The highly relativistic nature of the outflows is inferred from and constrained by the observations of GeV photons, which indicate the need for bulk Lorentz factors of 102 [118, 191, 24]. Such Lorentz factors result in synchrotron spectra which in the observer frame extend beyond 100 MeV, and inverse Compton (IC) scattering of such synchrotron photons leads to the expectation of GeV and TeV spectral components . While 18 GeV photons have been observed (e.g. ), TeV photons are likely to be degraded to lower energies by pair production, either in the source itself, or (unless the GRB is at very low redshifts) in the intervening intergalactic medium [73, 92].
Besides emitting in the currently studied sub-GeV electromagnetic channels, GRB are likely to be even more luminous in other channels, such as neutrinos, gravitational waves and cosmic rays. For instance, nucleons entrained in the fireball will have 100 GeV bulk kinetic energies in the observer frame, which can lead to inelastic collisions resulting in pions, muons, neutrinos and electrons as well as their anti-particles. The main targets for the relativistic baryons are other particles in the relativistic outflow and particles in the external, slower moving environment. The expected flux and spectrum of 1-30 GeV neutrinos and -rays resulting from pion decay due to interactions within the expanding plasma depends, e.g., on the neutron/proton ratio and on fireball inhomogeneities, while that due to interactions with the surrounding medium depends on the external gas density and its distribution; and both depend on the Lorentz factor. Massive progenitors offer denser targets for nuclear collisions and a larger photon density for p and interactions, leading to modification of the photon spectra. On the other hand GRB from NS-NS mergers would be characterized by neutron-rich outflows, leading to stronger 5-10 GeV neutrinos and photons from np collisions [17, 32, 416]. Photo-pion signatures of 100 GeV photons and 1014-1018 eV neutrinos may be expected to be relatively stronger in massive (high soft photon density) progenitors. Knowing what fraction of GRB, if any, arise from NS mergers is vital for facilitating interferometric gravitational wave detections, e.g. with LIGO. And, conversely, detection with LIGO would provide important clues as to whether short bursts are NS-NS (or NS-BH) mergers, or whether massive stellar collapses are asymmetric enough to produce substantial gravitational wave emission and serve as a test of the relationship between long GRB and supernovae.
The Fermi mechanism in shocks developing in the GRB outflow can also accelerate protons to observer-frame energies up to ~ 1020 eV [494, 492]. Internal shocks leading to the observed -rays have a high comoving photon density and lead to p photopion production and to 100 TeV neutrinos . In external shocks due to deceleration by the external medium, the reverse shock moving into the ejecta can produce optical photons (Section 5.2) which result in photopion production and 1019 eV neutrinos . Neutrinos in the TeV to EeV range may be easier to detect than those at ~10 GeV energies, due to their higher interaction cross section, with instruments currently under construction. Such neutrinos would serve as diagnostics of the presence of relativistic shocks, and as probes of the acceleration mechanism and the magnetic field strength. The flux and spectrum of 1019 eV neutrinos depends on the density of the surrounding gas, while the 1014 eV neutrinos depend on the fireball Lorentz factor. Hence, the detection of very high energy neutrinos would provide crucial constraints on the fireball parameters and GRB environment.
9.1. UHE photons from GRB
Ultra-high energy emission, in the range of GeV and harder, is expected from electron inverse Compton in external shocks  as well as from internal shocks  in the prompt phase. The combination of prompt MeV radiation from internal shocks and a more prolonged GeV IC component for external shocks  is a likely explanation for the delayed GeV emission seen in some GRB . (An alternative invoking photomeson processes from ejecta protons impacting a nearby binary stellar companion is ). The GeV photon emission from the long-term IC component in external afterglow shocks has been considered by [98, 523, 95, 488, 489]. The IC GeV photon component is likely to be significantly more important  than a possible proton synchrotron or electron synchrotron component at these energies. Another possible contributor at these energies may be 0 decay from p interactions between shock-accelerated protons and MeV or other photons in the GRAB shock region [55, 467, 137]. However, under the conservative assumption that the relativistic proton energy does not exceed the energy in relativistic electrons or in -rays, and that the proton spectral index is -2.2 instead of -2, both the proton synchrotron and the p components can be shown to be substantially less important at GeV-TeV than the IC component . Another GeV photon component is expected from the fact that in a baryonic GRB outflow neutrons are likely to be present, and when these decouple from the protons, before any shocks occur, pn inelastic collisions will lead to pions, including 0, resulting in UHE photons which cascade down to the GeV range [94, 17, 416]. The final GeV spectrum results from a complex cascade, but a rough estimate indicates that 1-10 GeV flux should be detectable  with GLAST  for bursts at z 0.1.
In these models, due to the high photon densities implied by GRB models, absorption within the GRB emission region must be taken into account [22, 272, 398, 364, 365]. One interesting result is that the observation of photons up to a certain energy, say 10-20 GeV with EGRET, puts a lower limit on the bulk Lorentz factor of the outflow, from the fact that the compactness parameter (optical depth to ) is directly proportional to the comoving photon density, and both this as well as the energy of the photons depend on the bulk Lorentz factor. This has been used by  to estimate lower limits on 300-600 for a number of specific bursts observed with EGRET. On the other hand, for GRB with 850, TeV photons can escape the source .
Long GRB have recently been shown to be associated with supernovae (Section 8.2). If GRB also accelerate cosmic rays, as suspected, then these could leave long-lasting UHE photon signatures in supernova remnants which were associated with GRB at the time of their explosion. One example may be the SN remnant W49B, which may be a GRB remnant. A signature of a neutron admixture in the relativistic cosmic ray outflow would be a TeV gamma-ray signature due to inverse Compton interactions following neutron decay  (see also ). Continued magnetic outflows upscattering companion photons may also signal GRB remnants . The imaging of the surrounding emission could provide new constraints on the jet structure of the GRB.
The recent detection of delayed X-ray flares during the afterglow phase of gamma-ray bursts (GRBs) with the Swift satellite (e.g. [528, 338, 360]) suggests an inner-engine origin of these flares, at radii inside the deceleration radius characterizing the beginning of the forward shock afterglow emission. Given the observed temporal overlapping between the flares and afterglows, one expects an inverse Compton (IC) emission arising from such flare photons scattered by forward shock afterglow electrons . The jet may also IC upscatter shock break-out X-ray photons . This IC emission would produce GeV-TeV flares, which may be detected by GLAST and ground-based TeV telescopes. The detection of GeV-TeV flares combined with low energy observations may help to constrain the poorly known magnetic field in afterglow shocks.
At higher energies, a tentative 0.1 TeV detection at the 3 level of GRB 970417a has been reported with the water Cherenkov detector Milagrito . Another possible TeV detection  of GRB 971110 has been reported with the GRAND array, at the 2.7 level. Stacking of data from the TIBET array for a large number of GRB time windows has led to an estimate of a ~ 7 composite detection significance . Better sensitivity is expected from the upgraded larger version of MILAGRO, as well as from atmospheric Cherenkov telescopes under construction such as VERITAS, HESS, MAGIC and CANGAROO-III [505, 342, 201, 198, 458, 123, 222]. However, GRB detections in the TeV range are expected only for rare nearby events, since at this energy the mean free path against absorption on the diffuse IR photon background is ~ few hundred Mpc [73, 92]. The mean free path is much larger at GeV energies, and based on the handful of GRB reported in this range with EGRET, several hundred should be detectable with large area space-based detectors such as GLAST [289, 523].
9.2. Cosmic rays from GRB
In the standard fireball shock model of the prompt -ray emission, say from internal shocks or magnetic dissipation, and also in the external afterglow shocks, the same acceleration mechanisms which lead to the non-thermal electron power laws implied by the observed photon spectra must also lead to proton acceleration. Using the shock parameters inferred from broad-band photon spectral fits, one infers that protons can be accelerated to Lorentz factors up to 1011 in the observer frame [494, 482], i.e. to so-called GZK energy of Ep ~ 1020 eV. This is interesting mainly for "baryonic" jets, where the bulk of the energy is carried by baryons, whereas in Poynting-dominated jets there would be much fewer protons to accelerate. Well below the GZK energy, protons interacting with the MeV photons present in GRB or with thermal nucleons are above the pion production threshold and can produce ultra-high energy neutrinos, as discussed below.
Discussions of GRB as cosmic ray sources are mainly oriented at exploring their contribution to the energy range above EeV (1018 eV; e.g. ), referred to as ultra-high energy cosmic rays, or UHECRs. (A model where GRB are responsible for CRs ranging from PeV to GZK is ). At EeV and higher energies the observed UHECR isotropy and the small expected magnetic deflection suggests an extra-galactic origin. The requirement that they are not attenuated by the cosmic microwave background through photomeson interactions constrains that they are originated within a volume inside a radius of 50-100 Mpc, the so-called "GZK" volume (e.g. ). Two broad classes of models suggested are the "top-down" scenarios, which attribute UHECR to decay of fossil Grand Unification defects, and the "bottom-up" scenarios, which assume UHECRs are accelerated in astrophysical sources. One of the most prominent candidate sources for bottom-up scenarios is GRBs [494, 482, 314] (two others are AGNs, e.g.  and cluster shocks, e.g. ). The most commonly discussed version of this scenario considers the UHECR to be protons accelerated in GRB internal shocks [494, 493, 492], while another version attributes them to acceleration in external shocks [482, 481, 99]. (For UHECR acceleration in alternative GRB models, see, e.g. [88, 117]).
The persuasiveness of this scenario is largely based on two coincidences, namely, the required condition to accelerate protons to GZK energies is similar to the requirement for generating the prompt observed gamma-rays in GRB, and the observed UHECR energy injection rate into the universe (~ 3 × 1044 erg Mpc-3 yr-1) is similar to the local GRB -ray energy injection rate [494, 482]. These coincidences have been questioned, e.g. [459, 445], but these objections have been resolved using new data and further considerations [492, 481], and GRBs remain a promising candidate for UHECRs. However, there are some caveats of principle. The internal shock scenario relies on the assumption that GRB prompt gamma-ray emission is due to internal shocks. Although this is the leading scenario, there is no strong proof so far, as is the case for the external shock (e.g., there are efficiency and spectrum issues, etc.). On the other hand, a Poynting flux dominated GRB model would have to rely on magnetic dissipation and reconnection, accelerating electrons and hence also accelerating protons- but details remain to be investigated. The external shock model would have to rely on a magnetized medium  to reach the desired cosmic ray energy (as expected in pulsar wind bubbles  in the supranova scenario , which however has become less likely since the almost simultaneous GRB 030329 / SN 2003dh and the more recent GRB 060218 / SN2006aj association).
Direct confirmation of a GRB orgin of UHECRs will be difficult. The next generation cosmic ray detectors such as the Pierre Auger Observatory  will have a substantially enhanced effective target area, which will greatly improve the cosmic ray count statistics. This will help to disentangle the two scenarios (top-down or bottom-up) and will reveal whether a GZK feature indeed exists. Within the bottom-up scenario, the directional information may either prove or significantly constrain the alternative AGN scenario, and may eventually shed light on whether GRBs are indeed the sources of UHECRs.
9.3. UHE neutrinos contemporary with gamma-rays
Internal shocks in the GRB jet take place at a radius ri ~ 2 i2 c t ~ 5 × 1012 t-3 3002 cm. Here i = 300 300 is the bulk Lorentz factor of the GRB fireball ejecta and t = 10-3 t-3 s is the variability time scale. Observed -rays are emitted from the GRB fireball when it becomes optically thin at a radius ri. Shock accelerated protons interact dominantly with observed synchrotron photons with ~ MeV peak energy in the fireball to produce a Delta resonance, p + . The threshold condition to produce a + is Ep E = 0.2 i2 GeV2 in the observer frame, which corresponds to a proton energy of Ep = 1.8 × 107 E, MeV-1 3002 GeV. The subsequent decays + n + n µ+ µ n e+ e µ µ produce high energy neutrinos in the GRB fireball contemporaneous with -rays [501, 388]. Assuming that the secondary pions receive 20% of the proton energy per interaction and each secondary lepton shares 1/4 of the pion energy, each flavor of neutrino is emitted with 5% of the proton energy, dominantly in the PeV range.
The diffuse muon neutrino flux from GRB internal shocks due to proton acceleration and subsequent photopion losses is shown as the short dashed line in Fig. 10. The flux is compared to the Waxman-Bahcall limit of cosmic neutrinos, which is derived from the observed cosmic ray flux . The fluxes of all neutrino flavors are expected to be equal after oscillation in vacuum over astrophysical distances.
Figure 10. Diffuse muon neutrino flux arriving simultaneously with the -rays from shocks outside the stellar surface in observed GRB (dark short-dashed curve), compared to the Waxman-Bahcall (WB) diffuse cosmic ray bound (light long-dashed curves) and the atmospheric neutrino flux (light short-dashed curves). Also shown is the diffuse muon neutrino precursor flux (solid lines) from sub-stellar jet shocks in two GRB progenitor models, with stellar radii r12.5 (H) and r11 (He). These neutrinos arrive 10-100 s before the -rays from electromagnetically detected bursts (with similar curves for µ, e and ) .
The GRB afterglow arises as the jet fireball ejecta runs into the ambient inter-stellar medium (ISM), driving a blast wave ahead into it and a reverse shock back into the GRB jet ejecta. This (external) reverse shock takes place well beyond the internal shocks, at a radius re ~ 4e2 c t ~ 2 × 1017 2502 t30 cm . Here e 250 250 is the bulk Lorentz factor of the ejecta after the partial energy loss incurred in the internal shocks and t = 30 t30 s is the duration of the GRB jet. Neutrinos are produced in the external reverse shock due to p interactions of internal shock accelerated protons predominantly with synchrotron soft x-ray photons produced by the reverse shock. The efficiency of pion conversion from p interactions in this afterglow scenario is much smaller than in the internal shocks .
In the case of a massive star progenitor the jet may be expanding into a wind, emitted by the progenitor prior to its collapse. In this case, the density of the surrounding medium, at the external shock radius, may be much higher than that typical ISM density of n 1 cm-3. For a wind with mass loss rate of 10-5 M yr-1 and velocity of vw = 103 km/s, the wind density at the typical external shock radius would be 104 cm-3. The higher density implies a lower Lorenz factor of the expanding plasma during the reverse shocks stage, and hence a larger fraction of proton energy lost to pion production. Protons of energy Ep 1018 eV lose all their energy to pion production in this scenario [502, 484, 80] producing EeV neutrinos.
9.4. Precursor neutrinos
As discussed before, the core collapse of massive stars are the most likely candidates for long duration GRBs, which should lead to the formation of a relativistic jet initially buried inside the star. The jet burrows through the stellar material, and may or may not break through the stellar envelope . Internal shocks in the jet, while it is burrowing through the stellar interior, may produce high energy neutrinos through proton-proton (pp) and photomeson (p) interactions . High energy neutrinos are produced through pion decays which are created both in pp and p interactions. The jets which successfully penetrate through the stellar envelope result in GRBs (-ray bright bursts) and the jets which choke inside the stars do not produce GRBs (-ray dark bursts). However, in both cases high energy neutrinos produced in the internal shocks are emitted through the stellar envelope since they interact very weakly with matter.
High energy neutrinos from the relativistic buried jets are emitted as precursors (~ 10-100 s prior) to the neutrinos emitted from the GRB fireball in case of an electromagnetically observed burst. In the the case of a choked burst (electromagnetically undetectable) no direct detection of neutrinos from individual sources is possible. However the diffuse neutrino signal is boosted up in both scenarios. The diffuse neutrino flux from two progenitor star models are shown in Fig. 10, one for a blue super-giant (labeled H) of radius R* = 3 × 1012 cm and the other a Wof-Rayet type (labeled He) of radius R* = 1011 cm. The Waxman-Bahcall diffuse cosmic ray bound , the atmospheric flux and the IceCube sensitivity to diffuse flux are also plotted for comparison. The neutrino component which is contemporaneous with the gamma-ray emission (i.e. which arrives after the precursor) is shown as the dark dashed curve, and is plotted assuming that protons lose all their energy to pions in p interactions in internal shocks.
Most GRBs are located at cosmological distances (with redshift z ~ 1) and individual detection of them by km scale neutrino telescopes may not be possible. The diffuse flux is then dominated by a few nearby bursts. The likeliest prospect for UHE detection is from these nearby GRBs in correlation with electromagnetic detection. Detection of ultrahigh energy neutrinos which point back to their sources may establish GRBs as the sources of GZK cosmic rays.
The detection of ultrahigh energy neutrinos by future experiments such as ICECUBE , ANITA , KM3NeT , and Auger  can provide useful information, such as particle acceleration, radiation mechanism and magnetic field, about the sources and their progenitors. High energy neutrino astrophysics is an imminent prospect, with Amanda already providing useful limits on the diffuse flux from GRB [457, 27] and with ICECUBE [3, 204, 189] on its way. The detection of TeV and higher energy neutrinos from GRB would be of great importance for understanding these sources, as well as the acceleration mechanisms involved. It could provide evidence for the hadronic vs. the MHD composition of the jets, and if observed, could serve as an unabsorbed probe of the highest redshift generation of star formation in the Universe.
9.5. Gravitational waves
The gamma-rays and the afterglows of GRB are thought to be produced at distances from the central engine where the plasma has become optically thin, r 1013 cm, which is much larger than the Schwarzschild radius of a stellar mass black hole (or of a neutron star). Hence we have only very indirect information about the inner parts of the central engine where the energy is generated. However, in any stellar progenitor model of GRB one expects that gravitational waves should be emitted from the immediate neighborhood of the central engine, and their observation should give valuable information about its identity. Therefore, it is of interest to study the gravitational wave emission from GRB associated with specific progenitors. Another reason for doing this is that the present and foreseeable sensitivity of gravitational wave detectors is such that for likely sources, including GRB, the detections would be difficult, and for this reason, much effort has been devoted to the development of data analysis techniques that can reach deep into the detector noise. A coincidence between a gravitational wave signal and a gamma-ray signal would greatly enhance the statistical significance of the detection of the gravitational wave signal [125, 239]. It is therefore of interest to examine the gravitational wave signals expected from various specific GRB progenitors that have been recently discussed, and based on current astrophysical models, to consider the range of rates and strains expected in each case, for comparison with the LIGO sensitivity. A general reference is , which also discusses GRB-related sources of gravitational waves.
Regardless of whether they are associated with GRBs, binary compact object mergers (NS-NS, NS-BH, BH-BH, BH-WD, BH-Helium star etc.) [466, 372, 76, 239, 419, 215, 148, 229] and stellar core-collapses [389, 144, 90, 229, 475, 476, 477] have been studied as potential gravitational wave (GW) sources. These events are also leading candidates for being GRB progenitors, and a coincidence between a GW signal and a gamma-ray signal would greatly enhance the statistical significance of the former . A binary coalescence process can be divided into three phases: in-spiral, merger, and ring-down [124, 229]. For collapsars, a rapidly rotating core could lead to development of a bar and to fragmentation instabilities which would produce similar GW signals as in the binary merger scenarios, although a larger uncertainty is involved. The GW frequencies of various phases cover the 10 - 103 Hz band which is relevant for the Laser Interferometer Gravitational-wave Observatory (LIGO)  and other related detectors such as VIRGO , GEO600  and TAMA300 . Because of the faint nature of the typical GW strain, only nearby sources (e.g. within ~ 200 Mpc for NS-NS and NS-BH mergers, and within ~ 30 Mpc for collapsars)  have strong enough signals to be detectable by LIGO-II. When event rates are taken into account [147, 28], order of magnitude estimates indicate that after one-year operation of the advanced LIGO, one event for the in-spiral chirp signal of the NS-NS or NS-BH merger, and probably one collapsar event (subject to uncertainties), would be detected . Other binary merger scenarios such as BH-WD and BH-Helium star mergers are unlikely to be detectable , and they are also unfavored as sources of GRBs according to other arguments .
A time-integrated GW luminosity of the order of a solar rest mass ( 1054 erg) is predicted from merging NS-NS and NS-BH models [239, 420, 321], while the luminosity from collapsar models is more model-dependent, but expected to be lower ([143, 318]; c.f. ). Specific estimates have been made of the GW strains from some of the most widely discussed current GRB progenitor stellar systems . The expected detection rates of gravitational wave events with LIGO from compact binary mergers, in coincidence with GRBs, has been estimated by [125, 126]. If some fraction of GRBs are produced by DNS or NS-BH mergers, the gravitational wave chirp signal of the in-spiral phase should be detectable by the advanced LIGO within one year, associated with the GRB electromagnetic signal. One also expects signals from the black hole ring-down phase, as well as the possible contribution of a bar configuration from gravitational instability in the accretion disk following tidal disruption or infall in GRB scenarios.
The most promising GW-GRB candidates in terms of detections per year are the DNS and BH-NS mergers  (Fig. 11), based on assumed mean distances from the formation rates estimated by . More recent rate estimates are in , and rates incorporating new information relating to Swift short GRB detections are in [29, 328]. Other binary progenitor scenarios, such as black hole - Helium star and black hole - white dwarf merger GRB progenitors are unlikely to be detectable, due to the low estimates obtained for the maximum non-axisymmetrical perturbations.
Figure 11. Gravitational wave strain from a double neutron merger (left) and a collapsar (right) compared to advanced LIGO sensitivity .
For the massive rotating stellar collapse (collapsar) scenario of GRB, the non-axisymmetrical perturbations are very uncertain, but may be strong [90, 144, 476], and the estimated formation rates are much higher than for other progenitors [144, 30], with typically lower mean distances to the Earth. For such long GRB the rate estimates must incorporate the beaming correction . This type of scenario is of special interest, since it has the most observational support from GRB afterglow observations. For collapsars, in the absence of detailed numerical 3D calculations specifically aimed at GRB progenitors, estimates were made  of the strongest signals that might be expected in the case of bar instabilities occurring in the accretion disk around the resulting black hole, and in the maximal version of the recently proposed fragmentation scenario of the infalling cores. Although the waveforms of the gravitational waves produced in the break-up, merger and/or bar instability phase of collapsars are not known, a cross-correlation technique can be used making use of two co-aligned detectors. Under these assumptions, collapsar GRB models would be expected to be marginally detectable as gravitational wave sources by the advanced LIGO within one year of observations. Figure 11 depicts the characteristic GW strains for the double neutrons star merger and the collapsar model.
Other calculations of massive stellar collapse GRB [476, 477] take into account MHD effects in the disk and BH. More general studies of massive stellar core collapse event gravitational wave emission are presented in , considering both core collapse SN and the progenitors of long GRB.
In the case of binaries the matched filtering technique can be used, while for sources such as collapsars, where the wave forms are uncertain, the simultaneous detection by two elements of a gravitational wave interferometer, coupled with electromagnetic simultaneous detection, provides a possible detection technique. Specific detection estimates have been made [229, 479] for both the compact binary scenarios and the collapsar scenarios.
Both the compact merger and the collapsar models have in common a high angular rotation rate, and observations provide evidence for jet collimation of the photon emission, with properties depending on the polar angle, which may also be of relevance for X-ray flashes. Calculations have been made  of the gravitational wave emission and its polarization as a function of angle expected from such sources. The GRB progenitors emit l = m = 2 gravitational waves, which are circularly polarized on the polar axis, while the + polarization dominates on the equatorial plane. Recent GRB studies suggest that the wide variation in the apparent luminosity of GRBs are caused by differences in the viewing angle, or possibly also in the jet opening angle. Since GRB jets are launched along the polar axis of GRB progenitors, correlations among the apparent luminosity of GRBs (L() -2 and the amplitude as well as the degree of linear polarization P degree of the gravitational waves are expected, P 4 L-2. At a viewing angle larger than the jet opening angle j the GRB -ray emission may not be detected. However, in such cases an "orphan" (see, e.g. [203, 531, 395]) long-wavelength afterglow could be observed, which would be preceded by a pulse of gravitational waves with a significant linearly polarized component. As the jet slows down and reaches ~ j-1, the jet begins to expand laterally, and its electromagnetic radiation begins to be observable over increasingly wider viewing angles. Since the opening angle increases as ~ -1 t1/2, at a viewing angle > j, the orphan afterglow begins to be observed (or peaks) at a time tp 2 after the detection of the gravitational wave burst. The polarization degree and the peak time should be correlated as P tp2.
Gravitational wave burst searches are underway with LIGO. The results from the third science run  searched for sub-second bursts in the frequency range 100-1100 Hz for which no waveform model is assumed, with a sensitivity in terms of the root-sum-square (rss) strain amplitude of hrss ~ 10-20 Hz-1/2. No gravitational-wave signals were detected in the eight days of analyzed data for this run. The search continues, as LIGO continues to be upgraded towards it ultimate target sensitivity.
Acknowledgements: Research partially supported through NSF AST 0307376 and NASA NAG5-13286. I am grateful for useful comments from two referees, as well as from P. Kumar, N. Gehrels, L. Gou, E. Ramirez-Ruiz, S. Razzaque, M.J. Rees, X.Y. Wang and B. Zhang.