source of Cassiopeia A (Cas A) has been observed, which indicates that the star has large mass and high effective temperature. We suspect that the compact object cools by the standard neutrino emission. We assume that the compact object contains quark matter with colour superconductivity and calculate cooling curves. Considering the Quark‐Hadron Mixed Phase, we obtain cooling curves which are found to be consistent with the observations.

We compare the Gibbs and Maxwell constructions for the hadron-quark phase transition in neutron and protoneutron stars, including interacting hyperons in the confined phase. We find that the hyperon populations are suppressed, and that neutrino trapping shifts the onset of the phase transition. The effects on the (proto)neutron star maximum mass are explored., Comment: 11 pages, 3 figures

We study the hadron‐quark phase transition with finite size effects at a finite temperature. For the hadron phase, we adopt a realistic equation of state with the Brueckner‐Hartree‐Fock theory including hyperons. Properties of the mixed phase are clarified by considering the finite size effects under the Gibbs conditions. We find that the equation of state gets closer to that given by the Maxwell construction. We also find that the equation of state becomes softer than that at a zero temperature for some density region. Moreover, the number of hyperons is suppressed by the presence of quarks. These are characteristic features of the hadron‐quark mixed phase, and should be important for many astrophysical phenomena such as mergers of binary neutron stars.